{"records":[{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4558c13aff05b581566b9ae405ddda1c7caea775564887b7e6ebbf5dbdec5a5.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2509.16034"},"id":"claim:thue-morse-reduced-abelian-odd","kind":"claim","scope":"This dossier deliberately anchors only the equality\n`rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1)` for every nonnegative integer\n`n`, proposed in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1 and\nprinted page 14 of the verified journal version. The shared Library note\nrecords the journal citation. The source sentence is:\n\n> Although it appears that rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for\n> nonnegative integers n, the problem of determining a full recursion for\n> rho^{ab,red}_t(n) seems to be challenging.\n\n`red(w)` collapses every maximal constant run; reduced abelian complexity\ncounts all length-`n` factors up to rearrangement of their reduced words\nwith equal reduced length. The full recursion and four further questions\nin the source paragraph, namely the nonzero sign of `rho(4n+2)-rho(4n)`, a\nrecursion for `rho(4n)`, equation (11), and non-k-automaticity of sequence\n(10), are deliberately out of scope.","status":"public-question-addressed","summary":"Take every factor of a fixed length in the infinite Thue-Morse word. Collapse each maximal run of equal symbols, then group the reduced words by their counts of zeros and ones. How does the number of classes change with the factor length?","title":"Section 3, Campbell-Currie-Rampersad, arXiv:2509.16034v1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=claim%3Athue-morse-reduced-abelian-odd"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fdc9ae4e0e2991808156ecd561bc91fb9e3b645d76e5d611a60d0b3f506c08f4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:a156216-diagonalghost","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's second comment in OEIS A156216, internal revision #3,\nreads:\n\n> More generally, if exp( Sum_{n>=1} C(n) * x^n/n ) equals a power series in x with integer coefficients, then exp( Sum_{n>=1} C(n)^n * x^n/n ) also equals a power series in x with integer coefficients (conjecture).\n\nFor a characteristic-zero field $K$ and $C:\\mathbb N\\to K$, put\n$H_C=\\sum_{n\\ge1}C_nX^n/n$ and $F_C=\\exp(H_C)$, interpreted formally.\nThe assertion is $F_C\\in\\mathbb Z[[X]]\\Rightarrow\nF_{(C_n^n)}\\in\\mathbb Z[[X]]$, with the rings of integer series understood\nthrough the canonical embedding into $K[[X]]$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| The sequence $C(n)$ | `claim` quantifies $K$, its field and characteristic-zero structures, and $C:\\mathbb N\\to K$ | Every binding field and sequence |\n| $\\sum_{n\\ge1}C_nX^n/n$ | `logarithmicInput` has coefficient zero at zero and $C_n/(n:K)$ otherwise | Entry $C_0$ is unused |\n| The formal exponential | `exponentialSum` substitutes the logarithmic input into `PowerSeries.exp K` | Ordinary coefficients; no analytic condition |\n| Integer input coefficients | The antecedent is the existing `HannaLogarithmicPowerClosure.IntegralSeries` | Canonical integer image |\n| $C(n)^n$ | The consequent uses the sequence $n\\mapsto C_n^n$ | Index-dependent natural power |\n| Integer output coefficients | The consequent uses the same `IntegralSeries` | No positivity or nonzero restriction |\n\nThe existing `HannaDiagonalDerivativeParity.step a 0` is used in the\nprivate differential-equation proofs; the public claim retains the literal\nformal exponential. No source clause is dropped.","title":"Hanna's diagonal ghost-power integrality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/a156216-diagonalghost/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c74a54a5bfdd555c77016b23806172313f94b066bbdac0201ef01c8833ab10f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:a320043-square-values-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stefano Spezia's sole COMMENTS conjecture in OEIS A320043, revision 43,\nsays: “Conjecture: a(n) is not a perfect square except for n = 1, 6 and 96.”\nThe source and its literal first FORMULA are cited in\n[the source note](../Library/ArithUnits/spezia2018a320043.md).\nFor $n\\ge1$ the sequence is\n\n$$\n A(n)=\\sum_{k=1}^{n}\\frac{(n+1-k)^2k}{\\gcd(n+1-k,k)^3}.\n$$\n\nThe assertion is $\\forall n\\in\\mathbb N,\\ 1\\le n\\Rightarrow\n\\forall m\\in\\mathbb N,\\ A(n)=m^2\\Rightarrow(n=1\\lor n=6\\lor n=96)$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| First FORMULA | `a`, the sum over `Finset.Icc 1 n` with the literal numerator and gcd-cubed denominator | Exact natural quotients on the source domain |\n| OFFSET begins at 1 | `1 ≤ n` in `claim` | The empty sum at zero is excluded |\n| Perfect square | Natural root `m`, equality `a n = m ^ 2` | All nonnegative square roots |\n| Except for 1, 6, 96 | Three alternatives in `claim` | No added exclusion or hypothesis |\n| Refutation | `result : ¬ claim` | Unchanged source assertion negated |\n\nFor every selected summand, put $u=n+1-k$, $v=k$ and $d=\\gcd(u,v)$.\nBoth $u$ and $v$ are positive. Writing $u=du_0$ and $v=dv_0$ gives\n$u^2v/d^3=u_0^2v_0$, so natural division equals the source's exact quotient.\nThis is a prose derivation of the encoding correspondence.","title":"A320043: an additional square value","url":"https://the-omega-institute.github.io/trureturing-pages/research/a320043-square-values-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f381e5dea0c01f603d85fc25b5af5271ac70e444af9d71a2df306a277992edc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aa2012","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gerardo Arizmendi and Octavio Arizmendi, *The graph energy game*,\narXiv:2012.14114v2, Conjecture 3.10, page 8:\n\n> For the energy game, the Shapley value is in the core.\n\nThe source defines the energy of a finite simple undirected graph as the\nsum of absolute adjacency eigenvalues, with multiplicity. Its game has\n$w_G(S)=\\mathcal E(G[S])$. For each vertex $i$ its allocation is\n\n$$\n\\phi_i(w_G)=\\sum_{S\\subseteq V\\setminus\\{i\\}}\n\\frac{|S|!(|V|-|S|-1)!}{|V|!}\n\\bigl(w_G(S\\cup\\{i\\})-w_G(S)\\bigr).\n$$\n\nThe core consists of all $x$ satisfying\n$\\sum_{i\\in S}x_i\\ge w_G(S)$ for every coalition $S$ and\n$\\sum_{i\\in V}x_i=w_G(V)$. The conjecture quantifies over every finite\nsimple graph, without connectedness, degree or bipartiteness assumptions.\n[Preregistration #21951](https://github.com/the-omega-institute/trureturing/issues/21951)\nbinds this universal proposition and its expected refutation.\nThe [Library note](../Library/GraphInvariants/arizmendi2020energygame.md)\nrecords the versioned source and the literal defining formulas.\n\n### Statement echo\n\nAll local names below belong to\n`D5.S3.Combinatorics.SpectralGraph.ArizmendiEnergyGameShapleyRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite simple undirected graph, every size | `claim`, `n : Nat`, `G : SimpleGraph (Fin n)` | Includes size zero; no extra graph hypothesis |\n| Energy of the induced coalition | `worth` | Reuses `CyclicExtensionRefutation.graphEnergy` with `MatrixKind.adjacency`, whose center is zero |\n| Every subset of the other players | `shapley`, powerset of `univ.erase i` | Includes empty and full predecessor sets |\n| Factorial coefficient and marginal contribution | `shapley` | Natural factorials and subtraction, followed by real coercions and real division |\n| Coalition payoff total | Finite sum in `core` | Inlined sum, with no duplicate payoff-total definition |\n| Every coalition inequality and efficiency | The two conjuncts of `core` | Singleton, empty and grand coalitions included |\n| Universal core membership | `claim` | Literal source conclusion |\n| Refutation | `result` | Negates the entire universal claim |\n\nThe explicit vertex labels $0,1$ in `Fin 52` choose a counterexample;\nthey add no hypothesis to the source claim. The private parameter $m\\ge50$\nrestricts a sufficient counterexample family, not the quantified claim.","title":"The Shapley allocation of a graph energy game can leave its core","url":"https://the-omega-institute.github.io/trureturing-pages/research/aa2012/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/789b5542c2b0ff324c246d73ef077317590cc0dae5e2199aa2f84bf486b89591.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ab2410","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ryan L. Acosta Babb, *A partial converse to the Riemann–Lebesgue lemma for Bessel–Fourier series of order zero*, arXiv:2410.17681v1, Section 4, Conjecture 1, states:\n\n> For all, $0\\leq\\alpha\\leq\\frac12$, $M\\in\\mathbb N$ and $x\\in[0,1]$, $K_M^\\alpha(x)\\geq0$.\n\nThe selected definition is the source's explicit weighted display\n\n$$\nK_M^\\alpha(x)=\\sum_{m=1}^M\\left(1-\\frac{m}{M+1}\\right)j_m^\\alpha J_0(j_mx)\\quad(M>0),\n\\qquad K_0^\\alpha(x)=1.\n$$\n\nHere $j_m$ is the $m$th positive zero of $J_0$. The printed partial sum is $S_m^\\alpha(x)=\\sum_{k=1}^m j_m^\\alpha J_0(j_kx)$. Its factor has index $m$, so the printed Cesàro equality does not follow. The explicit weighted display, rather than that equality, is the target fixed by [#22175](https://github.com/the-omega-institute/trureturing/issues/22175).\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Standard real $J_0$ | Existing `BesselRedheffer.Defs.nbessel 0` | Reused directly; no new Bessel definition. |\n| Positive zeros indexed from one | `j 1 = firstZero 0`; later `j` are infima of positive zeros strictly above the preceding entry | Literal binding recursive convention; first two infima identified with actual zeros in the proof of `root_brackets`. Index zero is unused. |\n| Printed $S_m^\\alpha$ | `S` | The printed index $m$ in the weight is preserved; this definition is separate from `K`. |\n| Weighted formula and exceptional $M=0$ | `K` | Natural indices $1$ through $M$, real weights and real powers; zero branch equals one. |\n| Every real $0\\leq\\alpha\\leq1/2$ | First quantifier and two hypotheses of `claim` | Both endpoints included. |\n| Every natural $M$ and real $0\\leq x\\leq1$ | Remaining quantifiers and hypotheses of `claim` | Includes zero modes and both interval endpoints. |\n| $K_M^\\alpha(x)\\geq0$ | Conclusion of `claim` | No extra hypothesis. |\n| Refutation | `result : Not claim` | Exact negation; witness $(\\alpha,M,x)=(1/2,2,999/1000)$. |","title":"Acosta Babb's weighted Bessel-kernel positivity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/ab2410/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57c9bbe40458570b830af7044c1fdb20feb69d04f5429423df61683bb72692d4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abbott-2019-sequential-weak-pointer-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. A. Abbott, R. Silva, J. Wechs, N. Brunner and C. Branciard (arXiv:1805.09364,\nquant-ph; Quantum 3, 194 (2019)) study sequential weak measurements without\npost-selection. For `n` observables `A_1, …, A_n` and a state `ρ`, the mean\nproduct of the pointer positions in the weak regime is\n`2^{−(n−1)} Tr[{A_1, {A_2, …, {A_{n−1}, A_n}…}} ρ]`. For two projection\nobservables (eigenvalues 0 and 1) the paper proves that this is at least\n`−1/8`, and states:\n\n> Interestingly, by numerically minimising the mean product of the pointer\n> positions for sequences of up to 5 projection observables, we were unable to\n> obtain a value smaller than −1/8, and we conjecture that this is in fact the\n> case for all n.\n\nIssue #11377 fixes the reading. A projection observable is an orthogonal\nprojection `P = P² = P*` on `ℂ^d` of any rank, as in the paper's two-observable\ntheorem. The formal `claim` is the conjecture for every dimension, every\nnumber of projections and every unit vector; the result refutes it.","title":"Three weak measurements of projections below minus one eighth","url":"https://the-omega-institute.github.io/trureturing-pages/research/abbott-2019-sequential-weak-pointer-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f85eea135fa136220c74bc80476616e8de07536ce0072a514630967263400c8e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abbott-wechs-costa-branciard-2017-lazy-two-causal-j-one-facet","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. A. Abbott, J. Wechs, F. Costa and C. Branciard, *Genuinely Multipartite Noncausality*, arXiv:1708.07663v2 (Quantum 1, 39, 2017), Section III.C, ask in `genuine_multi_noncausal.tex`, line 343:\n\n> It is interesting to ask if this inequality is tight (i.e., facet inducing) for all $N$.\n\nTheir conjecture is stated at line 344:\n\n> By explicitly enumerating the vertices of the 2-causal polytope for $N=4$ (of which there are $136\\,818\\,592$) we were able to verify that $J_1(4)\\ge 0$ is indeed also a facet, and we conjecture that this is true for all $N$.\n\nLet $U=\\{0,\\ldots,N-1\\}$ and represent binary inputs and outputs by their supports $T,A\\subseteq U$. A deterministic lazy two-causal strategy consists of a nonempty proper first group $K\\subsetneq U$ and a function $f$ on input supports such that $f(T)\\subseteq T$ for every $T$, and\n$T\\cap K=S\\cap K$ implies $f(T)\\cap K=f(S)\\cap K$ for every $T,S$. The complementary group's outputs may depend on the entire input. Its deterministic table is $P_f(T,A)=1$ when $f(T)=A$ and $0$ otherwise. Let $C_N$ be the real convex hull of these tables.\n\nIn these coordinates, Eq. (10) is\n$J_1(p)=\\sum_{\\varnothing\\ne K\\subsetneq U}p(K,K)-p(U,U)\\ge0$.\nLaziness makes $p(K,K)$ exactly the marginal event in the source's formula. Definition 2, the deterministic characterization, and the lazy convention occur at TeX lines 146–151, 203–205, and 213–214; Eq. (10) is at lines 333–337. See [the literature note](../Library/Observer/abbottwechscostabranciard2017gmnc.md).\n\nIssue [#23904](https://github.com/the-omega-institute/trureturing/issues/23904) fixes the reading: for every natural $N\\ge2$, $J_1$ is nonnegative on $C_N$, and $F_N=C_N\\cap\\{p\\mid J_1(p)=0\\}$ has affine dimension one less than $C_N$. This is proved. Precisely, the module uses $d(S)=\\dim_{\\mathbb R}\\operatorname{span}\\{p-q\\mid p,q\\in S\\}$ and asserts $d(F_N)+1=d(C_N)$.","title":"The lazy two-causal inequality $J_1$ induces a facet for every $N\\ge2$","url":"https://the-omega-institute.github.io/trureturing-pages/research/abbott-wechs-costa-branciard-2017-lazy-two-causal-j-one-facet/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/58801a2d3bfe7fef8460e63390d4858b65ee7355781c31fc22d333fa4151e69a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abdelkhalek-et-al-2015-optimal-state-independence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kais Abdelkhalek, René Schwonnek, Hans Maassen, Fabian Furrer, Jörg Duhme,\nPhilippe Raynal, Berthold-Georg Englert and Reinhard F. Werner,\n“Optimality of entropic uncertainty relations”, arXiv:1509.00398v1,\nsection V.E, Conjecture V.8 (Independence of the optimal states of\n$(\\alpha,\\beta)$), state:\n\n> If $\\rho$ is an optimal state for any unitary operator and any\n> $\\alpha,\\beta>\\frac12$ satisfying the duality relation (2), then\n> $\\rho$ is also an optimal state for all other dual pairs.\n\nThe relation is $1/\\alpha+1/\\beta=2$. Section II defines\n$f(\\rho)=(H_\\alpha(p_X^\\rho),H_\\beta(p_Y^\\rho))$ and\n$\\rho\\sqsubseteq\\sigma$ by both entropy coordinates being at most\nthose of $\\sigma$. Its definition of optimality is:\n\n> We call a state $\\rho$ optimal if $\\rho'\\sqsubseteq\\rho$ implies\n> $\\rho\\sqsubseteq\\rho'$, and hence $f(\\rho)=f(\\rho')$.\n\nThe formal claim quantifies over all natural dimensions, complex unitaries,\ncomplex positive trace-one density matrices, and both finite real dual pairs\nwith every order strictly above $1/2$. The source explicitly excludes\n$\\{1/2,\\infty\\}$. Issue [#11782](https://github.com/the-omega-institute/trureturing/issues/11782)\npreregisters this reading and the refutation before any Lean probe.","title":"Optimal quantum states depend on the Rényi orders","url":"https://the-omega-institute.github.io/trureturing-pages/research/abdelkhalek-et-al-2015-optimal-state-independence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a935bfd2927643eecf6954be49ebbdb158ad434c886eae2712363c0bb8e64a5a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abdesselam-2022-pgg-non-half-integer-exponent-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Abdelmalek Abdesselam, “Non-Abelian correlation inequalities and stable\ndeterminantal polynomials”, arXiv:2207.07603v2, Problem 3, printed p. 14:\n\n> In the light of investigations of spin models with non-integer number of\n> components N, as in [8], it would be interesting to see if Thm. 2.3 still\n> holds for P^{−η} where η is any positive real number instead of being\n> restricted to half integers.\n\nThe question concerns all of Theorem 2.3 (p. 10). For real symmetric positive\nsemidefinite q by q matrices A_j with positive definite sum, let\nP(x) = det(Σ_j x_j A_j). Given a parity homomorphism ρ from ℤⁿ to (ℤ/2ℤ)ᴸ,\nan even multi-index lies in its kernel. For every m ≥ 0, natural m by n matrix\nV with even 1_m V, signs ε_i in {−1,1}, and positive even natural padding u,\nthe proposed extension requires\n\n$$\n\\sum_{\\substack{\\alpha,\\beta\\in\\mathbb N^m\\\\\\alpha+\\beta=\\mathbf 1_m}}\n\\mathbf 1\\{\\alpha V\\text{ even}\\}\\,\\varepsilon^\\beta\nP(u+\\alpha V)^{-\\eta}P(u+\\beta V)^{-\\eta}\\geq 0\n\\quad\\text{for every real }\\eta>0.\n$$\n\nThe Lean `claim` retains these quantifiers and premises, including L = 0.\nRow-vector multiplication uses Mathlib `Matrix.vecMul`.\nThe single public settling theorem `result : ¬ claim` answers this question\nnegatively. Problem 3's subsequent question about characterizing distributions\nwith bilinear positivity is a separate open boundary.","title":"Abdesselam's Problem 3: the PGG inequalities at arbitrary positive exponents","url":"https://the-omega-institute.github.io/trureturing-pages/research/abdesselam-2022-pgg-non-half-integer-exponent-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5eaaabfe94a67def3b85e1458aa02764e2abe871130e8100a2c46e562ef60fa5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abdesselam-2022-xy-pgg-cycle-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Abdelmalek Abdesselam, “Non-Abelian correlation inequalities and stable\ndeterminantal polynomials”, arXiv:2207.07603v2, §4, Problem 2, printed p. 14:\n\n> For the XY model, or O(2) model, the GG inequalities were proved by\n> Ginibre [18]. What about the padded generalizations given by the PGG\n> inequalities?\n\nThe free O(2) measure is the product of the rotation-invariant probability\nmeasures on S¹. Its angle coordinates use normalized Haar measure on\nReal.Angle, with spin θ = (cos θ, sin θ). The basic observables are all pair\ninner products σᵢ·σⱼ, i < j. The parity map assigns one bit to each endpoint\nof each pair; the source's evenness means ρ(u) = 0.\n\nFor every p ≥ 1, the proposed assertion requires, for every m ≥ 0, every\nnatural row matrix V, every ε ∈ {−1,1}ᵐ, and every even natural padding u,\n\n$$\n\\int_{X^2} \\mathcal O(x)^u\\mathcal O(y)^u\n\\prod_{i=1}^{m}\\left[\\mathcal O(x)^{V_{i*}}\n+\\varepsilon_i\\mathcal O(y)^{V_{i*}}\\right]\n\\,d\\mu(x)\\,d\\mu(y) \\ge 0.\n$$\n\nThe Lean `claim` retains these quantifiers and the literal duplicated\nintegral. Its evenness premise is inline: `parity p (fun e => (u e : ℤ)) = 0`.\nThe sole settling theorem is `result : ¬ claim`.","title":"Abdesselam's Problem 2: padded general Ginibre inequalities for the XY model","url":"https://the-omega-institute.github.io/trureturing-pages/research/abdesselam-2022-xy-pgg-cycle-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb19a8b2e5992a63e70ce1f1e7d004ce5f684f6152b792372fdc1452e848cf0a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abed-nikolaeva-novikov-2026-af-density-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Abed, Nikolaeva and Novikov, *How characterisations of traces and central elements fail*,\narXiv:2610.34388v1, Section 11.2(5), ask:\n\n> Is a bounded on M_n(C) independently of rho?\n\nThe selected question fixes a positive dimension and varies every strictly positive\ndensity. Section 5.12 defines the quotient set using positive semidefinite A,\nHermitian B and nonzero X=A+iB. Section 3.1 uses the ordinary trace and\nmodulus |X|=(X*X)^(1/2), without a trace-one requirement.","title":"Density-independent Alhasan–Fawwaz boundedness","url":"https://the-omega-institute.github.io/trureturing-pages/research/abed-nikolaeva-novikov-2026-af-density-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/41dcc944847fc91b7a53b7694f0d572413a271703be2741a63ddc8aa3d9b7aa4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abellanet-vidal-muller-rigat-rico-sanpera-2026-negativity-reduction-inclusion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Abellanet-Vidal, G. Müller-Rigat, A. Rico and A. Sanpera, *Bounding the entanglement of a state from its spectrum*, arXiv:2604.02420v2, compare supersets of the states of Schmidt number at most χ.\n- **Negativity.** N(ρ) = (||ρ^Γ||_1 − 1)/2 (Definition 2, Eq. (7)). Schmidt number at most χ implies N(ρ) ≤ (χ − 1)/2 (Lemma 3).\n- **Reduction.** RED_κ(ρ) = Tr(ρ)1 − ρ/κ (Eq. (35)). Schmidt number at most κ implies [1 ⊗ RED_κ](ρ) ≥ 0 (Theorem 7).\n- **The question.** §5.4: \"the complete inclusions of the considered EFS_γ sets is only pending for the unknown case NEGFS_{(χ−1)/2} ?⊂ PREDFS_χ, which is also unknown for the non-FS case.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/abellanetvidalmullerrigatricosanpera2026spectrum.md).\n\nIssue [#17440](https://github.com/the-omega-institute/trureturing/issues/17440) fixes the reading of the non-FS case: for every density matrix ρ on C^N ⊗ C^M and every integer 1 ≤ χ ≤ min(N, M), does N(ρ) = (χ − 1)/2 imply [1 ⊗ RED_χ](ρ) = ρ_A ⊗ 1 − ρ/χ ≥ 0? The attained value is the stronger hypothesis, so a counterexample also refutes the sublevel reading N(ρ) ≤ (χ − 1)/2.\n\nThe non-FS inclusion is refuted.","title":"Negativity at most (χ − 1)/2 does not imply positivity under the χ-reduction map","url":"https://the-omega-institute.github.io/trureturing-pages/research/abellanet-vidal-muller-rigat-rico-sanpera-2026-negativity-reduction-inclusion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/545984e5c38d74de4d089e4251f075f91c3f0f505e67b74519a3907c8c206452.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abiad-equal-legged-spider-power-zero-forcing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Aida Abiad, Mary Flagg, Sina Ghasemi Nezhad and Sjanne Zeijlemaker,\n*The Zero Forcing Number of Graph Powers*, arXiv:2610.03892v1, section 3,\nConjecture 10: “Let m ≥ 2, k ≥ 1 and ℓ ≥ k+1. Then Z(S_m(ℓ)^k) = (m−1)k.”\nThe spider consists of m paths on ell vertices, joined at their first endpoints to a new center.\nIt has m*ell+1 vertices. The kth ordinary power joins distinct vertices at distance at most k.","title":"Zero forcing the ordinary powers of equal-legged spiders","url":"https://the-omega-institute.github.io/trureturing-pages/research/abiad-equal-legged-spider-power-zero-forcing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73b7eb9fbcad7176b785dd545b6b1d988e9bca0ba0b238d9919210974c120a56.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abiad-flagg-ghasemi-nezhad-zeijlemaker-2026-grid-power-zero-forcing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Abiad, Flagg, Ghasemi Nezhad and Zeijlemaker, *The Zero Forcing Number of Graph Powers*, arXiv:2610.03892v1, §3, Conjecture 12 (`conj:grid`):\n\n> Let b≥1, k≥1 and ℓ≥k+1. Then Z((P_b □ P_ℓ)^k)=bk.\n\nThe endpoint is refutation of this full universal statement as printed. It does not settle a corrected b≤ℓ conjecture or assert author intent.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Every natural b,k,ℓ | `∀ b k ell : ℕ` in `claim` |\n| b≥1, k≥1, ℓ≥k+1 | `1 ≤ b → 1 ≤ k → k + 1 ≤ ell →` |\n| P_n is the n-vertex path | Mathlib `pathGraph n` on `Fin n`, adjacency of consecutive values |\n| Cartesian product | Mathlib `pathGraph b □ pathGraph ell`, on `Fin b × Fin ell` |\n| Distinct vertices at base distance ≤k | `graphPower`: `u ≠ v ∧ G.edist u v ≤ (k : ℕ∞)` |\n| Ordinary unique-white-neighbor forcing | Frozen `ZeroForcingThree.Black` and `IsZeroForcing` |\n| Minimum initial cardinality | Frozen `ZeroForcingThree.zeroForcingNumber`, an infimum of attainable finite cardinalities |\n| Equality to bk | `zeroForcingNumber (...) = b * k` |\n| Negation of the entire assertion | `result : ¬ claim` |\n\nAll admissible grids are connected and finite: positive path factors are connected, and Cartesian products preserve reachability. Extended graph distance is used directly, so no disconnected-distance collapse or k=1-only power definition affects the universal statement. The coordinate correspondence is (i,j)↦(i+1,j+1). No source hypothesis is dropped or added. `Black.force` can also derive a vertex already black, which adds no vertex and does not alter the closure; when its target is white, its premises say precisely that the source is black and every other neighbor is black.","title":"abiad flagg ghasemi nezhad zeijlemaker 2026 grid power zero forcing","url":"https://the-omega-institute.github.io/trureturing-pages/research/abiad-flagg-ghasemi-nezhad-zeijlemaker-2026-grid-power-zero-forcing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dca39300709dfb887761aea0bc31cd258ba0628aa4ad6c46a807e71a50510ede.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:abuku-etal-2024-enforce-nimber-domination","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Abuku, Kimura, Kiya, Larsson, Saha, Suetsugu and Yamashita,\n*Enforce and selective operators of combinatorial games*, arXiv:2311.01006v2,\nSection 8, Problem 42, printed page 23 asks:\n\n> If a ruleset dominates another, does it also ‘nimber dominate’ that ruleset? That is, does it follow that all nimbers of the combined ruleset are the same as those of the dominating ruleset?\n\nThe source is also published in *International Journal of Game Theory* 53,\n1249–1273, DOI 10.1007/s00182-024-00910-6. The preregistration is\n[issue #19251](https://github.com/the-omega-institute/trureturing/issues/19251).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Positions and option maps, Definition 2 | Arbitrary type `X` and `fA fB : X → Set X` | No finite-board assumption in `claim`. |\n| Every play has a finite bound from its starting position, Definition 4 | `Short` quantifies over starts, bounds and sequences | The displayed source condition is used directly. |\n| Union ruleset is short, Definition 10 | `JointlyShort fA fB` | Both option maps share the same position type. |\n| Perfect-play outcomes, Definition 5 | `Outcome`, `IsOutcome fA OA` | P exactly when every option is N. |\n| Opponent enforces one ruleset before the move, Definition 13 | `IsEnforceOutcome fA fB OE` | P exactly when at least one ruleset has only N options. |\n| Domination, Definition 14 | Pointwise equality of `OE` and `OA` | Equality holds at every position. |\n| Least excluded natural, Definition 28 | `mex s = sInf sᶜ` | This totalizes the source's finite-set mex; empty complement gives 0 by the Nat sInf convention. The witness uses only finite sets. |\n| Ordinary nimbers, Definition 29 | `IsNimValue fA GA` | Mex of the ordinary option-value image. |\n| Enforce nimbers, Definition 33 | `IsEnforceNimValue fA fB GE` | Mex of the intersection of the two option-value images. |\n| All combined nimbers equal the dominating ruleset's nimbers, Problem 42 | `claim` concludes pointwise equality of `GE` and `GA` | `result` proves the negation of this universal implication. |\n\nThe four function parameters satisfy the source recursions; they introduce\nno extra restriction on the refuting board. The witness uses three positions\nand provides each of the four functions explicitly. No general uniqueness\nor existence theorem for these recursions is claimed.","title":"Ruleset domination does not imply nimber domination","url":"https://the-omega-institute.github.io/trureturing-pages/research/abuku-etal-2024-enforce-nimber-domination/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d918a6cca687742f3b83a5892f13e986af7ace9996225625497ac9865860065b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aceska-kaczanowski-2022-cross-frame-welch-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Aceska and M. Kaczanowski, *Cross-Frame Potential*, arXiv:2205.05613v3,\nSection 4.2, Conjecture 41, Eq. (21), page 17, states:\n\n> Let F be a frame for Fⁿ, and let G be one of its dual frames. Then\n> $\\mu(\\mathrm{Gr}(F,G))\\ge\\sqrt{\\frac{nk-n^2}{k^2(k-1)}}$.\n\nThe source permits real or complex Hilbert spaces and assumes $k\\ge n$.\nIt imposes no equal-diagonal, tightness or equal-norm condition. A real\ncounterexample therefore refutes its universal statement. The encoded\n`claim` quantifies all natural dimensions and family sizes, both real vector\nfamilies, `n ≤ k`, `2 ≤ k`, and the frame and dual-frame predicates.\nThe condition `2 ≤ k` ensures a nonempty off-diagonal maximum; the witness\nhas `n = 2` and `k = 3`, so this restriction does not remove the refutation.","title":"Refutation of the dual-frame cross-Gramian Welch bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/aceska-kaczanowski-2022-cross-frame-welch-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cef6a40e07c1af784feff19d97fa1a912137bc48d675b11e4e523f8c0bf6395d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aceska-kaczanowski-2022-exclusive-grassmannian-dual-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Aceska and M. Kaczanowski, *Cross-Frame Potential*, arXiv:2205.05613v3,\nSection 4.2, Conjecture 42, p. 17:\n\n> If a frame F for F^n forms an exclusive Grassmannian pair with one of its duals, then that dual must be the canonical dual frame of F.\n\nDefinition 1 requires k ≥ n vectors in an n-dimensional Hilbert space and\npositive finite frame bounds. Definition 3 requires both dual reconstruction\nidentities. Definition 34 minimizes the maximum off-diagonal cross-Gramian\nmagnitude over all dual frames. The explanation in Section 4.2 says that an\nexclusive pair has exactly one dual attaining this minimum. The canonical\ndual is S⁻¹F for the frame operator S.\n\nThe exact real finite-family reading is:\n\n```lean\n∀ n k (F G : Fin k → EuclideanSpace ℝ (Fin n)), n ≤ k →\n IsFrame F → IsDualFrame F G →\n (∀ H, IsDualFrame F H → mu F G ≤ mu F H) →\n (∀ H, IsDualFrame F H → mu F H ≤ mu F G → H = G) →\n G = canonicalDual F\n```\n\nIsDualFrame records the first reconstruction identity. Over the reals,\nits adjoint gives the second; reconstruction forces both finite families\nto span the space, hence to be frames. A real counterexample refutes the\nsource's assertion for real or complex Hilbert spaces. Zero-based indexing\nonly relabels the finite families. The empty off-diagonal maximum is zero;\nthe counterexample has three vectors and a nonempty off-diagonal set.","title":"An exclusive Grassmannian dual need not be canonical","url":"https://the-omega-institute.github.io/trureturing-pages/research/aceska-kaczanowski-2022-exclusive-grassmannian-dual-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca075d294e2efe17593e671ebb981c5f332819d046b9b0f1312685a6950146a5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:acharya-kypraios-guta-2016-pauli-fisher-ghz-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Acharya, T. Kypraios and M. Guţă, *Statistically efficient tomography of low rank states with incomplete measurements*, New J. Phys. 18 (2016), 043018; arXiv:1510.03229v2, Section 4, subsection “Pauli settings”, `Efficient-Ion-Tomography.tex`, line 360:\n\n> While the full dependence of $\\lambda_{min}$ on $r$ and $d$ is unclear, we conjecture that for any fixed rank, $\\lambda_{min}$ is larger than a fixed constant for all states of rank $r$, of arbitrarily many ions.\n\nHere $d=2^n$, $\\bar I$ averages the Fisher information over all $3^n$ product Pauli settings, and $\\lambda_{min}=\\lambda_{min}(G^{-1/2}\\bar I G^{-1/2})$, where $G$ is the Frobenius metric of the local rank-$r$ parametrisation. The measurement model, metric and Fisher matrix are in the same source, lines 114–121, 157–176 and 178–180; see also [the literature note](../Library/Estimation/acharyakypraiosguta2016lowranktomography.md).\n\nIssue [#24270](https://github.com/the-omega-institute/trureturing/issues/24270) fixes the tangent reading. For complex matrices on $n$ qubits, let $u_{s,o}$ be the product Pauli eigenvector, $p_\\rho(o\\mid s)=\\operatorname{Re}\\langle u_{s,o}|\\rho|u_{s,o}\\rangle$, and $\\bar I_\\rho(v)=3^{-n}\\sum_s\\sum_o(\\operatorname{Re}\\langle u_{s,o}|v|u_{s,o}\\rangle)^2/p_\\rho(o\\mid s)$, with zero-probability terms assigned zero.\n\n`IsRankTangent r ρ v` means that there exists a curve $\\gamma:\\mathbb R\\to M_{2^n}(\\mathbb C)$ with $\\gamma(0)=\\rho$, derivative $v$ at zero, and $\\gamma(t)$ a positive semidefinite trace-one matrix of rank $r$ for every real $t$. Differentiability is required only at zero.\n\n`claim` asserts that for every natural rank $r\\ge1$ there exists a real $c>0$ such that, for every $n\\in\\mathbb N$, every rank-$r$ density matrix $\\rho$ and every `IsRankTangent r ρ v`, one has $c\\operatorname{Re}\\operatorname{Tr}(v^2)\\le\\bar I_\\rho(v)$. The quantifier over $n$ includes zero. The module proves `result : ¬ claim`; its witnesses have rank one and $n>0$.","title":"GHZ phase tangents refute a rank-only lower bound for normalized Pauli Fisher information","url":"https://the-omega-institute.github.io/trureturing-pages/research/acharya-kypraios-guta-2016-pauli-fisher-ghz-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17e22ff67e94a0df88e4649adb53cc06b264c3a4b88a04696da28736e5d346df.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:achterberg-van-mieghem-2025-spectral-degree-exponent-submodularity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. A. Achterberg and P. Van Mieghem, *The spectral degree exponent of a graph*,\narXiv:2502.01815v1, Section 7, p. 15, asks:\n\n> Several other properties of the SDE have not yet been established, e.g. whether the SDE is submodular or if the SDE can be used successfully in applications to enhance or reduce assortativity.\n\nThis is the tier-2 external question preregistered in\nhttps://github.com/the-omega-institute/trureturing/issues/19958.\nThe settlement is **Refuted** for edge sets on a fixed vertex set.\n\nDefinition 2.1 and equation (2.1), p. 1, define the exponent by\n$\\lambda_1=((1/N)\\sum_i d_i^q)^{1/q}$ for a simple undirected nonregular graph.\nTheorem 2.2 places it in $[2,\\infty]$. Here every exponent used is a finite real\nnumber and $\\lambda_1$ is the supremum of the real adjacency spectrum.\n\n### Statement echo\n\n| Source clause or explicit interpretation | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Simple undirected graph on $N$ vertices | `SimpleGraph (Fin n)` | Unweighted finite graphs form a subclass of the source's graphs. |\n| Degree mean $((1/N)\\sum_i d_i^q)^{1/q}$ | `degreePowerMean` | Natural degrees and $n$ are cast to real numbers; both powers are real powers. |\n| Nonregular graph | `IsRegular`, negated in `IsSpectralDegreeExponent` | Regularity is exactly existence of a common natural degree, using `SimpleGraph.IsRegularOfDegree`. |\n| Largest adjacency eigenvalue | Real-spectrum supremum in `IsSpectralDegreeExponent` | The literal preregistered expression is inlined. The private correspondence `spectral_sup_eq_edgeMax` connects it to the reused `SpikeEdgeEstimate.edgeMax` in the proofs. |\n| Finite solution with $q\\ge2$ | `IsSpectralDegreeExponent G q` | Includes nonregularity, the lower bound and the defining equation. Infinite exponents are outside this restricted claim. |\n| Submodular on the possible edges of one vertex set | `claim` | All $n,G,H,q_G,q_H,q_U,q_I$ are quantified; all four defining equations precede $q_U+q_I\\le q_G+q_H$. |\n| Answer to the question | `result` | Negation of that whole universal proposition. |\n\nThe source does not specify the ground set. The fixed-vertex edge-set\ninterpretation is fixed by #19958. Refuting its finite-exponent restriction\nalso refutes a submodularity claim whose domain includes these instances.\nThe induced-vertex interpretation has separate numerical evidence below.\nUniqueness of the exponent and monotonicity of all degree means are not\nhypotheses of the Lean refutation.","title":"The spectral degree exponent is not submodular","url":"https://the-omega-institute.github.io/trureturing-pages/research/achterberg-van-mieghem-2025-spectral-degree-exponent-submodularity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f70018aab8d7e2423a9dc9d8ee5f1528177732715f0dd05152a4d949cc94ba04.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:adamaszek-2012-necklace-cycle-length","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Adamaszek, *Hard squares on cylinders revisited*, arXiv:1202.1655v2,\nSection 7, Conjecture 7.4, states:\n\n> The length of every cycle in the graph Neck(k, n) divides n−3k. In other\n> words, for every (k, n)-necklace N we have Tⁿ⁻³ᵏN = N.\n\nThe section assumes even positive circumference n and positive k. A necklace\nhas 2k distinct stones, each with signed vector −2, −1, 1 or 2. Consecutive\nstones have opposite directions. Away-facing gaps are odd; toward-facing gaps\nplus both vector lengths are odd. A toward-facing gap is at least three,\nand at gap three both vectors have length one. Necklaces are identified under\ncircle isometries. T performs simultaneous JUMP, then TURN\n(−2→1, −1→2, 1→−2, 2→−1), then FIX of facing gaps three by shortening\nlength-two vectors to one. Issue #11582 fixes this reading before the probe.","title":"Adamaszek's Conjecture 7.4 on necklace cycle lengths","url":"https://the-omega-institute.github.io/trureturing-pages/research/adamaszek-2012-necklace-cycle-length/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/608af462b4764a2285db1ff541d97079cf6186d0a1e0f2e269fbff9f3f581c9e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:adamopoulou-papamikos-2025-grassmann-measure-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Adamopoulou and G. Papamikos, *Entwining Yang–Baxter maps over Grassmann algebras*, Physica D 472, 134469 (2025), arXiv:2311.18673v1, construct birational entwining Yang–Baxter maps R_{a,b}, S_{a,b}, T_{a,b} with variables in the Grassmann algebra Γ(n) (Eqs. (17)–(19)).\n- **The map.** S_{a,b} (Eq. (18)) sends (x₁, x₂, χ₁, χ₂, y₁, y₂, ψ₁, ψ₂), with x, y even and χ, ψ odd, to u₁ = y₁²(1 + χ₁ψ₂)/(b x₁) + y₁(y₂ − a − ψ₁ψ₂)/b, u₂ = b/y₁, v₁ = x₁, v₂ = a + x₁x₂ − y₁/x₁ + χ₁χ₂, ξ₁ = ψ₁ − (y₁/x₁)χ₁, ξ₂ = ψ₂, η₁ = χ₁, η₂ = χ₂ + (y₁/x₁)ψ₂. Expanding every variable in a basis of Γ(n) gives a map on 2^{n+2} commutative coordinates.\n- **Measure preservation.** A map preserves the measure m when its Jacobian determinant is J = m(z)/m(S(z)) (§4). The authors verify m = 1/y₁ for n = 1 and m = 1/y₁₁² for n = 2.\n- **The conjecture.** \"We conjecture that in Γ(n) the map R_{a,b} is volume preserving for every n, while the maps S_{a,b} and T_{a,b} preserve measures of the form y₁₁^{−n} and x₁₁^{−n}, respectively\".\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/adamopoulou2025entwininggrassmann.md).\n\nIssue [#16775](https://github.com/the-omega-institute/trureturing/issues/16775) fixes the reading: the S clause at n = 3, with J the determinant of the derivative of the 32-coordinate map. Refuting one clause at one n refutes the conjecture, a conjunction over all n of clauses for R, S and T. The S clause is refuted.","title":"The Grassmann entwining Yang–Baxter map S over Γ(3) does not preserve the measure y₁₁⁻³","url":"https://the-omega-institute.github.io/trureturing-pages/research/adamopoulou-papamikos-2025-grassmann-measure-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc9377cbd06c6178b77ec110021306f70faadda5a613fd3a03f8835cb7739286.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:adamson-dudey-fleischmann-huch-prefix-normal-connectivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Duncan Adamson, Moritz Dudey, Pamela Fleischmann and Annika Huch,\n*Word Chain Generators for Prefix Normal Words*, arXiv:2508.19619v1,\nConclusion, pp. 17–18, ask:\n\n> Is there a path from any prefix normal generator to all others, just by swapping neighbouring numbers?\n\nThe institutional 18-page accepted author manuscript (AAM) poses the same\nquestion. For every positive n, the vertices are permutations of 1,...,n\nwhose complete flip-to-zero chains are prefix normal. Starting at the\nall-one word, each entry names the position to flip. All n+1 states,\nincluding both endpoints, must satisfy the comparison of every contiguous\nfactor with the prefix of the same length. A move exchanges entries at\nadjacent positions; their distinct values can be arbitrary. Every\nintermediate permutation must remain a valid generator. The requested\npath is finite and may have zero moves. No shortest-path, direction or\ncomplexity requirement is imposed.","title":"Connectivity of prefix normal generators","url":"https://the-omega-institute.github.io/trureturing-pages/research/adamson-dudey-fleischmann-huch-prefix-normal-connectivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7237f0a516d1b7335907808add1a1821eb93b62edaad114a789b56fe7ff25825.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:adamson-explicit-graph-outside-g2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source is the unnumbered request in the conclusion of the paper identified\nby `adamson2026twoword` in the Library. Its literal wording is:\n\n> We did not find an example of a graph G such that G ∉ G₂. By Remark 30 we know that such a graph exists. Finding such a graph remains an open problem.\n\nLet V = Fin 24 ⊕ Finset (Fin 24). The candidate U₂₄ has an edge from inl a to\ninr S exactly when a ∈ S, the same adjacency in reverse, and no within-part\nedges. Its endpoint states that there do not exist lists w,v over V such that\n(∀ z : V, count z w = 2), (∀ z : V, count z v = 2), and\n∀ x,y : V, x ≠ y → (U₂₄.Adj x y ↔ filter (z = x ∨ z = y) w =\nfilter (z = x ∨ z = y) v). These are Definitions 1 and 14 at k = 2.\nPositive counts give the full vertex set as the alphabet of both words.","title":"An explicit membership graph outside G₂","url":"https://the-omega-institute.github.io/trureturing-pages/research/adamson-explicit-graph-outside-g2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f142bf2ba65d61905aa0ca1bb26c09e7474b8792705768482f26bd7c81921699.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:adamson-flaherty-potapov-spirakis-2023-two-swap-greedy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Adamson, Flaherty, Potapov and Spirakis, *Structural and Combinatorial Properties\nof 2-swap Word Permutation Graphs*, arXiv:2307.01648v3, Section 5, page 23:\n\n> For the 2-swap graph specifically, we leave open the problem of determining the shortest path between two given words w and v. We conjecture that the simple greedy algorithm used to derive the upper bound in Lemma 4 can be used to find the shortest path between any pair of vertices.\n\nThe target is the greedy optimality conjecture, preregistered in\n[issue #19317](https://github.com/the-omega-institute/trureturing/issues/19317).\nFor words with the same Parikh vector, the formal claim asks whether some\nadmissible order makes the number of generated pairs with distinct indices equal\nthe configuration-graph distance. An admissible order omits any maximal-count\nreference symbol and lists every other symbol once in non-increasing count.\nAll tied choices are allowed. The count does not require that the generated\nswaps be legal at application time or reach the target, so refuting this\nnecessary condition also refutes the stronger shortest-path assertion.\n\n### Statement echo\n\n| Source clause | Formal counterpart and fidelity |\n| --- | --- |\n| Words in $\\Sigma^n$ | Functions `Fin n → Fin s`; zero-based relabelling of the source's positions and alphabet. |\n| Definition 1 exchanges positions $i For each k ∈ ℕ, the inclusion 𝒢_k ⊆ 𝒢_(k+1) is proper.\n\nThe source convention starts ℕ at 1. For a finite simple graph G on V,\nmembership requires actual words w,v over V, each containing every vertex\nexactly k times. For every distinct a,b, G.Adj(a,b) holds if and only if\ntheir ordered two-letter List projections in w and v are equal. Positive\ncounts give both full alphabets. Single-word alternation and unions of word\nrepresentation languages are different models.\n\nThe exact full target has two clauses for every positive k: inclusion for\nevery finite graph, and existence of a finite graph on one and the same\ncarrier with a (k+1)-uniform representation and no k-uniform representation.\nAn assumed nonuniversality theorem, a fixed-k result, or vertex extension\nalone does not settle either the full family or this named problem.","title":"Every positive-uniform adjacent class inclusion is proper","url":"https://the-omega-institute.github.io/trureturing-pages/research/adamson-positive-uniform-hierarchy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e41aa28966fc794eb739eb2df2e162a274ab88684581fcb6952565fa72cac0b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:agoh-alternating-numerator-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The sole target is the unnumbered Conjecture on final printed page 9 of\nINTEGERS 26 (2026), A9. Its exact source and provenance are in the linked\nLibrary note. For every nonzero real polynomial f splitting over the reals,\nthe asserted equivalence is:\n`(for every natural n, (X-1)^n divides num(Q_n(f))) iff\n(f is a monomial or every real root is simple and different from 1)`.\nQ_n is independently the literal rational sum in Equation (3.1), and num is\nits actual reduced `RatFunc.num`. The all-n condition is inside the equivalence.","title":"Agoh's alternating-numerator characterization","url":"https://the-omega-institute.github.io/trureturing-pages/research/agoh-alternating-numerator-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d82beeef6d772c7c2b9ec48c9a240d0d61132c51e51139bd928d279b7dcef665.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahanjideh-ekim-yildiz-2022-z-of-d-conjecture-6-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Ahanjideh, T. Ekim and M. A. Yıldız, *Maximum size of a triangle-free graph with bounded maximum degree and matching number*, arXiv:2207.02271v1, Conjecture 6.5:\n\n> For d ≥ 21 and odd, we have Z(d) = ⌊5(d + 1)/4⌋.\n\nThe journal locator is DOI 10.1007/s10878-024-01123-z. The quotations refer to version 1. The source defines $Z(d)$, for $d\\ge2$, as the least matching number of a triangle-free factor-critical graph that is regular of degree $d$ when $d$ is even, and almost regular of degree $d$ when $d$ is odd.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| “if d(w) = d − 1 for some w ∈ V(G) and d(v) = d for all v ∈ V(G) − w, then G is said to be almost d-regular.” | `FactorCriticalMatching.IsAlmostRegular` | One deficient vertex and every other degree are specified literally. |\n| “A graph G is said to be factor-critical if G \\\\ v has a perfect matching for all v ∈ V(G).” | `FactorCriticalMatching.IsFactorCritical` | Perfect subgraph matchings in each induced vertex complement. |\n| “For any d ≥ 2, let Z(d) be the smallest natural number n such that there exists a d-regular (if d is even) or almost d-regular (if d is odd) triangle-free and factor-critical graph G with ν(G) = n.” | `AhanjidehEkimYildizZ.Z` | Natural infimum of exactly those matching-number witnesses. |\n| “For d ≥ 21 and odd, we have Z(d) = ⌊5(d + 1)/4⌋.” | `AhanjidehEkimYildizZ.claim` and `.result` | Every odd natural degree at least 21; natural division is the floor. |\n\nFinite simple graphs have carrier `Fin n`; the matching number is the existing `DimerVirialFourthDifferenceRefutation.matchingNumber`. The formal definition of $Z$ is total on natural numbers and admits an empty carrier. The proved range has a deficient vertex and is nonempty. No statement about the paper's convention at even or small degrees is inferred from this total extension.","title":"The extremal matching number of almost regular triangle-free graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahanjideh-ekim-yildiz-2022-z-of-d-conjecture-6-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5cdd6c18f1c86f4a8f40aa8c2c2bc027e16d648c2f910d4c05c7f6be2af796e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahiable-kothakonda-winter-2026-appt-quadbit-maximum-purity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ahiable, Kothakonda and Winter, arXiv:2608.03390v2, Conjecture 6.7, page 29:\n\n> Let $\\mathcal P_{m,n}\\subseteq\\mathrm{APPT}_{m,n}$ be the inscribed absolute PPT polytope with $2\\leq m\\leq n,\\;n>2$. Then\n> $$\\max_{\\lambda\\in\\mathrm{APPT}_{m,n}}\\sum_{i=1}^{mn}\\lambda_i^2\n> =\\max_{\\lambda\\in\\mathcal P_{m,n}}\\sum_{i=1}^{mn}\\lambda_i^2$$\n> and occurs at the spectra given by Eq. (44).\n\nThe exact target here is its $m=n=4$ sector. A density matrix is a complex\nmatrix $\\rho$ on $\\mathbb C^4\\otimes\\mathbb C^4$ with $\\rho\\succeq0$ and\n$\\operatorname{Tr}\\rho=1$. Literal APPT requires\n$(U\\rho U^\\dagger)^\\Gamma\\succeq0$ for every global unitary $U$, where\n$\\Gamma$ transposes the second tensor factor.","title":"Maximum purity in the 4-by-4 absolutely PPT sector","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahiable-kothakonda-winter-2026-appt-quadbit-maximum-purity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb70d5b5862c1a68f8e7e3b7273d8a317f252c46d2d77de9dd7f614cd352fa56.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahiable-kothakonda-winter-2026-appt-qutrit-maximum-purity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ahiable, Kothakonda and Winter, arXiv:2608.03390v2, Conjecture 6.7, asks:\n\n> Let $\\mathcal P_{m,n}\\subseteq \\mathrm{APPT}_{m,n}$ be the inscribed absolute PPT polytope with $2\\leq m\\leq n,\\; n>2$. Then\n> $$\\max_{\\lambda\\in \\mathrm{APPT}_{m,n}}\\sum_{i=1}^{mn}\\lambda_i^2=\\max_{\\lambda\\in\\mathcal P_{m,n}}\\sum_{i=1}^{mn}\\lambda_i^2$$\n> and occurs at the spectra given by Eq. (44).\n\nThe resolved sector is $m=3$, every integer $n\\geq3$. For a positive semidefinite, trace-one matrix $\\rho$ on $\\mathbb C^3\\otimes\\mathbb C^n$, APPT means $(U\\rho U^\\dagger)^\\Gamma\\succeq0$ for every unitary $U$, with $\\Gamma$ the second-factor partial transpose. The purity is $\\operatorname{Re}\\operatorname{Tr}(\\rho^2)$.","title":"The exact maximum purity of absolutely PPT qutrit–qudit states","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahiable-kothakonda-winter-2026-appt-qutrit-maximum-purity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/538d48c815deec4cac542c70ae04b60bb8ea528f2a195111d033938236d52caa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahiable-kothakonda-winter-2026-eq31-absolute-separability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Ahiable, N. B. T. Kothakonda and A. Winter, *The geometry of absolute separability and other convex\nmatrix properties from spectrum*, arXiv:2608.03390v2, Section 7 (page 39):\n\n> Thus, another matter of interest is to resolve Conjecture 6.7 in the positive, and to show that the\n> linear condition in Eq. (31) is sufficient for absolute separability: because then the maximum purity\n> over all three sets must coincide.\n\nTheorem 6.2 (page 23) proves that a state on $\\mathcal M_m\\otimes\\mathcal M_n$ whose decreasingly ordered\neigenvalues satisfy $2\\lambda_{mn}+\\sum_{k=1}^{m-1}\\lambda_{mn-k}\\ge\\sum_{k=1}^{m-1}\\lambda_k$ (31) is\nabsolutely PPT. A state is absolutely separable when every $U\\rho U^\\dagger$, $U\\in\\mathcal U(mn)$, is\nseparable. The verbatim texts are in [the literature note](../Library/QuantumStates/ahiablekothakondawinter2026geometry.md).\nIssue [#14958](https://github.com/the-omega-institute/trureturing/issues/14958) preregisters the reading\n($2\\le m\\le n$, the repository's finite-sum separable cone) and the route.","title":"The linear spectral condition (31) and absolute separability","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahiable-kothakonda-winter-2026-eq31-absolute-separability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ab54d849b40fc400f7742d3adb108295869e00df62364f4260e0b41065637f4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahmed-kunjwal-2026-hamming-multipartite-partition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ahmed and Kunjwal, *Characterizing unitaries via quasi-process functions*,\narXiv:2610.00579v1, Conjecture V.1 in Section V.2, ask for genuine\nmultipartiteness of maximal-clique partitions whenever the number of\ncoordinates exceeds three. The surrounding definitions mean that every\ncoordinate direction must occur. Alphabet sizes are independent and at\nleast two. Definition V.1 defines adjacency by disagreement at exactly one\ncoordinate; maximal means inclusion-maximal, not maximum cardinality.\n\nThe exact formal statement is: for every natural `n` with `4 ≤ n`, every\n`d : Fin n → ℕ` with `∀ i, 2 ≤ d i`, and\n`V = (i : Fin n) → Fin (d i)`, there is a set `P : Set (Set V)` such that\n\n- Every `S ∈ P` is nonempty, is a full coordinate line, and satisfies\n `Maximal (graph d).IsClique S`.\n- For every `x : V`, there is exactly one `S` such that `S ∈ P ∧ x ∈ S`.\n- For every `i : Fin n`, some `x : V` has `line i x ∈ P`.\n\nThere is no equal-alphabet assumption, upper dimension bound, finite search\nbound, or assumed binary matching supplier.","title":"All-direction maximal clique partitions of mixed-alphabet Hamming graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahmed-kunjwal-2026-hamming-multipartite-partition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e543bcbe819a6ed9f02f657497faa650841e75cd6936e97eb810399370ba3f41.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ahn-corcoran-staudacher-2021-hypereclectic-non-shortening","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Ahn, L. Corcoran and M. Staudacher, *Combinatorial Solution of the Eclectic Spin Chain*,\narXiv:2112.04506v1, JHEP 03 (2022) 028, Appendix A, Eq. (A.4):\n\n> “We claim that the rank of A^(k) is always maximal:”\n>\n> “rank(A^(k)) = min(d_{S−k}, d_S).”\n\nHere $A^{(k)}$ is the matrix of $H^k:W_S^{L,M}\\to W_{S-k}^{L,M}$ in the elementary bases,\nand $d_S=\\dim W_S^{L,M}$. The settled statement is the rational encoding of the $K=1$\nstatic sector, for every $1\\le M\\le L$, every natural level $S$ and every $k\\le S$.\nThe rightmost letter is the fixed wall $3$; the other $L-1$ letters consist of $L-M$ ones\nand $M-1$ twos.","title":"Non-shortening in the one-wall hypereclectic spin chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/ahn-corcoran-staudacher-2021-hypereclectic-non-shortening/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b406e667f01d6a5003ba7f845d14297fe588a204f5dd415b51bb3a95d160abc1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:akella-2026-stabilizer-multi-entropy-n3-collapse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Akella, N. Iizuka and A. Miyata, *Genuine Multi-Entropy in the Toric Code*,\narXiv:2607.06050v2, state for stabilizer states the reduction\n$\\mathrm{GM}^{(4)}_{3}(A:B:C:D)=-\\left(a-\\tfrac19\\right)I_3$ (Eq. (3)), where $I_3$ is the Rényi\ntripartite information and $a$ the convention parameter of the genuine multi-entropy, note that\n\"this reduction is not a consequence of the Coxeter structure\", and ask in §6 \"Whether there is a\nsimilar counting argument for $n = 3$ case is an open question.\" The verbatim definitions are in\n[the literature note](../Library/QuantumStates/akella2026genuine.md).\n\nIssue [#13575](https://github.com/the-omega-institute/trureturing/issues/13575) reads Eq. (3) as a\ntheorem about every normalized pure qubit stabilizer state and every assignment of its qubits to\nfour labelled parties (parties may be empty), for every real $a$, together with positivity of every\nreplica partition function $Z_3^{(\\mathtt q)}$ so that every logarithm is defined.","title":"The n = 3 collapse of the genuine four-party multi-entropy for stabilizer states","url":"https://the-omega-institute.github.io/trureturing-pages/research/akella-2026-stabilizer-multi-entropy-n3-collapse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4588161733d699e15d471514f6d57250b30c71ff05e26176ec9d5cb962bd0d1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:akp-2012-two-centered-abelian-periods","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Avgustinovich, Karhumäki and Puzynina, *On abelian versions of critical factorization theorem*, RAIRO–ITA 46(1), Open problem 6.2, ask:\n\n> Let k be an integer. Does there exist a non-periodic word w and integers l₁, l₂, such that w contains a central abelian 2k-power of length l₁ or l₂ at every position?\n\nThe answer is negative for every $k\\geq1$, every finite alphabet and every pair of block lengths. The public theorem is `D5.S1.Words.TwoCenteredAbelianPeriods.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Biinfinite word over a finite alphabet | $w:\\mathbb Z\\to\\mathrm{Fin}(r)$, every natural $r$ | Alphabet labels are immaterial. |\n| Position between $w_i$ and $w_{i+1}$ | Integer cut $i$ | The first block starts at $i-lk+1$. |\n| Central abelian $2k$-power with period length $l$ | `CentralAbelianPower w k l i` | All $2k$ consecutive length-$l$ blocks have the same letter counts; $l>0$. |\n| Two lengths, one available at every position | The disjunction inside the universal integer-position hypothesis of `claim` | Both lengths are fixed globally. |\n| No non-periodic example | `result : claim` concludes `Periodic w` | A positive natural ordinary period is produced. |\n| Positive exponent | $1\\leq k$ | Nonempty centered powers; no assertion about $k=0$. |\n| Natural lengths may equal zero | Positivity is part of each central-power predicate | A zero length never satisfies that predicate, so only the other length remains. |","title":"Two centered abelian period lengths","url":"https://the-omega-institute.github.io/trureturing-pages/research/akp-2012-two-centered-abelian-periods/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd7477649e38c7876c7e9e016256ddfaa555c1ae3b5c4a21fbd5cd7973ad942e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aks2610","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alfuraidan, Khan and Sauras-Altuzarra, *Metric dimension and maximum eigenvalue\nmultiplicity of graph Laplacians*, arXiv:2610.26875v1, Problem 30, p. 27, ask:\n\n> Do there exist infinite families of connected twin-free graphs satisfying exactly one of the two equalities mm_L(G)=dim(G), mm_Q(G)=dim(G)? The computational census shows that finite twin-free examples occur in both directions.\n\nThe result answers this existence question affirmatively. Here $\\dim(G)$ is\nthe minimum size of a resolving set for actual shortest-path distances,\n$\\operatorname{mm}_L$ and $\\operatorname{mm}_Q$ maximize real eigenspace\ndimension over every real eigenvalue, and twin-freeness excludes both true\nand false twins. The signless Laplacian is $D(G)+A(G)$.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Finite simple graphs | `SimpleGraph (Fin (orders i))` in `claim` |\n| Connected | the connectivity conjunct for every natural index |\n| Twin-free, both kinds | `TwinFree`, defined using neighborhoods with the other vertex removed |\n| Metric dimension | minimum cardinality of an injective distance representation |\n| Maximum multiplicity | `maximumMultiplicity`, using actual real eigenspaces and all real eigenvalues |\n| Exactly one equality | disjunction of one equality and the other inequality |\n| Infinite family | all-natural indexing and pairwise nonisomorphism |\n\nThe concrete witness has $G_i=\\Theta_{3,3,6(i+1)+4}$ and order\n$6(i+1)+9$. The source permits either equality direction; the construction\nuniformly supplies the Laplacian direction. No source clause is dropped.\nThe fixed finite carrier only labels the graph; pairwise nonisomorphism\nexpresses infinitude independently of labels.","title":"An infinite twin-free theta family separating the Laplacian equalities","url":"https://the-omega-institute.github.io/trureturing-pages/research/aks2610/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72ed7802f1d0ad71ef022355e12ec183e43ac3f284d735c461e6dcb55ccf7ce6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aksoy-et-al-2023-asynchronous-iteration-spectral-radius-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. G. Aksoy et al., *Seven open problems in applied combinatorics*, Journal of Combinatorics 14(4), 559–601 (2023), arXiv:2303.11464v1, includes the section “Price of Asynchrony” by Carlos Ortiz Marrero and Stephen J. Young. Its Open Problem reads:\n\n> Let $M$ be a $n \\times n$ real matrix with $p(M)= \\lambda < 1$ partitioned into an $\\ell \\times \\ell$ array of blocks. Is it the case that for all elements $\\delta$ of $[k]^{\\ell \\times \\ell}$ with zero diagonals, $\\rho(M^{(\\delta)}) \\leq \\lambda^{\\nicefrac{1}{k+1}}?$ If not, is there some function $f \\colon (0,1) \\times \\N \\rightarrow (0,1)$ such that $\\rho(M^{(\\delta)}) \\leq f(\\lambda,k)$?\n\nThis quotation is from the downloaded v1 TeX, `Sections/ratfish.tex`, lines 75–77, md5 `08f1f56ce6f899219ee74094fe3c0e53`. The surrounding model, lines 48–49, specifies:\n\n> $\\tilde{x}_i^{(k+1)} = \\sum_j M_{ij}\\tilde{x}_j^{(k - \\delta_{ji})}$\n\nand says “we assume that $\\delta_{ii}=0$”. Here $p(M)$ in the question is read as the spectral radius $\\rho(M)$ of that model. The operator $M^{(\\delta,k)}$ acts on the newest iterate and its $k$ preceding iterates, updating the newest slot by this recurrence and shifting the remaining slots. The source uses $M^{(\\delta)}$ for a history operator with enough slots (lines 51–62).\n\nIssue [#21351](https://github.com/the-omega-institute/trureturing/issues/21351) fixes the formal reading: `claim` is `Q1 ∨ Q2`, where `Q1` is the universal power bound and `Q2` is the existence of the uniform bound strictly below one. Both alternatives are refuted, even with two scalar blocks, positive remote delays and $0<\\lambda<1$. The source and its locator are also recorded in [the literature note](../Library/Eigenstructure/aksoyetal2023asynchrony.md).","title":"Both proposed spectral-radius bounds for asynchronous iteration are false","url":"https://the-omega-institute.github.io/trureturing-pages/research/aksoy-et-al-2023-asynchronous-iteration-spectral-radius-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66bf1c867db78790e0973959c1081e6efbf4b27459ba73afc8bf8102106444bc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alai-2026-ghz-mermin-staircase","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Aaron Alai, *Exact minimum measurement dependence for faithful local\ndeterministic models of multipartite GHZ–Mermin correlations*,\narXiv:2608.00124v1, Section IX, Conjecture 1 (Staircase), PDF page 4:\n\n> With $R(n) := 2^{\\lfloor (n-1)/2 \\rfloor}$ the Mermin violation ratio and\n> $s(n) := (R+1)/(2R)$ the classical satisfiability,\n> $F_{\\min}(n) = \\frac{R}{2(R+1)} = \\frac{1}{4\\,s(n)},\n> \\qquad F_{\\min}\\cdot s = \\frac{1}{4}.$\n\nSection II specifies $n\\ge3$, all even-Y setting strings, and all $4^n$\ndeterministic X/Y answer tables. At each setting the real density is\nnonnegative and normalized. A faithful model reproduces the prescribed\nfull correlator and makes every nonempty proper-subset correlator zero.\nThe objective is $F=M/2$, where\n$M=\\max_{s,s'}\\sum_\\lambda|\\rho_s(\\lambda)-\\rho_{s'}(\\lambda)|$.","title":"Alai's Staircase for faithful GHZ–Mermin measurement dependence","url":"https://the-omega-institute.github.io/trureturing-pages/research/alai-2026-ghz-mermin-staircase/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04cf2918f616024ff19e3ad0e950f26688f4ff77f8b0d81846b9f742b3ed0ded.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:albalahi-das-ali-barman-hamza-2025-cdso-universal-vertex-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 4.1 states: “A graph minimizing (maximizing, respectively) the CDSO index (HSO index, respectively) among fixed-order connected graphs with cyclomatic number $\\ell(\\ge 1)$ has a vertex adjacent to all other vertices.” This dossier settles the CDSO half. The HSO half is outside the delivered claim.","title":"The order-seven unicyclic CDSO minimizer has no universal vertex","url":"https://the-omega-institute.github.io/trureturing-pages/research/albalahi-das-ali-barman-hamza-2025-cdso-universal-vertex-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a24f68eb1c6a2767e34b1b20cf8fd9607ef9fa9bdc42dff641672c548aba6030.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:albors-et-al-2024-bad-science-matrix-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Albors, Bhatti, Ganjoo, Guo, Kunisky, Mukherjee, Stepin and Zeng,\n*On the Structure of Bad Science Matrices*, arXiv:2408.00933v2,\nSection 1.4, ask: “We do not know whether this matrix is indeed optimal.”\nThe matrix is their displayed real $5\\times5$ matrix $A_*$, and the\nadmissible matrices have rows of Euclidean length one.\nThe journal reference is Involve 19 (2026), 443–460,\nDOI [10.2140/involve.2026.19.443](https://doi.org/10.2140/involve.2026.19.443).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Real square matrices and the uniform sign-cube average in Section 1 | `beta`, with `Matrix (Fin n) (Fin n) ℝ`, Boolean sign vectors and weight $2^{-n}$ | The function-space norm is the sup norm. The sign expression is literal; the proofs reuse `IsingUniquenessSets.sgn` through definitional equality. |\n| “rows are normalized to have ℓ²-length 1” | `UnitRows` | Every row's sum of squared real entries equals one. |\n| The displayed matrix of Section 1.4 | `Astar` | All 25 entries and the factor $1/(2\\sqrt3)$ agree with the source. |\n| “this matrix is indeed optimal” | `claim` and `result` | Every real $5\\times5$ matrix satisfying `UnitRows` has $\\beta(A)\\le\\beta(A_*)$; there is no extra hypothesis. |\n\nThe declarations belong to\n`D5.S3.Combinatorics.Hamming.BadScienceFiveOptimality`.","title":"The exact five-dimensional bad science optimum","url":"https://the-omega-institute.github.io/trureturing-pages/research/albors-et-al-2024-bad-science-matrix-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c8f77864d8da2edc98e713d8c44aaf543e18310c67618b19fc0594f575b80bf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:albright-et-al-2023-kemeny-edge-derivative-conjecture-3-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Albright, K. P. Hadaway, A. H. Pomerance, J. Jeffries, K. J. Lorenzen\nand A. K. Nix, *On the Edge Derivative of the Normalized Laplacian with\nApplications to Kemeny's Constant*, arXiv:2211.01495v2, Conjecture 3.5,\nprinted page 19:\n\n> For a given n, the derivative of Kemeny's constant on simple graphs of order n is smallest on the path on n vertices with respect to the non-edge connecting the ends of the path. The value of this derivative is O(n³).\n\nThe minimality sentence is refuted under the literal conventions in\n[preregistration #19527](https://github.com/the-omega-institute/trureturing/issues/19527).\nThe [Library note](../Library/GraphInvariants/albright2023kemenyedgederivative.md)\nrecords the source definitions. The path's own cubic growth sentence is not\nnegated by this settlement.\n\n### Statement echo\n\nAll short declaration names below belong to\n`D5.S3.Combinatorics.SpectralGraph.AlbrightKemenyEdgeDerivativeRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A(t)=A+tB$, with $B$ indicating one unordered pair | `paramAdj` | The two matrix units have weight one; degrees change with $t$ |\n| $L=I-D^{-1/2}AD^{-1/2}$ | `normLap` | $D$ contains the row sums of the perturbed matrix |\n| $K=\\sum_{\\lambda_L\\ne0}1/\\lambda_L$ | `kemeny` | Nonzero real characteristic roots, counted with multiplicity; the proof identifies them spectrally |\n| Simple graphs of order $n$ | `claim`: $G$ on `Fin n`, $2\\le n$ | Connectedness is the domain where the source's random-walk constant is defined |\n| Edge or non-edge with distinct ends | `claim`: $a\\ne b$ | The witness uses a missing edge, so also refutes a non-edge-only comparison |\n| Path on $n$ vertices, endpoint perturbation | `SimpleGraph.pathGraph n`, endpoints $0,n-1$ | Bounds follow from $2\\le n$ |\n| Smallest derivative | `claim`: two `HasDerivAt` premises at zero and $d_P\\le d_G$ | Derivative existence is required; no default value for a nondifferentiable function |\n| Universal minimality fails | `result` | Negates this entire proposition |\n| The path value is $O(n^3)$ | Outside `claim` | The refutation makes no contrary assertion about the path value |\n\nThe instance class `DecidableRel G.Adj` supplies computation on the finite\ngraph and imposes no mathematical restriction. Real parameters allow the\nordinary two-sided derivative; the proof establishes positive degrees and\nthe inverse update on a neighborhood of zero. No source clause within the\nminimality proposition is dropped.","title":"The path does not minimize Kemeny's edge derivative","url":"https://the-omega-institute.github.io/trureturing-pages/research/albright-et-al-2023-kemeny-edge-derivative-conjecture-3-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23288da4f81a729fb4c62c5b8e1c63db71415bd0d1ec4909689e8879602a7ca3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alcala-arroyo-2025-lazy-invertible-generation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Alcalá-Arroyo and A. Castillo-Ramirez (arXiv:2510.14841, cs.FL with\nnlin.CG; Theoret. Comput. Sci. 1076 (2026) 115965) study the lazy cellular\nautomata over `A^G`: a local map on a finite neighborhood `S ∋ e` returns the\ncentral value `z(e)` on every pattern `z` except one, the unique active\ntransition `p`, where it writes a symbol different from `p(e)`. Motivated by the\ngeneration of the full transformation monoid of a finite set by its\npermutations and its idempotents of defect one, they ask:\n\n> If $\\mathcal{L}(G,A)$ is the set of all lazy cellular automata over $A^G$,\n> prove or disprove the following:\n> $\\text{CA}(G,A) = \\langle \\text{ICA}(G,A) \\cup \\mathcal{L}(G,A) \\rangle$.\n\nIssue #11773 fixes the reading: every group `G` and every finite alphabet `A`\nwith at least two symbols, cellular automata and lazy automata as in the paper,\n`ICA(G, A)` the units of the monoid `CA(G, A)` under composition, and `⟨C⟩` the\nsubmonoid generated by `C`. The result disproves the equality.","title":"Lazy and invertible cellular automata do not generate all cellular automata","url":"https://the-omega-institute.github.io/trureturing-pages/research/alcala-arroyo-2025-lazy-invertible-generation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e995844b70feed5b493678ee688437eed31cae34028e2b76928998b4b5eef83.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alcaraz-ramos-2024-conformal-tower-lowest-mode","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. C. Alcaraz and L. M. Ramos, *Conformally invariant free parafermionic quantum chains with multispin interactions*, arXiv:2402.09936v3, Physical Review E 109, 044138, propose the leading prediction\n$M\\varepsilon_i\\to\\pi A^{(N)}(x_s^{(N)}+i-1)$ for each fixed $i\\ge1$.\nTheir conjectured constants for $N=3$ and $M\\equiv0\\pmod3$ are $A^{(3)}=2\\sqrt3$ and $x_s^{(3)}=7/6$.\nThe prediction is **Refuted** for homogeneous couplings, already at $i=1$ along $M=3m$.\n[Preregistration #19220](https://github.com/the-omega-institute/trureturing/issues/19220) fixes the source and Lean conventions; its corrected scaled constant is $7\\pi/(3\\sqrt3)$.\nThe [Library note](../Library/QuantumContext/alcarazramos2024conformaltower.md) gives the source quotations and locators. Printed Eqs. (26)–(27), (38), (39), (41) correspond to TeX labels (3.11)–(3.12), (4.7), (4.8), (4.10).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $P_M=P_{M-1}-z\\lambda_M^N P_{M-N}$, with $P_M=1$ for $M\\le0$ | `poly`, specialized to $N=3$, $\\lambda_j=1$, with index shift $\\operatorname{poly}(n)=P_{n-2}$ | All three initial values and the recurrence are literal |\n| $P_M$ for nonnegative $M$ | `P M = poly (M+2)` | Natural-index representation of the source recurrence |\n| $\\varepsilon_i=1/z_i^{1/N}$ over the positive roots | `quasienergy M i`: roots with multiplicity, positive-root filter, decreasing root order, real exponent $-1/3$ | Increasing quasienergy order; index $i-1$ uses truncated natural subtraction |\n| Every fixed $i\\ge1$ | `claim`: every natural `i` with `1 ≤ i` | No mode is dropped from the conjecture |\n| Leading behaviour with $A^{(3)}=2\\sqrt3$, $x_s^{(3)}=7/6$ | `claim`: `Tendsto` of $3m\\varepsilon_i(3m)$ to $\\pi(2\\sqrt3)(7/6+i-1)$ | The product $3m$ is natural before the real cast; $i$ is cast to real |\n| $M\\equiv0\\pmod3$ | The sequence $M=3m$ | Exact nonnegative residue class |\n| Refutation | `result : ¬ claim` | The contradiction uses $i=1$ only |\n\nAbsent root indices are assigned zero in `quasienergy`, as fixed by the binding conventions. A positive proposed limit forces the selected quasienergy to be positive eventually; the proof then obtains an actual positive root. This totalization adds no hypothesis to the claim and supplies no spurious counterexample at absent indices.\nEvery binder and conjunct implements a row above; no source clause is weakened or dropped.","title":"The Alcaraz–Ramos lowest-mode conformal prediction fails","url":"https://the-omega-institute.github.io/trureturing-pages/research/alcaraz-ramos-2024-conformal-tower-lowest-mode/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eecdc6efc56ac326c91d02dd1c24ef91839e9f96b370c6c1272a757426415944.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alekseyev-halving-partitions-nonregularity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Max A. Alekseyev, *Halving Partitions and Hyperbinary Expansions*,\n[arXiv:2611.01243v1](https://arxiv.org/html/2611.01243v1), Definition 1 and\nsection 6, asks:\n\n> This raises the question of whether a(n) is 2-regular: does there exist a finite family of integer sequences containing a(n) such that, for each sequence b(n) in the family, both b(2n) and b(2n+1) are integer linear combinations of sequences in the family [16]?\n\nThis dossier anchors exactly that existence question. The negative answer is\n`D5/S1/Recurrence/Partitions/HalvingPartitionNonregularity.result`.\nThe resolution kind `Refuted` refers to the proposed existence of such a family.\n\n### Statement echo\n\nFor every natural total n, the source carrier consists of every finite ordered\nlist of positive integers summing to n. With virtual predecessor p0=n, each\nsuccessive part belongs to the value set consisting of the lower and upper\ninteger halves of its predecessor. Equal halves give one value. Repeated parts,\narbitrary finite lengths and repeated terminal ones are retained. At zero the\nempty list is the sole object.\n\nThe count a(n) is the cardinality of that independently defined carrier. Its\ninteger cast admits no positive finite dimension d, vector sequence F and fixed\ninteger matrices M0,M1 satisfying, for every n including zero,\n\n$$\nF(n)_0=a(n),\\qquad F(2n)=M_0F(n),\\qquad F(2n+1)=M_1F(n).\n$$\n\nThe same theorem proves both directions of the matrix equivalence with finite\nindexed families for every integer sequence, including arbitrary finite index\ntypes with a distinguished member. Duplicate and dependent constituent sequences\nare allowed. The distinguished member makes the family nonempty; no dimension\nbound, basis assumption or invertibility condition enters the statement. The\nresult also identifies the actual-list count with the filtered Composition\ncardinality for every n and proves a(0)=1.","title":"Halving partitions admit no finite integral decimation family","url":"https://the-omega-institute.github.io/trureturing-pages/research/alekseyev-halving-partitions-nonregularity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b37cfd280e625a0e85d0cf0f32236f5a71ef7aca2fa7488c1e2963cb404b2ca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alexandersson-jal-quemener-2025-rook-eulerian-interlacing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alexandersson, Jal and Quemener, *Real-rootedness of rook-Eulerian polynomials*,\narXiv:2502.05939v1, Section 3.3, Conjecture 28:\n\n> For Ferrers boards, the polynomial $R(\\lambda,\\alpha;t)$ is real-rooted.\n> Moreover, $R_{\\lambda_1}(\\lambda,\\alpha;t),R_{\\lambda_1-1}(\\lambda,\\alpha;t),\\dotsc,R_2(\\lambda,\\alpha;t),R_1(\\lambda,\\alpha;t)$ forms an interlacing sequence.\n\nHere $\\lambda_1\\le\\cdots\\le\\lambda_n$ is the board, $\\alpha$ is a vector of\nnonnegative integer multiplicities summing to $n$, and $W$ consists of words\nwith those multiplicities satisfying $0 (**Schatten norms of non-integer order**) We used Cor. IV.6 to resolve the spin alignment problem for Schatten norms of integer order. It is of interest to know if a similar approach can be used in cases where the order of the norm is not an integer. Suppose that $A_{0}, A_{1}, B_{0}, B_{1} \\geq 0$ satisfy $\\lambda(A_{0}) = \\lambda(B_{0})$, $\\lambda(A_{1}) = \\lambda(B_{1})$. For a string $s \\in \\{0, 1\\}^{*}$, define the ordered products $\\Pi_s(A_{0}, A_{1}) := \\Pi_{i \\in [|s|]} A_{s(i)}$ and $\\Pi_s(B_{0}, B_{1}) := \\Pi_{i \\in [|s|]} B_{s(i)}$. If for all strings $s$, $\\operatorname{tr}(\\Pi_s(A_{0}, A_{1})) \\geq |\\operatorname{tr}(\\Pi_s(B_{0}, B_{1}))|$, does it hold that $\\lVert A_{0} + A_{1} \\rVert_{p} \\geq \\lVert B_{0} + B_{1} \\rVert_{p}$ for all $p \\in [1,\\infty)$? If so, then the overlap lemma may be used to prove the spin alignment conjecture for all Schatten norms, and by extension for the von Neumann entropy as well.\n\nThe quotation follows the PDF, with mathematical notation typeset in Markdown. The source defines $\\lambda$ using eigenvalues with multiplicity (`main.tex`, line 117), and defines $\\lVert R\\rVert_p=(\\operatorname{tr}|R|^p)^{1/p}$ with $|R|=\\sqrt{R^*R}$ (lines 274–278). See [the literature note](../Library/Eigenstructure/alhejjiknill2024spinalignment.md).\n\nIssue [#21843](https://github.com/the-omega-institute/trureturing/issues/21843) fixes the reading: for every natural size $d$, four positive semidefinite complex $d\\times d$ matrices with $A_0$ and $B_0$ having equal characteristic polynomials, and likewise $A_1$ and $B_1$, satisfy the norm comparison for every real $p\\ge1$ whenever every ordered-word trace of the $A$ pair is real and dominates the absolute value of the corresponding $B$ trace. The universal implication is refuted.","title":"Word-trace domination does not control fractional Schatten norms","url":"https://the-omega-institute.github.io/trureturing-pages/research/alhejji-knill-2024-fractional-schatten-alignment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6eeff3c34eb7ccbe934d2a02d9fee1a966a84c1074cd128d0074d46de31c45df.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ali-2018-qutrit-collective-dephasing-invariant-negativity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Ali, *Freezing dynamics of entanglement and nonlocality for qutrit-qutrit (3 ⊗ 3) quantum systems*, arXiv:1810.01123v1, published in Mod. Phys. Lett. A 34, 1950102 (2019). The conjecture is in the Conclusions:\n\n> We have found no evidence for time-invariant entanglement for qutrit-qutrit states and based on our results, we conjecture that under current dynamics, this feature of time-invariant entanglement may not exist for qutrit-qutrit systems.\n\nThe feature, the Kraus operators of the collective dephasing and the entanglement measure are quoted in [the literature note](../Library/QuantumContext/ali2019qutritfreezing.md).\n\nIssue [#19382](https://github.com/the-omega-institute/trureturing/issues/19382) fixes the reading:\n- **Hypotheses:** a rate Γ > 0 and a density matrix ρ₀ whose negativity is positive and constant along Φ_{Γ,t} for all t ≥ 0.\n- **Conclusion:** ρ₀ is stationary.\n\nThe conjecture is refuted.","title":"Collective dephasing of two qutrits admits nonstationary states with constant positive negativity","url":"https://the-omega-institute.github.io/trureturing-pages/research/ali-2018-qutrit-collective-dephasing-invariant-negativity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/810cb41cae7b86dc86cfe900b9784a0bbce475cd25bf9200437f0e075ac8add4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aliabadi-2025-matchability-subspace-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mohsen Aliabadi, *Conditions for matchability in groups and field extensions II*,\nDiscussiones Mathematicae General Algebra and Applications 45 (2025), 135–157,\nsection 5.1, Conjecture 38, printed page 154 / PDF page 20, states:\n\n> Let $K\\subset L$ be a field extension and $A,B\\subset L$ be a $K$-subspace\n> of $L$ of finite positive dimensions. Assume further that for every\n> $a,b\\in L$ and every nontrivial proper finite intermediate subfield\n> $K\\subset M\\subsetneq L$, we have\n>\n> $$\\dim_K(aM\\cap A)+\\dim_K(bM\\cap B)\\le [M:K]+1.$$\n>\n> Then for all $K$-subspaces $S\\subset A$ and $T\\subset B$, the following holds\n>\n> $$\\dim_K\\langle ST\\rangle\\ge\\dim_K S+\\dim_K T-1.$$\n\nThe same statement is Conjecture 5.1 in arXiv:2309.14664v1, PDF page 14,\nand v2/v3, PDF page 15. Section 2.2 defines the brackets as the $K$-linear\nspan of the actual product set in $L$. The\n[source note](../Library/Factorization/aliabadi2025matchabilityii.md)\nrecords the primary versions, DOI and locators. The sole target is the\nwhole original universal assertion. Preregistration:\n[#23409](https://github.com/the-omega-institute/trureturing/issues/23409).\n\n### Statement correspondence\n\n| Source clause | Formal counterpart and witness |\n| --- | --- |\n| Arbitrary field extension $K\\subset L$ | `claim` quantifies fields with a `K`-algebra structure on `L`. The witness is the actual field `AdjoinRoot (X^7-C 2)` over $\\mathbb Q$, with the canonical injective algebra map and degree seven. |\n| Finite positive-dimensional $A,B$ | Both finite-dimensional hypotheses and both positive-finrank hypotheses occur in `claim`; $A=B=\\operatorname{powerSpan}(6)$ has dimension six. |\n| Every $a,b\\in L$, including zero | Both unrestricted scalar binders occur in `intersectionPremise`; `translate` maps the underlying submodule of $M$ by actual multiplication by the scalar. |\n| Every finite nontrivial proper intermediate field | `IntermediateField K L`, `⊥ < M`, `M < ⊤`, and `FiniteDimensional K M`. Neither endpoint is included. Prime degree proves that no witness field has such an $M$. |\n| Both intersection dimensions and $[M:K]+1$ | Submodule infima and finranks in `intersectionPremise` express the exact displayed inequality. No assumed certificate replaces it. |\n| Every allowed $S\\subset A$, $T\\subset B$ | `claim` quantifies all submodules with `S ≤ A` and `T ≤ B`, including the zero subspaces. The counterexample has both containments strict, so it also refutes the strict reading of the printed subset symbols. |\n| Actual $\\langle ST\\rangle$ | `Submodule.span K ((S : Set L) * (T : Set L))`, using field pointwise multiplication. |\n| Uncapped lower bound | `finrank K S + finrank K T - 1 ≤ finrank K (span ...)`; natural subtraction gives the same inequality truth value as integer subtraction, including zero dimensions. |\n| Whole-claim negation | The sole public theorem `result : ¬ claim` has no extra mathematical premise. |\n\nThe matching definitions in the introduction use equal dimensions and\n$1\\notin B$ when defining matchability. Conjecture 38 states neither\nmatchability nor these conditions as an added premise. The witness\nnonetheless has $A=B$ and $S=T$, with $1$ absent from all four spaces.\nIt also has $S Does every finite nonabelian group admit a multiset A of cardinality |G|\n> satisfying the abelianization obstruction but not quotient-realizable?\n\nThe following paragraph says that a positive answer would characterize finite\nabelian groups as precisely those finite groups for which the abelianization\nobstruction is the only obstruction. The official journal article, DOI\n[10.61091/um128-18](https://doi.org/10.61091/um128-18), has the same question,\nglobal conventions and relevant definitions. The source note is\n[aliabadi2026nonabeliantwist](../Library/FiniteGroups/aliabadi2026nonabeliantwist.md).\n\nFor every finite group G with multiplication not commutative, let n=|G| and\nlet pi:G->G/[G,G] be the canonical abelianization map. There exists a\nmultiset A of elements of G, counted with multiplicity, such that |A|=n\nand the multiplicity-counted product of pi(a) over A is 1, but there are\nno bijections b,c:{1,...,n}->G for which A is the multiset of\nb(i)*c(i)^(-1). Repeated identity entries are allowed. No\nproduct-one ordering in G, identity-free condition, group-family,\nsolvability or parity restriction is imposed.\n\nDefinition 2.1 supplies exactly this two-bijection meaning of quotient\nrealizability. Proposition 2.3 supplies the necessary abelianization\ncondition: the product is taken after projection, with multiplicities,\nin the abelian group G/[G,G]. It does not assert an unordered product in G.","title":"Aliabadi's finite nonabelian quotient-multiset problem","url":"https://the-omega-institute.github.io/trureturing-pages/research/aliabadi-2605-16478v2-problem-7-6/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d2c91a169919d429c0b619ed10d0e5cacc491f14ede22e2ff88fe81c84ebba30.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aliabadi-zerbib-2022-matroid-matching-question-2-7","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Aliabadi and S. Zerbib, “Matchings in matroids over abelian groups”, arXiv:2202.07719v3, published in *Journal of Algebraic Combinatorics* (2024), Question 2.7, ask:\n\n> Let $G$ be an abelian torsion-free group. Let $M$ be a matroid over $G$ and $N$ be a sparse paving matroid over $G$, both of the same rank $n$, so that $|E(M)| = |E(N)| - 1$ or $|E(M)| = |E(N)|$. If $0 \\notin E(N)$, is $M$ matched to $N$?\n\nThe [source note](../Library/GraphInvariants/aliabadizerbib2024matroidmatching.md) records the DOI and arXiv locator. The preregistration is [#19681](https://github.com/the-omega-institute/trureturing/issues/19681). The answer is **Refuted** by the settling declaration `D5/S3/Combinatorics/Matroids/AliabadiZerbibMatroidMatching.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Abelian torsion-free group | `G`, `[AddCommGroup G]`, `[IsAddTorsionFree G]` in `claim` | The group is arbitrary within the stated class |\n| Finite loopless matroids | `M.Finite`, `N.Finite`, `M.Loopless`, `N.Loopless` | The source standing convention is explicit in the proposition |\n| Same positive rank | `0 < M.eRank`, `M.eRank = N.eRank` | Rank is Mathlib's extended natural rank |\n| Sparse paving target | `IsSparsePaving N` | Both the matroid and its dual are paving |\n| Ground-size alternatives | `M.E.ncard + 1 = N.E.ncard ∨ M.E.ncard = N.E.ncard` | Both alternatives are retained |\n| Exclusion of zero | `(0 : G) ∉ N.E` | The source's zero condition is literal |\n| Matching of ordered bases | `BasesMatched` and `IsMatchedTo` | Injective `Fin n` enumerations and Mathlib bases are used |\n| Negative answer | `result : ¬ claim` | The proposition is refuted by an exact integer witness |","title":"Aliabadi–Zerbib Question 2.7 has a negative answer","url":"https://the-omega-institute.github.io/trureturing-pages/research/aliabadi-zerbib-2022-matroid-matching-question-2-7/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/56eb2203ee99a984e18a38858d3861e134740bd021e8588a42e5c57a11c19727.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alikhani-ghanbari-dehghanizadeh-2024-elliptic-sombor-energy-integer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alikhani, Ghanbari and Dehghanizadeh, *Elliptic Sombor energy of a graph*,\narXiv:2404.18622v1, Conjecture 3.9, states:\n\n> There is no graph with integer-valued elliptic Sombor energy.\n\nFor a simple graph, the elliptic Sombor matrix has entry\n`(d_i + d_j) sqrt(d_i^2 + d_j^2)` on an adjacent pair and zero otherwise;\nthe elliptic Sombor energy is the sum of the absolute values of its eigenvalues.","title":"Two four-cycles sharing a vertex refute elliptic Sombor Conjecture 3.9","url":"https://the-omega-institute.github.io/trureturing-pages/research/alikhani-ghanbari-dehghanizadeh-2024-elliptic-sombor-energy-integer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab48d9955124bae799ecfdfc5bbdc425bd39b4670e8a66a196f3b6f22806c73d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alikhani-ghanbari-henning-clique-gluing-problem-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Saeid Alikhani, Nima Ghanbari and Michael A. Henning, *Strong domination number\nof graphs from primary subgraphs*, Quaestiones Mathematicae 49(2), 195–211,\nProblem 1, p. 202, asks whether, for every $r\\ge2$,\n\n$$\\gamma_{st}(G)\\ge\\gamma_{st}(G_1)+\\gamma_{st}(G_2)-1-\\Sigma_{X,Y}.\\tag{4}$$\n\nThe finite connected graphs $G_1,G_2$ have order at least $r+1$ and contain\nenumerated $r$-cliques $A,B$. The graph $G$ identifies corresponding clique\nvertices. Here $X_i=N_{G_1}(u_i)\\setminus A$,\n$Y_i=N_{G_2}(v_i)\\setminus B$, and\n$\\Sigma_{X,Y}=\\sum_i|X_i|+\\sum_i|Y_i|$. A strong dominating set contains,\nfor every vertex outside it, an adjacent vertex of at least that vertex's degree.\nThe strong domination number is its minimum cardinality.\n\nThe settlement is **Proved** for Inequality (4), under exactly these hypotheses;\nthe preregistration is [#19327](https://github.com/the-omega-institute/trureturing/issues/19327).\nThe source's request for an optimum using finer parameters is outside this claim.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every $r\\ge2$ | `claim` quantifies over `r` with `2 ≤ r` | No upper cutoff. |\n| Two finite simple connected graphs | `V1`, `V2`, their finite instances, `G1`, `G2`, and connectedness hypotheses | Finite vertex labels and decidable equality add representation, not restrictions on finite graphs. |\n| Order at least $r+1$ | The two `Fintype.card` inequalities | Literal order hypotheses. |\n| Enumerated $r$-cliques | Embeddings `A`, `B` from `Fin r`, with `IsClique` on their ranges | Zero-based labels correspond to source indices plus one. |\n| Identify $u_i$ with $v_i$ | `embedRight`, `glue` on the sum/subtype vertex type | Edges are the union of the factor edge images; identified clique edges occur once. |\n| Strong domination | `IsStrongDominatingSet`, `strongDominationNumber` | The degree comparison is literal; the natural infimum is attained. |\n| External-neighbour sum | `X`, `sigmaXY` | Neighbours shared by several clique vertices contribute once for each such vertex. |\n| Inequality (4) | `result : claim` | Subtraction is in integers; no additional assumptions. |","title":"Strong domination under clique gluing","url":"https://the-omega-institute.github.io/trureturing-pages/research/alikhani-ghanbari-henning-clique-gluing-problem-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad1b89da43a2b16c1b8628da6d8db17ea5fb0f57766568efe4eb6a63b55a5163.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alikhani-griswold-2026-connected-minus-four-family","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Saeid Alikhani and Max Griswold, *On the Integer Domination Root Conjecture*,\narXiv:2608.00109v1, Section 4 item 3 asks:\n\n> Does there exist an infinite family of graphs G satisfying D(G, -4) = 0?\n\nThe source uses finite simple undirected connected graphs. For ordinary\ndomination, a subset S dominates when each vertex belongs to S or has a\nneighbor in S. The polynomial counts each such subset exactly once:\n\n$$D(G,x)=\\sum_{S\\text{ dominates }G}x^{|S|}.$$\n\nThe exact result is: for every natural N there exist n>N and a simple graph G\non Fin n such that G is connected and D(G,-4)=0. Unbounded orders ensure\ninfinitely many isomorphism classes. Connectedness applies to G; the\ndominating subsets need not themselves be connected. There is no additional\ndegree, planarity, cyclomatic-number, or root-multiplicity hypothesis.\n\nThe declaration is\n`D5/S3/Combinatorics/Graph/Domination/MinusFourReassembly.result`.\nIts ordinary dominating subsets are exactly `DominatingSetAverage.domSets`;\nthe summands are integer powers of the integer minus four. Both existential\nwitnesses follow the universal N, and the inequality is strict.","title":"Connected graphs with ordinary domination root minus four","url":"https://the-omega-institute.github.io/trureturing-pages/research/alikhani-griswold-2026-connected-minus-four-family/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/41c191df82a76a3682f0cdf42d4029b85b49602001221b0de89a3346c6ea3cd2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alikhani-griswold-2026-minus-four-minimum-order","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Saeid Alikhani and Max Griswold, *On the Integer Domination Root Conjecture*,\narXiv:2608.00109v1, Section 4 item 2, printed page 5 asks:\n\n> Is n=33 the minimal order of a graph admitting -4 as a domination root?\n\nSection 1 uses finite nonempty connected simple undirected graphs. An\nordinary dominating subset S contains or neighbors every vertex, and each\nsubset is counted exactly once:\n\n$$D(G,x)=\\sum_{S\\text{ dominates }G}x^{|S|}.$$\n\nConnectedness applies to G; the dominating subsets need not induce connected\ngraphs. There is no degree, planarity, cyclomatic-number or\nroot-multiplicity restriction.\n\nThe complete positive claim includes both attainment at order 33 and\nexclusion at every smaller positive order. Its exact Lean definition is:\n\n~~~lean\ndef claim : Prop :=\n (∃ G : SimpleGraph (Fin 33),\n G.Connected ∧ ∑ S ∈ domSets G, (-4 : ℤ) ^ S.card = 0) ∧\n (∀ n : ℕ, 1 ≤ n → n < 33 →\n ∀ G : SimpleGraph (Fin n), G.Connected →\n ∑ S ∈ domSets G, (-4 : ℤ) ^ S.card ≠ 0)\n~~~\n\nHere domSets is DominatingSetAverage.domSets. The designated result is\nliterally \"theorem result : ¬ claim\", with no parameters or extra hypotheses.\nIts declaration GID is\nD5/S3/Combinatorics/Graph/Domination/AlikhaniGriswoldMinOrderRefutation.result.","title":"Refutation of order-33 minimality for domination root minus four","url":"https://the-omega-institute.github.io/trureturing-pages/research/alikhani-griswold-2026-minus-four-minimum-order/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b322f2df18e9cd4e4cc9b8c4316467fcadcaa08c779768cee3638754d2d77518.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:allen-meyer-2017-negativity-region-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. W. Allen and D. A. Meyer, *Polynomial Monogamy Relations for Entanglement Negativity* (Phys. Rev. Lett. 118, 080402 (2017); arXiv:1502.04807), study the triples of squared negativities (N²_A|C, N²_A|B, N²_A|BC) of pure three-party states, with N(ρ) = ‖ρ^{T_A}‖₁ − 1.\n- **Three qubits.** They describe the achievable region by the states d|000⟩ + a|101⟩ + b|110⟩ + c|111⟩.\n- **Qudits.** They generalize these states to the family (13), d|000⟩ + Σ_{j=1}^{D−1} (a|j0j⟩ + b|jj0⟩ + c|jjj⟩).\n- **The conjecture.** \"we conjecture that it is the entire achievable set of negativities, (N²_A|C, N²_A|B, N²_A|BC), for D > 2 as well\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/allen2017polynomialmonogamy.md).\n\nIssue [#18420](https://github.com/the-omega-institute/trureturing/issues/18420) fixes the reading for D = 3: every unit three-qutrit vector has the negativity triple of some normalized family state. The conjecture is refuted.","title":"The Allen–Meyer family does not give the entire three-qutrit negativity region","url":"https://the-omega-institute.github.io/trureturing-pages/research/allen-meyer-2017-negativity-region-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39a592842dc64ef1af1c70755cec813c14fc740e0295e83ecbc251d851849dc9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:allouche-shallit-2021-sixfold-evil-odious","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Allouche and Shallit, *Additive properties of the evil and odious numbers and similar sequences*, arXiv:2112.13627v3, Section 4, Conjecture 12; Funct. Approx. Comment. Math. 70(1) (2024), 55–69, DOI 10.7169/facm/2108:\n\n> (a) Both $r_6(n)$ and $s_6(n)$ are eventually strictly increasing.\n>\n> (a) $r_6(n)\n> (b) $s_6(n) Question 6. Can it be determined whether or not the number of stochastically\n> recurrent states dominates the set of stable states? I.e. can it be decided\n> |Sto(K_{m,n})| ≶ n^{m−1} m^n / 2 ?","title":"Question 6 of Alofi and Dukes: do the stochastically recurrent states of K_{m,n} dominate?","url":"https://the-omega-institute.github.io/trureturing-pages/research/alofi-2024-stochastic-sandpile-bipartite-half/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a623830a4978ad77a2c78b931093077c8fde3a04ab0e1049baca5b4b7fe8bb50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:aloy-mullerrigat-tura-fadel-2024-three-outcome-pibi-validity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Aloy, G. Müller-Rigat, J. Tura and M. Fadel, *Deriving three-outcome\npermutationally invariant Bell inequalities*, arXiv:2406.11792v1,\nEntropy 26 (2024), 816, Section III.A:\n\n> To give a concrete example, we propose for any $N>3$ the five 3PIBIs shown in Tab. III.\n\n> At this point, for each conjectured inequality we have to prove that it is indeed valid for arbitrary number of parties $N$, or at least for all $N$ larger than a minimum number.\n\nThere are $N$ parties, inputs $x_i\\in\\{0,1\\}$ and outcomes\n$a_i\\in\\{0,1,2\\}$. The one-party observables sum marginal probabilities;\nthe two-party observables sum products over ordered distinct parties.\nThe five symmetrized observables and all six coefficients of each\nTable III row are retained literally in `P1`, `P2`, `Pt0`, `Pt00`,\n`Pt01`, `Pt10`, `Pt11`, `bell` and `table`.\n\n| Row | $(\\alpha_1,\\alpha_2,\\alpha_3,\\alpha_4,\\alpha_5,\\beta_c)$ |\n| --- | --- |\n| 1 | $(1,1,0,-2,0,0)$ |\n| 2 | $(1,1,-2,-2,2,0)$ |\n| 3 | $(-2,1,2,2,0,4)$ |\n| 4 | $(-6,1,4,4,2,12)$ |\n| 5 | $(-6,1,4,0,0,24)$ |","title":"Validity of the five three-outcome PI Bell inequalities","url":"https://the-omega-institute.github.io/trureturing-pages/research/aloy-mullerrigat-tura-fadel-2024-three-outcome-pibi-validity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e110d4812b81eda63f046a9ec9cd431115581f4231a1aed2bbb8cb77cf42153d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:alsing-birrittella-2024-log-negativity-sandwich","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. M. Alsing and R. J. Birrittella, *Numerical Evidence for a Bipartite Pure State Entanglement Witness from Approximate Analytical Diagonalization*, arXiv:2404.13725v1, Section IV, p. 11, Equation (22), state:\n\n> we have numerically found that 𝓛𝓝_a^avg(ρ_ab) ≤ LN_e(ρ_ab) ≤ LN_a(ρ_ab) when the mixed state is written as a pure state decomposition (PSD) ρ_ab = Σ_i p_i |ψ_i⟩⟨ψ_i|, and in addition the quantum amplitude matrix C_i of each pure state component |ψ_i⟩ = Σ_nm C_i,nm |n,m⟩ is uniformly generated (over the Haar measure) as a positive Hermitian matrix, C_i = C_i† ≥ 0.\n\n> Further, we find that the second inequality is surprisingly saturated, i.e. LN_e(ρ_ab) = LN_a(ρ_ab) in Eq.(22), if the uniformly randomly generated C_i matrices above are real, i.e. C_i = C_i^T.\n\nThe settling theorem proves both inequalities and the real-amplitude equality for every finite ensemble of positive semidefinite complex amplitude matrices, with nonnegative weights summing to one and $\\operatorname{tr}(C_iC_i^\\dagger)=1$. Different amplitudes need not commute. Zero weights and singular amplitudes are allowed.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite bipartite amplitudes $C_i$ | `d K : ℕ`, `C : Fin K → Matrix (Fin d) (Fin d) ℂ` in `claim` | Every finite dimension and ensemble size; no sampling restriction. |\n| $\\rho=\\sum_i p_i|\\psi_i\\rangle\\langle\\psi_i|$ | `ensembleState` | `Function.uncurry` is the literal amplitude vector; `Matrix.vecMulVec` forms its outer product, with real weights embedded in $\\mathbb C$. |\n| Probabilities and normalized pure components | Nonnegative weights, sum one, and trace normalization in `claim` | State and ensemble normalization made explicit. |\n| Positive Hermitian amplitudes | `(C i).PosSemidef` | Includes Hermitian symmetry and permits zero eigenvalues. |\n| Exact logarithmic negativity | `exactLogNegativity` | `Real.logb 2` of the existing trace norm of the existing second-factor partial transpose. |\n| Equations (11a)–(11b) | `pureApproxNegativity`, `pureApproxLogNegativity` | Literal half-sum over ordered unequal pairs and base-two logarithm. |\n| Equations (17b)–(17c) | `mixedApproxNegativity`, `mixedApproxLogNegativity` | Literal mixed entry difference, half-sum and logarithm. |\n| Average of the pure logarithms | First inequality in `claim` | Weighted sum of logarithms, with the logarithm inside the ensemble average. |\n| Upper inequality and real-case equality | Second inequality and guarded equality in `claim` | For Hermitian matrices, transpose symmetry expresses real entries. |\n\nThe public definition bodies are the conventions in [#19208](https://github.com/the-omega-institute/trureturing/issues/19208). The universal positive-semidefinite class contains the randomly generated source class. No simultaneous diagonalization, invertibility or positive-weight assumption is added.","title":"The Alsing–Birrittella logarithmic-negativity sandwich","url":"https://the-omega-institute.github.io/trureturing-pages/research/alsing-birrittella-2024-log-negativity-sandwich/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9119910a2d741e8ba863fe2d58e0e4b150dfedccb4b3c9abd761e85567da9979.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:altizio-2025-geometric-sum-disk","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David Altizio, *Zeros of Stern polynomials in the complex plane*,\narXiv:2511.03847v1, Section 4.3, p. 20, Remark 4, states:\n\n> One might suspect that |(z)_n| ≥ n|z| more generally whenever z ∈ B.\n> However, numerical evidence suggests that the value min_{z∈B} |(z)_n/z|\n> approaches 2π as n → ∞. It seems that this inequality is true only for\n> n = 5, 6, and 7, but we do not have a proof.\n\nSection 2 defines $(z)_n=1+z+\\cdots+z^{n-1}$ for positive integers $n$;\nSection 3 defines $B=\\{z\\in\\mathbb C:|z-2|\\le1\\}$.\nThe settled statement is\n\n$$\n\\forall n\\in\\mathbb N,\\quad n>0\\Longrightarrow\n\\left[\\left(\\forall z\\in B,\\ n|z|\\le\\left|\\sum_{k=0}^{n-1}z^k\\right|\\right)\n\\Longleftrightarrow n\\in\\{5,6,7\\}\\right].\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive integer $n$ | Natural $n$ with $0 Finally, we state a conjecture for the greatest component of $\\nu(\\rho)\\otimes\\lambda(\\rho)$ that allows us to obtain a nonoptimal but nontrivial bound for the tensor-product majorization relation, which depends only on the dimension of the Hilbert space.\n\n> Conjecture 5. Let $\\rho\\in\\mathcal S(\\mathcal H_d)$. Then $\\nu_1(\\rho)\\lambda_1(\\rho)\\ge\\frac1d$. (20)\n\n> For $d=2$ the conjecture is clearly valid, whereas for $d=3$ we have numerical evidence of its validity.\n\nHere $\\mathcal S(\\mathcal H_d)$ is the set of density operators on the $d$-dimensional Hilbert space, with a fixed incoherent basis. Section 2.2, Eqs. (9)–(12), PDF p. 3, defines the pure-state squared-amplitude vector $\\mu(|\\psi\\rangle\\langle\\psi|)$, its decreasing rearrangement $\\mu^\\downarrow$, and\n\n$\\mathcal U^{\\rm psd}(\\rho)=\\{\\sum_jp_j\\mu^\\downarrow(|\\psi_j\\rangle\\langle\\psi_j|):\\rho=\\sum_jp_j|\\psi_j\\rangle\\langle\\psi_j|\\}$,\n\nwhere the ensembles are finite, their weights are nonnegative and sum to one, and their vectors have norm one. The source defines $\\nu(\\rho)=\\bigvee\\mathcal U^{\\rm psd}(\\rho)$ in the majorization lattice; Section 2.3, PDF p. 4, defines $\\lambda(\\rho)$ as the decreasingly ordered spectrum. Thus the subscript $1$ denotes the greatest component in each vector. The definitions are also recorded in [the literature note](../Library/Entropy/alvarezlosadaportesibosyk2023complementarity.md).\n\nIssue [#23165](https://github.com/the-omega-institute/trureturing/issues/23165) fixes the formal reading: for every $d\\ge1$, every positive semidefinite complex $d\\times d$ matrix $\\rho$ of trace one, and every decreasing, nonnegative vector $\\nu$ of sum one that majorizes every member of $\\mathcal U^{\\rm psd}(\\rho)$, one has $\\nu_1\\lambda_1(\\rho)\\ge1/d$. This is proved. Applying it to the source's supremum uses that supremum's upper-bound property; the module does not construct the majorization lattice.","title":"The coherence vector and spectrum satisfy the product bound in every positive dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/alvarez-losada-portesi-bosyk-2023-coherence-spectrum-product-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9265685289f0d9f9d98612b57aecd93dfa3381ec0ac166258ae248de9929b557.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ambainis-2024-random-access-code-two-setting-advantage","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ambainis, D. Kravchenko, S. Sazim, J. Bae and A. Rai, *Quantum advantages in (n,d)↦1 random access codes*, arXiv:1510.03045v4, New J. Phys. 26 (2024) 123023, Section 5:\n\n> We conjecture that the quantum advantages would appear in $(n,d) \\mapsto 1$ RACs in general and leave it as an open question. The problem also addresses minimal resources, i.e., two measurement settings, to achieve quantum advantages in information theory.\n\nThe scenario, the figure of merit and the classical strategies with shared randomness are quoted in [the literature note](../Library/QuantumContext/ambainis2024racadvantage.md).\n\nIssue [#19414](https://github.com/the-omega-institute/trureturing/issues/19414) fixes the reading. For all n, d ≥ 2, some pure-state quantum strategy decoded by two projective measurements has an average success probability strictly above that of every classical strategy with shared randomness. The conjecture is proved.","title":"Every (n,d)↦1 random access code has a quantum advantage with two measurement settings","url":"https://the-omega-institute.github.io/trureturing-pages/research/ambainis-2024-random-access-code-two-setting-advantage/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cbaada46a67f57e00aa52156e30b9f474ff2c9204d6e56457ff90a7c3c849c20.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:amdeberhan-chen-moll-sagan-2014-reciprocal-tail-floor-conjecture-4-6","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Amdeberhan, Chen, Moll and Sagan, *Generalized Fibonacci polynomials and\nFibonomial coefficients*, Annals of Combinatorics 18 (2014), 541–562,\nConjecture 4.6, first clause:\n\n> If $s>t\\geq1$ with $(s,-t)\\ne(2,-1)$ and $n,r\\in\\mathbb P$ then\n> $\\left\\lfloor\\left(\\sum_{k=n}^{\\infty}1/\\{rk\\}_{s,-t}\\right)^{-1}\\right\\rfloor\n> =\\{rn\\}_{s,-t}-\\{r(n-1)\\}_{s,-t}-1$.\n\nThe source defines $\\mathbb P$ as the positive integers and\n$\\{0\\}_{s,t}=0$, $\\{1\\}_{s,t}=1$,\n$\\{j+2\\}_{s,t}=s\\{j+1\\}_{s,t}+t\\{j\\}_{s,t}$.\nThe squared-reciprocal clause is a separate question.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Integral coefficients and the three recurrence cases | `U` | Literal recursive definition over integers and natural indices. |\n| $s>t\\geq1$ | `claim`'s `t < s` and `1 ≤ t` | Same inequalities. |\n| $(s,-t)\\ne(2,-1)$ | The excluded-pair hypothesis | Same pair and signs. |\n| $n,r\\in\\mathbb P$ | Natural indices with `1 ≤ n`, `1 ≤ r` | Exactly positive indices. |\n| Tail starting at $n$ | Real reciprocal of `U s (-t) (r * (n + k))`, summed over natural $k$ | Reindexing, with the integer denominator cast to the reals. |\n| Convergence presupposed by the source | Explicit `Summable` conjunct | Prevents a divergent `tsum` from satisfying the identity vacuously. |\n| Reciprocal, integer floor and difference endpoint | Second conjunct of `claim` | Natural subtraction at $n-1$ is ordinary subtraction because $n\\geq1$. |","title":"Floors of reciprocal generalized Fibonacci tails","url":"https://the-omega-institute.github.io/trureturing-pages/research/amdeberhan-chen-moll-sagan-2014-reciprocal-tail-floor-conjecture-4-6/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c9fd56832c0ea6b5dd79e770e3c66db70f6e98839230bb2a29678f8d7e25333.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andoyo-2026-pisano-question-5-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jason D. Andoyo, *(a,b)-Fibonacci-Legendre Cordial Graphs and k-Pisano-Legendre Primes*, Question 5.2, asks:\n\n> Suppose that $a$ and $b$ are integers with $(a,b)\\ne(0,0)$. For every $k\\in\\chi(a,b)$, is $|\\vartheta_{(a,b)}(k)|=\\aleph_0$?\n\nThe exact locator is [the versioned preprint, Question 5.2](https://arxiv.org/html/2601.10561v1#S5.Thmquestion2). The [journal text](https://www.technoskypub.com/wp-content/uploads/2026/04/acm-090111.pdf) has DOI 10.62072/acm.2026.09011. The existing Library citation `andoyo2026pisanolegendrecollision` identifies this same article and is reused.\n\nStatement echo: $F_0=a$, $F_1=b$, $F_{n+2}=F_{n+1}+F_n$; $\\pi_p(a,b)$ is the least positive scalar period, valid at every index including zero. For $j\\in\\{-1,0,1\\}$, $N_j$ counts indices $0\\le n<\\pi_p(a,b)$ with Legendre symbol $j$, with multiplicity. Set $K=N_1-N_{-1}-N_0$ in the integers, $\\chi(a,b)=\\{K(p,a,b):p\\text{ odd prime}\\}$, and $\\vartheta_{(a,b)}(k)=\\{p:p\\text{ odd prime},K(p,a,b)=k\\}$. The Lean `claim` asserts that every such attained fiber is infinite, equivalently countably infinite because it is a subset of the natural numbers. Seeds are signed and may be nonprimitive; primes dividing both are included.","title":"Andoyo Question 5.2: infinitude of exact prime fibers","url":"https://the-omega-institute.github.io/trureturing-pages/research/andoyo-2026-pisano-question-5-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/519f273ceb5c9566cae97802ecfc3cadcd523a9320c40ef5f3ecd683ff7e83b4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andoyo-2026-pisano-question-5-3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 5, Question 5.3 asks:\n\n> Are there values of a₁, a₂, b₁, and b₂ for which χ(a₁,b₁)=χ(a₂,b₂)?\n\nThe preceding paragraph requires a₁,b₁,a₂,b₂∈ℤ, distinct ordered pairs,\nand neither pair equal to (0,0). Definitions 2.6 and 2.7 specify the least\npositive scalar period at every nonnegative index, the three index counts\nfor Legendre values 1,−1,0, and their integer difference over all odd primes.","title":"Andoyo's equal Pisano sets question","url":"https://the-omega-institute.github.io/trureturing-pages/research/andoyo-2026-pisano-question-5-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ead3767fbd3b4b61a0e598bd20683c50799c2584e8e23201e947b8360f4491ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andrade-severini-2018-star-graph-unitary-similarity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. M. Andrade and S. Severini, *Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs*, arXiv:1808.01154v2, state Claim 1 after defining the star-graph maps. In the source TeX `qgl.tex`, lines 522–530 read:\n\n> The quantum evolution maps $U^{S}_{S_n}(k)$ and $U^{G}_{S_n}(k)$ are unitarily similar.\n>\n> Given the properties of these maps, there are strong reasons to believe that this claim works for every $n$. Although a proof for a general $n$ is not known, we checked this for $n=2,3,4,5$ by using Specht's theorem \\cite{Book.2012.Horn}.\n\nThe formal problem defines `scatteringMatrix` and the diagonal `propagationMatrix` on the index type `Fin (n - 1) × Fin 2`. It asks whether, for every $n : \\mathbb N$ with $2 \\le n$, every $r : \\mathrm{Fin}(n-1) \\to \\mathbb C$, every $r1,t1 : \\mathbb C$, every $k : \\mathbb R$, and every $\\ell : \\mathrm{Fin}(n-1) \\to \\mathbb R$, there is a $w$ in `Matrix.unitaryGroup (Fin (n - 1) × Fin 2) ℂ` such that\n\n$w\\, (\\operatorname{scatteringMatrix}(r,r1,t1)\\,\\operatorname{propagationMatrix}(k,\\ell))\\,\\operatorname{star}(w) = \\operatorname{propagationMatrix}(k,\\ell)\\,\\operatorname{scatteringMatrix}(r,r1,t1)$.\n\nThe source identifies these products as $U^S_{S_n}(k)=S_{S_n}(k)D_{S_n}(k)$ and $U^G_{S_n}(k)=D_{S_n}(k)S_{S_n}(k)$ at lines 383–405 and 472–474.","title":"Andrade–Severini star-graph evolution maps are unitarily similar","url":"https://the-omega-institute.github.io/trureturing-pages/research/andrade-severini-2018-star-graph-unitary-similarity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86f9405cee025a4fedc2c4e583e9c069ec531a14fb016ec0631df2a55ce76a6d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andreadakis-2024-algebra-otoc-long-time-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Andreadakis, E. Dallas, P. Zanardi, *Long-time Quantum Scrambling and\nGeneralized Tensor Product Structures*, Phys. Rev. A 109, 052424 (2024),\narXiv:2312.13386v2, the Conjecture of Section IV:\n\n> Let $H$ be an NRC Hamiltonian and $\\mathcal{A}_f$ be the equivalence class of isomorphic algebras of the form $\\mathcal{A}\\cong \\oplus_{J=1}^{d_\\mathcal{Z}} \\mathds{1}_{n_J} \\otimes \\mathcal{L}(\\mathbb{C}^{d_J} )$. Then, the algebra corresponding to a gTPS (\\cref{structure_H}) for which the Hamiltonian eigenstates have the form $\\{\\ket{\\psi_J}\\otimes\\ket{\\phi_J} \\, \\vert \\, J=1,\\dots d_\\mathcal{Z}; \\, \\psi_J = 1,\\dots, n_J; \\, \\phi_J = 1,\\dots , d_J \\}$ minimizes the $\\mathcal{A}$-OTOC LTA over the family of isomorphic algebras. The minimum value is $\\overline{G_{\\mathcal{A}}}^{NRC}_{min}=1-\\frac{1}{d} \\left( \\sum_J d_J + \\sum_J n_J - d_\\mathcal{Z} \\right).$\n\nThe long-time average under the non-resonance condition (NRC) is Eq. (6)\n(Proposition 1 (ii)): for the eigenbasis $\\{\\phi_k\\}$ of $H$ and\n$\\Pi_k=|\\phi_k\\rangle\\langle\\phi_k|$,\n$$\\overline{G}^{NRC}=1-\\frac1d\\sum_{\\mathcal X\\in\\{\\mathcal A,\\mathcal A'\\}}\n\\Big(\\operatorname{Tr}(R^{(0),\\mathcal X}R^{(1),\\mathcal X'})-\\tfrac12\n\\operatorname{Tr}(R^{(0),\\mathcal X}_DR^{(1),\\mathcal X'}_D)\\Big),$$\nwith $R^{(0),\\mathcal X}_{lk}=\\|\\mathbb P_{\\mathcal X}(|\\phi_k\\rangle\n\\langle\\phi_l|)\\|_2^2$, $R^{(1),\\mathcal X}_{kl}=\\langle\\mathbb\nP_{\\mathcal X}(\\Pi_k),\\mathbb P_{\\mathcal X}(\\Pi_l)\\rangle$,\n$\\mathbb P_{\\mathcal A}=\\oplus_J\\frac{\\mathbb 1_{n_J}}{n_J}\\otimes\n\\operatorname{Tr}_{n_J}$ and $\\mathbb P_{\\mathcal A'}=\\oplus_J\n\\operatorname{Tr}_{d_J}\\otimes\\frac{\\mathbb 1_{d_J}}{d_J}$. The paper\nproves the minimum for Hamiltonians that are block diagonal in the sectors,\nand for $d_\\mathcal Z=1$ over all NRC Hamiltonians (Proposition 2); its\nSection IV A reports numerical evidence up to $d=40$.","title":"The minimum of the long-time algebra OTOC","url":"https://the-omega-institute.github.io/trureturing-pages/research/andreadakis-2024-algebra-otoc-long-time-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6306f1b3cc2c423b04a6951fa163683e48aaedee5db9603d1cc10a586979e635.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andrews-el-bachraoui-c24-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"George E. Andrews and Mohamed El Bachraoui, *Certain positive q-series and inequalities for two-color partitions*,\narXiv:2507.09276v1, Section 2, Conjecture 2. With\nΣ_n C'(k,m,n) qⁿ = Σ_{j≥0} q^{m(2j+1)} (q^{2j+2}, q^{2j+2k}; q²)_∞ / (q^{2j+1}; q²)_∞² (equation (1.2)), the\nconjecture states that C'(2,4,n) ≥ 0 for every n ≥ 0. In contrast, the paper notes that C'(2,5,n) changes sign.","title":"Positivity of the Two-Color Partition Series C'(2,4,n)","url":"https://the-omega-institute.github.io/trureturing-pages/research/andrews-el-bachraoui-c24-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28095193a6a9c3aa41a0e0c0941956d0cc6b308ecfeb6c159fd40ab25c989631.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andrews-el-bachraoui-d22-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"George E. Andrews and Mohamed El Bachraoui, *Certain positive q-series and inequalities for two-color partitions*,\narXiv:2507.09276v1, Section 3, Conjecture 3. With\nΣ_n D'(k,m,n) qⁿ = Σ_{j≥0} q^{m(2j+2)} (q^{2j+4}, q^{2j+2+2k}; q²)_∞ / (q^{2j+3}; q²)_∞² (equation (1.3)), the\nconjecture states that D'(2,2,n) ≥ 0 for every n ≥ 0. Equivalently, among the two-color partitions of Definition 2\nwith k = m = 2, those with an odd number of even parts above the smallest part never outnumber those with an even\nnumber.","title":"Positivity of the Two-Color Partition Series D'(2,2,n)","url":"https://the-omega-institute.github.io/trureturing-pages/research/andrews-el-bachraoui-d22-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbe0004bbd068fa5b030f2648ce8c57079d0d3b1c0220069cf2fef31f99cdf0d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andrews-el-bachraoui-d23-sign-pattern","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"George E. Andrews and Mohamed El Bachraoui, *Certain positive q-series and inequalities for two-color partitions*,\narXiv:2507.09276v1, Section 3, Conjecture 4. With\nΣ_n D'(k,m,n) qⁿ = Σ_{j≥0} q^{m(2j+2)} (q^{2j+4}, q^{2j+2+2k}; q²)_∞ / (q^{2j+3}; q²)_∞² (equation (1.3)), the\nconjecture states that the only negative coefficients of Σ_n D'(2,3,n) qⁿ occur at n = 10 and n = 22.","title":"The Negative Coefficients of the Two-Color Partition Series D'(2,3,n)","url":"https://the-omega-institute.github.io/trureturing-pages/research/andrews-el-bachraoui-d23-sign-pattern/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d519e1b688f0e80bf8483539d351029d26d7411fde3e3147efcc046d7ff7e892.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:andrews-el-bachraoui-dprime-eventual-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"George E. Andrews and Mohamed El Bachraoui, *Certain positive q-series and inequalities for two-color partitions*,\narXiv:2507.09276v1, Conjecture 5: “If k>m, then sum of D'(k,m,n) q^n is eventually positive.” The source defines\npositive to mean that all coefficients are nonnegative. Its parameters k and m are positive integers.\n\nThe exact assertion is: for every natural k and m with 1 ≤ m < k, there exists a natural N depending on k and m,\nsuch that D'(k,m,n) ≥ 0 for every natural n ≥ N. Equation (1.3) defines this coefficient by\n\nΣ_n D'(k,m,n) q^n = Σ_{j≥0} q^{m(2j+2)} (q^{2j+4};q²)_∞ (q^{2j+2+2k};q²)_∞ / (q^{2j+3};q²)_∞².\n\nThe original `AndrewsElBachraouiDefs.dCoeff` uses precisely these factors and degree n, without a coefficient shift.\nFor m ≥ 1, omitted summands and product factors have degree above n. The finite truncation therefore retains the\nsource coefficient exactly.","title":"Eventual positivity of the complete D-prime family","url":"https://the-omega-institute.github.io/trureturing-pages/research/andrews-el-bachraoui-dprime-eventual-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2078a21cf4f5bf32eacb6ee7500e61c333296b31fa92c26de0a38840f13adfb4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:angelinos-2022-bform-averaged-enumerator","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Angelinos, D. Chakraborty and A. Dymarsky (arXiv:2206.14825, hep-th; JHEP\n11 (2022) 118) build Narain CFTs from codes over `F_p × F_p`. For prime `p` the\ngenerating matrix is `(I | B^T)` with `B` antisymmetric mod `p` and `B_ii = 0`;\nthe codewords are `(r, B^T r)`, `r ∈ Z_p^c`. Averaging the full enumerator\npolynomial over the `p^{c(c−1)/2}` such `B` at `x_ab = t_a t_b`, `t_a = t_{−a}`,\nthey state:\n\n> We conjecture the form of corresponding averaged enumerator polynomial based\n> on invariance under MacWilliams identity and explicit checks for\n> sufficiently small n and prime p\n\nwith eq. (barP):\n`P̄ = t_0^{2c} + (Σ_{k=0}^{p−1} (Σ_{a,b} cos(2πkab/p) t_a t_b)^c − p t_0^c (Σ_a t_a)^c) / p^c`.\n\nIssue #11296 fixes the reading: `p` prime, `c ≥ 0`, `t : Z/p → ℂ` even, and in\nthe cosine `k, a, b` are the representatives `0, …, p − 1`.","title":"The averaged enumerator of B-form codes","url":"https://the-omega-institute.github.io/trureturing-pages/research/angelinos-2022-bform-averaged-enumerator/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6f0a1f59631ca1decfdd5bb70b6ce6ee2210c43f4966df0254c8fdac9fa881f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:anholcer-2025-global-coalition-universality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Anholcer, Besharati, Mojdeh and Samadzadeh, *Global coalitions in graphs*, arXiv:2509.15386v4, Section 7 Problem 58, PDF p. 29, asks whether every graph is realisable as a global coalition graph. The earlier alias is Besharati, Mojdeh and Samadzadeh, *Global coalition sets in graphs*, arXiv:2509.15386v1, Section 6 Problem 4, PDF p. 19. Both aliases identify one target.\n\n### Original question and definitions\n\nThe following is the complete question and its conventions as recorded in preregistration [#23084](https://github.com/the-omega-institute/trureturing/issues/23084):\n\n> The current complete question asks whether every finite simple undirected graph H of order at least two is isomorphic to the global coalition graph of some finite simple undirected host G of order at least two and some actual gc-partition of G. A gc-partition need not maximize the number of blocks. There is no connectedness, isolate-free, positive-edge-count or independent-block restriction.\n>\n> For the same vertex set, S dominates G precisely when every vertex is in S or adjacent to a member of S. Global domination means domination of both G and its genuine loopless complement. Two actual subsets A,B form a global coalition precisely when they are disjoint, neither is individually globally dominating, and their union globally dominates. The partition consists of nonempty pairwise-disjoint actual subsets covering all host vertices; every block has another distinct block in that same partition forming such a coalition. There is no singleton-dominating exception. The global coalition graph has these actual blocks as vertices and adjacency exactly the full global-coalition relation. Realization means genuine graph isomorphism.\n>\n> Current v4 places the order>=2 convention after Definition 2. V1 has no corresponding global lower-order restriction; its unrestricted full question is preserved separately, not silently narrowed. The proposed two-vertex witness obeys both versions and they count as one target. V3 withdrew v2 because it had been uploaded without consulting all authors and contained incorrect results; current v4 is restored and retains the question.\n\nThe formal claim is the literal closed assertion\n```lean\ndef claim : Prop :=\n ∀ (V : Type) [Finite V], 2 ≤ Nat.card V → ∀ H : SimpleGraph V,\n ∃ (W : Type) (_ : Finite W) (G : SimpleGraph W), 2 ≤ Nat.card W ∧\n ∃ π : GCPartition G, Nonempty (globalCoalitionGraph G π ≃g H)\n```\n\nThe model retains actual finite Type-valued target and host types, native simple graphs and native loopless complements, genuine `Setoid.IsPartition` nonempty/disjoint-cover structure, and a distinct actual coalition partner for every actual block. The v1 unrestricted-order alias is a source boundary, not a second theorem or credit.","title":"Global-coalition graph universality","url":"https://the-omega-institute.github.io/trureturing-pages/research/anholcer-2025-global-coalition-universality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92e17b9e9e10e0ecd762c75867280153908e655b58e66511ffb4e331eb161e31.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:anholcer-hypergraph-alon-tarsi","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjectures 4 and 28 of Anholcer et al., *Electronic Journal of Combinatorics*\n33(3), P3.74 (arXiv:2501.00157v2), assert AT(p) ≤ 2 ceil(ed(H)) + 1 for every\nhypergraph polynomial, respectively every fully unbalanced hypergraph polynomial.\nThe field is arbitrary; multiple edges are allowed. Each edge has at least two\nvertices and every incident coefficient is nonzero. Full unbalance requires two\nunequal incident coefficients on each edge. Density counts induced occurrences\nover nonempty vertex subsets, and AT uses nonzero maximum-total-degree monomials.","title":"The fixed-polynomial hypergraph Alon-Tarsi bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/anholcer-hypergraph-alon-tarsi/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e66b6f7fd195ef7e32cf93ad6a08931c23ee75bffe2b85139d16cf9cbe96a68b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:antipin-2021-four-qubit-genuinely-entangled-subspace-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. V. Antipin, *Construction of genuinely multipartite entangled subspaces and the associated bounds on entanglement measures for mixed states*, arXiv:2104.09664v1 (J. Phys. A: Math. Theor. 54, 505303 (2021)), constructs genuinely entangled subspaces (GESs): subspaces in which every nonzero vector is entangled across every bipartition of the parties.\n- **The construction.** §III.B.2 starts from the seven orthonormal vectors (35a)–(35g) of ℂ⁴ ⊗ ℂ⁴, each of the form √λ_i |p_i⟩⊗|q_i⟩ + √(1 − λ_i) |p′_i⟩⊗|q′_i⟩ with 0 < λ_i < 1. It splits each ququart into two qubits by the substitutions (39) for the first factor and (40) for the second. This gives a seven-dimensional subspace S_λ of four qubits A₁A₂B₁B₂.\n- **The conjecture.** With every λ_i = 1/2 the source obtains a GES. Then: \"Actually, we've done calculations for several different sets of values of λ_i and each time obtained a GES. We can conjecture that eqs. (35) together with eqs. (39) and (40) determine a GES for any set of values of λ_i.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/antipin2021genuinely.md).\n\nIssue [#17193](https://github.com/the-omega-institute/trureturing/issues/17193) fixes the reading: S_λ is a GES for every λ ∈ (0, 1)⁷. The conjecture is refuted.","title":"Antipin's four-qubit subspaces are not genuinely entangled for every parameter","url":"https://the-omega-institute.github.io/trureturing-pages/research/antipin-2021-four-qubit-genuinely-entangled-subspace-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8c1d3505eed7b514701da09144ea1e7f630e52cb3d5e821b68e92174758729f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:apte-parekh-sud-2025-token-graph-cut-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Apte, Parekh and Sud, *Conjectured Bounds for 2-Local Hamiltonians via Token Graphs*, arXiv:2506.03441v2, Appendix D.2, Conjecture 7, page 39, states:\n\n> In contrast, the following bound was shown to hold on our dataset\n>\n> For all $G$ and for all $1\\le k\\le\\lfloor n/2\\rfloor$, $\\lambda_{\\min}(A(F_k(G)))\\ge-(C(G)+M_k(G))/2$.\n\nHere $F_k(G)$ is the token graph, $C(G)$ is the maximum cut weight, and $M_k(G)$ is the maximum matching weight among matchings with at most $k$ edges. The settlement is **Refuted** by a unit-weight simple graph with $n=200$ and $k=100$.\n\n### Statement echo\n\n| source clause | Lean counterpart | fidelity |\n| --- | --- | --- |\n| Finite graph $G$ on $n$ vertices | `claim` quantifies `SimpleGraph (Fin n)`. | A labelled finite simple graph with unit edge weights is within the source's universal range. |\n| Token graph $F_k(G)$ | Frozen `TokenGraphLaplacianRadiusRefutation.tokenGraph` | Vertices are $k$-subsets; adjacency moves one token along an original edge. |\n| Maximum cut $C(G)$ | `crosses` and `maxCut` | Exactly one endpoint is in the cut; each unordered unit-weight edge is counted once; the supremum ranges over all cuts. |\n| Matching weight with at most $k$ edges | `maxMatchingLe` | Edge subsets have pairwise disjoint endpoint sets and cardinality at most $k$; weight equals cardinality. |\n| Minimum adjacency eigenvalue | `lambdaMin` applied to the real adjacency matrix | Minimum of the finite real Hermitian eigenvalue family; the zero extensions for empty or non-Hermitian matrices are unused. |\n| $1\\le k\\le\\lfloor n/2\\rfloor$ | The two hypotheses of `claim` | Natural division by two gives the floor. |\n| The proposed lower bound | The conclusion of `claim` | Cut and matching cardinalities are cast to real numbers before division by two. |\n| Failure of the universal assertion | `result : ¬ claim` | The instance uses the five-cycle blow-up with block size 40. |\n\nDecidable adjacency is an implementation instance, not a restriction of the finite graphs. Connectedness and bipartiteness are not extra hypotheses. Public definitions display the literal source notions; the phase vector, block occupancies and other proof constructions are private.","title":"apte parekh sud 2025 token graph cut bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/apte-parekh-sud-2025-token-graph-cut-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1dcda9591c782d07ca86437d3e95d9ddd0229836b929e9f04eab98f4ac8afb73.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arai-2023-advantage-quantum-non-simulability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Arai, *The Diversity of Entanglement Structures with Self-Duality and Non-Orthogonal State Discrimination in General Probabilistic Theories*, Ph.D. thesis, arXiv:2302.12361v1, Section 5.2 (Open Problems), Theme B:\n\n> Similarly, it is considered that AQ measurement might be non-simulable because AQ measurement also has an extraordinary performance. This is an open problem.\n\nHere:\n- **DOVM:** a two-outcome measurement $\\{M_1,M_2\\}$ on $\\mathcal H_A\\otimes\\mathcal H_B$ whose effects are block positive (nonnegative on product vectors) and sum to $I$.\n- **AQ:** one effect satisfies $\\lambda_1(M_i)<0$ and $1+\\lambda_1(M_i)<\\lambda_d(M_i)<1$.\n- **Domain:** the states $\\rho$ with $\\mathrm{tr}\\,\\rho M_i\\ge0$ for both $i$.\n- **Quantum $n$-simulable:** some POVM $\\{N_i\\}$ on $(\\mathcal H_A\\otimes\\mathcal H_B)^{\\otimes n}$ has $\\mathrm{tr}\\,\\rho^{\\otimes n}N_i=\\mathrm{tr}\\,\\rho M_i$ on the domain.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/arai2023diversity.md).\n\nIssue [#20035](https://github.com/the-omega-institute/trureturing/issues/20035) fixes the reading: no AQ DOVM is quantum $n$-simulable for any $n$, with the simulation identity required only on trace-one domain states, which makes the statement stronger than the literal domain. This is proved.","title":"Advantage-quantum measurements are not quantum n-simulable","url":"https://the-omega-institute.github.io/trureturing-pages/research/arai-2023-advantage-quantum-non-simulability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86741c7544b05dc7fc52262508c97657e77c6e23745a7314c119fa629c619e34.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:araoka-2026-yang-baxter-automaton-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Araoka and T. Tokihiro (arXiv:2602.17148, nlin.SI; Math. Phys. Anal. Geom.\n29, 30 (2026)) build a cellular automaton over a finite field `F` from the\nR-matrix `R(x, y) = (y + f(x + y), x − f(x + y))`. In characteristic 2 the\nYang–Baxter equation for `R` reduces to\n`f(x) + f(x + f(y)) = f(x + f(y + f(x)))`. The automaton runs `R` along a row\nof `N` cells, `R(x_i, y_{i−1}) = (x_i', y_i)` with `y_0 = b`, and the helical\nboundary condition feeds `y_N` back as the next boundary value. The paper\nstates:\n\n> Conjecture. The cellular automaton thus constructed over a finite field of\n> order 2^n has a period that is a divisor of the order of the field.\n\nand proves it for orders 4 and 8. Issue #11405 fixes the reading: every finite\nfield of characteristic 2, every bijective solution `f` of the reduced\nequation, every `N`, and \"period divides `q`\" as `step^q = id` on the pair of\ncell values and boundary value.","title":"The Yang–Baxter cellular automata over F_{2^n} have period dividing 2^n","url":"https://the-omega-institute.github.io/trureturing-pages/research/araoka-2026-yang-baxter-automaton-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef80c11f534e940ed105c6f79f1336cab2ed59a0368774f6309b206c60d03f4b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:araujo-2026-orthodox-idempotent-ordering-complete-mapping-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Araújo, Bentz, Cameron, Hendrey, and Kinyon, *Complete Mappings of\nSemigroups*, arXiv:2608.25092v1 (2026), define a complete mapping in the\nabstract on printed page 1:\n\n> A complete mapping of a semigroup S is a bijection α : S → S such that the\n> map θ : S → S defined by xθ = x · xα is also a bijection.\n\nSection 2 defines regularity on printed page 6:\n\n> An element a of a semigroup S is said to be regular if there exists b ∈ S\n> such that aba = a. If every element of a semigroup is regular, then the\n> semigroup itself is said to be regular.\n\nSection 14 defines E-semigroups and orthodox semigroups on printed page 59:\n\n> A semigroup S is said to be an E-semigroup if the set E(S) of idempotents is\n> a subsemigroup of S. A regular E-semigroup is said to be orthodox.\n\nTheorem 14.3 on printed page 59 states:\n\n> **Theorem 14.3.** Let S be a finite E-semigroup with a complete mapping. Then\n> there exists an ordering c1, . . . , cn of all elements of S such that\n> c1 · · · cn is an idempotent.\n\nThe following remark on printed page 59 records the zero case:\n\n> We remark that for semigroups with zero this conclusion can be uninformative:\n> nothing rules out the possibility that the product of elements given by\n> Theorem 14.3 is zero, even if restricted to the non-zero elements. For\n> instance, for the 3-element commutative idempotent semigroup S = {0, 1, 2}\n> with 1 · 2 = 0, let α = θ = id_S and note that Pα = {{0}, {1}, {2}}.\n\nSection 15 then says on printed page 60:\n\n> The converse of Theorem 14.3 leads to the following question. An affirmative\n> answer would generalize the Hall–Paige conjecture.\n\nProblem 15.5 on printed page 60 asks:\n\n> **Problem 15.5.** Let S be an orthodox semigroup with an ordering c1, . . . ,\n> cn of all elements of S such that c1 · · · cn is an idempotent. Must S have a\n> complete mapping?\n\nThe formal reading makes six decisions: (i) an ordering of all elements is a\nduplicate-free finite list containing every element; (ii) `c₁ ⋯ c_n` is the\nleft fold; (iii) idempotent means `p * p = p`; (iv) orthodox means regular and\nclosed under multiplication of idempotents; (v) a complete mapping is a\nbijection whose product map is also bijective; and (vi) the question is the\nuniversal claim over `S : Type`, so one counterexample answers No.","title":"Problem 15.5 answered negatively by a five-element Clifford semigroup","url":"https://the-omega-institute.github.io/trureturing-pages/research/araujo-2026-orthodox-idempotent-ordering-complete-mapping-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59d17493efa5245ec58deac0957b36899ac834a559cf29b047393a19b3f269ed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:araujo-orlando-reffert-2020-quantum-crystal-yangian-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Araujo, D. Orlando and S. Reffert, *Quantum crystals, Kagome lattice, and plane partitions fermion-boson duality* (Phys. Rev. D 103, 026020 (2021); arXiv:2005.09103), study the quantum crystal melting Hamiltonian H = −J(Γ + Π) + V(√q O₊ + O₋/√q) on plane-partition states.\n- Γ and Π add and remove boxes in all possible ways.\n- O₊ and O₋ count the boxes that can be added and removed.\n- A fermion-boson correspondence, Eq. (46), defines bosonic states |Λ(N K)⟩⟩ = Σ_Λ χ_Λ[C(K)]|Λ⟩ through characters of symmetric groups; it preserves the number of boxes.\n\nSection VI conjectures: \"The growth operators Γ, Π and O±(Γ, Π) are linear combinations of the generators of the algebra Y[gl̂(1)], and the states |N K⟩⟩ fulfill the Y[gl̂(1)] MacMahon representation.\"\n- In that representation e_j adds and f_j removes a box on the bosonic states, Eq. (49), and φ_j acts diagonally on them, Eq. (50).\n- The authors conclude that \"O±(Π, Γ) ∈ B since they can be written as linear combinations of φ_j\".\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/araujoorlandoreffert2021quantumcrystals.md).\n\nIssue [#18238](https://github.com/the-omega-institute/trureturing/issues/18238) fixes the reading: the consequences of the conjecture at the three-box bosonic state of the identity class of S₃. The conjecture is refuted.","title":"The quantum crystal potential is not a Yangian combination on the bosonic states","url":"https://the-omega-institute.github.io/trureturing-pages/research/araujo-orlando-reffert-2020-quantum-crystal-yangian-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d1b0f99d184d17444ae477daa94756cec32e74f38d7717545d6b77fc3e34f11.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-bourne-cube-2143-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier deliberately anchors only the unnumbered counting conjecture\nin Section 5, \"Further directions and open questions\", page 13 of\narXiv:2505.05218v3. The worker extracted this sentence from the PDF; line\nwrapping is removed and mathematical glyphs are transcribed into inline LaTeX:\n\n> For example, based on the ideas similar to the ones in this paper, we conjecture\n> that the number of permutations that avoid the chain $(312,321 : \\varnothing : 2143)$\n> (i.e, those with the property that $\\pi$ avoids $\\{312,321\\}$ and $\\pi^3$\n> avoids $2143$) is equal to the number of compositions $d = (d_1,d_2,\\ldots,d_m)$\n> of $n$ so that all $d_i$ are 1 or 3, except for at most one.\n\nThus, for each positive `n`, count permutations of `n` avoiding 312 and 321\nwhose cubes avoid 2143, and compare with ordered compositions of `n` having\nat most one part outside `{1,3}`. The paper's broader questions about other\npatterns and higher powers are deliberately out of scope.","title":"Archer-Bourne cube-avoidance counting equality","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-bourne-cube-2143-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0465a60a62e0773677f0d9ddd0a3269b6603cb605dcb4fdb4f1fb86b73366d09.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-bourne-square-tetranacci-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kassie Archer and Noel Bourne, *Pattern avoidance in compositions and powers of permutations*, arXiv:2505.05218v3,\nSection 5, printed page 14:\n\n> For example, we conjecture that if we take a_n := a_n(312, 54321 : 132) then a_n = a_{n−1} + a_{n−2} + a_{n−3} +\n> a_{n−4} + n − 1 for n ≥ 6.\n\nHere a_n(312, 54321 : 132) is the number of permutations π of [n] that avoid 312 and 54321 and whose square π ∘ π\navoids 132.","title":"The Archer–Bourne Square Recurrence Fails at n = 11","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-bourne-square-tetranacci-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b26971d8b63dc9a8d9b4ef6cc70f90ff19d9c6849408c7dfd262262720c904b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-cyclic-4123-1324-tetranacci","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves and Millie Jeske, *Pattern-restricted\ncyclic permutations with a pattern-restricted cycle form*, arXiv:2408.15000v1, Section 4:\n\n> we conjecture that |A°_n(4123;1324)| is the (n−2)nd Tetranacci number\n\nHere `A°_n(4123; 1324)` is the set of cyclic permutations of `1, …, n` whose one-line notation avoids\n`4123` and all of whose cycle forms, the rotations of the cycle word, avoid `1324`.","title":"Cyclic Permutations Avoiding 4123 with 1324-Avoiding Cycle Forms","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-cyclic-4123-1324-tetranacci/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52f1ef9eb889c9dfe5743b41399b62332455900d171777533a4ff128f2a9544d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-cyclic-4132-1324-padovan","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves and Millie Jeske, *Pattern-restricted\ncyclic permutations with a pattern-restricted cycle form*, arXiv:2408.15000v1, Section 4:\n\n> we conjecture that … |A°_n(4132;1324)| is the (3n)th Padovan number\n\nHere `A°_n(4132; 1324)` is the set of cyclic permutations of `1, …, n` whose one-line notation avoids\n`4132` and all of whose cycle forms, the rotations of the cycle word, avoid `1324`.","title":"Cyclic Permutations Avoiding 4132 with 1324-Avoiding Cycle Forms","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-cyclic-4132-1324-padovan/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/117ffb89db8b744badea2c8552c316408f060f28040ccf8d3e767818ba1e7aa2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-laudone-theta-cube-231-312","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kassie Archer and Robert P. Laudone, *Pattern avoidance and the fundamental bijection*,\narXiv:2407.06338v1, Section 5, Conjecture 5.6:\n\n> For σ ∈ {231, 312}, the generating functions F_σ^k(x) are rational. A few examples of conjectured\n> generating functions for these patterns are found below. F_σ^3(x) = 1/(1 − x − x^2 − 2x^3)\n\nHere F_σ^k(x) = Σ f_n^k(σ) x^n and f_n^k(σ) counts the permutations of [n] avoiding σ and fixed by the\nk-th iterate of the fundamental bijection θ.","title":"Avoiders of 231 and 312 Fixed by the Third Iterate of the Fundamental Bijection","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-laudone-theta-cube-231-312/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69ce246d957cf1afe1f46bb9426bc084d682af3254d9c95bf07c64664feb0a41.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:archer-laudone-theta-iterate-132","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kassie Archer and Robert P. Laudone, *Pattern avoidance and the fundamental bijection*,\narXiv:2407.06338v1, Section 4, Conjecture 4.5:\n\n> For n ≥ 2, t_n^2(132) = k^3 + 3k^2 + 2k − 1 (n = 3k), k^3 + 4k^2 + 4k (n = 3k + 1),\n> k^3 + 5k^2 + 7k + 2 (n = 3k + 2). The conjectured values for k ≥ 3 are found in the table below:\n> 3n − 4, 2n − 1, n + 2 and 5 for k = 3, 4, 5 and k ≥ 6.\n\nHere t_n^k(σ) counts the permutations π of [n] such that π, θ(π), …, θ^k(π) all avoid σ.","title":"Permutations Whose First Iterates under the Fundamental Bijection Avoid 132","url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-laudone-theta-iterate-132/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38465ef0ec24902371ba981a2ff5ce5a801a8782190c40c9be4e83fcd91b0eee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arias-di-giulio-keski-vakkuri-tonni-2023-charge-sector-majorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Arias, G. Di Giulio, E. Keski-Vakkuri and E. Tonni, *Probing RG flows, symmetry resolution and quench dynamics through the capacity of entanglement*, arXiv:2301.02117v1, JHEP 03 (2023) 175, Section 7, “Symmetry-resolved related issues”, ask (`CDraft-Final.tex`, line 4140):\n\n> For instance, given two density matrices such that $\\rho\\succ \\sigma$, we may ask whether this majorization order survives once we decompose $\\rho$ and $\\sigma$ according to (\\ref{eq:decompositionrho}). One can try to diagnose this aspect using $M_A^{(n)}(q)$ in (\\ref{SR-M_n}), given that these quantities are entanglement monotones also in a fixed charge sector. We leave this analysis for future investigations.\n\nThe referenced equation is the normalized charge-block decomposition of Section 6, equation (6.1), TeX lines 3550–3569: $\\rho_A=\\bigoplus_q p(q)\\rho_A(q)$, with $p(q)=\\operatorname{Tr}(\\Pi_q\\rho_A)$ and $\\operatorname{Tr}\\rho_A(q)=1$. Section 6 obtains this decomposition for a reduced state commuting with its subsystem charge; the registered comparison uses one common subsystem charge operator. See [the literature note](../Library/Entropy/ariasdigiuliokeskivakkuritonni2023capacity.md).\n\nIssue [#22533](https://github.com/the-omega-institute/trureturing/issues/22533) fixes the sectorwise reading: does $\\rho\\succ\\sigma$ imply $\\rho(q)\\succ\\sigma(q)$ for every charge whose weight is positive in both states? This implication is refuted already for finite diagonal states.\n\nThe formal `claim` quantifies over every $n\\in\\mathbb N$, charge map $c:\\mathrm{Fin}\\,n\\to\\mathbb Z$ and real vectors $r,s:\\mathrm{Fin}\\,n\\to\\mathbb R$ with nonnegative entries and $\\sum_i r_i=\\sum_i s_i=1$. Put $p_r(q)=\\sum_{i:c(i)=q}r_i$, the module's reused `pushforward`, and let $\\operatorname{sec}_q(r)$ be the descending list of $r_i/p_r(q)$ for all indices with $c(i)=q$, the module's `sectorSpectrum`. Zero eigenvalues retain their multiplicities. The claim says that $\\operatorname{desc}(r)\\succ\\operatorname{desc}(s)$ implies $\\operatorname{sec}_q(r)\\succ\\operatorname{sec}_q(s)$ for every integer $q$ with $p_r(q)>0$ and $p_s(q)>0$.\n\nFor equal-length descending lists $x,y$, the reused `Majorizes` requires every prefix sum of $x$ to be at least the corresponding prefix sum of $y$ for $1\\le k\\le |x|$, and equal total sums. This is the direction used in the source's definition of majorization (TeX line 830).","title":"Majorization need not survive normalization in a common charge sector","url":"https://the-omega-institute.github.io/trureturing-pages/research/arias-di-giulio-keski-vakkuri-tonni-2023-charge-sector-majorization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c627ace4deafd8e19b8184048109fc871f703ace98a2d3fc3a894d55a2b67153.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arndt-2017-a293452-torus-column-sandpile","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A293452 (Joerg Arndt, 2017) counts the iterations of the sandpile of\nA249872 on the `n × k` torus:\n\n> Let the lattice be c[i,j], 0 <= i,j < n. Fill each cell except c[0,0] with\n> 4 grains of sand. Until all c[i,j] < 4, do the following: Find a c[i,j] >= 4.\n> (According to Knuth, it does not matter which cell is chosen, the result\n> will be the same.) Decrement the chosen cell by 4 and increment its 4\n> neighbors by 1. c[0,0] is never increased, sand grains placed here are lost.\n\nand its FORMULA section states\n\n> Conjecture: T(n,1) = A023855(n).\n\nwith A023855 `a(n) = 1*(n) + 2*(n-1) + ... + (n+1-k)*k, k = floor((n+1)/2)`.\n\nIssue #10122 fixes the readings. The printed identity is false at every `n`\n(`T(1,1) = 0` but `A023855(1) = 1`; `T(2,1) = 1` but `A023855(2) = 2`): the\ncolumn `T(n,1) = 0, 1, 2, 7, 10, 22, 28, 50, 60, 95` of the entry is A023855\nshifted by one place, so the statement settled is the corrected identity\n`T(n,1) = A023855(n − 1)` for `n ≥ 2`. On the `n × 1` torus the four\nneighbours of `(i, 0)` are `(i ± 1, 0)` and the cell itself twice. The number\nof iterations is read, following Knuth's remark, as the length of every legal\ntoppling sequence that ends in a final state, and the existence of such a\nsequence is part of the statement.","title":"The toppling count of the n × 1 torus sandpile","url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6c927506f736c0da2ba6760874ccfafb33007113ac265d5920337aeac9d4478b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arunachalam-johnston-russo-2015-generalized-choi-absolutely-ppt","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Arunachalam, N. Johnston and V. Russo, *Is absolute separability determined by the partial\ntranspose?*, Quantum Inf. Comput. 15(7–8), 694–720 (2015); arXiv:1405.5853v3. Section 7 (p. 21):\n\n> Other open problems in this work include proving that all of the generalized Choi maps\n> $\\Phi_{b,c}$ in Section 5.3 are incapable of detecting absolutely PPT entanglement (including\n> those in the light gray region of Figure 3) …\n\nHere $\\Phi_{b,c}(X)=\\tfrac12\\begin{pmatrix}ax_{11}+bx_{22}+cx_{33}&-x_{12}&-x_{13}\\\\-x_{21}&cx_{11}+ax_{22}+bx_{33}&-x_{23}\\\\-x_{31}&-x_{32}&bx_{11}+cx_{22}+ax_{33}\\end{pmatrix}$\nwith $a=2-b-c$, positive but not completely positive exactly when $(b,c)\\ne(0,0)$, $b,c\\ge0$ and\n$b+c\\le1$ or $bc\\ge(b+c-1)^2$ (Section 5.3, p. 14). The verbatim texts are in\n[the literature note](../Library/QuantumChannels/arunachalam2015absolute.md).\nIssue [#14738](https://github.com/the-omega-institute/trureturing/issues/14738) preregisters the\nreading and the proof plan.","title":"No generalized Choi map detects absolutely PPT entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/research/arunachalam-johnston-russo-2015-generalized-choi-absolutely-ppt/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/79e45e48b01e0ce6311fcf96a1381fc0f22462c3c247869bc037c91e8e00ac94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arunachalam-molina-russo-2013-hedging-majority-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Arunachalam, A. Molina and V. Russo, *Quantum hedging in two-round prover-verifier interactions*, arXiv:1310.7954v5, TQC 2017, LIPIcs 73 (2018) 5:1–5:30, Section 5:\n\n> However, it still remains open to determine the conditions under which Bob can always win at least $k$ out of $n$ repetitions for some $k > 1$.\n\n> Based on our current numerical evidence, it is possible that Bob cannot perfectly hedge more than $k = n/2$ games.\n\nThe game, the shared state u_α = α|00⟩ + √(1 − α²)|11⟩, Alice's measurement {P₀,θ, P₁,θ} with P₁,θ = v_θ v_θ* and v_θ = cos θ|00⟩ + sin θ|11⟩, and the parallel repetition with one joint channel Φₙ are quoted in [the literature note](../Library/QuantumContext/arunachalam2017quantumhedging.md).\n\nIssue [#19590](https://github.com/the-omega-institute/trureturing/issues/19590) fixes the reading. Bob perfectly hedges k games if some channel on his n registers makes Alice record at least k wins with certainty. For all n and k with n < 2k ≤ 2n, all α ∈ (0, 1] and all θ, this is possible exactly when α² = cos²θ, that is, when a single game can already be won with certainty. The conjecture is proved.","title":"Bob cannot perfectly hedge more than n/2 of n parallel games","url":"https://the-omega-institute.github.io/trureturing-pages/research/arunachalam-molina-russo-2013-hedging-majority-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/855d4d62d86a2cebaf0b63e460cfead5b8dc5d94a122c0593c8a16a7b7e78cf7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arxiv-2506-09912-shkolnikov-sandpile-limit-divisible","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mikhail Shkolnikov, *Algebraic Limits of Sandpiles*, arXiv:2506.09912v3, Section 3, states that the image of the direct limit of rectangular sandpile groups may be a rational subtorus or a divisible group. The settled question is the divisibility alternative: for every positive rectangle, every circle-valued sandpile function on it, and every positive integer $k$, a later rectangular stage contains a sandpile function whose $k$-fold sum equals the signed reflected image of the original function.\n\n### Statement echo\n\nThe Lean declaration `D5.S3.StatisticalMechanics.Sandpiles.ShkolnikovSandpileLimitDivisible.claim` preserves the six binding conventions from issue #20011: the carrier is `AddCircle (1 : ℝ)`; `InRect` is the open integer rectangle; `IsSandpile` is zero extension plus the four-neighbour equation; `unfold` is the parity-reflected coordinate; `embed` is the signed reflected zero extension; and the quantifiers require positive $p,q,k$, a sandpile $f$, and a later $g$ with the exact equality. The theorem `result` proves this proposition without weakening its quantifiers.","title":"arxiv 2506 09912 shkolnikov sandpile limit divisible","url":"https://the-omega-institute.github.io/trureturing-pages/research/arxiv-2506-09912-shkolnikov-sandpile-limit-divisible/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37319925be596a722ce510dd530a6ec6bd075f6de59b8d0823221354f27b029f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arxiv-2510-07530-gallardo-rahavandrainy-collatz-length-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gallardo and Rahavandrainy, *A variant of Collatz's Conjecture over Binary\nPolynomials*, arXiv:2510.07530v1, §1, state: “However, several computations\nsuggest that here also r_A is bounded by a polynomial in deg(A) (see Section 3).”\nThe question is preregistered in [#20129](https://github.com/the-omega-institute/trureturing/issues/20129).\nFor every nonzero $A\\in\\mathbb F_2[x]$, the theorem `rA_quadratic` proves\n$r_A\\le 2+\\lfloor\\deg(A)^2/4\\rfloor$. The theorem `result` proves the\nliteral polynomial-bound prediction, using $p(X)=X^2+2\\in\\mathbb N[X]$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Nonzero binary polynomial $A$ | `A : Polynomial (ZMod 2)`, `hA : A ≠ 0` | Same coefficient ring and domain. |\n| $M_1=x^2+x+1$; removal of all linear factors | Existing `M`, `oddPart` | Reused at GallardoRahavandrainyBinaryCollatz. |\n| $A_0=A$, $A_{2k}=1+M_1A_{2k-1}$ for $k>0$ | `evenTerm A k` | Literal conditional expression. |\n| $A_{2k+1}$ | Existing `oddSeq A k` | The natural index is half the source's odd index, rounded down. |\n| Least positive $m$ with terminal pair $(x(x+1),1)$ for all $k\\ge m$ | `rA A` | Natural infimum of precisely that set, plus one; the proof constructs a member. |\n| Bound by a polynomial in degree | `claim`, `result` | Existential natural-coefficient polynomial; no extra hypothesis. |\n| Explicit stronger estimate | `rA_quadratic` | $r_A\\le2+\\lfloor\\deg(A)^2/4\\rfloor$. |\n\nThe natural degree agrees with polynomial degree on the nonzero domain.\nDefinitions are total at zero; the theorem excludes zero. The existential\npolynomial witness is an upper envelope of the sharper floor estimate.","title":"A polynomial bound for binary-polynomial Collatz length","url":"https://the-omega-institute.github.io/trureturing-pages/research/arxiv-2510-07530-gallardo-rahavandrainy-collatz-length-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/42a625cad3325fcf43612b3ddb0a60eb7a08d8a27f00158b11baefc907166373.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:arxiv-2610-00765-cube-recursion-family","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Matijasevick, Rodriguez and Salazar, *Perfect squares in reciprocal-sum\nsequences and primes that are inert in quadratic fields*, arXiv:2610.00765v1,\nSection 7, question 2, asks:\n\n> For the recursions of this paper the exponent m=2 is forced. Is there a natural family of rational recursions for which c a_n+b_n is always a cube, other than those conjugate to z -> z^3?\n\nThe reduced-fraction convention is inherited from Problem 1.1: a_n,b_n are\ncoprime positive integers, starts are positive integers, c is positive rational,\nand indices are n>=2. “Natural” is undefined. The explicit minimal reading here\nis a uniform autonomous rational family over Q of fixed degree, parameterized\nby a nonnegative integer, with one formula and a single c independent of the\nparameter, initial value and index. An unstated stricter author intent is not\nresolved by this reading.\n\nFor every t in N, including zero, put\n\n$$F_t(x)=\\frac{x+t}{(x+t+1)^3-(x+t)},\\qquad x_n=F_t^{\\circ n}(x_0).$$\n\nThe proved endpoint is: for every positive rational x_0, the intrinsic reduced\nnumerator a_n and denominator b_n are positive and coprime for every n>=0;\nfor every n>=1,\n\n$$a_n+b_n=\\bigl(a_{n-1}+(t+1)b_{n-1}\\bigr)^3.$$\n\nThus c=1 works uniformly and the inherited integer starts and n>=2 follow.\nEach map has degree exactly 3 and no M in PGL_2(C) has\nM F_t M^{-1}(z)=z^3 on the entire projective line. No assertion at n=0 of\na cube sum, or classification of all cube recursions, is made.","title":"A uniform family of reduced-coordinate cube recursions","url":"https://the-omega-institute.github.io/trureturing-pages/research/arxiv-2610-00765-cube-recursion-family/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8341c39096b4e1d3b5c5842e169363784a0b0f4c6b989fef990c925b039533aa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:asgarli-yip-2021-pseudo-paley-two-dimensional-cliques","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 5.9 of S. Asgarli and C. H. Yip, “The subspace structure of maximum cliques in pseudo-Paley graphs from unions of cyclotomic classes”, arXiv:2110.07176v5 (2024), asks for the classification of two-dimensional cliques containing $1$ in $PP(p^4,p+1,I)$:\n\n> Let $V$ be a $2$-dimensional subspace in $mathbb F_{p^4}$, such that $1\\in V$. Then $V$ is a clique in $PP(p^4,p+1,I)$ for some $I$ if and only if $V=\\mathbb F_p\\oplus a\\mathbb F_p$, where $a=g^{(p+1)k}$ and $k$ is odd.\n\nThe formal convention is `GaloisField p 4`, with `ZMod p` as the prime field, `cyclotomicClass`, `connection`, and `PP` as defined in the settling module. The quantified claim ranges over odd primes, primitive roots, two-dimensional subspaces containing $1$, and index sets contained in `Finset.range (p+1)` with cardinality $(p+1)/2$.","title":"Asgarli–Yip two-dimensional pseudo-Paley cliques","url":"https://the-omega-institute.github.io/trureturing-pages/research/asgarli-yip-2021-pseudo-paley-two-dimensional-cliques/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2af704faff6e57f923f270fee667356456afa3e205c5b27ebd24cb9e044d32db.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:atashpendar-mestel-roscoe-ryan-2018-kappa-alternating-minimizer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Atashpendar, Mestel, Roscoe and Ryan, *A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics*, arXiv:1807.11609v1, printed page 7, state Conjecture IV.4:\n\n> The alternating subsequence of length m, i.e., x = 1010..., minimizes the autocorrelation coefficient κ².\n\nEquation (11) on the same page defines\n\n$$\\kappa^2(x)=\\sum_{1\\le r,s\\le m}\\binom{r+s-2}{r-1}\\binom{2m-r-s}{m-r}[[x_r=x_s]].$$\n\nThe universal minimization statement is false. At length $m=348$, the word $x=(011001)^{58}$ has strictly smaller coefficient than the alternating word $a=1010\\ldots$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Binary word of length $m$ | `x : Fin m → Bool` in `claim` | `false` is 0 and `true` is 1; each position is counted once. |\n| Every positive length | The natural binder $m$ and hypothesis $1\\le m$ | No restriction on positive lengths. |\n| Equation (11) | Both literal double sums in `claim` | Positions are shifted down by one; the binomials become $\\binom{r+s}{r}\\binom{2m-2-r-s}{m-1-r}$. |\n| $1010\\ldots$ | `decide (i.val % 2 = 0)` inside the left sum | Even zero-based positions carry 1. |\n| Minimizes | The weak inequality against every word $x$ | All words of the same length are included. |\n| Refutation | `result : ¬ claim` | The strict instance at length 348 contradicts the universal inequality. |\n\nNatural subtraction in the binomial arguments is exact on these indices. The binders represent source lengths, positions and words; there is no additional mathematical premise. The claim keeps the source expression literal, while the proof uses the frozen `D5.S3.Entropy.DeletionChannelEntropyOrdering.kappa2`, whose body is that expression.","title":"Alternating binary words need not minimize positional autocorrelation","url":"https://the-omega-institute.github.io/trureturing-pages/research/atashpendar-mestel-roscoe-ryan-2018-kappa-alternating-minimizer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dca6a7747a796b4cce70540315efc11dfc905531fb585b47c09af2915de34249.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:atashpendar-mestel-roscoe-ryan-2018-kappa2-entropy-ordering","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Atashpendar, D. Mestel, A. W. Roscoe and P. Y. A. Ryan, *A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics*, arXiv:1807.11609v1, Section V, state:\n\n> Beyond establishing this maximization, proving the entropy ordering of x strings determined by κ2(x) for finite n and m represents another open problem.\n\nSection IV-E.3 and Table I specify the direction: larger $\\kappa^2$ predicts smaller posterior entropy. The finite claim is false at the source's own parameters $n=8$, $m=5$: for $u=00100$, $v=01110$, $\\kappa^2(u)=450>410=\\kappa^2(v)$, but $H_8(u)>H_8(v)$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Section III-A.f: the number of masks projecting $y$ to $x$ | `embeddings` | `List.sublists` retains index-subsequence multiplicities; filtering and taking the length counts masks. |\n| Equation (1): $\\mu_{n,m}=\\binom{n}{m}2^{n-m}$ | `D5.S3.Arith.FibonacciAtomic.CommonPrediction.Capacity.pref` | Reuses the frozen binomial factor, with natural coefficients and subtraction; `claim` restricts $mm$, and every pair of binary words of length $m$; strict $\\kappa^2$ comparison predicts weak entropy comparison. |\n| Negation of that ordering | `result` | A counterexample at $m=5,n=8$ refutes the universal statement. |\n\nThe added binders are representations of the source's lengths and strings, not extra hypotheses. `Bool` encodes binary letters (`false` is zero); `List.ofFn` reads increasing positions. The domain excludes only zero output length and the no-deletion case, neither of which is needed for the counterexample.","title":"Positional autocorrelation does not determine finite deletion-channel entropy ordering","url":"https://the-omega-institute.github.io/trureturing-pages/research/atashpendar-mestel-roscoe-ryan-2018-kappa2-entropy-ordering/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7aa911dcaab6ff2d8f3832a6d330c37b6eb3aeecd7fb215650ddcf4b1141ae2c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:avan-frappat-ragoucy-2026-peakon-transfer-expansion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Avan, L. Frappat and E. Ragoucy, *Quadratic Poisson brackets for the Camassa–Holm peakons*, arXiv:2512.11066v3, J. Phys. A **59**, 245202 (2026), study the two-parameter deformation of the Camassa–Holm peakon Lax matrix.\n- Its matrices are A, T and S of Eqs. (2.1)–(2.3), and its Lax matrix is L(z) = S + Λ(z)T + ξ(z)A with ξ² = Λ² + γΛ + 1 (5.18).\n- The transfer matrices are τ_m(z) = tr L(z)^m.\n- Proposition 5.1 gives their degree-one term in Λ, h^(m) = m tr((T + γA/2)(S + A)^{m−1}) (5.17).\n\nConjecture 5.2 asserts that τ_m(z) = Σ_j Λ^j Σ a_j(k_1, …, k_j) h^(k_1)⋯h^(k_j), summed over partitions k_1 ≥ ⋯ ≥ k_j of m (5.20), with a_1(m) = 1 and the values a_m(1, …, 1) = 1 and a_{j+1}(m − j, 1, …, 1) = m/(m − j) for 0 ≤ j < m − 1 (5.21). It leaves \"the other coefficients … to be determined\", and the conclusion calls for a proof. The verbatim statements are in [the literature note](../Library/FluidDynamics/avan2026peakonquadratic.md).\n\nIssue [#16323](https://github.com/the-omega-institute/trureturing/issues/16323) fixes the reading: for every N and m ≥ 1, there are coefficients independent of N and of all parameters with these values such that (5.20) holds. It also preregisters the explicit coefficients a_j(k) = m (j − 1)! / (∏_i k_i · ∏_r e_r!), where e_r counts the parts equal to r.","title":"The peakon transfer matrices expand in the Hamiltonians h^(k) with explicit universal coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/avan-frappat-ragoucy-2026-peakon-transfer-expansion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb3fc47ed244ad717d06fae3821dbadadaf20775e232c641d257fdb4ff425705.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:avitan-factor-gelbwaser-klimovsky-2026-mpemba-triplet-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"I. Avitan, R. Factor and D. Gelbwaser-Klimovsky, *Necessary conditions for the Markovian Mpemba effect*, arXiv:2603.04567v1, study relaxation under a rate matrix M with detailed balance.\n- **Slow-mode criterion.** The Mpemba effect occurs iff the slow-mode coefficient |b_2| of the initial Gibbs state Π(β_in) has a maximum at some β_in ≠ β_b (p. 2).\n- **Three-level conditions.** For three levels they derive the necessary condition (3) on the rates.\n- **Triplets.** For N levels they apply (3) to every triplet of levels, keeping the original rates.\n- **The conjecture** (p. 4): \"we conject that if for an N−level system, all its triplets violate conditions (3), then the N−level system does not experience the Mpemba effect.\"\n\nThe follow-up arXiv:2609.09988v1 (§II) restates the conjecture as shown.\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/avitanfactorgelbwaserklimovsky2026mpembatriplets.md).\n\nIssue [#17871](https://github.com/the-omega-institute/trureturing/issues/17871) fixes the reading:\n- the slow-mode coefficient is ℓ·Π(β) for a left eigenvector ℓ of the slowest nonzero eigenvalue, assumed simple;\n- the effect is two initial temperatures on the same side of β_b, the farther one with the smaller |ℓ·Π(β)|. This implies the source's criterion.\n\nThe conjecture is refuted.","title":"A four-level system whose triplets all violate the three-level conditions, yet shows the Markovian Mpemba effect","url":"https://the-omega-institute.github.io/trureturing-pages/research/avitan-factor-gelbwaser-klimovsky-2026-mpemba-triplet-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/510e28332d05981ac93849d28d15eb0d6ab6cca0fcb24acf13738d00bcf5339b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ayad-bouchenna-base-b-reverse-multiple-divisors","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mohamed Ayad and Rachid Bouchenna, *Which Integer Divides the Reverse of Any\nof Its Multiples?*, INTEGERS 25 (2025), #A37, printed p. 9, Problem 1:\n\n> Let B ≥ 2 be any number base. Are the divisors of B² − 1 the only positive\n> integers satisfying the property P*_B ?\n\nSection 4.1 on printed p. 8 defines the base-B reverse m*_B by reversing\nthe positional digits of the positive integer m. The property P*_B means\nthat n divides m*_B for every positive multiple m of n. The exact statement\nis: for all natural B ≥ 2 and all natural n > 0,\n`(∀ m : ℕ, 0 < m → n ∣ m → n ∣ Nat.ofDigits B (Nat.digits B m).reverse)\n↔ n ∣ B ^ 2 - 1`.","title":"ayad bouchenna base b reverse multiple divisors","url":"https://the-omega-institute.github.io/trureturing-pages/research/ayad-bouchenna-base-b-reverse-multiple-divisors/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08117c7c29008e39b4e0e2873ead51615b744a0d0a59f68f94f1714c579fe8b2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ayyer-mallick-2010-annihilation-maximal-degeneracy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ayyer and K. Mallick, *Exact results for an asymmetric annihilation process\nwith open boundaries*, J. Phys. A 43, 045003 (2010), arXiv:0910.0693v2,\nafter Eq. (79), p. 14:\n\n> We also conjecture that the Markov matrices are maximally undiagonalizable in\n> the sense that each eigenvalue seems to have exactly one eigenvector independent\n> of the number of times the eigenvalue appears (this implies that the Jordan\n> blocks are of the maximum possible size). In other words, the degeneracy seems\n> to be so strong that the Markov matrix M_L which is of size 2^L has only 2L\n> eigenvectors.\n\nA. Ayyer and V. Strehl, *The spectrum of an asymmetric annihilation process*,\narXiv:1008.2134v1, FPSAC 2010, DMTCS Proc. AN, 461–472,\nDOI 10.46298/dmtcs.2884, Conjecture 1, p. 466:\n\n> The matrix M_L is maximally degenerate in the sense that it has exactly 2L\n> eigenvectors.\n\nThe model has $L$ Boolean sites, bulk transitions $10\\to01$ and $11\\to00$\nat rate $1$, a symmetric left-site flip at rate $\\alpha$, and right-site\nremoval at rate $\\beta$. This is the papers' specialization $\\lambda=1$.\nThe matrix entry $M_L(t,s)$ is the rate from $s$ to $t$ off the diagonal;\neach diagonal entry is the negative sum of the other entries in its column.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| $L$ sites, empty or occupied | `Fin L → Bool` | `true` denotes a particle; index $0$ is site $1$. |\n| $10\\to01$, $11\\to00$, rate $1$ | `bulkMove`, `bulkCount` | Clear the occupied site and toggle its right neighbour; sum over all bonds. |\n| Left-site $0\\leftrightarrow1$, rate $\\alpha$ | `leftCount`, `offRate` | Both directions have the same rate at $\\lambda=1$. |\n| Right-site $1\\to0$, rate $\\beta$ | `rightCount`, `offRate` | Removal requires an occupied last site. |\n| Target-row/source-column matrix, columns sum to zero | `genMat` | Diagonal is the negative sum of all off-diagonal outgoing rates. |\n| One independent eigenvector at each eigenvalue | `claim` | Every $L\\ge1$, positive real $\\alpha,\\beta$, and real $\\mu$ has $\\dim\\ker(M_L-\\mu I)\\le1$. |\n| Negative answer | `result : ¬ claim` | $L=5$, $\\alpha=1/3$, $\\beta=2/5$, $\\mu=-2$. |\n\nThe source assertion includes eigenvalues without restricting their scalar\nfield. A two-dimensional real eigenspace already refutes it. Quantifying over\nall real $\\mu$ also includes numbers outside the spectrum, whose kernels have\ndimension zero. The definitions at $L=0$ are total extensions; `claim` requires\n$L\\ge1$. The extension of `bulkMove` to coinciding indices is never used by a\nbulk transition. None of these conventions weakens the counterexample.","title":"Ayyer–Mallick maximal degeneracy of the annihilation-process generator","url":"https://the-omega-institute.github.io/trureturing-pages/research/ayyer-mallick-2010-annihilation-maximal-degeneracy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55636ab8c34631d0abf433126db031933be915637891f99d3101ee703ead76a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ayyer-misra-2023-half-density-most-likely-occupation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ayyer and S. Misra, *An exactly solvable asymmetric K-exclusion\nprocess*, arXiv:2310.03343v2, Section 4.5, page 21:\n\n> We focus on when t ≪ 1 and t ≫ 1. From the discussion in Sections 4.2\n> and 4.3, we know that the most likely number of particles is either k or k + 1.\n> The question we want to address is which one is most likely.\n\nThe question is AM-1 of [#18570](https://github.com/the-omega-institute/trureturing/issues/18570).\nWrite $L=2r\\ge4$, $K\\ge1$, $0\\le k0$ such that, whenever $t>0$ and either\n$t<\\varepsilon$ or $t>1/\\varepsilon$, the unique modal occupation is\n$k$ if $2k+12$ and half-integer density | `claim`: `r`, `K`, `k`, `hr` | $L=2r$, $r\\ge2$, $K\\ge1$, $k1/\\varepsilon$. |\n| Which adjacent occupation is most likely | `claim`: three conjuncts | Strict comparison with every $b\\le K$ outside the stated mode or tied pair. |\n| Answer to the question | `result` | The full `claim`, with no additional hypothesis. |\n\nThe public definitions follow the binding source conventions of #18570.\nThey introduce no numerical certificate or experimental premise.","title":"Half-density modes of the asymmetric K-exclusion process","url":"https://the-omega-institute.github.io/trureturing-pages/research/ayyer-misra-2023-half-density-most-likely-occupation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66aca2315a3c888883c0c8f11bcb49e3c24ef09a674988a3f2296f55a85bb11f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:b164660-chebfloor","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Anthony Browne's final FORMULA in OEIS A164660, revision 19, reads:\n\n> a(n+1) = Product_{k=1..n} ( 1-2*(floor(k^n/n)-floor((k^n -1)/n)) ) = (-1)^(A003557(n)) for n>0 (conjecture). - _Anthony Browne_, May 29 2016\n\nThe sequence, introduced by Wolfdieter Lang, defines:\n\n> a(n) = numerator(Sum_{m=1..n+1} IT(n,m)), n>=0, with IT(n,m):= A164658(n,m)/A164659(n,m) (coefficient triangle from the indefinite integral Integral_{x} T(n,x), n>=0, in lowest terms).\n\nLet $T_m$ be the first-kind Chebyshev polynomial over $\\mathbb Q$ and put\n$S_m=\\sum_{j=1}^{m+1}[X^{j-1}]T_m/j$ and $a(m)=\\operatorname{num}(S_m)$,\nwhere the numerator is reduced and signed with positive denominator.\nLet $R(n)=\\operatorname{rad}(n)$, including $R(1)=1$.\nFor every positive natural $n$, the theorem proves\n\n$$\na(n+1)=\\prod_{k=1}^{n}\\left(1-2\\left(\\left\\lfloor\\frac{k^n}{n}\\right\\rfloor-\\left\\lfloor\\frac{k^n-1}{n}\\right\\rfloor\\right)\\right)=(-1)^{n/R(n)}.\n$$\n\nThe division and subtraction inside each floor are rational; $n/R(n)$ is natural integer division.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Integrated coefficient triangle and row sum | `rowSum` | The coefficient of degree $j-1$ is divided by $j$ and summed over the inclusive interval $1\\le j\\le m+1$. The integrated coefficient is inlined. |\n| Reduced signed numerator | `a` | `Rat.num` retains the positive-denominator convention. |\n| Literal floor product | `floorProduct` | Rational subtraction precedes integer floor; the inclusive range is $1\\le k\\le n$. |\n| $n>0$, both equalities | `claim`, `result` | All positive naturals, including one; no extra hypothesis. |\n| A003557 exponent | Existing natural radical | $n/\\operatorname{rad}(n)$, including the value at one. |","title":"Browne's Chebyshev numerator and floor product","url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb29715ca682ad7479a872ced5b012d5a9e9d854e00ec7346aba4dd2a78d6c7e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:badhani-das-2026-noisy-unitary-conditional-min-entropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Badhani and S. Das, *Conditional channel entropy sets fundamental limits on thermodynamic quantum information processing*, arXiv:2604.01217v2, Section VI.1, conjecture after Proposition 10:\n\n> We conjecture that Proposition 10 can be extended to all unitaries with added white noise. We leave this as an open problem.\n\nProposition 10 assumes $|B'|=|B|=2$. The statement here retains that hypothesis and takes $A'=A=\\mathbb C^d$, for every positive natural $d$. For every unitary $U$ on $\\mathbb C^d\\otimes\\mathbb C^2$ and every $p\\in[0,1]$, put $N_p(Z)=p\\operatorname{tr}(Z)I/(2d)+(1-p)UZU^\\dagger$. Then\n\n$$\nS_\\infty[A\\mid B]_{N_p}=S_\\infty(R_AA\\mid R_BB)_{\\Phi^{N_p}}-\\log_2 d.\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Arbitrary positive input/output dimension on A | The natural dimension and positivity binder in `claim` | No upper bound on $d$. |\n| Qubit input/output conditioning | `Matrix.unitaryGroup (Fin d × Fin 2) ℂ` | Both B systems have dimension two. |\n| Arbitrary unitary channel | The universally quantified unitary in `claim` | No selected matrix or spectrum. |\n| Added white noise | `noisyUnitary` | Literal replacement-plus-unitary expression. |\n| $p\\in[0,1]$ | The two real inequalities in `claim` | Both endpoints included. |\n| Choi operator/state | `choiN`, `choiState` | Index order $(R_A,A),(R_B,B)$, normalization $1/(2d)$. |\n| State/channel conditional entropy | `condMinEntropy`, `channelCondMinEntropy` | Density states and quantum channels range over the existing repository carriers. The channel formula inlines its matrix-unit Choi expression. |\n| Subtract $\\log|A'|$ | The base-two logarithm term in `claim` | $|A'|=d$. |\n| Universal equality | `result` | Full claim, without additional hypotheses. |\n\nThe public definitions retain the expressions prescribed in [#19060](https://github.com/the-omega-institute/trureturing/issues/19060). `Dmax` uses the real logarithm followed by coercion to `EReal`; at zero input it gives zero rather than an extended logarithm of zero. The settlement uses trace-one states and positive lower bounds on feasible costs, so that zero-input extension is outside its entropy-scaling argument.","title":"Badhani–Das noisy-unitary conditional min-entropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/badhani-das-2026-noisy-unitary-conditional-min-entropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83af45eddb6affb7eb69e8f60761869a3dc6083c28943c7ff638232ac35f9eb1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:badica-2026-jaccard-arbitrary-lattices","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Bădică and A. Bădică, *On the Triangle Inequality for the Jaccard Distance in Arbitrary Lattices*, arXiv:2608.18194v1, Section 8, asks whether the generalized Jaccard distance\n\n$$d_{f,J}(A,B)=1-\\frac{f(A\\wedge B)}{f(A\\vee B)}$$\n\nsatisfies the triangle inequality on arbitrary lattices under either of two sets of hypotheses: strict positivity, monotonicity, supermodularity and log-submodularity of $f$; or strict positivity, monotonicity and supermodularity of both $f$ and $1/f$. These are the second and fourth questions, preregistered in [#20617](https://github.com/the-omega-institute/trureturing/issues/20617).\n\n### Statement echo\n\n| source clause | Lean counterpart |\n| --- | --- |\n| Arbitrary lattice, without distributivity or relative complementation | Universe-polymorphic $L$ with only a `Lattice L` instance in `claim` and `claimSufficiency` |\n| Strict positivity | `IsStrictlyPositive`: $\\forall X,\\ 0 Conjecture 1. If d = 2^r and r > 1, the minimal number of MUBs required for bound entanglement detection equals m = d/2 + 1 = 2^{r−1} + 1.\n\nThe preceding sentence specifies the witness $W^\\Gamma(\\mathcal M_m,s)$.\nEquation (4) defines $W(\\mathcal M_m,s)=((d+m-1)/d)I-B(\\mathcal M_m,s)$;\npartial transposition conjugates the second factor of each equal-column\ntensor projector. The computational term pairs $\\ell$ with $\\ell+s$\nmodulo $d$. The source's Eqs. (2)–(3) establish nonnegative expectation\non separable states. This source result supplies the interpretation of\nnegative expectation as entanglement detection; this module does not\nformalize separability or the separable-state bound.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $d=2^r$, $r>1$ | `claim` quantifies over natural `r` with `2 ≤ r` and dimension `Fin (2 ^ r)` | The same integer range |\n| Fewer than $2^{r-1}+1$ MUBs cannot detect bound entanglement | Every natural `m` below the threshold in `claim` | Necessity half of the minimum equality |\n| Pairwise mutually unbiased orthonormal bases | `IsMUBFamily`: unitary matrices and all distinct-basis squared overlaps equal to $1/d$ | Columns are the basis vectors |\n| Canonical basis $B_0$ | `IsMUBFamily` sets basis zero to the identity whenever `0 < m` | Source convention; no nonempty-family assumption is added |\n| Source witness and modular shift | `witnessGamma` with every `s : Fin d` | Literal Eq. (4) after partial transposition; column projectors use `Matrix.vecMulVec` directly |\n| PPT density matrix with negative expectation | `DetectsBoundEntanglement` uses the existing `IsDensity` and `partialTransposeB`, and the real trace is strictly negative | Literal binding convention in #16454 |\n| Refutation of the universal necessity statement | `result : ¬ claim`, using $r=3$, $m=4$, $s=1$ | One counterexample refutes the equality; optimality of four is not asserted |\n\nThe four public definitions have complete defining formulas in the\nBlueprint. Natural subtraction in $r-1$ is harmless under $2\\le r$.\nThe predicates are total at zero dimension and zero family size, but\nthe refutation uses dimension eight and four bases. The computational\nbasis convention is from the source, and positivity, normalization and\nstrict negativity all remain requirements.","title":"Four MUBs detect a PPT state in dimension eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/bae-bera-chruscinski-hiesmayr-mcnulty-2022-four-mub-bound-entanglement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a015f289834174f96facdac15b4f245626814be7213689affca9b917264e5a3f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baek-jo-nha-2026-maximal-coherence-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Baek, Jo and Nha, *Maximal coherence of quantum measurement and the resource theory of sharpness*, arXiv:2607.05847v1, asks whether $C_D^{\\max}$ remains monotone under general CP unital maps for non-dichotomic POVMs. Table I marks the corresponding entry under any fuzzifying operations as open.","title":"baek jo nha 2026 maximal coherence monotonicity","url":"https://the-omega-institute.github.io/trureturing-pages/research/baek-jo-nha-2026-maximal-coherence-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60e979450e63fbeaa7463d9978dd9b93e6228a5fd6f43664c6b7394703525e03.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bahturin-centrone-pereira-2026-peak-group-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bahturin, Centrone and Pereira, *Quantum regularity of finite dimensional semisimple algebras*, arXiv:2608.21936v1, Conjecture 31, asks whether every peak group satisfies exactly one of solvability and the stated Frobenius-complement alternative. The preregistration is [#19254](https://github.com/the-omega-institute/trureturing/issues/19254).\n\nThe settling declaration is `D5.S3.FiniteGroups.BahturinCentronePereiraPeakGroups.BCP31.result`. It proves the negation of the literal `claim`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A peak group has a greatest irreducible-character degree for divisibility. | `BCP31.IsPeak` and `BCP31.degreeSet` | The degree set is defined by simple finite-dimensional complex representations, and every degree divides the selected degree. |\n| A Frobenius complement is nontrivial, proper, and has trivial intersection with each conjugate outside it. | `BCP31.IsFrobeniusWithComplement` | The subgroup-map expression is the literal conjugation condition. |\n| The complement quotient contains the specified `SL(2, 5) × T` normal subgroup, with cyclic Sylows and excluded primes. | `BCP31.ComplementCondition` and `BCP31.AlternativeTwo` | The quantified finite group $T$ and all three structural conditions are retained. The cyclic-Sylow condition is written directly as a quantification over primes and Sylow subgroups; Mathlib’s `IsZGroup` states the corresponding property and is not on the proof dependency path. |\n| Exactly one of solvability and the Frobenius alternative holds for every finite peak group. | `BCP31.claim` | The result proves the full negation, including the exclusive-or conjunction. |","title":"bahturin centrone pereira 2026 peak group refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/bahturin-centrone-pereira-2026-peak-group-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a28af9500f075760e2ad298f78bd8d4ab49cd9175d2088da7570fb2b57c4f091.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bai-2007-residual-sum-monotone","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y.-K. Bai, D. Yang, Z. D. Wang, *Multipartite quantum correlation and\nentanglement in four-qubit pure states*, Phys. Rev. A 76, 022336 (2007),\narXiv:quant-ph/0703098v2:\n\n> Nevertheless, we conjecture that the correlation $M$ is an entanglement\n> monotone\n\nHere $M=\\sum_k\\tau_k-2\\sum_{p>q}C_{pq}^2$ (Eq. (7)) for a four-qubit pure\nstate, with the linear entropies $\\tau_k=2(1-\\mathrm{tr}\\,\\rho_k^2)$ of the\none-qubit reduced states and Wootters' concurrences\n$C_{pq}=\\max(\\sqrt{\\lambda_1}-\\sqrt{\\lambda_2}-\\sqrt{\\lambda_3}-\\sqrt{\\lambda_4},0)$\nof the six two-qubit reduced states, $\\lambda_1\\ge\\dots\\ge\\lambda_4$ the\neigenvalues of $\\rho_{pq}(\\sigma_y\\otimes\\sigma_y)\\rho_{pq}^\\ast(\\sigma_y\n\\otimes\\sigma_y)$. An entanglement monotone does not increase on average\nunder LOCC; in particular, for every local instrument $\\{K_j\\}$ on one qubit,\n$\\sum_jp_jM(\\phi_j)\\le M(\\psi)$, where $p_j=\\|K_j\\psi\\|^2$ and\n$\\phi_j=K_j\\psi/\\sqrt{p_j}$.","title":"Monotonicity of the residual-correlation sum of four qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/bai-2007-residual-sum-monotone/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/494a513f355634d923a743a5d586cdd90387b83331aab2e224d91f992f276ff9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baio-chruscinski-horodecki-messina-sarbicki-2019-fixed-marginal-negativity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Baio, D. Chruściński, P. Horodecki, A. Messina and G. Sarbicki, *Bounds on the entanglement of two-qutrit systems from fixed marginals* (Phys. Rev. A 99, 062312 (2019); arXiv:1904.07650), study the largest entanglement of a bipartite state whose marginals are fixed.\n- For two qubits the maximum of the negativity N(ρ) = (‖ρ^Γ‖₁ − 1)/2 is attained by an explicit state (12).\n- For two qutrits with ordered marginals ρ_A = diag(1−λ₁−λ₂, λ₁, λ₂) and ρ_B = diag(1−μ₁−μ₂, μ₁, μ₂) (Eq. (24)), they maximize the negativity within the circulant family (22). They obtain three quasidistillable states, Eqs. (27)–(29).\n- From numerical optimization they conclude: \"all the above results strongly support our conjecture that our quasidistillable states (27, 28, 29) are legitimate candidates for two-qutrits MEMS with respect to fixed marginals\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/baiochruscinskihorodeckimessinasarbicki2019fixedmarginals.md).\n\nIssue [#18345](https://github.com/the-omega-institute/trureturing/issues/18345) fixes the reading: in the region of Eq. (27), the state (27) maximizes the negativity over all two-qutrit states with the marginals (24). The conjecture is refuted.","title":"The quasidistillable candidates are not two-qutrit negativity maximizers for fixed marginals","url":"https://the-omega-institute.github.io/trureturing-pages/research/baio-chruscinski-horodecki-messina-sarbicki-2019-fixed-marginal-negativity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3dcea70e74830a75bfc226767fc5ba2d4406e602f83dda1fde38452a0affa833.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bajnok-beatty-original-conjecture-4-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Bajnok and W. K. Beatty, *On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups*, arXiv:2406.04045v1, Section 4, Conjecture 4.1:\n\n> Let k and s be positive integers. The group Z2 × Z2k has an s-spanning set of size 2 if, and only if, one of the following holds:\n>\n> - s is odd and k ≤ (s² − 1)/2;\n> - s is even and k ≤ (s² − s)/2;\n> - s ≡ 0 mod 4, k is even, and (s² − s)/2 < k ≤ s²/2; or\n> - s ≡ 2 mod 4, k ≡ 2 mod 4, and (s² − s)/2 < k ≤ s²/2.\n\nThe signed span consists of $xa+yb$ for integers $x,y$ with $|x|+|y|\\le s$, including the zero coefficients. A set of size two is represented by two distinct elements $a,b$.","title":"The original Bajnok–Beatty Conjecture 4.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/bajnok-beatty-original-conjecture-4-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e110006adca8f02cb5427db7ad414ff971ba5ab773875aaba112161723abc24.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bajnok-berson-just-2022-perfect-restricted-bases-question-6","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Bajnok, C. Berson and H. A. Just, *On Perfect Bases in Finite Abelian Groups*, Involve 15 (2022), 525–536, arXiv:2211.13675v1, Question 6:\n\n> Are there any noncyclic groups that have a perfect restricted s-basis of size s + 1?\n\nThe paper works with finite abelian groups and positive integers $s$. Its definition is:\n\n> A is a perfect restricted s-basis of G if each element of G can be written uniquely as the sum of at most s distinct elements of A.\n\nUniqueness disregards the order of the summands. The empty subset contributes the empty sum $0$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite abelian group $G$ | `G : Type`, `AddCommGroup G`, `Fintype G` | Additive notation; no order or prime-order assumption. |\n| Positive integer $s$ | `s : ℕ`, `1 ≤ s` | Exact positive range. |\n| A subset $A$ of size $s+1$ | `A : Finset G`, `A.card = s + 1` | Distinct elements, without an ordering. |\n| At most $s$ distinct summands | A unique `T : Finset G` with `T ⊆ A`, `T.card ≤ s` and sum equal to the given element | Includes the empty sum. |\n| No noncyclic example | `result : claim`, concluding `IsAddCyclic G` | Universal negative answer to Question 6. |\n| Decidable equality | `DecidableEq G` | Computational representation of finsets; every finite group admits it classically. |","title":"Perfect restricted bases of size s + 1 force cyclicity","url":"https://the-omega-institute.github.io/trureturing-pages/research/bajnok-berson-just-2022-perfect-restricted-bases-question-6/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9e05e12a248d79274650691fc66b68ba61fba6f928f54be7ec8df0e882e9970.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bakhshinezhad-mehboudi-roch-i-carceller-tavakoli-2024-odd-prime-unentangled-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Bakhshinezhad, M. Mehboudi, C. Roch i Carceller and A. Tavakoli, *Scalable entanglement certification via quantum communication*, PRX Quantum 5, 020319 (2024), arXiv:2401.00796v2, certify entanglement through a symmetric prepare-and-measure task.\n- **The task.** For an odd prime d, Alice and Bob receive x, y ∈ {0, …, d − 1}² and send d-dimensional messages; Charlie receives z ∈ {0, …, d}, may measure the two messages jointly, and outputs c. He wins when c = w_z, with w_z = x₁ + y₁ − 2z(x₀ − y₀) for z ≠ d and w_d = x₀ − y₀ mod d (Eq. (27)), and R_d is the average success probability (Eq. (28)).\n- **Without shared entanglement.** p(c | x, y, z) = tr(α_x ⊗ β_y M_c|z) (Eq. (19)); sending x₀ and y₀ achieves R_d = 2/(d + 1).\n- **The conjecture.** After see-saw searches at d = 3 and d = 5: \"Conjecture 1. For every odd prime d, the largest average success probability achievable in a quantum model without shared entanglement is R_d = 2/(d + 1). (29)\" With shared entanglement and product measurements R_d = 1, so the bound would certify every state with a non-trivial maximally entangled fraction.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/bakhshinezhad2024scalable.md).\n\nIssue [#16770](https://github.com/the-omega-institute/trureturing/issues/16770) fixes the reading: messages are d × d density matrices, Charlie's measurement for each z is any d-outcome POVM on the joint d²-dimensional space, and shared classical randomness is a finite mixture of strategies with one common index. The conjecture is proved for every odd prime d.","title":"Without shared entanglement the odd-prime communication task succeeds with probability at most 2/(d + 1)","url":"https://the-omega-institute.github.io/trureturing-pages/research/bakhshinezhad-mehboudi-roch-i-carceller-tavakoli-2024-odd-prime-unentangled-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f17439683e3fb35d119a65b9e5bca200bcb676f4a3704ae8ae1f913026bcca2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala-a258899-totient-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala's COMMENTS conjecture for OEIS A258899 and A258903 states:\n\n> Conjecture: the sequence a(n) taken modulo a positive integer k is eventually periodic with the period dividing phi(k).\n\nThe two sequences use the FORMULA fields\n\n$$a_{899}(n)=\\sum_{d\\mid n}2^d\\,\\frac{n!}{d!(n/d)!},\\qquad\na_{903}(n)=\\sum_{d\\mid n}2^d3^{n/d}\\,\\frac{n!}{d!(n/d)!}$$\n\nfor $n\\ge1$, with value $1$ at index zero. The settling declaration states the two eventual-period assertions literally, with natural-number division and `Nat.ModEq`.\n\n### Statement echo\n\nFor each positive natural modulus $m$, the result supplies an onset $N$, a positive natural period $T$ dividing `Nat.totient m`, and the congruence for every $n\\ge N$, separately for A258899 and A258903.","title":"Bala's A258899 and A258903 totient-period conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-a258899-totient-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/030aa4681e54192ddb631cde244d1419fbc6f1b245d3b5ef1c0b9912f6aa116a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala-machine3626-exact-halting-time","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS A393210, conjecture of September 11, 2026, revision 18:\n\n> Conjecture: a(n) = Sum_{d | n} m(d)^2 * phi(d), where m(n) = Sum_{d | n} (-1)^(d+1) * moebius(d) = A069517(n) and phi(n) denotes Euler's totient function A000010(n). Cf. A296306.\n\nThe sequence has offset 1. The equivalent conjectural relation in A069517,\nrevision 31, is the same question, counted once. Here a(n) is the actual\nnumber of transitions of the one-sided two-state two-symbol machine 3626\non every positive natural input n in binary. Its half-tape extends\ninfinitely leftward, blank zero beyond the most significant bit. The head\nstarts in state A at the right edge on the least significant bit. Its\nrules are A0 → A0L, A1 → B1R, B0 → A1L, B1 → B0R. A simultaneous read,\nwrite, state change and move counts as one transition. Moving right off\nthe edge terminates the run, and that final transition is counted.\nInput zero is excluded. There is no time cap, tape bound or counter width.\nBoth divisor sums range over positive divisors; the signed Moebius\narithmetic is in the integers.","title":"The exact halting time of Wolfram machine 3626","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/499e980a2a30885be49508ee3a7de07b634bfc97f8be70cf01279afe5d207b20.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala126156","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A126156, revision 110, NAME:\n\n> Expansion of e.g.f. sqrt(sec(sqrt(2)*x)), showing coefficients of only the even powers of x.\n\nThe third COMMENTS entry states:\n\n> Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic with the period dividing phi(k), where phi(n) = A000010(n). An example is given below. - _Peter Bala_, Mar 01 2026\n\nThe source has offset zero. For a rational formal power series $A$ with\n$A(0)=1$ and $A(x)^2\\cos(\\sqrt2x)=1$, put\n$a_A(n)=(2n)![x^{2n}]A$. The assertion includes existence of such a series\nand says that, for every such $A$ and every positive natural modulus $k$,\nthere are natural numbers $N,p$ with $p>0$, $p\\mid\\varphi(k)$ and\n$a_A(n+p)-a_A(n)=kz$ for some integer $z$ at every $n\\ge N$.\nThe delivered `claim` is this positive universal assertion; `result` negates it.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Rational expansion of $\\cos(\\sqrt2x)$ | `cosSqrtTwo` | At even degree $j$, $(-2)^{j/2}/j!$; at odd degree, zero. |\n| Square root with constant term one | `IsSourceEGF` | Both the constant-coefficient condition and the square equation. |\n| Even-power EGF coefficients; offset zero | Factorial coefficient expressions in `claim` | The expressions are literal; proofs use the existing `IntegralEGFComposition.eCoeff`. |\n| Every positive integer modulus | Natural `k` and `0 < k` | Positive integers have the same domain. |\n| Eventually | Natural `N`; every natural `n` with `N ≤ n` | No onset is fixed or bounded. |\n| Period dividing the totient | Natural `p`, `0 < p`, `p ∣ Nat.totient k` | Every positive divisor is retained. |\n| Reduced modulo $k$ | Existential integer multiple of $k$ in the rational field | The coefficients' integrality is derived, not assumed. |\n| The sequence exists | Existential source-series conjunct in `claim` | A preregistered existence conjunct, not an extra hypothesis on `result`. |","title":"The A126156 eventual totient-period conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala126156/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/124d60ce9174b46b875f03eccbff98ebf149fc7e80db2f4186ec815b3dbf1441.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala193236","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala's FORMULA line in OEIS A193236 states: “Conjectures: a(5*n+4) == 0 (mod 5) and a(7*n+5) == 0 (mod 7) (both checked up to n = 200).” The source is preregistered in [#22113](https://github.com/the-omega-institute/trureturing/issues/22113). The companion claim is [#22112](https://github.com/the-omega-institute/trureturing/issues/22112).\n\nWrite $T_j=j(j+1)/2$ and $F(x)=\\sum_{j\\ge0}(-3)^j(2j+1)x^{T_j}$. The question concerns the constant-one reciprocal cube root $A(x)=F(x)^{-1/3}$ and every natural index, without a finite cutoff.\n\n### Statement echo\n\n| Source clause or convention | Formal counterpart | Scope |\n| --- | --- | --- |\n| The displayed source series | `BalaA193236ModFiveSeven.F` | Its degree-$m$ coefficient sums the literal source weights over $j Conjecture: a(n) is odd iff the 1 bits in the binary expansion of n are separated by at least three 0's, that is, iff n belongs to A048718.\n\nThe offset includes $n=0$. The rational quotient is interpreted by exact division;\nits natural realization is explicit before oddness is tested.\n[Preregistration #22428](https://github.com/the-omega-institute/trureturing/issues/22428)\nfixes the statement and conventions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The displayed factorial quotient | `a` | All five natural factorials are cast to $\\mathbb Q$; division is exact. |\n| Every natural index, including zero | `claim` | Universal natural quantifier, with no added hypotheses. |\n| The quotient is a natural integer | `claim` | A natural witness realizes the rational value. |\n| Odd iff at least three zero bits separate ones | `spacedBits`, `claim` | Every occupied pair $i Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.\n\nThe index $n$ is any natural number, including zero.\nThe prerequisite integrality and the exact exponent bound are part of the source's statement in\n[preregistration #18606](https://github.com/the-omega-institute/trureturing/issues/18606).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Factorial quotient with coefficient 2520 and shift 2 | `BalaA361034Parity.a` | Literal source expression, using natural-number division. |\n| Every natural index, including zero | `BalaA361034Parity.claim` | Universal quantifier over the naturals. |\n| Odd exactly when the shifted index is a power of two | `BalaA361034Parity.claim` | Both directions of the iff; the exponent is natural and at least 1. |\n| Quotient is an integer | Private `BalaA361034Parity.integrality` | Denominator divides numerator; its hypotheses are discharged in the settling proof. |\n| Conjecture holds | `BalaA361034Parity.result` | Complete claim without extra assumptions. |\n| Parameters $C,j$ in the shared proof | `BalaA361034Parity.quotient_odd` | Internal generalization, instantiated with $C=2520$ and $j=2$; no added premise in `result`. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A361034 parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3610-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/820992fc40eefe51b19e0778e04951f835d43079ba91e97cedc78f9554bbb5f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala361035-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala's [OEIS A361035](https://oeis.org/A361035) defines\n$a(n)=9979200(4n)!/(n!(n+3)!^3)$ and states:\n\n> Conjecture: a(n) is odd iff n = 2^k - 3 for some k >= 2.\n\nThe index $n$ is any natural number, including zero.\nThe prerequisite integrality and the exact exponent bound are part of the source's statement in\n[preregistration #18607](https://github.com/the-omega-institute/trureturing/issues/18607).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Factorial quotient with coefficient 9979200 and shift 3 | `BalaA361035Parity.a` | Literal source expression, using natural-number division. |\n| Every natural index, including zero | `BalaA361035Parity.claim` | Universal quantifier over the naturals. |\n| Odd exactly when the shifted index is a power of two | `BalaA361035Parity.claim` | Both directions of the iff; the exponent is natural and at least 2. |\n| Quotient is an integer | Private `BalaA361034Parity.integrality` | Denominator divides numerator; its hypotheses are discharged in the settling proof. |\n| Conjecture holds | `BalaA361035Parity.result` | Complete claim without extra assumptions. |\n| Parameters $C,j$ in the shared proof | `BalaA361034Parity.quotient_odd` | Internal generalization, instantiated with $C=9979200$ and $j=3$; no added premise in `result`. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A361035 parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala361035-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/41d9b55fc4b5bc8033fe92592b5587dfa17b054d7e651600781b439873b3d90d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala3634-nondivisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"$3\\nmid b(3n+1)$ for every $n\\in\\mathbb N$, including zero.\nHere $a(n)$ and $b(n)$ are the integer coordinates of\n$\\prod_{k=0}^n(1+k\\sqrt{-3})$.\nPreregistration #18635 maps to `D5.S3.Arith.Congruence.BalaA363412.result` (Proved).","title":"Three-adic nondivisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-nondivisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca9edcae8dd466b948c1717d24de805405477f2c134f566c7c591ce026190d82.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala3634-paired-valuations","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"$v_3(b(3n))=v_3(b(3n-1))$ for every $n\\in\\mathbb N$ with $n\\ge1$.\nHere $a(n)$ and $b(n)$ are the integer coordinates of\n$\\prod_{k=0}^n(1+k\\sqrt{-3})$.\nPreregistration #18636 maps to `D5.S3.Arith.Congruence.BalaA363412Valuation.result` (Proved).","title":"Paired three-adic valuations","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-paired-valuations/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/feba8f4b226c5b722052d490a59dc07969244a9af375afc37756d84c5722fd71.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala3634-type-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"$\\forall p\\le101$, the primes dividing no $a(n)$ are exactly $\\{3,11,17,23,29,41,47,53,59,83,89,101\\}$, with the term index $n$ unbounded.\nHere $a(n)$ and $b(n)$ are the integer coordinates of\n$\\prod_{k=0}^n(1+k\\sqrt{-3})$.\nPreregistration #18634 maps to `D5.S3.Arith.Congruence.BalaA363411TypeOne.result` (Proved).","title":"Type-one primes through 101","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-type-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3194aa56a7ea4bfc555c6a97e672497db0b58e49d9a584c717628066c8cbf5dd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bala3634-valuations","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Bala's OEIS A363411 asks whether the real coordinate of\n$$P_n=\\prod_{k=0}^{n}(1+k\\sqrt{-3})$$\nsatisfies $v_5(a(n+2))=A079998(n)$, where $a(n)=\\operatorname{Re}(P_n)$ and\nA079998 is the characteristic function of the multiples of five. The companion\nA363411 type-one statement lists the primes at most $101$ that divide no real\nterm. The companion A363412 statements concern\n$b(n)=\\operatorname{Im}(P_n)/\\sqrt3$: $3\\nmid b(3n+1)$ and\n$v_3(b(3n))=v_3(b(3n-1))$ for $n\\ge1$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $P_n=\\prod_{k=0}^{n}(1+k\\sqrt{-3})$ in $\\mathbb Z[\\sqrt{-3}]$ | D5.S3.Arith.Congruence.BalaA363411.P | Literal finite product over Finset.range (n + 1) |\n| A363411 real coordinate | BalaA363411.a | Zsqrtd.re of P |\n| A363412 imaginary coordinate divided by $\\sqrt3$ | BalaA363411.b | Zsqrtd.im coordinate |\n| $v_5(a(n+2))=1$ when $5\\mid n$, and $0$ otherwise | BalaA363411.claim and result | Exact all-$n$ quantifier |\n| Type-one primes through $101$ | BalaA363411TypeOne.claim and result | Exact finite prime bound and all-term quantifier |\n| $3\\nmid b(3n+1)$ | BalaA363412.claim and result | Exact all-$n$ quantifier |\n| $v_3(b(3n))=v_3(b(3n-1))$ for $n\\ge1$ | BalaA363412Valuation.claim and result | Exact positive-index condition |","title":"Bala's OEIS A363411 and A363412 valuation conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-valuations/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68a65e5f943023fc80894ae163cb5f79f836d61bb4243919287b73d9edcc2410.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:balanced-composition-paths","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A026010 is defined by\n\n> a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative\n> integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2\n\nand carries a comment by Gus Wiseman dated Mar 17 2018, still standing at\nrevision #55 of Oct 13 2025:\n\n> Conjecture: a(n) is the number of integer compositions of n + 2 in which the\n> even parts appear as often at even positions as at odd positions (confirmed up\n> to n = 19).","title":"Balanced Compositions and Nonnegative Walks from Height Two","url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c2fdc068883cd45df35a994002f0965a56acd3bc1c5f32549095dce8a908f53.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ball-galvin-hyry-weingartner-2022-unimodal-coefficient-permutations","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Ball, D. Galvin, C. Hyry, and K. Weingartner, *Independent set and\nmatching permutations*, Journal of Graph Theory 99(1) (2022), 40–57,\nSection 4, Question 4.6 (arXiv:1901.06579v2), asks:\n\n> Which unimodal permutations of [n] can arise via (1) from the coefficient sequence of a polynomial of the form (1+r₁x)(1+r₂x)⋯(1+rₙx) with rᵢ real and nonnegative?\n\nEquation (1) means $a_{\\pi(1)}\\le\\cdots\\le a_{\\pi(n)}$, excluding the\nconstant coefficient. Equation (4) means that the inverse positions\nincrease up to the final value $m=\\pi(n)$ and decrease afterwards.\n\nThe formal reading takes the polynomial to have degree $n\\ge1$, so\nall $r_i>0$. If zero parameters are allowed without the degree\ncondition, trailing coefficients vanish; choosing every $r_i=0$\nmakes all nonconstant coefficients zero and every permutation arises\ntrivially. The full-degree reading agrees with the source's restriction\nto unimodal permutations.","title":"Ball–Galvin–Hyry–Weingartner: unimodal coefficient permutations","url":"https://the-omega-institute.github.io/trureturing-pages/research/ball-galvin-hyry-weingartner-2022-unimodal-coefficient-permutations/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5db25d5ffea9aa8029759e23da5d45b0a523c67648ba1404b61d541ce515c67a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ballantine-regular-symmetric-image-conjecture-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ballantine, Beck, Merca and Sagan, arXiv:2409.11268v3, Section 5,\nConjecture 16 (printed page 19), state:\n\n> The value of r_{d,2}(n) - r_{d,3}(n) for d = 2, 3, 4, and 5 are\n> shown in the columns of the following table. These values depend on the\n> congruence class of n modulo 2, 6, 4, and 10, respectively. The first\n> column of the table gives the congruence class for n.\n\nHere `pre_k(lambda)` is the partition whose parts are the products indexed by\nthe `k`-element position subsets of the parts of `lambda`. The set\n`ImP_k(n)` is the image of the partitions of `n` having at least `k` parts,\nwith equal images counted once. The value `r_{d,k}(n)` counts the members of\nthat image having no part divisible by `d`.\n\nFor `d = 5` and `n` congruent to 4 modulo 10, the printed table gives\n`3 floor(n/10) + 1`. Thus its entry at `n = 4` is one.","title":"Refutation of the regular elementary symmetric image table","url":"https://the-omega-institute.github.io/trureturing-pages/research/ballantine-regular-symmetric-image-conjecture-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/722c5d5b27bc7ca6edb60b4c54738a13e92364a07bb063ef5e8cf623482ad71f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ballesteros-et-al-2025-qdicke-closest-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ballesteros, I. Gutierrez-Sagredo, J. de Ramón and J. J. Relancio, *Deformations of the symmetric subspace of qubit chains*, J. Phys. A 58, 405301 (2025), arXiv:2503.23554v3, subsection “Entanglement of q-symmetric states”, TeX member `newsubmission__1_.tex`, l. 1258:\n\n> Therefore, the closest separable state to a $q$-symmetric state for two qubits is also $q$-symmetric. If we conjecture that a similar relation holds for arbitrary $q$-Dicke states, it would mean that we can construct whole new families of states whose multipartite entanglement has an analytic expression.\n\nThe optimization in that subsection is the ordinary overlap modulus over product vectors with every local factor normalized (l. 1234–1243). Write $|0\\rangle=|\\downarrow\\rangle$ and $|1\\rangle=|\\uparrow\\rangle$. For $q>0$, the three-qubit vectors at l. 581–582 are\n\n$|D^1_3\\rangle_q=(\\sqrt q\\,|001\\rangle+|010\\rangle+q^{-1/2}|100\\rangle)/\\sqrt{q+1+q^{-1}}$,\n\n$|D^2_3\\rangle_q=(\\sqrt q\\,|011\\rangle+|101\\rangle+q^{-1/2}|110\\rangle)/\\sqrt{q+1+q^{-1}}$.\n\nHere $[3]_q=q+1+q^{-1}$ follows from the source's q-number at l. 469, with its undeformed convention at $q=1$. The source identifies $\\mathrm{Sym}_q\\mathcal H$ with the complex span of $|000\\rangle$, $|D^1_3\\rangle_q$, $|D^2_3\\rangle_q$ and $|111\\rangle$ (l. 586). These definitions are recorded in [the literature note](../Library/LieTheory/ballesterosetal2025qsymmetric.md).\n\nIssue [#21970](https://github.com/the-omega-institute/trureturing/issues/21970) fixes the reading: for every real $q>0$, there exist unit $a,b,c\\in\\mathbb C^2$ such that $a\\otimes b\\otimes c$ maximizes $|\\langle D^1_3|_q(a\\otimes b\\otimes c)\\rangle|$ over all unit local factors and belongs to $\\mathrm{Sym}_q\\mathcal H$. This three-qubit, one-excitation instance of the conjecture is refuted.","title":"A closest product vector to a q-Dicke state need not be q-symmetric","url":"https://the-omega-institute.github.io/trureturing-pages/research/ballesteros-et-al-2025-qdicke-closest-product/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/569ca85e2d3fcb53297730acd3fa34209ec8179b3280fc115450a1609b17d899.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bao-harper-2020-bit-thread-ingleton-multiflow","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ning Bao and Jonathan Harper, *Bit threads on hypergraphs* (2020), arXiv:2012.07872v1, Conjecture 1, Section 3.2, `BT_Hyper.tex` lines 419–424:\n\n> For a given geometry consider a linear combination of entropies $R(S)$. If $R(V)\\geq0$ for all multiflows and there exists a multiflow such that $R(V)\\neq0$, then for all max multiflows\n>\n> $R(S)=R(V).$\n\nThe source is the TeX member `BT_Hyper.tex` from the [v1 source archive](https://arxiv.org/src/2012.07872v1), md5 `e001e89a911f72e1614de54e82230780`. Lines 409–416 define the same linear combination on entropies and subflows; the footnote at line 417 expressly includes ordinary 2-graphs among the geometries. The definitions and source locators are collected in [the literature note](../Library/Entropy/baoharper2020hypergraphs.md).\n\nIssue [#22654](https://github.com/the-omega-institute/trureturing/issues/22654) fixes the finite-graph reading. Let $X$ be any finite vertex set, $k$ any natural number, $b:\\mathrm{Fin}\\,k\\hookrightarrow X$ an injective boundary labelling, and $c:X\\times X\\to\\mathbb R$ a symmetric capacity function with $c(u,u)=0$ and $c(u,v)\\ge0$. For each boundary region $I$:\n\n- $S_I$ is the minimum of $\\sum_{u\\in W,\\,v\\notin W}c(u,v)$ over cuts $W\\subseteq X$ containing precisely the boundary vertices labelled by $I$.\n- A multiflow consists of real pair flows $f_{ij}$ and thread numbers $N_{ij}$, using $i (2) What is the value of λ̄₁₀ in Lemma 3.2 (4) for n ∈ ℕ?\n\nLemma 3.2(4) states:\n\n> Let λ̄₁₀ = sup_{a∈(0,1/2]} {λ₁₀(a)} for n ∈ ℕ₀. Then λ̄₁₀ = 2 if n = 0.\n\nEquation (3.2), specialized to $c=1$, gives\n\n$$g_{6,k}(a)=a(a)_k(1-a)_k,\\qquad\n(a)_k=\\prod_{j=0}^{k-1}(a+j).$$\n\nWrite $S_n(a)=\\sum_{k=0}^n g_{6,k}(a)/((k+1)(k!)^2)$.\nEquation (3.6) is\n\n$$\\lambda_{10}(a)=\\frac{a}{1-a}\n\\frac{\\sin(\\pi a)+\\pi(1-a)\\cos(\\pi a)\n-\\pi(1-a)^2\\sum_{k=0}^n\\frac{g'_{6,k}(a)}{(k+1)(k!)^2}}\n{\\sin(\\pi a)-\\pi(1-a)\\sum_{k=0}^n\\frac{g_{6,k}(a)}{(k+1)(k!)^2}}.$$\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.EllipticTailExponent`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $g_{6,k}(a)$ | `Defs.g6` | Exactly $a(a)_k(1-a)_k$, with empty products equal to one. |\n| $S_n(a)$ | `Defs.S` | The finite sum through index $n$, inclusive; `deriv (S n)` is its real derivative. |\n| $\\lambda_{10}(a)$ in (3.6) | `Defs.lambda10 n a` | The same trigonometric quotient, with truncation order explicit. |\n| $a\\in(0,1/2]$ | `Set.Ioc 0 (1 / 2)` | The left endpoint is excluded and the right endpoint included. |\n| $\\bar\\lambda_{10}=2$ for all $n\\in\\mathbb N_0$ | `Defs.claim`, `Result.result` | `IsLUB (lambda10 n '' Set.Ioc 0 (1 / 2)) 2` for every `n : ℕ`, including zero. |\n| Positive denominator and strict upper bound | `Result.denominator_pos`, `Result.lambda10_lt_two` | Both hold at every point of the stated interval. |\n| Limit at zero from the right | `Result.tendsto_lambda10` | `Tendsto (lambda10 n) (𝓝[>] 0) (𝓝 2)` for each natural $n$. |","title":"The supremum of the elliptic remainder parameter exponent","url":"https://the-omega-institute.github.io/trureturing-pages/research/bao-wang-qiu-2022-elliptic-tail-exponent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c4553675b5354c91892ef3978a1b8af89550706063940729a050caf243ef929.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bar-lev-2026-pareto-code-length-monotonicity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bar-Lev, arXiv:2603.17154v2, Section VII-A, Conjecture 2, asks whether every\nrank-`k` length-`n` code over `F_q` admits a rank-`k` length-`n+1` code with\nboth actual expected file-retrieval times no larger, whenever `n>=k`,\n`s1,s2>0`, `s1+s2=k`, and `max(s1,s2)>=2`. The expected time is the minimum\nnumber of independent uniform-with-replacement draws required for the\nsampled columns to span the whole coordinate file. Issue #11618 fixes this\ntarget before the proof probe.","title":"The finite-length Pareto conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/bar-lev-2026-pareto-code-length-monotonicity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/53da59397a56de1b3de0266133b94acb8251045aef7a346f6bb075c03065f7f8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bar-lev-2026-universal-hyperbolic-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bar-Lev, arXiv:2603.17154v2, Section V-B, Conjecture 1, asserts\n`s1/E1+s2/E2<=1` for every finite field, every positive two-file\npartition `s1+s2=k` with `max(s1,s2)>=2`, every length `n>=k`, and every\nrank-`k` generator. Each physical column index is drawn independently,\nuniformly and with replacement. `Ei` is the expectation of the minimum\ntime at which the sampled-column span contains the entire coordinate\nfile. Duplicate indices remain separate draws. Preregistration #11729\nfixes this complete statement and a nonzero-column refutation criterion.","title":"The universal hyperbolic retrieval bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/bar-lev-2026-universal-hyperbolic-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37fe2eb9b85616adf4d4bad38e23bf30ce02d9a29bc5e59521ff93ebc70fd240.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barato-seifert-2015-waiting-time-randomness-skewness-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. C. Barato and U. Seifert, *Skewness and Kurtosis in Statistical Kinetics*, Phys. Rev. Lett. 115, 188103 (2015), arXiv:1510.03279v1, ask in `skewkurtstatkinarxiv.tex`, line 358, at the end of the multicyclic discussion:\n\n> It remains as an open question whether this property is true for arbitrary networks that do not consist of a single cycle.\n\nThe preceding numerical investigation concerns a four-state network with two product-forming routes and its off-pathway specialization. At lines 350–352 the authors report that constraining the skewness to be at most two prevents the randomness parameter from exceeding one. The question asks whether the superior detection of departures from a single cycle by higher moments extends to arbitrary networks.\n\nThe source defines cycle completion at an irreversible product-release transition (Eq. `fullreaction`, lines 110–117). For completion-time moments $t_n$, its definitions at lines 133–149 are\n$$\n\\sigma^2=t_2-t_1^2,\\qquad\n\\mathcal R=\\frac{\\sigma^2}{t_1^2},\\qquad\n\\mathcal S=\\frac{t_3-3t_1\\sigma^2-t_1^3}{\\sigma^3}.\n$$\n\nThe source's kurtosis is $\\mathcal K=(t_4-4t_3t_1+6\\sigma^2t_1^2+3t_1^4)/\\sigma^4$ (line 149); it is not used by the settlement.\n\nIssue [#26046](https://github.com/the-omega-institute/trureturing/issues/26046) fixes the implication to be refuted: for every finite network with nonnegative off-diagonal rates, a positive marked rate $p\\to r$, every state able to reach $p$, and positive variance, $\\mathcal S\\le2$ implies $\\mathcal R\\le1$. Moments use the first-passage matrix definition below. This implication is refuted.\n\nThe [literature note](../Library/StatisticalMechanics/baratoseifert2015skewness.md) records the source and its scope.","title":"Skewness at most two does not force randomness at most one in multicyclic enzyme networks","url":"https://the-omega-institute.github.io/trureturing-pages/research/barato-seifert-2015-waiting-time-randomness-skewness-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8030d48f71aba390da43f3a952ebd7a5a0845d711f9c963e9305adc3019e1a9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barber-gillespie-taylor-2023-purification-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Barber, N. I. Gillespie and J. M. Taylor, *Post-selection-free preparation of high-quality physical qubits*, Quantum 7, 994 (2023), arXiv:2209.05391v3, Section 3.2, PDF page 20 (`purification.tex`, lines 859–862), ask:\n\n> We conjecture that it should be possible to prove monotonicity under some reasonable set of conditions. For instance, suppose that $\\big|\\binom n 0 + \\cdots + \\binom n e\\big| = 2^{n-k}$ and $p_0 < 0.5$. Is $p_{\\text{out}}(p_0,p_I,p_C,p_T)$ an increasing function of $p_0$, $p_I$, $p_C$ and $p_T$? We leave this as an open question.\n\nTheir $(n,k,e)$ purification circuit consists of CNOT and Toffoli gates and sends every input bitstring of Hamming weight at most $e$ to a bitstring zero in $k$ nominated output positions (Section 2, line 203). Each input bit independently starts at one with probability $p_0$. Each idle qubit depolarises independently with probability $p_I$, and each CNOT or Toffoli independently depolarises its entire support with probability $p_C$ or $p_T$. Depolarisation replaces that support by uniformly random bits (Section 2.1, lines 219–243). The quantity $p_{\\text{out}}$ is the expected number of erroneous output bits divided by $k$ (lines 711–713).\n\nIssue [#22506](https://github.com/the-omega-institute/trureturing/issues/22506) fixes the nondecreasing reading: for all natural $n,k,e$ with $1\\le k\\le n$ and $1\\le e\\le n$, every gate list executed one gate per round, and every set of exactly $k$ outputs, purification through weight $e$ together with $\\sum_{j=0}^{e}\\binom nj=2^{n-k}$ implies separate nondecreasing dependence on each noise coordinate. Both parameter vectors must lie in $0\\le p_0<1/2$, $0\\le p_I,p_C,p_T\\le1$, and the other three coordinates are held fixed in each comparison. This is `claim`. Its refutation already occurs in the preparation coordinate. One gate per round is an allowed subclass of the source's scheduled circuits.\n\nThe definitions and source locators are also in [the literature note](../Library/Hamming/barbergillespietaylor2023purification.md).","title":"Tight purification circuits need not have monotone output error","url":"https://the-omega-institute.github.io/trureturing-pages/research/barber-gillespie-taylor-2023-purification-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15c8c0c95c8dac14e6f2a9d40283c66ab489eb57139aa28cdbb20b4c7486b3da.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barczy-2025-unrestricted-jensen-converse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mátyás Barczy and Zsolt Páles, *Jensen convex functions and doubly stochastic matrices*, arXiv:2510.03715v1, Remark 2.2(i), ask:\n\n> However, we do not know whether the assertion in part (i) of Theorem 2.1 holds for every doubly stochastic matrix having rank at least 2. We leave this question as an open problem.\n\nThe referenced assertion is Theorem 2.1(i), equation (2.1), with Jensen convexity defined in Definition 1.1. The complete target is the following closed universal assertion:\n\nFor every positive integer $n$, every nonempty open real interval $I$ (including unbounded intervals and $I=\\mathbb R$), every arbitrary function $f:I\\to\\mathbb R$, and every real $n\\times n$ matrix $P=(p_{ij})$ satisfying\n\n$$\np_{ij}\\geq0,\\qquad \\sum_jp_{ij}=1,\\qquad\n\\sum_ip_{ij}=1,\\qquad \\operatorname{rank}_{\\mathbb R}P\\geq2,\n$$\n\ndoes\n\n$$\nf\\!\\left(\\frac1n\\sum_{j=1}^n x_j\\right)\n\\leq\\frac1n\\sum_{i=1}^n f\\!\\left(\\sum_{j=1}^np_{ij}x_j\\right)\n\\qquad\\text{for every }(x_1,\\ldots,x_n)\\in I^n\n$$\n\nimply\n\n$$\nf\\!\\left(\\frac{u+v}{2}\\right)\\leq\\frac{f(u)+f(v)}2\n\\qquad\\text{for every }u,v\\in I?\n$$\n\nThe rank condition replaces the original special column-span premise; it is not conjoined with it. There is no continuity, measurability, boundedness, rational-entry or rational-input assumption. The function is defined on the original interval itself, with no ambient-extension datum. Convexity of the interval puts each mean and each matrix row output in its domain.\n\nThe answer is negative. The exact result is the negation of this whole universal assertion:\n\n$$\n\\neg\\Bigl[\\forall n\\in\\mathbb N_{>0}\\;\\forall I\\subseteq\\mathbb R\n\\text{ nonempty open interval}\\;\\forall f:I\\to\\mathbb R\\;\n\\forall P\\in\\mathbb R^{n\\times n},\\quad\n\\bigl(P\\text{ doubly stochastic}\\ \\land\\ \\operatorname{rank}_{\\mathbb R}P\\geq2\\bigr)\n\\Longrightarrow\n\\Bigl(\\bigl[\\forall x\\in I^n,\\ f(\\tfrac1n\\sum_jx_j)\n\\leq\\tfrac1n\\sum_i f(\\sum_jp_{ij}x_j)\\bigr]\n\\Longrightarrow\n\\bigl[\\forall u,v\\in I,\\ f(\\tfrac{u+v}2)\\leq\\tfrac{f(u)+f(v)}2\\bigr]\\Bigr)\\Bigr].\n$$","title":"The unrestricted rank-only Jensen converse is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/barczy-2025-unrestricted-jensen-converse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c31ac9bf30023af73fb8494dfde6c4e37e741a11f454dbf16bb57a1041f94dc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bardakov-2022-t1-groupoid-reduct-question","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"V. Bardakov, B. Chuzinov, I. Emel'yanenkov, M. Ivanov, T. Kozlovskaya and V. Leshkov, *Set-theoretical\nsolutions of simplex equations*, arXiv:2206.08906v1, Section 9.2 (Мат. труды 27(1) (2024), pp. 51–53;\nSiberian Adv. Math. 34(1) (2024) 1–40), define a first tetrahedral 4-groupoid ($T_1$-groupoid)\n$(X,*,\\circ,\\triangleleft,\\triangleright)$ by four identities and a reduced $T_1$-groupoid $(X,*,\\circ)$ by three\nidentities, both giving elementary 1-solutions of the tetrahedron equation. They write: \"It is not clear\nif there is a $T_1$-groupoid such that forgetting about $x\\triangleleft y$ and $x\\triangleright y$ will not yield\na reduced $T_1$-groupoid.\" The verbatim statements are in\n[the literature note](../Library/StatisticalMechanics/bardakov2024simplex.md).\n\nIssue [#14643](https://github.com/the-omega-institute/trureturing/issues/14643) reads the question as\nwhether the universal statement \"the reduct $(X,*,\\circ)$ of every $T_1$-groupoid is a reduced\n$T_1$-groupoid\" fails.","title":"A two-element T1-groupoid whose reduct is not a reduced T1-groupoid","url":"https://the-omega-institute.github.io/trureturing-pages/research/bardakov-2022-t1-groupoid-reduct-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06510e8ac79e4fc57875f892068c320813c1c02da7beafbb26f3cf0cc8a54359.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bardakov-2024-simplex-fixed-point-free-permutations","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"V. Bardakov, B. Chuzinov, I. Emel'yanenkov, M. Ivanov, T. Kozlovskaya and\nV. Leshkov, *Set-theoretical solutions of simplex equations*,\narXiv:2206.08906v1, Section 4.2 (Question 4.22 of Siberian Adv. Math.\n34(1) (2024) 1–40, and of its Russian original in Мат. труды 27(1) (2024)),\nask:\n\n> 2) We know that the permutation $P_{12}$ is a solution of the YBE. For which $n > 2$ there are non-identity permutations without fixed points that gives solutions of the \\SE?\n\nThe $n$-simplex equation $R_{\\overline 1}\\cdots R_{\\overline{n+1}} =\nR_{\\overline{n+1}}\\cdots R_{\\overline 1}$ acts on $X^N$, $N=n(n+1)/2$, with\nthe multi-indices given by the rows of the matrix $MI_n$; a permutation $s$\ngives the simple map $(x_1,\\dots,x_n)\\mapsto(x_{s(1)},\\dots,x_{s(n)})$.\nThe precise question, preregistered in issue #11874, asks for the set of\n$n>2$ for which some permutation without fixed points gives a solution on\nevery set.","title":"Fixed-point-free permutation solutions of the n-simplex equation","url":"https://the-omega-institute.github.io/trureturing-pages/research/bardakov-2024-simplex-fixed-point-free-permutations/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f581a4d19da5e44441ac066d32100d19c56c6d348d54ee72d5f95a9c55634b7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barei-2026-remark-3-5-minimal-idempotent","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Barei, with Muse Spark via meta.ai and review by Nicolás Jaramillo Torres,\nstates in Remark 3.5:\n\n> Given a finite-dimensional evolution algebra E, we are yet to find idempotent subspaces of dimension greater than two which do not properly contain a nontrivial idempotent subspace.\n\nThe reading fixed in [#12879](https://github.com/the-omega-institute/trureturing/issues/12879)\nis an existence question over ℂ. An evolution algebra is a vector space with a\nbilinear, possibly nonassociative multiplication and a natural basis whose\nmixed products vanish. For every complex linear subspace U, let\n$U^2=\\operatorname{Span}_{\\mathbb C}\\{xy:x,y\\in U\\}$. Idempotent means\n$U^2=U$, and nontrivial means nonzero. V is not required to be an evolution\nalgebra in its own right. Minimality ranges over all complex linear subspaces,\nnot only coordinate subspaces or subspaces defined over ℚ.\n\nThe precise positive answer is: there exist such an E and V with\n$\\dim_{\\mathbb C}V=3$, $V^2=V$, and\n$U\\le V$, $U\\ne0$, $U^2=U$ implying $U=V$ for every U.\nThe displayed remark is an unresolved search, not an asserted universal\nnonexistence theorem. Barei's Example 2.1 is literature and is excluded.","title":"A three-dimensional minimal idempotent subspace","url":"https://the-omega-institute.github.io/trureturing-pages/research/barei-2026-remark-3-5-minimal-idempotent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32ea47cbb11a90443240ffb77785bc205935bfec5f7dcdd550cf31a85cc3872e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baricz-mehrez-2018-bessel-redheffer-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Á. Baricz and K. Mehrez, *Redheffer type bounds for Bessel and modified\nBessel functions of the first kind*, arXiv:1701.08446v1, §3,\nConjecture 1, inequality (3.2):\n\n> If ν > −1 and |x| < j_{ν,1}, then the following inequalities are valid (j_{ν,1}²/(j_{ν,1}² − x²))^{ϑ_ν} ≤ 𝒥_{ν+1}(x)/𝒥_ν(x) ≤ (j_{ν,1}²/(j_{ν,1}² − x²))^{η_ν}, where ϑ_ν = j_{ν,1}²/(4(ν+1)(ν+2)) and η_ν = 1 are the best possible constants.\n\nThe source states the left-hand inequality as the conjecture; the\nright-hand inequality is a known result. The source proves the lower\nbound in the case $\\nu=-1/2$. Only the lower bound is refuted here.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.BesselRedheffer`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $\\mathcal J_\\nu(x)$ | `Defs.nbessel` | The real sum $\\sum_{k\\ge0}(-x^2/4)^k/[k!(\\nu+1)_k]$ with ascending Pochhammer evaluations. |\n| $j_{\\nu,1}$ | `Defs.firstZero` | The infimum of $\\{x>0:\\mathcal J_\\nu(x)=0\\}$; existence, positivity and attainment are proved for $\\nu\\ge30$. |\n| $\\vartheta_\\nu$ | Exponent in `Defs.claim` | Exactly $j_{\\nu,1}^2/[4(\\nu+1)(\\nu+2)]$, using the real power. |\n| Left-hand inequality of (3.2) | `Defs.claim` | Every real $\\nu>-1$ and every real $x$ with $|x| Motivated by Theorems 1 and 2 we conjecture the following.\n>\n> Conjecture. The function ν ↦ I_ν(u)K_ν(u) is log-convex on (−1, ∞) for all u > 0 fixed.\n\n### Formal reading\n\nLean names below have prefix\n`D5.S3.Analytic.BesselProductOrderLogConvexity.Result`, except the reused\nfirst-kind function.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| First-kind modified Bessel function | `D5.S3.Analytic.BesselRatioLogConcavity.Result.besselI` | Existing standard Gamma-denominator power series. |\n| Second-kind function at noninteger orders | `besselK` | Standard connection formula DLMF 10.27.4 using the reused first-kind function. |\n| Product at fixed positive argument | `P` | The first-kind function times the second-kind function. |\n| Necessary three-point inequality | `claim` | Every u>0, ν>−1 and h>0 with ν, ν+h and ν+2h noninteger satisfies P(u,ν)P(u,ν+2h)≥P(u,ν+h)². |\n| Refutation | `result` | Exactly ¬ claim. |\n\nFor a positive function, convexity of its logarithm implies the midpoint\ninequality after exponentiation. Restricting the three orders to\nnonintegers gives a weaker assertion than the conjecture, where the\nconnection formula defines the standard second-kind function. Refuting\nthis weaker assertion therefore refutes the conjecture.","title":"Refutation of order log-convexity for the modified Bessel product","url":"https://the-omega-institute.github.io/trureturing-pages/research/baricz-ponnusamy-2013-bessel-product-order-log-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec2cb91840bf9497eacf82cc1e7bf15774d9d8275f8ccffe2bf28c97bba70c1d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baricz-ponnusamy-vuorinen-2011-gamma-gamma-nbu","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Á. Baricz, S. Ponnusamy and M. Vuorinen, *Functional inequalities for\nmodified Bessel functions*, Expo. Math. 29(4) (2011), 399–414,\nDOI 10.1016/j.exmath.2011.07.001, arXiv:1009.4814v2,\nQuestion 8, printed p. 13:\n\n> Question 8. Is it true that the gamma-gamma distribution satisfies the NBU property?\n\nThe preceding definition, printed p. 13, reads:\n\n> By definition a life distribution satisfies the new-is-better-than-used (NBU) property if u ↦ log F̄(u) is sub-additive, i.e.\n>\n> $$\\overline F(u_1+u_2)\\le\\overline F(u_1)\\overline F(u_2)$$\n>\n> for all u₁, u₂ > 0.\n\nThe density, §4, printed p. 10, is introduced as follows:\n\n> The probability density function fₐ,ᵦ,α : (0, ∞) → (0, ∞) of the three parameter gamma-gamma random variable is defined by (see [21])\n>\n> $$f_{a,b,\\alpha}(u)=\\frac{2(ab)^{(a+b)/2}u^{(a+b)/2-1}}{\\Gamma(a)\\Gamma(b)\\alpha^{(a+b)/2}}K_{a-b}\\!\\left(2\\sqrt{\\frac{ab}{\\alpha}u}\\right),$$\n>\n> where a, b > 0 are the distribution shaping parameters, Kν stands for the modified Bessel function of the second kind, and α > 0 is the mean of the gamma-gamma random variable.\n\n### Formal reading\n\nLean names have prefix `D5.S3.Analytic.GammaGammaNBU.Result`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Modified Bessel function of the second kind | `besselK` | DLMF 10.32.9: $\\int_0^\\infty e^{-z\\cosh t}\\cosh(\\nu t)\\,dt$, for real $\\nu$ and $z>0$. |\n| Source density | `density` | The displayed density with `Real.Gamma`, real powers and nonnegative square root. |\n| Survival function | `survival` | The Lebesgue integral of the density over $(u,\\infty)$. |\n| Universal NBU assertion | `claim` | Every $a,b,\\alpha>0$ and every $u_1,u_2>0$. |\n| Refutation | `result` | Exactly $\\neg\\mathrm{claim}$. |\n\nOnly positive arguments occur in the counterexample. The total real\nfunctions outside the source domains impose no additional hypotheses\nand do not change the universally quantified positive-domain assertion.","title":"A counterexample to the gamma-gamma NBU assertion","url":"https://the-omega-institute.github.io/trureturing-pages/research/baricz-ponnusamy-vuorinen-2011-gamma-gamma-nbu/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d401d676c8b796ec1e4a3ece8a4d55850f77ac4c106b0ccdad5fb7d40e7f6c9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barrese-prescribed-ordered-level-enumeration","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kenneth Barrese, *Weighted File Placements on Singleton Boards*,\n[arXiv:2312.10855v1](https://arxiv.org/pdf/2312.10855v1), Section 2.2, asks:\n\n> This suggests the open question, given an ordered set of level numbers:\n> (l_1, l_2, . . . , l_n), how many unique Ferrers boards have those specific level numbers.\n\nSection 5 repeats this question. For every positive m and n, count every weakly increasing\nn-tuple of column heights between zero and mn in the fixed n-column, mn-row rectangle,\nwith a prescribed top-down vector l of n nonnegative level totals. A level has m rows;\nits total counts actual occupied cells. Leading zero columns and empty top levels are retained.\nThe input board need not be singleton, and no permutation of l is allowed.\n\nThe same prescribed-level counting problem has an earlier all-width formulation in Kenneth\nBarrese, Nicholas Loehr, Jeffrey Remmel and Bruce E. Sagan, *Bijections on m-level Rook Placements*,\n[arXiv:1508.05944v1](https://arxiv.org/pdf/1508.05944v1), Section 6.2:\n\n> If C is a singleton board, then perhaps counting the number of Ferrers boards B\n> with l(B) = C would be a good start to this problem, but this too remains open.\n\nFor every positive m and every nonempty positive singleton target C, count all normalized\nFerrers boards B of arbitrary width with l(B)=C. Definition 4 reverses the nonempty\nbottom-up level totals. Here normalization uses positive weakly increasing column-height lists:\nSection 4.1 aligns lower-right cells and treats leading zero padding as another notation\nfor the same board. Arbitrary horizontal translations as distinct unbounded absolute-grid\nsubsets are a different carrier. The bibliographic header identifies the later 2023 question;\nthe original 2015 formulation and its governing conventions are identified explicitly here.","title":"Ferrers Boards with Prescribed Ordered Level Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/barrese-prescribed-ordered-level-enumeration/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ea0eb1368e54a801d6350088a1f2f680c8ac73e116939b0515c6072f2fe220a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barriga-2026-dft-ten-mode-edge-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Barriga et al., *N-dimensional discrete Fourier transform via bosonic\nHamiltonian*, arXiv:2609.05644v1, realize the discrete Fourier transform $F_N$\nwith a single waveguide array, $\\Phi^{\\rm out}e^{-i\\mathcal C}\\Phi^{\\rm in}=F_N$.\nThe third conjecture of §III.D states that the remaining solutions for\nnon-prime $N$ have between $N\\ell/2$ and $|E(K_{N-1})|-\\ell$ edges, and the\nsame section fixes $\\ell=5$ for $N=10$:\n\n> Third, all the other possible solutions for non-prime $N$ are in the\n> following range of edges\n> $\\lvert E(R_{\\ell})\\rvert \\leq \\lvert E\\rvert < \\lvert E(K_{N-1})\\rvert-\\ell$\n\nIssue #13297 fixes the reading, taking the most restrictive notion of solution.\n`claim`: every real symmetric $10\\times10$ coupling matrix with nonnegative\noff-diagonal entries, connected support, and diagonal unimodular phases\n$\\Phi^{\\rm out},\\Phi^{\\rm in}$ with $\\Phi^{\\rm out}e^{-iH}\\Phi^{\\rm in}=F_{10}$\nhas at least 25 edges. The propagator is the frozen\n`ProjectionProbabilityFlow.hamiltonianPropagator`.","title":"A 23-edge coupling realizes the ten-mode Fourier transform","url":"https://the-omega-institute.github.io/trureturing-pages/research/barriga-2026-dft-ten-mode-edge-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebecfcf83744c02545811fa9fe3b1668fe54f8a9275f108d858537ac830dc8bc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barriga-2026-polygonal-fourier-couplings","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Barriga et al., *N-dimensional discrete Fourier transform via bosonic\nHamiltonian*, arXiv:2609.05644v1, §II.B, model an array of $N$ waveguides on a\nregular polygon by a real circulant coupling matrix with zero diagonal and\n$C_l=C_{N-l}$, evolve it by $U=e^{-i\\mathcal C}$, and ask for diagonal phase\nshifters with $\\Phi^{\\rm out}U\\Phi^{\\rm in}=F_N$, $F_N=(\\omega^{jk})/\\sqrt N$,\n$\\omega=e^{-2\\pi i/N}$. They find analytic solutions for $N=4,5,6$ and numerical\nones up to $N=31$, and write:\n\n> Whether the polygonal model can always yield solutions to obtain $F_N$ from\n> Eq.~\\eqref{eq:Coupling_polygon} or not is beyond our theoretical efforts.\n\nThe verbatim statements are in\n[the literature note](../Library/QuantumChannels/barriga2026dftbosonic.md).\nIssue [#13954](https://github.com/the-omega-institute/trureturing/issues/13954)\nreads the question for every $N\\ge1$, with exact equality (no column\nrelabelling) and, in addition, strictly positive couplings.","title":"Positive polygonal couplings realize every discrete Fourier transform","url":"https://the-omega-institute.github.io/trureturing-pages/research/barriga-2026-polygonal-fourier-couplings/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9a63dbe52454678a6b70c6a73b5ab5ed37f34bec31041e516aaba87c766e217.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barry-2013-riordan-pascal-kernel-determinant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Barry defines the generalized Pascal matrix on printed page 1:\n\n> In this note, we investigate some properties of the two-parameter generalized\n> Pascal triangle M^{(m)}(a, b) given by the Riordan array\n> M^{(m)}(a, b) = ( 1/(1 − ax) , x(1 + bx)/(1 − ax)^m ).\n\nThe row polynomials are defined on printed page 19:\n\n> We can regard the matrix M^{(m)}(a, b) as the coefficient array of a family of\n> polynomials P_n(x; m, a, b). We have\n> P_n(x; m, a, b) = Σ_{k=0}^{n} T^{(m)}_{n,k}(a, b) x^k.\n\nThe comparison that introduces the kernel appears later on printed page 19:\n\n> It is interesting to compare the matrices defined by the quotient above with\n> those defined by Σ_{j=0}^{n} P_j(x; m, a, b)P_j(y; m, a, b). In the case\n> of monic orthogonal polynomials, such expressions lead to the same matrices.\n> However, M^{(m)}(a, b) is not in general the coefficient array of a family of\n> monic orthogonal polynomials, and hence the two family of matrices will in\n> general differ. In the case m = 2, a = b = 1, we can conjecture the following.\n\nConjecture 32 is stated on printed page 20:\n\n> **Conjecture 32.** The matrix D̃_n(2, 1, 1) with generating function\n> Σ_{k=0}^{n} P_j(x; 2, 1, 1)P_j(y; 2, 1, 1) is given by\n> M_n^{(2)}(a, b)^t M_n^{(2)}(a, b). We then have\n> |D̃_n(2, 1, 1)| = 1 for n ≥ 0. In the above the notation M_n denotes the\n> matrix formed from the first (n + 1) rows and columns of M.\n\nThe page-20 lower summation index is printed as `k=0`, while its summand uses\n`P_j`. The unambiguous comparison display on page 19 uses `j=0,...,n`, so the\nformal sum uses that index. The right side is read at `(a,b)=(1,1)`, as fixed\nby the sentence immediately preceding the conjecture and by its left side.\nThe generic reading fails already at `n=1`: `Dtilde_1 = [[2,1],[1,1]]`, while\n`M_1(a,b)^t M_1(a,b) = [[1+a^2,a],[a,1]]`.\n\nThe phrase \"with generating function\" uses the coefficient-array convention\nfrom the same section: entry `(i,k)` is the coefficient of `x^i y^k`. Section\n2 defines a Riordan array entry by `T_{n,k} = [x^n] g f^k`, with columns\nindexed from zero. Thus the quantified claim is, for every natural `n`,\n`Dtilde_n = M_n^t M_n` and `det Dtilde_n = 1`.","title":"Barry's Riordan-Pascal kernel determinant","url":"https://the-omega-institute.github.io/trureturing-pages/research/barry-2013-riordan-pascal-kernel-determinant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/855adb3f106047bf8e4e3b22b9105264c4b934f9ddedee9198c92a991889935f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:barry-kevrekidis-2013-alternating-vortex-polygon-spectrum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. M. Barry and P. G. Kevrekidis, *Few-particle vortex cluster equilibria in Bose–Einstein condensates: existence and stability*, J. Phys. A: Math. Theor. 46, 445001 (2013), arXiv:1306.1545v2, Section 4.1, Conjecture 4.4 (`BEC_VFP6.tex`, lines 630–631):\n\n> For $N\\geq 6$, the $N$-gon has $\\frac{N}{2}-2$ distinct pairs of real eigenvalues.\n\nThe setting is the precessing point-vortex system of Section 2, Eqs. (1)–(2), with $b=2$:\n\n$$\\dot x_j=-S_j\\Omega y_j-\\sum_{k\\ne j}S_k\\frac{y_j-y_k}{(x_j-x_k)^2+(y_j-y_k)^2},\\qquad\n\\dot y_j=S_j\\Omega x_j+\\sum_{k\\ne j}S_k\\frac{x_j-x_k}{(x_j-x_k)^2+(y_j-y_k)^2}.$$\n\nSection 4.1 specifies the charges:\n\n> We will assume hereafter that the vortices are of alternating charge, namely $S_j=(-1)^{j+1}$, $j=1,2, \\dots,N$.\n\nThis is `BEC_VFP6.tex`, lines 506–507. The polygon has vertices $z_j=e^{2\\pi i j/N}$, $1\\le j\\le N$ (line 515); the even polygon is an equilibrium at $\\Omega=1/2$ (Lemma 4.2, lines 524–532). Thus the conjecture concerns even $N\\ge6$ in this equilibrium family.\n\nIssue [#26188](https://github.com/the-omega-institute/trureturing/issues/26188) fixes the reading. For every natural $m\\ge3$, let $J_{2m}$ be the real Cartesian Jacobian of this field at the unit regular $2m$-gon. Every real root of its characteristic polynomial is accompanied by its negative, and the set of nonzero real roots has exactly $2(m-2)$ elements. Pairs count distinct nonzero values, without algebraic multiplicity. This statement is proved.\n\nThe source and encoding are described in [the literature note](../Library/FluidDynamics/barrykevrekidis2013vortex.md).","title":"The alternating vortex polygon has exactly $N/2-2$ distinct real eigenvalue pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/barry-kevrekidis-2013-alternating-vortex-polygon-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65dc807e30d39c7bc60b21a174eaf630a10f21c9fabc42fef3c1368ab6696e21.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bartoli-stanica-reduced-critical-point-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Daniele Bartoli and Pantelimon Stănică, *Reduced polynomial lifts of APN\npermutations over Galois rings and effective non-APN bounds*,\narXiv:2608.30808v1 (31 August 2026), Conjecture 2 in Section 2 states:\n\n> For every q = 2^m, the reduced representative f ∈ F_(q)[x] of every APN\n> permutation of F_(q) has a critical point in F_(q).\n\nHere APN means differential uniformity exactly two. The reduced representative\nhas degree less than q and the critical point is a finite rational a with\n`f'(a) = 0` in the same field. The dossier addresses this exact universal\nstatement. It does not assert a result about unnormalized polynomial lifts,\naffine-invariance, Galois-ring permutations, or non-rational ramification.","title":"Bartoli--Stănică reduced critical-point conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/bartoli-stanica-reduced-critical-point-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/208db59f7c7542de8a7f1b5678f77495fc76f6b42ff62d6673aa43211553cbac.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:base-phi-negative-prefix-trident","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Write the canonical base-phi expansion as `beta(N) = beta^+(N) . beta^-(N)`, and,\nfor a word `w` of length `m`, let `R_{.w}` be the increasing sequence of natural\nnumbers `N` whose first `m` negative-position digits are `w`, that is\n`d_{-1}...d_{-m}(N) = w`. The paper defines three families `V_F, V_G, V_H`\nthrough first-difference words `x_F, x_G, x_H` arising from three Sturmian\nmorphisms.\n\nThe conjecture, quoted from arXiv:2305.08349v1:\n\n> “Let \\(\\beta(N)=\\beta^+(N)\\cdot\\beta^-(N)\\) be the base phi expansion of the\n> number \\(N\\). Let \\(w\\) be a word of length \\(m\\). Let \\(R_{\\cdot w}\\)\n> be the sequence of occurrences of numbers \\(N\\) such that the first \\(m\\)\n> digits of \\(\\beta^-(N)\\) are equal to \\(w\\), i.e.,\n> \\(d_{-1}\\ldots d_{-m}(N)=w\\). Then there exist two Lucas numbers \\(a\\) and\n> \\(b\\) such that either \\(R_{\\cdot w}=V_F\\), or \\(R_{\\cdot w}=V_G\\), or\n> \\(R_{\\cdot w}=V_H\\). A second possibility is that \\(R_{\\cdot w}\\) is a\n> union of three of such sequences.”\n\nFormal target, in two parts. Port the paper's parameterized definitions\nfaithfully, then prove that every admissible negative prefix cylinder has an\noccurrence set represented by one trident component `V_F`, `V_G`, or `V_H`, or a\nunion of three such components with Lucas parameters. For the repository's depth\nreading that is done and frozen; see Gap. For the paper's reading, which is a\ndifferent predicate on the same words, it is open, and that is what this entry\ntracks. Neither part may be weakened to mere eventual periodicity or occurrence.\n\nThe paper states the obstruction:\n\n> “However, this does not work. The reason is that the \\(\\beta^-(N)\\) words do\n> not occur in lexicographical order, in contrast with the \\(\\beta^+(N)\\)\n> words.”\n\nIt adds that some occurrence sequences are Lucas-Wythoff and some are not,\nalthough they remain close to that form. It exhibits the first `V_G`, the first\n`V_H`, and a three-component case, but not the general classification.","title":"Classify negative base-phi prefix occurrence sequences","url":"https://the-omega-institute.github.io/trureturing-pages/research/base-phi-negative-prefix-trident/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81c152b9088716653e167f5d485945da0ca917adb5c9fd08f594da02545030ec.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bassler-heinrich-kliesch-2024-global-zz-gate-time-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Baßler, M. Heinrich and M. Kliesch, *Time-optimal multi-qubit gates: Complexity, efficient heuristic and gate-time bounds*, arXiv:2307.11160v2 [quant-ph]; Quantum 8, 1279 (2024), Conjecture 16 (`conj:gate-time`), TeX source `J_shaping.tex`, lines 919–927:\n\n> The optimal gate time of $\\operatorname{\\mathsf{GZZ}}(A)$ with $A \\in \\mathrm{Sym}_0 (\\mathbb{R}^n)$ is tightly upper bounded by\n>\n> $\\vec 1^T \\vec \\lambda^* \\leq \\|M\\|_{\\ell_\\infty} \\cdot \\begin{cases} n\\,, &\\text{for odd } n\\,,\\\\ n-1\\,, &\\text{for even } n\\,. \\end{cases}$\n\nThe wording and equation are quoted from the downloaded v2 source (`J_shaping.tex`, md5 `3c751c31a7a40505b6013390b841878a`); its macros for the gate, $\\mathrm{Sym}_0$, $\\mathbb R$ and the norm are expanded for rendering. The source also states, at line 349:\n\n> Unfortunately, we were not able to prove this result and state the challenges in \\cref{app:proof_problems}.\n\nHere $\\mathrm{Sym}_0(\\mathbb R^n)$ consists of real symmetric matrices with zero diagonal (line 240). The effective coupling is $M=A\\oslash J$, with $M_{ij}=A_{ij}/J_{ij}$ when $i\\ne j$ and $J_{ij}\\ne0$, and zero otherwise; the physical interpretation assumes $\\operatorname{nz}(A)\\subseteq\\operatorname{nz}(J)$ (lines 250–263). The source's LP minimizes $T=\\sum_m\\lambda_m$ over $\\lambda_m\\ge0$ with\n\n$M_{ij}=\\sum_{m\\in\\{-1,+1\\}^n,\\ m_n=+1}\\lambda_m m_i m_j\\quad(i= 6`. The precise resolution here is its nonvacuous reading:\n\n```lean\n∀ n : Nat, 2 ≤ n →\n ((∃ R : Fin n → Fin n → Prop, IsTournament R ∧ LinkIrregular R) ↔ 6 ≤ n)\n```\n\n`IsTournament R` requires irreflexivity and exactly one directed edge between\neach two distinct vertices. `DirectedLink R v` is `Subrel R` on the carrier\n`{w // R v w ∨ R w v}`, the union of the out- and in-neighbors.\n`LinkIrregular R` requires `IsEmpty (DirectedLink R u ≃r DirectedLink R v)`\nfor every two distinct vertices `u` and `v`. Thus the isomorphisms excluded\nare arbitrary bijections preserving and reflecting the directed relation.\nIn a tournament the neighbor union is exactly the complement of the vertex,\nso these links are the actual vertex-deleted directed tournaments.\n\nThe restriction `2 ≤ n` is essential. At orders zero and one the pairwise\ncondition is vacuous and a tournament exists, while `6 ≤ n` is false.\nThe result does not prove the literal unrestricted equivalence at those\norders. It proves both directions for every order in the stated domain.","title":"Bastien–Khormali link-irregular tournament existence","url":"https://the-omega-institute.github.io/trureturing-pages/research/bastien-khormali-link-irregular-tournaments/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0775db35e95b8b4c16907de289b74ebc3877fa9d5660750136dbd4d8e8a94a83.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:basu-kashyap-2019-unique-indecomposable-basis","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pranab Basu and Navin Kashyap, *The Lattice Structure of Linear Subspace Codes*,\narXiv:1911.00721v1, Section 6, Conjecture 6.1 (page 23; TeX label `UIB`):\n\n> A linear code $\\mathcal U$ in $\\mathbb P_q(n)$ has a unique indecomposable basis\n> if and only if $\\mathcal U$ is closed under intersection.\n\nA linear subspace code contains the zero subspace and carries an abelian,\nexponent-two group operation $\\boxplus$ whose identity is zero and whose\ntranslations preserve $d_S(X,Y)=\\dim X+\\dim Y-2\\dim(X\\cap Y)$.\nA nonzero codeword is indecomposable when no two codewords of strictly smaller\nambient dimension have it as their $\\boxplus$-sum. A basis is an unordered finite\nset of codewords giving exactly one finite-subset-sum representation of each\ncodeword, with respect to the same operation. Thus the basis belongs to the\ncode's $\\mathbb F_2$-space, rather than to the ambient $\\mathbb F_q$-space.","title":"A Unique Indecomposable Basis Without Intersection Closure","url":"https://the-omega-institute.github.io/trureturing-pages/research/basu-kashyap-2019-unique-indecomposable-basis/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ef12e5dbb8cfa80fe8be37258a62346f1245f0547dad2ea7ce14ae20148e5fa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bathory-2025-commutator-logarithm-branch-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Bathory, *Commutator calculus and symbolic differentiation of matrix functions* (2025 preprint; SIAM J. Matrix Anal. Appl., 2026), arXiv:2505.07987v3, Conjecture 3.7. The following is the verbatim TeX text of `cc_arxiv_1_.tex`, line 506 (conjecture environment at lines 505–507), MD5 `3f5e6f983ebd4bc8e0036ae4a4b880db`:\n\n> With a~suitable generalization of Definition~\\ref{Def}, all identities of Theorem~\\ref{Tform}, Lemma~\\ref{Had}, and Corollaries~\\ref{corpow},~\\ref{Cor} (except \\eqref{eq1}, \\eqref{eq2}) continue to hold for every $\\mathbf{G}\\in\\mathbb{C}^{d\\times d}$ and for any $\\mathbf{A}\\in\\mathbb{C}^{d\\times d}$ such that $\\log\\mathbf{A}$ exists.\n\nThe references in this quotation denote Definition 2.1, Theorem 3.2, Lemma 3.4, Corollaries 3.5 and 3.6, and the exceptions (3.55) and (3.56). The source and operator definitions are also documented in [the literature note](../Library/Eigenstructure/bathory2025commutator.md).\n\nFor a matrix $G$, write $L_G(X)=GX$, $R_G(X)=XG$, and $\\operatorname{ad}_G=(L_G-R_G)/2$, following Eq. (1.1), TeX line 38. Lemma 3.4 assumes $G\\in\\mathbb R^{d\\times d}_{\\rm sym}$ and $A=e^G$; its Eq. (3.42), at TeX line 385, expresses $\\operatorname{ad}_G$ as $(\\log L_A-\\log R_A)/2$.\n\nIssue [#20790](https://github.com/the-omega-institute/trureturing/issues/20790) fixes the complex extension to be settled. Put $M=\\mathbb C^{2\\times2}$. The claim is that there exist two functions $\\Lambda_L,\\Lambda_R:M\\to\\operatorname{End}_{\\mathbb C}(M)$ such that, for every $G\\in M$,\n\n$\\displaystyle \\frac12(L_G-R_G)=\\frac12\\bigl(\\Lambda_L(e^G)-\\Lambda_R(e^G)\\bigr).$\n\nBoth operator-valued functions depend only on $A=e^G$. No continuity, analyticity, or logarithm identities are imposed on them. This reading of the conjecture is refuted.","title":"The complex matrix exponential does not determine the halved commutator","url":"https://the-omega-institute.github.io/trureturing-pages/research/bathory-2025-commutator-logarithm-branch-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b5c31d5d62ac40f1cb460fa407c51456c04133af5e8fb357ec830c8201b4de4b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baumeler-tselentis-wolf-2025-source-digraph-polytope","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Baumeler, Tselentis and Wolf, *Graphical Tests of Causality*, arXiv:2511.22552v1,\nafter Definition 13, state: “We know that $\\widetilde Q_n^{\\rm src}=Q_n^{\\rm src}$\nfor $n\\leq4$. In general, this identity is unsolved.” The settlement is\n$\\widetilde Q_n^{\\rm src}=Q_n^{\\rm src}$ for every natural $n\\geq2$.\nThe external question and literal conventions are preregistered in\nhttps://github.com/the-omega-institute/trureturing/issues/20672.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Section 2: vertices $[n]$, arcs $(s,r)$ with $s\\ne r$ | `Fin n` and the off-diagonal pair subtype, inlined in the definitions | All loopless directed arcs, with sender first and receiver second. |\n| Section 2: adjacency is one exactly on the arc set | The inlined conditional in `sourcePolytope` | Every arc set is a finite set; the empty digraph is included. |\n| Theorem 3: some vertex has in-degree zero | `HasSource` | A vertex is never the receiver of an arc in the set. |\n| Definition 8: convex hull of source-digraph adjacency vectors | `sourcePolytope` | Real convex hull of precisely those vectors. |\n| Definition 13 (58): nonnegative coordinates | First conjunct of `kefalopodaPolytope` | Every arc coordinate. |\n| Definition 13 (59): coordinate upper bounds for $n\\geq3$ | Second conjunct | No upper-bound hypothesis is added for $n=2$. |\n| Definitions 9, 13 (60): every fixed-point-free function gives a sum at most $n-1$ | Third conjunct | One incoming arc per receiver; the bound is real. |\n| Identity in every dimension | `claim`, proved by `result` | Every natural $n\\geq2$; no bounded enumeration hypothesis. |\n\nThe arc subtype and adjacency expression remain inline, while the proof uses\nMathlib's `Set.indicator` for that same adjacency function.","title":"The source-digraph polytope","url":"https://the-omega-institute.github.io/trureturing-pages/research/baumeler-tselentis-wolf-2025-source-digraph-polytope/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe7bd00c0ea14770dee6b00662c3efebfe7324aa84f6d6f97dcf99c9f6ca8f14.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baumer-2024-triangle-inequality-l1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Bäumer, V. Gitton, T. Kriváchy, N. Gisin and R. Renner (arXiv:2405.08939,\nquant-ph; Phys. Rev. A 111, 052453 (2025)) build Bell-type inequalities for the\ntriangle network with four outcomes per party. They penalise asymmetry by\n\n> Δ_l = Σ_{X ∈ {111,112,123}} Σ_{{a,b,c} ∈ I_X} |M_X − p(a,b,c)|^l,\n> M_X = (1/|I_X|) Σ_{{a,b,c} ∈ I_X} p(a,b,c),\n\nwith `I_X` the outcome triples of type `X` (4, 36 and 24 of them), and from\nneural-network estimates they state:\n\n> … getting that approximately s₁₁₁(p) − 0.475 Δ_{l=1}(p) ≤ 0.289 (eq. ineq_l1),\n> s₁₁₁(p) − 5.211 Δ_{l=2}(p) ≤ 0.316 (eq. ineq_l2), should both hold for all\n> local models.\n\nIssue #11263 fixes the reading. Locality is eq. `trilocal` with the frozen\n`IsTriangleLocal`, `s₁₁₁` is the frozen `s111`, and `deltaL1` is `Δ_{l=1}`\nsummed over the three outcome types. The formal `claim` is eq. `ineq_l1` with\nits printed constants for every local `p`; the result refutes it.","title":"A local counterexample to the triangle inequality ineq_l1","url":"https://the-omega-institute.github.io/trureturing-pages/research/baumer-2024-triangle-inequality-l1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7bf4fa1329e246842c8211e82ae99420289eb3baa50f65ab63d4703e8e54dfe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:baumer-2024-triangle-symmetric-local-quarter","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Bäumer, V. Gitton, T. Kriváchy, N. Gisin and R. Renner (arXiv:2405.08939,\nquant-ph; Phys. Rev. A 111, 052453 (2025)) study classical (local) models in\nthe triangle network with four outcomes per party. Their best local fully\nsymmetric construction has `p(A = B = C) = 1/4`, and their conclusion states:\n\n> Open problem. Does there exist a distribution p(A,B,C) that is local in the\n> triangle network and fully symmetric such that p(A=B=C) > 1/4?\n\nIssue #11256 fixes the reading. Locality is eq. `trilocal`: sources uniform on\n`[0, 1]` and measurable conditional distributions `p_A(a|β,γ)`, `p_B(b|γ,α)`,\n`p_C(c|α,β)`. Full symmetry is invariance under every permutation of the\nparties and every joint relabelling of the four outcomes. The formal `claim` is\nthe negative answer, `s₁₁₁(p) ≤ 1/4` for every local fully symmetric `p`; the\nresult refutes it, so the answer is yes.","title":"A fully symmetric triangle-local distribution with p(A = B = C) above 1/4","url":"https://the-omega-institute.github.io/trureturing-pages/research/baumer-2024-triangle-symmetric-local-quarter/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1cbee297333e69a0c0e5be9d8be656c6c5b26e850f868f98082bc548258792e1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bavaresco-murao-quintino-2022-classical-permutation-sequential-advantage","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Bavaresco, M. Murao and M. T. Quintino, *Unitary channel discrimination beyond group structures: Advantages of sequential and indefinite-causal-order strategies* (Journal of Mathematical Physics 63, 042203, 2022; arXiv:2105.13369v3), ask in the Conclusion, `main.tex`, line 514:\n\n> Assuming the analog of a unitary channel in classical information theory to be a classical channel that maps deterministic probability distributions into deterministic probability distributions, i.e., channels that can be described by a permutation matrix, it would be interesting to investigate whether their discrimination could also be enhanced by sequential strategies.\n\nThe quotation is from the v3 TeX source (`main.tex`, md5 `7b7f57add94cbe20dcb088bf7c16f071`); see also [the literature note](../Library/Entropy/bavarescomuraoquintino2022unitarydiscrimination.md).\n\nIssue [#22253](https://github.com/the-omega-institute/trureturing/issues/22253) fixes the existential reading: there are a finite alphabet, a finite permutation ensemble with a probability vector, and a two-use sequential strategy whose average success exceeds a real number $c$ bounding every two-use parallel strategy. Both strategies query the same unknown permutation twice. This is proved.","title":"Classical permutations admit a strict two-query sequential advantage","url":"https://the-omega-institute.github.io/trureturing-pages/research/bavaresco-murao-quintino-2022-classical-permutation-sequential-advantage/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f47eeb7a0eef45e42e1fdf0af69f88b39815e183d7a33b1c66b951c0408eb7ab.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:beccaria-hagendorf-2012-staggered-m1-endpoint-density","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Beccaria and Hagendorf, arXiv:1206.4194v2, §3.2.3, printed page 13:\n\n> As similar pattern is found by probing if a particle is present on the last site. The data is consistent with $\\rho_n^{(N)}(y)=y^{-2}\\rho_n^{(1)}(y)$ for finite $n\\leq8$. We conjecture this to hold for arbitrary system sizes.\n\nThe chain has $N=3n$, $n\\geq1$, nearest-neighbour exclusion, the two inaccessible empty boundary sites, Jordan–Wigner fermionic signs, and staggering II $(y,y,1)$. The preregistration is issue #12308. The quantified formal claim covers every real $y\\ne0$ and every nonzero zero-energy state, so it contains the source's ground-state assertion without an existence or uniqueness premise.","title":"Beccaria–Hagendorf endpoint-density conjecture for the open staggered M1 chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/beccaria-hagendorf-2012-staggered-m1-endpoint-density/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1875e61f4b08dcef1e486e3cd07c5b9fc23af17a5a05d45356bf20b7800364ae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:belkin-allen-clark-2025-permuted-brickwork-even-depth","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Belkin, Allen and Clark, arXiv:2510.23726v2, Section 5.2, ask:\n\n> Like the brickwork, this architecture can be shown to have a PSD vectorization if the depth is odd. It is unclear if the vectorization is PSD at even depths.\n\nThe source defines a complete layer by a uniformly random perfect matching,\nwith independent Haar-random two-site gates on its pairs. Permuted brickwork\nrequires every adjacent pair of matchings to have connected union. Its\nsecond-moment vectorization is\n$\\mathbb E[\\overline U\\otimes\\overline U\\otimes U\\otimes U]$.\nThe bars are entrywise conjugation. Matrix vectorization is column-first.\n\nIssue #13744 preregisters the negative answer on four sites: for every integer\n$q\\ge2$ and every even $d\\ge2$, the depth-$d$ vectorization is not positive\nsemidefinite. The public Lean `claim` states this negative answer and\n`result : claim` proves it; `Refuted` refers to even-depth PSD, not to `claim`.\nThe three matchings are $A=\\{01,23\\}$, $B=\\{02,13\\}$ and $C=\\{03,12\\}$.\nEqual matchings have disconnected union and distinct matchings form a\nfour-cycle, so the connected-block condition is exactly no adjacent repeats.","title":"Even-depth permuted brickwork has a negative moment direction","url":"https://the-omega-institute.github.io/trureturing-pages/research/belkin-allen-clark-2025-permuted-brickwork-even-depth/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ac6c33c568cda56eedc0dbf34835762a062345023624d4616013cf95bc91010.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:beluhov-2026-conjecture-3-dimension-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nikolai Beluhov, *Diamond Determinants and Somos Sequences*, arXiv:2602.24239v2, Section 10, page 24, states:\n\n> For every proper type 𝐧 of order n, it holds that dim Ω⊠ = ⌊n/2⌋.\n\nThe formal claim uses the paper's stated parity-only gauge space, the rational-function field in three independent coefficients, and the kernel of the displayed operator over the constrained monomial subspace.","title":"Refutation of Beluhov's invariant-kernel dimension conjecture at type (1,3,6)","url":"https://the-omega-institute.github.io/trureturing-pages/research/beluhov-2026-conjecture-3-dimension-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7c8b8a00c9a0000d54edd0ba9610c3a57c889a5abb312105a2f6522280f5811.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:benedito-sierra-2025-geometric-tangle-ansatz-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Benedito and G. Sierra, *Visualizing Three-Qubit Entanglement*,\narXiv:2505.23638v2, p. 7, Eq. (15), state:\n\n> “τ(r⃗) = 1 − |r⃗|²/3 − d(r⃗, V_line) · F(r⃗) where |ψ⟩ ∈ GHZ excluding type 5 and F(r⃗) ≥ 0.”\n\nThe canonical state is\n`λ₀|000⟩ + λ₁ e^{iφ}|100⟩ + λ₂|101⟩ + λ₃|110⟩ + λ₄|111⟩`, with\n`0 ≤ λⱼ ≤ 1`, `Σ λⱼ² = 1` and `0 ≤ φ ≤ π` (p. 3, Eqs. (3)–(4)).\nThe vector `r⃗ = (r_A,r_B,r_C)` contains the lengths of the three reduced\none-qubit Pauli vectors. `V_line = {(t,t,t)}` is the main diagonal in ℝ³,\nand `d` is its Euclidean distance, defined as the infimum of distances to points on that line. The canonical three-tangle is\n`τ(ψ) = 4|Hdet(t_ijk)| = 4λ₀²λ₄²` (p. 5, Eq. (10)).\n\nIssue #11500 preregisters the existence of a nonnegative real function F\nsatisfying the equation for every normalized canonical GHZ state that is\nnot type 5. The formal `claim` retains exactly this state domain.\nType 5 means every λⱼ ≠ 0 and every Jₖ ≠ 0. The J invariants are the\nAcín invariants cited by the paper, with μⱼ = λⱼ² and\nΔ = |λ₁λ₄ exp(iφ) − λ₂λ₃|²:\nJ₁ = Δ, J₂ = μ₀μ₂, J₃ = μ₀μ₃, J₄ = μ₀μ₄ and\nJ₅ = μ₀(Δ + μ₂μ₃ − μ₁μ₄). Their canonical definitions are Eq. (23)\nof arXiv:quant-ph/0003050, the paper's cited classification source.\nThe witness has λ₁ = 0, which excludes it from the full type-5 predicate\nwithout evaluating any J invariant.\n\nThe complex tensor `amplitudes` gives all eight coefficients of the state.\n`jointDensity` is its outer product, with entries\n`t_ijk conjugate(t_i'j'k')`. `rhoA`, `rhoB` and `rhoC` are finite-sum\npartial traces using the frozen `partialTraceLeft` and `partialTraceRight`.\nThe frozen `bloch` map extracts\n`(2 Re ρ₀₁, −2 Im ρ₀₁, Re ρ₀₀ − Re ρ₁₁)`.\n`blochLengths` takes the Euclidean lengths of these actual vectors.\n`cayley` is the full complex hyperdeterminant quartic;\n`tangle` is `4 ‖cayley (amplitudes ψ)‖`, with no coordinate shortcut\nin the definition of the claim.\n\nThe source and the formal distance definition agree as follows:\n\n| Source definition | Lean definition |\n| --- | --- |\n| “the distance from that point to the straight line spanned by the main diagonal is the length of the vector connecting it to a point on the line such that this vector is perpendicular to the line” (arXiv v2, Appendix B, before the displayed distance formula) | `V_line : Set (EuclideanSpace ℝ (Fin 3)) := Set.range (fun t : ℝ => WithLp.toLp 2 ![t,t,t])`; `euclideanVector r := WithLp.toLp 2 ![r.1,r.2.1,r.2.2]`; `distanceToDiagonal r := Metric.infDist (euclideanVector r) V_line` |\n\nThe vector embedding and the diagonal set both live in `EuclideanSpace ℝ (Fin 3)`.\nThe local distance step in `result` applies `Metric.infDist_zero_of_mem` with\nrange witness `t = 1/2`.","title":"A type-4c diagonal state refutes the geometric tangle ansatz","url":"https://the-omega-institute.github.io/trureturing-pages/research/benedito-sierra-2025-geometric-tangle-ansatz-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b78c008026dfa6012c692a3aaed14d97d08b6656eb53cbbe030595783d512ce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:benjamini-helman-tov-zhukovskii-2023-noisy-voter-repetition-variance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"I. Benjamini, H. Helman Tov and M. Zhukovskii, *Global information from local observations of the noisy voter model on a graph*, Annals of Probability 51(5), 1963–1992 (2023), Section 2.2, asks:\n\n> May Var(ζ₁ + … + ζ_t) be sublinear? We do not have an answer. However, we prove that it is either linear in t or O(ln t).\n\nThe quotation is from arXiv:2207.01224v2, p. 6. The answer is no. For a connected locally finite simple graph with a neighbour at every vertex, noise $0<\\varepsilon<1$, observation vertex $u$ of degree $D$ and lag $d\\ge1$, put $\\zeta_i=\\mathbf{1}\\{X_i(u)=X_{i+d}(u)\\}$ and $W_t=\\sum_{i=1}^{t}\\zeta_i$. The formal settlement is `D5.S3.StatisticalMechanics.NoisyVoter.RepetitionVarianceLinear.result : claim`. Preregistration: [#16646](https://github.com/the-omega-institute/trureturing/issues/16646). The [Library note](../Library/StatisticalMechanics/benjaminihelmantovzhukovskii2023noisyvoter.md) supplies the publication locator and quotations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Connected graph, possibly infinite | `G : SimpleGraph V`, `G.LocallyFinite`, `Model.connected` | No finite-vertex restriction; finite neighbour sets make uniform choices meaningful |\n| A vertex has a neighbour to copy | `Model.hasNeighbor`; `Value G (Coord.choice k v)` is the neighbour subtype | Excludes an isolated one-vertex graph whose uniform copying law is undefined |\n| Independent fair initial opinions | `Coord.initial`, `coordinateLaw`, `Model.independent`, `Model.laws` | Bernoulli one-half |\n| Independent fair fresh opinions and Bernoulli noise controls | `Coord.fresh`, `Coord.refresh`, `coordinateLaw`, `Model.ε`, its strict bounds | True means refresh |\n| Independent uniform neighbour choices | `Coord.choice`, neighbour-subtype value and uniform finite law | The choice is an actual neighbour |\n| Synchronous recursion | `nvm` | Initial, refresh and copying cases mirror Section 1 |\n| Repetition indicator at lag $d$ | `W`, sum over `Finset.Icc 1 t`, positive lag in `claim` | No normalization; the observation reaches time $t+d$ |\n| Every graph, vertex and probability model | The closed quantifiers of `claim` | Arbitrary vertex type and measurable probability space |\n| Variance is never sublinear | `result`, negated little-o in `claim` | Private `model_explicitBound` supplies the explicit lower bound for every natural $t$ |\n| Encoding requirements | Decidable equality, measurable-space and local-finiteness instances | Decidable equality is representational; no additional graph hypothesis |","title":"The repetition-count variance of the noisy voter model","url":"https://the-omega-institute.github.io/trureturing-pages/research/benjamini-helman-tov-zhukovskii-2023-noisy-voter-repetition-variance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef2e454a26623170f583dcb33629c71067d34449941dc0bf34626a1c3c9abef8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:benyi-mansour-ramirez-weak-ascent-210","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Beáta Bényi, Toufik Mansour and José L. Ramírez, *Pattern Avoidance in Weak Ascent Sequences*,\narXiv:2309.06518v4 (Discrete Math. Theor. Comput. Sci. 26:1, 2024), Section 3, Conjecture 3.1:\n\n> The sequence w_210(n) coincides with the sequence A117106.\n\nThe paper adds that A117106 enumerates permutations avoiding the vincular pattern 2-41-3. Here\nw_210(n) counts the weak ascent sequences of length n with no indices i < j < k and e_i > e_j > e_k.","title":"Weak Ascent Sequences Avoiding 210 and the Semi-Baxter Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/benyi-mansour-ramirez-weak-ascent-210/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8bb2cc45696976c06d3e6bd9081d9cd52b6b96c0afafc80f38076ba87f343a7d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bera-2026-tomiyama-diagonal-k-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Bera, B. Bhattacharya and D. Chruściński, *Tomiyama-type maps with a diagonal perturbation*,\narXiv:2604.18600v1, study $\\Phi_{\\alpha,\\beta}=(1-\\alpha-\\beta)\\,\\mathrm{id}+\\alpha\\,\\tau_0+\\beta\\,\\Delta$\non $M_d$, where $\\tau_0(X)=\\mathrm{tr}(X)\\,I/d$ and $\\Delta$ keeps the diagonal. Conjecture 2.4\nstates that the $k$-positive maps of the family form the quadrilateral\n$\\mathcal P_k=\\mathrm{conv}\\{\\Psi_0,\\Psi_1,\\Psi_2,\\mathcal T_k\\}$ with parameter points $(0,0)$,\n$(0,\\tfrac d{d-1})$, $(\\tfrac d{d-1},-\\tfrac1{d-1})$ and $(\\tfrac{kd}{kd-1},0)$, and §4 states that\nthe full proof is missing. The verbatim statements are in\n[the literature note](../Library/QuantumChannels/bera2026tomiyama.md).\n\nIssue [#13894](https://github.com/the-omega-institute/trureturing/issues/13894) reads the\nconjecture for $2\\le k\\le d$, with $k$-positivity of $\\Phi$ meaning that $\\mathrm{id}_k\\otimes\\Phi$\npreserves positive semidefiniteness, and the convex hull taken in the $(\\alpha,\\beta)$ plane.","title":"The k-positive region of the diagonal-perturbed Tomiyama maps","url":"https://the-omega-institute.github.io/trureturing-pages/research/bera-2026-tomiyama-diagonal-k-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db0a0bcc22c5aa7549105ae152366613174f59439a81d8d603bcd248e6bbd939.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bera-sarbicki-chruscinski-2023-gcd3-equator","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Bera, G. Sarbicki and D. Chruściński, *Optimizing positive maps in the matrix algebra M_n*, arXiv:2309.09621v1, Conjecture 2, Eq. (13), asserts that when $1\\le k Finding maps $\\phi$ giving CSS-T codes with better parameters than the choice $\\phi=I$ or a permutation is an interesting open question.\n\nFor a binary CSS code (C₁, C₂) and a linear map φ on C₁ satisfying the triple-product parity condition of the source's Proposition 1, the CSS-T code (C₁^{Nφ}, C₂^{Nφ}) has C^{Nφ} = {(x, φ(x)) : x ∈ C}. Every such φ gives length 2n and dimension k = dim C₁ − dim C₂. The definitions are quoted in [the literature note](../Library/QuantumContext/berardini2025csst.md).\n\nIssue [#19814](https://github.com/the-omega-institute/trureturing/issues/19814) fixes the reading. The question asks for some CSS code and some admissible φ for which the φ-extension has strictly larger CSS distance than the identity extension. Such a pair is constructed.","title":"A CSS-T doubling map with better distance than the identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/berardini-et-al-2024-csst-doubling-map-improvement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2fd1f247c01eefc704b5a779346a54edda85346d2c807512d739c2c13bac12ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bergman-question15","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bergman, *Thoughts on Eggert's Conjecture*, Question 15, asks:\n\n> Suppose X is a finite subset of a commutative semigroup S with zero, n a\n> positive integer such that the n-th power map is one-to-one on X and does not\n> take any nonzero element of X to 0, and F a field of characteristic 0. Must\n> every nonzero element of the span FX of X in F_0 S have nonzero n-th power?\n\nThe semigroup is associative and commutative with absorbing zero, with no\nidentity or finiteness assumption. The field is arbitrary of characteristic\nzero. The subset may be empty or contain zero. The algebra is the free vector\nspace on the nonzero semigroup elements, with the semigroup zero sent to the\nzero vector.\n\n| Source clause | Lean meaning |\n| --- | --- |\n| arbitrary commutative semigroup with zero | `S : Type`, `SemigroupWithZero S`, `IsMulCommutative S` |\n| characteristic-zero field | `F : Type`, `Field F`, `CharZero F` |\n| finite subset | `X : Set S`, `X.Finite` |\n| positive exponent | index `n : ℕ` represents exponent `n+1` |\n| positive power without identity | `positivePower a 0 = a`, followed by repeated multiplication by `a` |\n| injective power map on X | `Set.InjOn (fun s => positivePower s n) X` |\n| nonannihilation on nonzero X | `∀ s ∈ X, s ≠ 0 → positivePower s n ≠ 0` |\n| contracted algebra | `Contracted F S = {s : S // s ≠ 0} →₀ F` |\n| contracted zero | `delta F S 0 = 0` |\n| linear span | `Submodule.span F (delta F S '' X)` |\n| claimed conclusion | every nonzero vector in this span has nonzero positive power |\n\nThe utility claim is closed over ordinary small carriers (`Type`), including\narbitrary infinite semigroups and fields. The algebra definitions remain\nuniverse-polymorphic. The finite counterexample lies in these universes and\ntherefore also refutes the unrestricted mathematical question; no\nfinite-carrier restriction is imposed on the universal claim.","title":"Bergman's Question 15","url":"https://the-omega-institute.github.io/trureturing-pages/research/bergman-question15/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d72af2995db83f1b90c1ba16952403ef1f0dcda32cd502f002c51bc0d6a5a484.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bergman-question19-question16-field-condition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bergman, *Thoughts on Eggert's Conjecture*, arXiv:1206.0326v2,\nContemporary Mathematics 609 (2014), 1–17, Question 16, asks:\n\n> Let R be a finite-dimensional nilpotent commutative algebra over an\n> algebraically closed field F, let V be a subspace of R, and let n be a\n> positive integer such that every nonzero element of V has nonzero n-th\n> power. Must dim R ≥ n dim V?\n\nQuestion 19 explicitly asks whether, for Question 16, algebraic closedness\ncan be dropped or weakened to “F is infinite”. The target is only this\nfield-condition branch, retaining associativity and commutativity. Convention 2\nassumes associativity unless explicitly stated otherwise; an identity is not\nrequired. All dimensions and all linear structures use the same F.\n\nThe affirmative infinite-field assertion is quantified over every infinite\nfield F, every finite-dimensional F-vector space R with an actual F-bilinear\nmultiplication satisfying associativity and commutativity, every linear\nsubspace V, and every positive integer n. Common global nilpotence and\nnonvanishing n-th powers of every nonzero vector in V are its antecedents.\nThe conclusion is n dim_F V ≤ dim_F R.\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| every infinite field | `F : Type`, `[Field F]`, `[Infinite F]` |\n| F-vector space | `R : Type`, `[AddCommGroup R]`, `[Module F R]` |\n| finite dimension | `[FiniteDimensional F R]` |\n| F-bilinear multiplication | `μ : R →ₗ[F] R →ₗ[F] R` |\n| associativity | `∀ x y z, μ (μ x y) z = μ x (μ y z)` |\n| commutativity | `∀ x y, μ x y = μ y x` |\n| common nilpotence | `CommonNilpotent μ` |\n| actual F-linear subspace | `V : Submodule F R` |\n| positive exponent | `n : ℕ`, `0 < n` |\n| all nonzero vectors | `∀ v ∈ V, v ≠ 0 → … ≠ 0` |\n| n-th power without identity | `(List.replicate (n-1) v).foldr (fun x y => μ x y) v` |\n| dimension bound | `n * Module.finrank F V ≤ Module.finrank F R` |\n| exact negation | closed `claim : Prop`, designated `result : ¬ claim` |\n\n`CommonNilpotent μ` means that there is k in N such that every list xs of\nlength k and every terminal z have `xs.foldr μ z = 0`. The terminal is an\nadditional factor: the common vanishing length is k+1, always positive.\nUnder associativity this is global algebra nilpotence, equivalently the\ncorresponding product-span being zero. It is stronger than pointwise\nnilpotence. For positive n the displayed power has exactly n factors;\nfor n=1 it is v. The closed assertion uses ordinary small carriers, including\narbitrary infinite carriers. The counterexample lies in these universes and\nalso refutes the unrestricted mathematical assertion.","title":"Bergman's Question 19: the field condition in Question 16","url":"https://the-omega-institute.github.io/trureturing-pages/research/bergman-question19-question16-field-condition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1df55961c22db0cd59ffe3aa5f85e2058650af5f59e80a0aafbae63ce919516.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bernards-guhne-2022-hybrid-nonlocality-depth-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Bernards and O. Gühne, *Bell inequalities for nonlocality depth*,\narXiv:2205.04250v2, Phys. Rev. A 107, 022412 (2023), Section III.F,\nEq. (28), define\n\n$$F_n=\\sum_{\\ell=1}^n(-1)^{1+\\lceil\\ell/2\\rceil}\\ell\n(1\\ldots1\\,2\\ldots2)\\le n2^{n-2}.$$\n\nThe bracket contains $n-\\ell$ settings 1 and $\\ell$ settings 2,\nand sums distinct party permutations. Section V states:\n\n> For (k, m) models with m > 2, we have a conjecture for the classical\n> bound. Proving this bound or finding a counterexample remains an open problem.\n\nIssue #12316 preregisters this Tier 1 question, its source, full quantifiers,\ninterpretation and literature check. The encoded claim is the sharp bound\nfor every $k,m\\ge2$ with $n=k+m$, including $m>2$.","title":"Bernards–Gühne hybrid classical bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/bernards-guhne-2022-hybrid-nonlocality-depth-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12f38da391f7f286062bc777835652f67fbb26ddece15bae47c6f6153150e564.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bernards-guhne-2023-nonlocality-depth-quantum-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Bernards and O. Gühne, *Bell inequalities for nonlocality depth*,\narXiv:2205.04250v2, Phys. Rev. A 107, 022412, Section III.F, Eqs. (28)–(29),\ndefine the permutation-symmetric Bell functional\n\n$$F_n=\\sum_{\\ell=1}^n(-1)^{1+\\lceil\\ell/2\\rceil}\\ell\n(1\\ldots1\\,2\\ldots2).$$\n\nThe bracket sums distinct party permutations, with $n-\\ell$ settings 1\nand $\\ell$ settings 2. The source states:\n\n> In the following, we discuss numerical findings concerning the Bell inequalities F_n for n ≤ 8 parties. We find, that the quantum bound of the inequalities is F_n = √2 n 2^{n−2} (29) up to numerical precision, which is achieved by choosing A₁ = B₁ = … = σ_x and A₂ = B₂ = … = σ_z\n\nFor every integer $n\\ge3$, the maximum over all finite local dimensions,\nbinary observables and density matrices is exactly\n$Q_n=\\sqrt2\\,n2^{n-2}$. This is a uniform extension and proof of the numerical\nfinding; the source does not state a general-$n$ conjecture.\nThe preregistration is [#16626](https://github.com/the-omega-institute/trureturing/issues/16626).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Every party number $n\\ge3$ | `claim`, quantified `n` and lower bound | All such natural numbers |\n| Finite local dimensions | `d`, local carriers `Fin (d j)` | Arbitrary finite dimensions, including empty carriers |\n| Two binary measurements at each party | `A j s`, literal Hermitian and shift conditions | Hermitian matrices with both shifts positive semidefinite; general binary POVMs |\n| Full correlators and distinct party permutations | Entrywise product in `bellOp`, Boolean setting strings | Entrywise tensor product; Hamming weight supplies the coefficient and multiplicity |\n| Arbitrary states | Positive semidefinite `ρ` with trace one in `claim` | Literal body of the existing `GHZMeasureBiseparableBound.IsDensity`; no duplicate predicate |\n| Exact quantum maximum | `IsGreatest` in `claim`; `result` | Universal upper bound and attained value |\n| Source's X/Z optimum | X/Y phase-adjusted GHZ realization in `result` | Equivalent local measurement plane; the explicit X/Z conjugation is a prose identification |\n| Shift and involution hypotheses of the auxiliary estimate | `Upper.unweighted_bound` | Auxiliary inputs discharged by dilation and party decomposition inside `result`; no extra assumption in `claim` |","title":"Exact quantum bound of the Bernards–Gühne nonlocality-depth family","url":"https://the-omega-institute.github.io/trureturing-pages/research/bernards-guhne-2023-nonlocality-depth-quantum-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/006099425f018b13274842d08e5e8e81866d99c523f19837ee3baeb16d2bb4bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bertuzzo-2026-simplex-coverage-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bertuzzo–Ravagnani–Yaakobi, arXiv:2603.06489v1, Section 3,\nConjecture 3.2, asks whether the q-ary simplex code solves Problem B.\nThe predecessor is arXiv:2507.20639v1, Section III, its unnumbered\nsimplex-optimizer paragraph. Issue\nhttps://github.com/the-omega-institute/trureturing/issues/11799\npreregisters the complete assertion before its mathematical probes.\n\nFor every finite field K of cardinality q and every k at least two,\nset n=(q^k-1)/(q-1). For every rank-k k-by-n physical generator G\nand every actual simplex generator S with exactly one nonzero\nrepresentative of each projective line, prove E[T_S] <= E[T_G].\nT is the original minimum time at which the sampled columns span K^k;\neach physical Fin n position is drawn independently, uniformly and\nwith replacement. Zero, repeated and scalar-parallel competitor\npositions remain in the alphabet without conditioning or renormalization.","title":"The original simplex coverage optimizer","url":"https://the-omega-institute.github.io/trureturing-pages/research/bertuzzo-2026-simplex-coverage-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3761c9bc1e000fce2947d3e1b50ab04e169d7f45ab24dfc5b63f2f89924c675f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:beyer-uola-luoma-strunz-2022-calibrated-visibility-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Beyer, R. Uola, K. Luoma and W. T. Strunz, *Joint measurability in\nnonequilibrium quantum thermodynamics*, arXiv:2111.02854v2,\n*Physical Review E* 106, L022101 (2022), p. 4, state:\n\n> The ansatz for the effects W^U_ab in Eq. (6) is a priori not the most general one [54]. However, numerical simulations suggest that our visibility bounds are tight and we conjecture that our construction is optimal to satisfy condition (ii) for the given model Eq. (5)\n\nThe Supplemental Material repeats:\n\n> Therefore, we conjecture that the proposed form of the joint POVM is optimal for the given noise model if one wants to satisfy the fluctuation condition (ii).\n\nFor $d\\ge2$ and $0<\\lambda,\\gamma<1$, set\n$A_a=\\lambda\\Pi_a+(1-\\lambda)I/d$ and\n$B_b=\\gamma\\Pi'_b+(1-\\gamma)I/d$, with $B_b^U=U^\\dagger B_bU$.\nIf every unitary admits a parent POVM satisfying the average and fluctuation\nconditions, then\n\n$$\\gamma\\le\\frac{2\\kappa}{d+2\\kappa-d\\kappa},\\qquad\n\\kappa=2\\sqrt{\\lambda+(1-\\lambda)/d}\\sqrt{(1-\\lambda)/d}\n +(d-2)(1-\\lambda)/d.$$\n\nThe [Library note](../Library/QuantumChannels/beyer2022jointmeasurability.md)\nrecords conditions (i)–(ii), Eq. (5), Eqs. (8)–(9), and both conjecture passages.\nIssue [#16507](https://github.com/the-omega-institute/trureturing/issues/16507)\nis the first-tier preregistration of the literal conventions and universal necessity statement.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Eq. (5), initial white-noise effects | `effA` | Literal scalar-weighted matrix unit plus identity |\n| Eq. (5), final white-noise effects | `effB` | Literal scalar-weighted matrix unit plus identity |\n| Arbitrary relative energy bases | `effBU` and unitary quantifier in `claim` | Standard matrix units with arbitrary unitary rotation cover every relative basis |\n| Lüders instrument | `luders` | The positive square root is `CFC.sqrt` |\n| Condition (i), all energy assignments | `AverageCondition` | All real functions on both outcome sets |\n| Condition (ii), every diagonal state and outcome | `FluctuationCondition` | PSD, trace one, zero off-diagonals, and every outcome pair |\n| A parent POVM for every unitary | `claim` and frozen `ConcealmentKernelNecessityRefutation.IsPOVM` | Every effect is PSD and the effects sum to identity |\n| $d\\ge2$ and strict visibility bounds | Binders and antecedents in `claim` | Every natural dimension and every pair of real visibilities in the source's range |\n| Eqs. (8)–(9) | `kappa`, `visibilityBound`, `result` | Necessity of the literal bound |\n\nThe finite index type represents $d$ outcomes and a $d$-dimensional Hilbert\nspace. The universal unitary quantifier absorbs the choice of the final energy\nbasis. PSD and trace-one hypotheses define a density matrix; the zero-entry\nhypothesis defines its diagonality. These are source conditions, not additional\nrestrictions on admissible parents. The public effects are literal source\nexpressions; the proof uses the frozen Tomiyama noise map `phi` through the\nprivate consumed correspondence `source_effect_eq_phi`.","title":"Optimal calibrated visibility for joint quantum work measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/beyer-uola-luoma-strunz-2022-calibrated-visibility-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab9d4bcbda6db09aecaa0816e9731cb80003045dc7d62997daa5a881487b7ec4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhagat-kulkarni-larsson-murali-conjecture-4-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 4 of arXiv:2510.24280v2 states:\n\n> Consider any subtraction set S, and suppose both players act friendly in case of indifference. Then each player's PSPE utility is never worse than if both players have antagonistic tie-breaking rules.\n\nThe paper orders outcome pairs coordinatewise. In the formal reading, a finite nonempty duplicate-free list of positive natural numbers represents S, and both coordinates of the AvA outcome are required to be at most the corresponding FvF coordinates for every natural heap.","title":"Refutation of Bhagat, Kulkarni, Larsson and Murali Conjecture 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhagat-kulkarni-larsson-murali-conjecture-4-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9d2855aca1623b0424cbcc039216b2d9826fb2aadabbff3ab468405f3ec4c94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhanja-2025-restricted-sumset-question","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Bhanja, *Sumsets with a minimum number of distinct terms*, Integers 25\n(2025), A12, arXiv:2307.03977v2, §4 “Further Remarks”, printed p. 11:\n\n> Another sumset that may come immediately to the reader’s mind is the sumset\n> h(≤r) A, which contains those elements of the group G that are the sum of h elements\n> of A with at most r distinct elements. The sumset hA is a special case of h(≤r) A\n> for r = h.\n\n> Question 1. Is it always the case that the minimum size of both the sumsets hA\n> and h(≤r) A are the same in any abelian group G, where the minimum runs over the\n> subsets of G of size (say) k?\n\n### Formal reading\n\nNames have prefix `D5.S3.FiniteGroups.BhanjaRestrictedSumset.Result`.\n`hSumset h A` is Mathlib's pointwise $h\\mathbin{\\bullet}A$, with exactly\n$h$ summands. `restrictedSumset h r A` is the image of functions\n$f:\\operatorname{Fin}h\\to G$ taking values in $A$ whose image has at most\n$r$ elements, under $f\\mapsto\\sum_i f(i)$.\n`minSize G k h` and `minRestrictedSize G k h r` are infima in the natural\nnumbers of the corresponding cardinalities over $k$-element finsets.\nThe claim requires a $k$-element subset to exist, so these infima are\nactual minima. It quantifies over all abelian groups, $h\\ge1$, $r\\ge2$,\nand all attainable cardinalities $k$. Allowing $k=0$ adds only a case in\nwhich both sumsets are empty for $h\\ge1$.","title":"Unequal minima for sumsets with at most two distinct terms","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhanja-2025-restricted-sumset-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec32dd8be875a4021fb15793d0c1be3e3c401ded54fd9839feb13521c4c501be.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhargava-2026-facemagic-module","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Manjul Bhargava's arXiv:2609.26762v1, [Section 6.2](https://arxiv.org/html/2609.26762v1#S6.SS2) asks: **What is the structure of the R-module U_n(R) of face magic hypercubes of dimension n over R when the characteristic is even and positive?** The source concerns unrestricted entries and one common sum on every two-coordinate face. The source question is recorded in [preregistration #19560](https://github.com/the-omega-institute/trureturing/issues/19560).","title":"All-ring coordinates for unrestricted face-magic hypercubes","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhargava-2026-facemagic-module/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/387fb3e3730b835606e955d6d42bb0082fd5d4aa3b8ada3ce09f870513e91f56.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhatia-jain-2016-stolarsky-mean-matrices-class-a","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Bhatia and T. Jain, *Mean matrices and conditional negativity*,\nElectronic Journal of Linear Algebra 29 (2016), 206–222, section 2.9,\ndefines the Stolarsky means by\n\n> Sγ (a, b) = ((aγ − bγ)/(γ(a − b)))^{1/(γ−1)}, −∞ < γ < ∞\n\nwith their continuous extensions at equal inputs and at parameters\nzero and one. Class A means positive entries and exactly one positive\neigenvalue, counted with multiplicity. The cited source states:\n\n> This family does not yield readily to our analysis. We have some evidence for the following: **Conjecture 1.** Let λ1, λ2, . . . , λn be positive numbers. The matrix [Sγ(λi, λj)] is in the class A if γ ≥ 2 but this is not always so when 0 ≤ γ < 2.\n\nThe exact formal target quantifies over every real parameter at least\ntwo and every nonempty positive finite family, allowing repeated\nnodes. For each parameter from zero inclusive to two exclusive it\nasks for some positive finite family outside class A. The source\nproves parameters three and four in Theorems 2.10 and 2.11 and relates\nparameters zero, one half, and two to earlier sections.","title":"Bhatia–Jain: Stolarsky mean matrices in class A","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhatia-jain-2016-stolarsky-mean-matrices-class-a/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/91da82dc1e975d016898fd88afd910ced72a5ff30c4b111c53aa43a4caa105da.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhattacharjee-mandal-bhattacharya-2026-sublattice-chain-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bhattacharjee–Mandal–Bhattacharya, arXiv:2610.02833v1, Question 9.3:\n\n> Is f(n) = O(n)? Is f(n) = ⌈(n + 1)/2⌉ for every n?\n\nSection 1 specifies:\n\n> a k-colouring of 2^[n] is a map χ : 2^[n] → [k], and a family ℱ ⊆ 2^[n] is monochromatic if χ is constant on ℱ. … it is a sublattice if it is closed under both union and intersection. … f(n) = min_χ max{|ℒ| : ℒ ⊆ 2^[n] is a monochromatic sublattice}, where the minimum is over all 2-colourings.\n\nSublattices here are nonempty. The Lean definition uses finite families of\nsubsets of `Fin n`, constant Bool colour, and closure under both union and\nintersection. The extremal function is literally a minimum over all colourings\nof the maximum cardinality of such a family. The universal `claim` concerns\nonly the second question, using `(n + 2) / 2` for the ceiling.","title":"The ceiling formula for monochromatic sublattices is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhattacharjee-mandal-bhattacharya-2026-sublattice-chain-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d69cf0e2c06151985350102bba0bb2f425c14f28976c39d4bfedb17b8125e0e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhattacharjee-mandal-bhattacharya-2026-union-closed-count-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bhattacharjee–Mandal–Bhattacharya, arXiv:2610.02833v1, Question 9.2:\n\n> Is there a constant C such that U(n,N) < 2^{N−1} for all large n and all N ≥ n^{C log₂ log₂ n}?\n\nProposition 5.4 defines the labelled count and its proposed application:\n\n> Let U(n,N) be the number of union-closed families ℱ ⊆ 2^[n] with |ℱ| = N. If U(n,N) < 2^{N−1}, then F(n) < N.\n\nSection 1 defines union-closed by requiring A ∪ B ∈ ℱ whenever A,B ∈ ℱ.\nThe Lean definition counts finite families of subsets of `Fin n`, with no\nquotient by ground-set permutations. The claim quantifies over a real constant\nC, a natural cutoff, every dimension above that cutoff, and every natural N\nmeeting the real-power threshold. Its result is the literal negation of that\nclaim. The logarithms and real power are Mathlib's total functions; the proof\nuses only dimensions at least 24, where both logarithms have positive\narguments and the usual interpretation applies.","title":"The proposed union-closed counting threshold is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhattacharjee-mandal-bhattacharya-2026-union-closed-count-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e3cad0ebb49f86e3aa75910fb268ba73b7d837653e47159019c27daa6146165.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhattacharjya-monterde-pal-2024-double-blow-up-pgst","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bhattacharjya, Monterde and Pal, *Quantum walks on blow-up graphs*,\narXiv:2308.13887v2, J. Phys. A 57 (2024) 335303, Section 8, Conjecture 1:\n\n> Let n = 2^t r − 1, where t ≥ 2 and r is an odd prime number. If u is a multiple of 2^{t−1}, then PGST does not occur between (0, u) and (1, u) in the double blow-up of P_n.\n\nSection 12 asks whether this conjecture is true and, if not, for which\nparameters pretty good state transfer occurs. The source uses\n$U(s)=\\exp(isA)$. PGST means the existence of a sequence of real times\nwhose transition-entry moduli converge to one. Source vertex $u=j+1$\nis represented by $j:\\mathrm{Fin}(n)$; the blow-up carrier is\n$\\mathrm{Fin}(2)\\times\\mathrm{Fin}(n)$ and adjacency ignores the copy coordinate.\nIssue [#14649](https://github.com/the-omega-institute/trureturing/issues/14649)\npreregisters BMP-1, the literal definitions, the tier-1 classification,\nthe witness and the literature readings.","title":"Pretty good state transfer in double blow-ups of paths","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhattacharjya-monterde-pal-2024-double-blow-up-pgst/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db8860e19350ddbab79d414131da75fe0514e5f38a27b1202080fe7a63644010.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhattacharya-2015-relaxed-hardy-no-signalling-optimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. S. Bhattacharya, A. Roy, A. Mukherjee and R. Rahaman, *Witnessing\nGenuine Mutipartite Non-locality*, arXiv:1507.07327v1, Section 3,\nconjecture:\n\n> The optimal value of the probability of success, $q$ in(\\ref{relH}) is $\\frac{1}{3}$ for any arbitrary dimension for any number of parties in GNST.\n\nHere (relH) are the relaxed Hardy conditions for $N$ parties with inputs\n$u$, $v$ and outcomes $1,\\dots,d$: $P(1\\dots1|u\\dots u)=q>0$; for every $r$,\nwith $v$ at $r$ only, every outcome string with $1$ elsewhere and a value\nother than $d$ at $r$ has probability $0$; and for a fixed $j$ and every\n$i\\ne j$, with $v$ at $i$ and $j$, the string with $d$ at $i$ and $j$ and $1$\nelsewhere has probability $0$. GNST is the theory of no-signalling boxes.\nThe paper's Observation reports the value $1/3$ for $n=3,4$ and\n$d=2,\\dots,5$.","title":"The no-signalling optimum of the relaxed multipartite Hardy test","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhattacharya-2015-relaxed-hardy-no-signalling-optimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/852dc9b41db68d97ab09f49dc7a536f4468a0a4f6c6923fd8a04191051973412.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bhoria-eyyunni-santra-2026-overpartition-threshold-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bhoria, Eyyunni and Santra, *On the number of missing integers in partitions*, arXiv:2604.12557v1, Section 4, Question 3, asks:\n\n> Is there an overpartition analogue of Theorem 2.9, namely, a concise identity for the number of integers less than or equal to the largest part that appear less than k times in all overpartitions of n.\n\nLet $\\mathcal O_n$ contain every partition of $n$ together with an arbitrary subset of its distinct positive part sizes. A chosen size has its first occurrence overlined. Write $m_r(o)$ for its total multiplicity, $S(o)$ for the chosen subset, and $L(o)$ for its largest part, with $L(o)=0$ for the empty partition. Define\n\n$$\nU_k(n)=\\sum_{o\\in\\mathcal O_n}\\#\\{r:1\\le r\\le L(o),\\ m_r(o)0,\\ n\\ge0).\n$$\n\nThe equality is in the integers. Every legal marking pattern is retained, including marking every distinct size. For $k>n$ the sum is empty. The case $n=0$ is a harmless extension supplied here, not a quotation from the paper. The source asks for an analogue; it does not propose the displayed formula as a conjecture.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| All overpartitions of $n$ | `Overpartition n` | A native `Nat.Partition n` and any subset of its support. |\n| Integers at most the largest part | `largest`, `U` | The closed positive interval $[1,L(o)]$. |\n| Appearing fewer than $k$ times | `multiplicity`, `U` | Total multiplicity includes the overlined occurrence; absent sizes have multiplicity zero. |\n| A concise identity for every threshold | `result` | Every natural $k>0$ and every natural $n$, with no bounded enumeration or extra restriction. |\n| Counts on the right side | `T`, `N` | All occurrences and actual nonoverlined occurrences, respectively. |","title":"The overpartition threshold identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/bhoria-eyyunni-santra-2026-overpartition-threshold-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7dec1f43ce61fae0953a1ade3208b5c3dbae3ac8830ef0862afc11cb855fa054.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bianchi-collet-magnanini-2022-ergm-ghs-marginal","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Bianchi, F. Collet and E. Magnanini, *The GHS and other correlation inequalities for the two-star model*, arXiv:2107.08889v2 (ALEA 19 (2022), 1679–1695), Section 4, closing paragraph, conjecture:\n\n> Since $\\text{Cov}_{n;\\boldsymbol{\\beta}}(x_i,x_j) \\geq 0$, due to the GKS inequality \\eqref{GKS_g}, we can conjecture that the necessary and sufficient condition for the GHS inequality to hold in the present general setting is again only the requirement $\\mathbb{E}_{n;\\boldsymbol{\\beta}}(x_i) \\geq 1/2$.\n\nThis is the verbatim sentence at line 1322 of the downloaded v2 source, `GHS_arXiv-replacement.tex` (md5 `b353bfa014348d3c29f4b9c509b5665d`). The general setting is specified at lines 1248–1287; the model and third Ursell function are defined at lines 286–312 and 577–589, respectively. The definitions and source locators are also in [the literature note](../Library/StatisticalMechanics/bianchicolletmagnanini2022ghs.md).\n\nIssue [#22390](https://github.com/the-omega-institute/trureturing/issues/22390) fixes the following reading. For every $n,k\\in\\mathbb N$, take a nonempty family $h_j$ of finite simple graphs on $\\mathrm{Fin}(m_j)$, indexed by $j\\in\\mathrm{Fin}(k+1)$, with $h_0=K_2$, and real coefficients $\\beta_j$ with $\\beta_j\\geq0$ for every $j\\ne0$. The coefficient $\\beta_0$ is unrestricted. The source indexes this family from one instead of zero. On the labeled simple graphs $G$ with vertex set $\\mathrm{Fin}(n)$, put\n\n$t(h_j,G)=|\\operatorname{hom}(h_j,G)|/n^{m_j}$, $H(G)=n^2\\sum_{j=0}^k\\beta_jt(h_j,G)$ and $\\mu(G)=e^{H(G)}/\\sum_{G'}e^{H(G')}$.\n\nWrite $\\mathcal E_n=E(K_n)$, $X_e(G)=\\mathbf1_{e\\in E(G)}$ and\n\n$u_3(e,f,g)=\\mathbb E(X_eX_fX_g)-\\mathbb E(X_e)\\mathbb E(X_fX_g)-\\mathbb E(X_f)\\mathbb E(X_eX_g)-\\mathbb E(X_g)\\mathbb E(X_eX_f)+2\\mathbb E(X_e)\\mathbb E(X_f)\\mathbb E(X_g)$.\n\nThe proposition `claim` asserts, for every such choice,\n\n$(\\forall e\\in\\mathcal E_n,\\ \\mathbb E(X_e)\\geq1/2)\\ \\Longleftrightarrow\\ (\\forall e,f,g\\in\\mathcal E_n,\\ u_3(e,f,g)\\leq0)$.\n\nThe triple quantifier includes repeated edges. The equivalence is refuted by its sufficiency direction.","title":"Edge marginals at least one half do not imply GHS for general exponential random graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/bianchi-collet-magnanini-2022-ergm-ghs-marginal/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/570c9c56a012d21a685644844b7f7cf1c6d72558c555bf7aaad871dec8badd63.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bibak-del-santo-dakic-2024-classical-network-coherence-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Bibak, F. Del Santo and B. Dakić, *Quantum coherence in networks*, arXiv:2407.11122v1 [quant-ph]; Phys. Rev. Lett. 133, 230201 (2024), subsection \"Generalization of the inequality to $n$ parties\", `main.tex`, lines 277–278:\n\n> We obtained classical bound $C_n^{\\text{cl}}$ by substituting $p_{\\text{cl}}$ (Eq.~\\eqref{pncl}) into the expression for $C_n$ and performing extensive numerical optimization. For all tested values of $n \\in \\{2, \\ldots, 10\\}$, we consistently obtained $\\text{max}(C_n^{\\text{cl}}) = \\frac{1}{4}$. Consequently, we conjecture that $\\text{max}(C_n^{\\text{cl}}) = \\frac{1}{4}$ for all $n$.\n\nThe quotation preserves the source's words and mathematical content, with inline TeX math delimiters rendered as dollar delimiters and the source line break joined. The source read is `main.tex` from the v1 source archive, md5 `c47930e35162d18bc08d65fb1201100d`.\n\nIssue [#21125](https://github.com/the-omega-institute/trureturing/issues/21125) fixes the reading: for every integer $n\\ge2$, maximize over $\\gamma\\in[0,1]$ and binary probability laws $p_i,q_i$ the merit of\n\n$r(a)=\\gamma\\prod_{i=0}^{n-1}p_i(a_i)+(1-\\gamma)\\prod_{i=0}^{n-1}q_i(a_i).$\n\nEach local law is nonnegative and sums to one. This is the mathematical mixture in the source's equation `pncl`, lines 248–253. Define $E=\\{a:\\sum_i a_i\\equiv0\\pmod2\\}$ and $O=\\{a:\\sum_i a_i\\equiv1\\pmod2\\}$. The reading of equation `cn`, lines 263–271, is\n\n$C_n(r)=\\frac12\\left|\\sum_{\\substack{a,b\\in E\\\\a\\ne b}}r(a)r(b)-\\sum_{\\substack{a,b\\in O\\\\a\\ne b}}r(a)r(b)\\right|.$\n\nThe pairs are ordered, and both strings in each pair have the stated parity. The factor $1/2$ counts each unordered pair once. The conclusion is an attained maximum of $1/4$, as expressed by `IsGreatest`, rather than an unevidenced supremum. The restriction $n\\ge2$ is explicit: at $n=1$, each parity class has only one string and the merit is zero.\n\nThe source definitions and quotations are also collected in [the literature note](../Library/QuantumContext/bibakdelsantodakic2024coherence.md).","title":"The classical network coherence maximum is $1/4$ for every $n\\ge2$","url":"https://the-omega-institute.github.io/trureturing-pages/research/bibak-del-santo-dakic-2024-classical-network-coherence-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f62287f63cb32409fdcebf92e320c8eece8f1c4e1418191b817a359ed818198.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bilyk-et-al-2022-averaged-kernel-characterization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Bilyk, D. Ferizović, A. Glazyrin, R. Matzke, J. Park and O. Vlasiuk,\n*Potential theory with multivariate kernels*, Math. Z. 301 (2022),\nSection 5, paragraph before Theorem 5.3, verbatim:\n\n> We suspect that the property that σ minimizes I_K is equivalent to the fact that U_K^{σ^{n−2}} is conditionally positive definite, i.e. the two-input energy I_{U_K^{σ^{n−2}}} is minimized by σ. This conjecture is supported by all the examples known to us.\n\nThe setting is the unit sphere in real Euclidean space, normalized surface\nmeasure σ, and continuous symmetric rotationally invariant kernels with n ≥ 3.\nDefinitions 2.1–2.2 give the measure and energy conditions; (5.1) gives invariance.\n\n| Source object | Formal reading |\n|---|---|\n| S^{d−1} | `Defs.S d`, the unit sphere subtype in `EuclideanSpace ℝ (Fin d)` with its Borel structure |\n| σ | `Defs.sigma d`, radial spherical measure of volume scaled to mass one |\n| K | `(Fin (m+3) → Defs.S d) → ℝ`, where n=m+3 |\n| Continuity, symmetry, rotational invariance | `Defs.IsAdmissible`; the counterexample is invariant under the full orthogonal group and hence under the special orthogonal group |\n| I_K(μ) | `Defs.energy K μ`, the integral against μ in every input |\n| σ minimizes I_K | `Defs.IsMinimizer K (Defs.sigma d)`, over all Borel probability measures |\n| U_K^{σ^{n−2}} | `Defs.avgKernel K`, integrating the first n−2 inputs |\n| Conditional positive definiteness | `Defs.IsCondPosDef`, the quadratic form on differences of two finite positive measures of equal total mass; Jordan decomposition covers all finite signed measures of mass zero |\n| Universal equivalence | `Defs.claim`, for all d≥2 and n≥3; its orthogonally invariant subclass already contains the refuting family |","title":"The averaged-kernel characterization fails","url":"https://the-omega-institute.github.io/trureturing-pages/research/bilyk-et-al-2022-averaged-kernel-characterization-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33bbd9580926d271fa6620e94bff423ade141cafe440d22b20356b9a695b53b9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bingham-2026-caterpillar-principal-specialization-family","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Aram Bingham, Kyle Celano, Patricia Commins, Rosa Orellana, Jianping Pan, and Mary Claire Simone,\n*Principal specializations of the t-chromatic symmetric function*, arXiv:2610.31785v1,\nRemark 3.11, first question:\n\n> Do (5,j,j,4,3) and (4,j,j,5,3) exhibit this phenomenon for all j?\n\nThe single target is the conjunction, for every positive integer j, that the actual caterpillars\nof Definition 2.1 associated with these two compositions have unequal ordinary second principal\nspecializations and equal ordinary third principal specializations. Equality and inequality are\nthose of formal integer polynomials in X. Equation (2.1) uses colors 0 through k−1 and sums\nover all proper colorings; every color need not occur. The coefficient inclusion into the source's\ncomplex polynomial ring is injective. The t refinement is outside this question.","title":"Equal third and unequal second principal specializations for a caterpillar family","url":"https://the-omega-institute.github.io/trureturing-pages/research/bingham-2026-caterpillar-principal-specialization-family/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68ba65a672f3538fe3eac2fb055850ecc680d361fa4a8ca18c0c860c35d433c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bingham-2026-caterpillar-principal-specialization-uniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bingham, Celano, Commins, Orellana, Pan and Simone, *Principal specializations of the\nt-chromatic symmetric function*, arXiv:2610.31785v1, [Remark 3.11](https://arxiv.org/html/2610.31785v1#S3.Thmtheorem11), second question:\n\n> And if so, do they continue to be the only pairs of compositions which do?\n\nThe assertion is that for every n at least one and every two source-admissible positive\ncompositions of n, unequal ordinary second and identical ordinary third principal\nspecializations imply that their unordered graph-isomorphism pair is the displayed base\npair {(5,4,3),(4,5,3)} or {(5,j,j,4,3),(4,j,j,5,3)} for some positive integer j.\nAll graph relabelings, reversals and pair exchanges are included. There is no size cutoff.\nThe first all-j question is a separate result.\n\nDefinition 2.1 admits (1), and every nonempty positive composition whose two endpoint\nparts are at least two, including length-one stars. The actual graph has one spine\nvertex per part and a_i−1 pendant leaves at the ith spine vertex. Definitions 2.4,\n2.7 and 2.9 and equation (2.1) give the ordinary specialization as the sum over all\nproper colorings into {0,...,k−1} of X raised to the sum of vertex colors. Surjectivity,\nweights, the t refinement and numerical evaluation of X are absent.","title":"The exhaustive-pairs question for caterpillar principal specializations","url":"https://the-omega-institute.github.io/trureturing-pages/research/bingham-2026-caterpillar-principal-specialization-uniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4182c4329c7862b644bae0fbf3fd1ae25777d6085a12a1edeaaae29d54d65672.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:biswas-shankar-sivasubramanian-p1-matchings","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sucharita Biswas, Umesh Shankar and Sivaramakrishnan Sivasubramanian, *Matchings and shape-Wilf-Equivalence of sets\nof patterns of length three I: Triples*, arXiv:2609.08562v1, Section 6, Question 1: \"Can we enumerate the matchings\nthat avoid the set of patterns P1 = {123, 132, 213} and P13 = {132, 213, 321}?\" Matching patterns follow Section 4\nand Figure 1: three arcs form an occurrence only when all three left endpoints precede all three right endpoints, and\nthe label records the complement of the order of the right endpoints. This dossier settles the P1 clause and records\nan all-size structural scan bridge, finite continuation recurrence and exact all-size enumeration for the P13 clause.\nThe P13 result concerns the original ordinary matching carrier, including empty and disconnected matchings.","title":"Generating Function of Perfect Matchings Avoiding {123, 132, 213}","url":"https://the-omega-institute.github.io/trureturing-pages/research/biswas-shankar-sivasubramanian-p1-matchings/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1aea5e6f34bbc79541418aef5ae2331ae007f48819ac1311bf5c159314c7c6a0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bkm-double-root","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Christian Ballot, Clark Kimberling and Peter J. C. Moses, *Linear Recurrences\nOriginating From Polynomial Trees*, The Fibonacci Quarterly 55(5), 15–27,\nTable 5, gives $q(x)=x^2-2x+1$, initial counts\n$1,1,2,4,7,11,16,22,29,37,46$, and recurrence coefficients $3,-3,1$.\nSection 5 states: “Most of the entries in the column headed ‘linear recurrence’\nare conjectured as ‘eventual’; i.e., the recurrence applies after some unspecified\nnumber of initial terms.” The selected assertion and literal Lean conventions\nare specified in [the problem statement](https://github.com/the-omega-institute/trureturing/issues/23638).\nThe Library note records the published PDF quotations and verified locator.\nA source arXiv identifier has not been located.\n\nStart at $0$ in $\\mathbb Z[x]$, applying $p\\mapsto(p+1)\\bmod q$ or\n$p\\mapsto xp\\bmod q$, with monic remainders. Let $S_n$ contain the distinct labels\nreached within $n$ edges. Put $g(0)=\\{0\\}$ and $g(n+1)=S_{n+1}\\setminus S_n$,\nand $G_n=|g(n)|$. The assertion is\n$\\exists N\\geq3\\;\\forall n\\geq N,\\;G_n=3G_{n-1}-3G_{n-2}+G_{n-3}$,\nwith the recurrence arithmetic over $\\mathbb Z$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Table 5 polynomial $x^2-2x+1$ | `q` | Literal integer polynomial. |\n| Root $0$; children $p+1$ and $xp$ modulo $q$ | `addChild`, `mulChild`, `seen` | Literal monic-remainder operations and finite recursion. |\n| Remove repeated labels | `seen`, `generation` | Finsets remove same-generation duplicates; set difference removes every earlier label. Parent order does not change the label sets. |\n| $G_n=|g(n)|$, including the root | `G` | Natural cardinality of the finite generation. |\n| Eventual recurrence $3,-3,1$ | `claim`, `result` | Existential threshold, every tail index, and integer arithmetic are retained; the proof chooses $N=3$. |\n\nThe pair coordinates occur only in private proof definitions; the public sequence\nretains the source's polynomial operations. Natural predecessor subtraction is\nused only under $n\\geq N\\geq3$ in the recurrence. The zero root is a separate case.","title":"The double-root row of Ballot–Kimberling–Moses Table 5","url":"https://the-omega-institute.github.io/trureturing-pages/research/bkm-double-root/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/25e9867730cf97e6177ef762b974b6007b046c39094b7e656fd6dac6c8b10aa0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:blecher-2025-reversible-four-by-four-nonuniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. P. Blecher, *Commutativity of operator algebras*, arXiv:2509.17673v2, Example 5.2, ends its construction of the CAR algebra $A={\\rm Span}\\{U,V,UV\\}\\subseteq M_4$ with the sentence in `3commfarx.tex`, line 684:\n\n> We conjecture that up to unitary equivalence $A$ is the only noncommutative reversible $4 \\times 4$ matrix algebra.\n\nThe preregistered formal reading takes a complex submodule $b$ of $M_4(\\mathbb C)$ to be a matrix algebra when it is closed under multiplication, noncommutative when it contains a noncommuting pair, and reversible when some $m>0$ and complex-linear map $\\rho:M_4(\\mathbb C)\\to M_m(\\mathbb C)$ reverse multiplication on $b$ and preserve every block operator norm. `UnitarilyEquivalent b carAlgebra` means that some $q\\in\\operatorname{Matrix.unitaryGroup}(\\operatorname{Fin} 4,\\mathbb C)$ satisfies $x\\in b$ exactly when $q^*xq\\in carAlgebra$ for every $x$. Thus `claim` is the universal statement that every such $b$ is unitarily equivalent to the CAR submodule `carAlgebra`.","title":"Blecher's reversible four-by-four algebra is not unique up to unitary equivalence","url":"https://the-omega-institute.github.io/trureturing-pages/research/blecher-2025-reversible-four-by-four-nonuniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7936390b2263025b11f92007085c8a6bfe20ed36472726f86645ca2ca0b7ee0a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:blitvic-kammoun-steingrimsson-2026-packing-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Blitvić–Kammoun–Steingrímsson, *Permutations, Pattern Covers, Posets and\nPositivity: Part I: Patterns and Posets*, arXiv:2610.29846v1, Section 7\nitem 4, Conjecture 7.1:\n\n> For overlap $m\\le K/2$, the maximum is obtained when prefix and suffix\n> are interleaved and minimal, that is, for patterns with prefix\n> $135\\ldots(2m-1)$ and suffix $246\\ldots(2m)$.\n\nFor every $K\\ge2$, $1\\le m We conjecture that $s_{\\mathbb C}(2,4)=2/\\sqrt{13}$.\n\nDefinitions 2.13, 2.15 and 2.16 specify one joint POVM with the original effects\nas marginals, white noise $E_i(s)=sE_i+(1-s)I/2$, the maximum feasible\n$s\\in[0,1]$, and the minimum of these degrees over all four-tuples.\nIssue [#14687](https://github.com/the-omega-institute/trureturing/issues/14687)\npreregisters this literal statement and its literature boundary.","title":"The exact compatibility degree of four dichotomic qubit measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/bluhm-2025-four-qubit-compatibility-degree/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e5d54f9a06952b3215c243f5451ba96278680ff7f211ef096353cad530add5c6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bochniak-chelstowski-kasprzak-soltan-2025-quantum-twin-vertices","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bochniak, Chełstowski, Kasprzak and Sołtan, “Quantum Mycielskians: symmetries, twin vertices and distinguishing labelings”, arXiv:2502.10521v2, Question 3.17, p. 13:\n\n> Do the above-mentioned graphs possess quantum twin vertices?\n\nThe graphs are $K_2\\mathbin\\square K_5$, $C_4\\mathbin\\square C_3$ and the tesseract $Q_4$. Definition 3.8 requires two distinct actions of a compact quantum group whose compositions with the adjacency operator agree. Issue [#19959](https://github.com/the-omega-institute/trureturing/issues/19959) fixes the stronger algebraic claim: for each graph and every unital complex algebra $B$, no two distinct unital complex algebra homomorphisms $\\mathbb C^V\\to B$ agree after adjacency.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Classical function algebra and adjacency | `adjacencyOperator`, with function carrier $V\\to\\mathbb C$ and `Matrix.toLin'` of `SimpleGraph.adjMatrix` | Literal preregistered expression |\n| Two different actions agreeing after adjacency | `HasTwinsIn` retains distinctness and agreement, allowing arbitrary `AlgHom` targets | A strengthening: excluding all such homomorphisms excludes actions |\n| $K_2\\mathbin\\square K_5$ | `prism` | Mathlib complete graphs and Cartesian product |\n| $C_4\\mathbin\\square C_3$ | `cycleProduct` | Mathlib cycle graphs and Cartesian product |\n| Tesseract | The literal Hamming-distance-one graph on `Fin 4 → Bool` in `claim` | The proof uses `Hypercube.hypercube` through the private equality `tesseract_eq_hypercube`; no duplicate graph definition |\n| Negative answer for the three graphs | `claim`, proved by `result` | Every complex algebra $B$, without commutativity or finite-dimensionality assumptions |\n| General twin-free hypothesis | `algHom_eq_of_adjacency_eq` assumes injectivity of adjacency rows | A hypothesis of the general lemma, discharged for all three graphs in `result`; not an extra settlement premise |\n\nThe compact quantum group structure and the action-to-homomorphism passage are mathematical interpretation, not additional Lean declarations in this module.","title":"Quantum twins in three finite classical graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/bochniak-chelstowski-kasprzak-soltan-2025-quantum-twin-vertices/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00193e1bd9ffa7837d18cb464438b724973d43ebea804a112224485f424f889f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bodor-koniorczyk-2017-fidelity-measurement-roc-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Bodor and M. Koniorczyk, *Receiver operation characteristics of quantum state discrimination* (J. Russ. Laser Res. 38, 150–163, 2017; arXiv:1603.06801v1), ask whether the cumulative points of the fidelity measurement lie on the optimal receiver operating characteristic (ROC) curve. The v1 TeX member `Bodor_Koniorczyk_QuantumROC_submit_1.tex`, lines 940–944, in the subsection “The quantum Bhattacharyya coefficient”, states:\n\n> This obviously defines $d+1$ points in the feasible ROC domain, where $d$ stands for the dimension of the joint support of the two density operators. (It is an interesting open question whether these points are actually on the optimal ROC curve.)\n\nThe observable in the preceding equation, labelled `eq:fidelitymeasurable`, is\n$$M=(\\sqrt{\\rho})^{-1}\\sqrt{\\sqrt{\\rho}\\,\\sigma\\sqrt{\\rho}}\\,(\\sqrt{\\rho})^{-1},$$\nwhere $\\rho=\\varrho_N$ and $\\sigma=\\varrho_P$. For an ordered orthonormal eigenbasis $(a_i)_{i=0}^{d-1}$, put $q_i=\\operatorname{Re}(a_i^\\dagger\\rho a_i)$ and $p_i=\\operatorname{Re}(a_i^\\dagger\\sigma a_i)$, ordered by nonincreasing $p_i/q_i$, and define $Q_k=\\sum_{i More dramatically, do there exist strongly polynomial families $G_q$, $H_n$ such that the depth function of $\\hom(G_q,H_n)$ is constant while the degree function is a polynomial of positive degree?\n\nThe binding assertion in [#21296](https://github.com/the-omega-institute/trureturing/issues/21296) reads depth as maximal uniform integer constant depth. Definition 1.2 is downward closed in depth and does not select a pointwise greatest function. The assertion requires one integer $d\\geq-1$ for which the coefficient condition holds at every nonnegative parameter, and excludes every larger integer constant. Pointwise greatest depth and maximality on every tail are separate assertions.\n\n### Statement echo\n\nAll Lean names below belong to `D5.S3.Combinatorics.ExtremalDensity.BogartJimenezSalazarConstantDepthHom` unless an owner is stated.\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Definition 1.1: numerical on every integer | `Numerical` | Integer inputs and integer outputs, cast to rational numbers; negative inputs included. |\n| Definition 1.2: eventually nonnegative degree and the set $S$ | `TwoLevel` | Numerical degree, eventual nonnegativity and guards $0\\leq g(q)$. |\n| Definition 1.2(a): degree and descending coefficients | `TwoLevel` | Nonzero $f_q$, degree $g(q)$, and polynomial coefficient functions; coefficients beyond the degree are zero. |\n| Definition 1.2(b): extension outside $S$ | `TwoLevel`, `constantDepth` | Zero polynomial and depth $-1$ outside $S$. |\n| Definition 3.2: every fixed finite graph and every natural parameter | `StronglyPolynomial` | Fin-labelled finite simple graphs and the existing `ThreeClassHomomorphismLimit.homCount`; all adjacency-preserving maps. |\n| Question 3.19: existence of both graph families and their counting polynomials | `claim`, proved by `result` | Independent strong polynomiality of both families and equality for every $q,n\\in\\mathbb N$. |\n| Positive degree and constant depth | `claim` | Positive natural degree of $g$, a finite integer depth at least $-1$, and failure of every larger uniform integer depth. |\n\nThe finiteness encoding by `Fin` is a labelling choice; no injectivity, connectivity or positive-parameter assumption is added. Rational polynomial coefficients suffice for polynomial integer-count functions. The eventual-nonnegative and guarded coefficient bodies are inlined literally, without duplicate aliases; `Polynomial.reverse` is used inside the private proof through `reverse_coeff`.","title":"Strongly polynomial graph families with constant depth","url":"https://the-omega-institute.github.io/trureturing-pages/research/bogart-jimenez-salazar-2026-strongly-polynomial-question-3-19/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f6faaae7a52c9c19744952d120ead665ec8d0a48d5b3fc811386cb0ef3a8f0d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bohac-2025-qram-count-statistic-insufficiency","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Bohac, *Bias-Class Discrimination of Universal QRAM Boolean Memories*, arXiv:2512.17503v1, asks in Remark 5.3, “Open question: do collective measurements help?” The source TeX `bias_class_discrimination.tex`, lines 454–457, states:\n\n> The $t$-copy induced state $\\rho_p^{(t)}$ generally does not factor as $\\rho_p^{\\otimes t}$, so in principle a collective Helstrom measurement on $\\rho_{p_0}^{(t)}$ vs.\\ $\\rho_{p_1}^{(t)}$ could outperform any separable strategy.\n>\n> An interesting open problem is to characterize the optimal error exponent for distinguishing these \\emph{exchangeable} but potentially correlated mixtures, and to determine whether the ``$\\ket{+}$-count'' statistic is sufficient for optimal discrimination in the exact-weight prior model.\n\nThe module formalizes the second question through `claim`. Its `weightClass n m` is the finite set of Boolean memories $f : \\mathrm{Fin}(2^n) \\to \\mathrm{Bool}$ having exactly $m$ true addresses. Its `copyState n t m` averages the $t$-fold product of one `addressState n f` over that class, with the same $f$ in every copy. For every $n \\ge 1$, $t \\ge 1$, $m_0,m_1 \\le 2^n$, every $e : \\operatorname{Matrix}(\\mathrm{Fin}(t) \\to \\mathrm{Fin}(2^n))(\\mathrm{Fin}(t) \\to \\mathrm{Fin}(2^n))\\,\\mathbb{C}$ with `e.PosSemidef` and `(1 - e).PosSemidef`, and `successProb` as defined in the module, `claim` requires a Boolean rule $g : \\mathrm{Fin}(t+1) \\to \\mathrm{Bool}$ such that\n$\\operatorname{successProb}(n,t,m_0,m_1,e) \\le \\operatorname{successProb}(n,t,m_0,m_1,\\operatorname{countTest}(n,t,g))$.\nThis is the equal-prior special case: if the count were sufficient for every prior, it would in particular satisfy this equal-prior assertion.","title":"The plus-count statistic is insufficient for exact-weight discrimination","url":"https://the-omega-institute.github.io/trureturing-pages/research/bohac-2025-qram-count-statistic-insufficiency/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea7e464cf5219d27cbecbd57b82798a5b904ed37a435dc7119c1e7be7f849d00.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bona-maga-richey-2026-half-frequency-balanced-borders","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Miklós Bóna, Balázs Maga and Jacob Richey, *Letter frequency in shifts of finite type with one forbidden word*, arXiv:2606.06655v2, Section 6, Conjecture 6.1: “A word $w$ has $\\rho^w = 1/2$ if and only if $w$ has balanced borders. The same holds with $\\rho^w$ replaced by $q^w$.”\n\nOnly the first sentence is settled. For every nonempty binary word $w$, let $\\Omega_m^w$ be the length-$m$ binary words without $w$ as a factor and let\n\n$$\n\\rho_m^w=\\frac{\\sum_{u\\in\\Omega_m^w}|u|_1}{m|\\Omega_m^w|}.\n$$\n\nA border is a nonempty prefix that is also a suffix, including $w$ itself. Balanced borders means $2|b|_1=|b|$ for every border $b$. Since the source defines $\\rho^w$ as the limit if it exists, the literal settlement is $\\rho_m^w\\to1/2$ if and only if every border is balanced; no independent convergence assumption is added.","title":"Half letter frequency characterizes balanced borders","url":"https://the-omega-institute.github.io/trureturing-pages/research/bona-maga-richey-2026-half-frequency-balanced-borders/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e9885848f271c35bdadd1241e49aabd99f5a55bcef7917c7621c5a7e0faa17e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bonetto-jardon-kojakhmetov-2024-signature-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bonetto and Jardón Kojakhmetov, *On the Eigenvalues of Graphs with Mixed\nAlgebraic Structure*, arXiv:2408.00487v1, Section 3, Conjecture 1 states:\n\n> The signature of the matrix M(ε) is a monotonic decreasing function of ε.\n\nFor every universe-polymorphic finite vertex type V with Fintype and\nDecidableEq instances, let F,G,H be simple undirected loopless graphs on\nV with disjoint G,H and G union H = F. Set M(u)=L_G+u A_H over the reals,\nusing the ordinary combinatorial Laplacian and 0/1 adjacency matrix.\nFor every real 00$ and $(\\alpha/\\gamma,\\beta/\\gamma)$ belongs to the rational convex hull of $(0,0),(1/2,0),(1,1),(0,2)$. Natural numbers include zero, as in Equation (8) and the boundary vertices. The result does not quantify over independent root permutations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| The two directions of Section 5 | `Move` | Left mutates the second pair; right the first |\n| Mutation formulas (5), rational division | `step` | Both real and shadow coordinates are literal source expressions |\n| Root and mandatory right, left moves, followed by every finite word | `path`, universal word and node quantifiers in `claim` | `List.scanl` includes the initial node |\n| All six coordinates in $\\mathbb N$, Equation (8) | `IsNatNode` | Six natural witnesses cast to $\\mathbb Q$ |\n| Nonzero initial natural shadow triple | The three natural binders and nonzero hypothesis in `claim` | Excludes the triple with no projective point |\n| Four projective vertices in Conjecture 1 | `quadrilateral` | Rational convex hull in the chart $\\gamma=1$ |\n| Tree positivity exactly in that region | `result : claim` | Both implications, including the boundary |\n| Positive chart denominator | The $0<\\gamma$ conjunct of `claim` | Follows from the inequalities and nonzero triple; no extra restriction on the region |","title":"Bonin's shadow Markoff positivity region","url":"https://the-omega-institute.github.io/trureturing-pages/research/bonin-2024-shadow-markoff-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ca1a20a845428e8a271ce9bff730141a10c21a86401dd7067f71d80abf631dc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bors-panario-wang-2023-arithmetic-complexity-question-6-3-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alexander Bors, Daniel Panario and Qiang Wang, *Functional graphs of generalized\ncyclotomic mappings of finite fields*, arXiv:2304.00181v1, Question 6.3.1\n(arXiv page 189; published memoir page 224):\n\n> Is it true that lim inf_{m→∞} AC(T_m)/(log² m/log log m) > 0?\n\nHere $T_m$ is the partition of $\\mathbb Z/m\\mathbb Z$ into singletons and\n$\\mathrm{AC}(T_m)$ is the least number of congruences with positive moduli\ndividing $m$ whose truth values separate all residues. The source is documented\nin [the Library note](../Library/Factorization/borspanariowang2023cyclotomic.md).\nThe mathematical preregistration is [#19381](https://github.com/the-omega-institute/trureturing/issues/19381).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Correspondence |\n| --- | --- | --- |\n| $x\\equiv b\\pmod a$, $a\\ge1$, $a\\mid m$ | `MCongruence` and `MCongruence.Holds` | Positive natural modulus, integer residue, and `Int.ModEq`; positive natural and integer moduli describe the same tests. |\n| Standard representatives $0,\\ldots,m-1$ | Natural representatives with bounds in `SpansTrivial` | Equality of all test truth values forces equality of representatives. |\n| $K$ congruences span the singleton partition | `Fin K` family in `SpansTrivial` | Each possible sign system is a truth vector; each nonempty block has one representative. |\n| Smallest admissible $K\\in\\mathbb N_0$ | `acTrivial` | The natural infimum of separating-family sizes; the set is nonempty for $m>0$. |\n| $\\liminf_{m\\to\\infty}\\mathrm{AC}(T_m)/(\\log^2m/\\log\\log m)>0$ | `claim`, proved by `result` | Natural index at infinity, real natural logarithms and real division, with the ratio embedded in `EReal`. |\n\nThe result has no hypotheses. Values at $m=0,1$ use the total Lean operations;\nchanging a finite initial segment does not change the lower limit at infinity.","title":"Arithmetic complexity of singleton partitions","url":"https://the-omega-institute.github.io/trureturing-pages/research/bors-panario-wang-2023-arithmetic-complexity-question-6-3-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb3a34e522e5d2cf76fea9762e0fbf742af5937ebd896013d05c30ba531a71eb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bosyk-et-al-2016-unified-entropy-triangle-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. M. Bosyk, G. Bellomo, S. Zozor, M. Portesi and P. W. Lamberti, *Unified entropic measures of quantum correlations induced by local measurements*, Physica A 462 (2016) 930–939, DOI 10.1016/j.physa.2016.06.131, arXiv:1604.00329v1, Concluding remarks, `Unifiedarxiv.tex`, lines 944–951:\n\n> In the general case, we only proof this for the von Neumann and Tsallis with entropic index of order $2$, which follows from the contractivity property under a projective measurement of quantum relative entropy and Hilbert Schmidt distance, respectively. We provide numerical counterexamples where the local contractivity property of unilocal disturbances fails in a wide range of the entropic index of R\\'enyi and Tsallis entropies, but it remains open if the triangle-like inequality is fulfilled for other entropic measures.\n\nFor $q>0$, $q\\ne1$ and $s\\ne0$, the source defines\n$S_{(q,s)}(\\rho)=((\\operatorname{Tr}\\rho^q)^s-1)/((1-q)s)$ and\n$D^{\\Pi^K}_{(q,s)}(\\rho)=[S_{(q,s)}(\\Pi^K(\\rho))-S_{(q,s)}(\\rho)]/(\\operatorname{Tr}\\rho^q)^s$.\nThe correlations $D^K_{(q,s)}$ are the minima over complete local rank-one projective measurements without postselection, for $K\\in\\{A,B,AB\\}$. These definitions occur at lines 187–194, 371–394 and 418–433; the triangle-like inequality (38) is at lines 697–701. See also [the literature note](../Library/Entropy/bosyketal2016unifiedcorrelations.md).\n\nIssue [#24210](https://github.com/the-omega-institute/trureturing/issues/24210) fixes the universal reading: for every admissible pair $(q,s)$ and every density matrix on a product of finite-dimensional complex spaces,\n$D^{AB}_{(q,s)}(\\rho)\\le D^A_{(q,s)}(\\rho)+D^B_{(q,s)}(\\rho)$.\nThis reading is refuted at $(q,s)=(2,-1)$ by the two-qubit state\n\n$\\rho_0=\\frac1{16}\\begin{pmatrix}6&1&1&0\\\\1&2&0&1\\\\1&0&2&1\\\\0&1&1&6\\end{pmatrix}$,\n\nin coordinate order $00,01,10,11$.","title":"The unified-entropy correlation triangle inequality is not universally valid","url":"https://the-omega-institute.github.io/trureturing-pages/research/bosyk-et-al-2016-unified-entropy-triangle-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/870d14da87b892bd5770730899c1a31d23146f38e35321c64d444f9963624f50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:botteron-et-al-2024-nonlocal-box-orbit-right-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Botteron, A. Broadbent, R. Chhaibi, I. Nechita and C. Pellegrini,\n*Algebra of Nonlocal Boxes and the Collapse of Communication Complexity*,\narXiv:2312.00725v3, Conjecture 15 (Dyck paths), ask:\n\n> We conjecture that the same result actually holds without the tilde, i.e. the right multiplication $P^{\\boxtimes k}$ gives the highest CHSH-value of $\\mathrm{Orbit}^{(k)}(P)$, as observed numerically.\n\nThe source uses the wiring $\\mathsf W_{\\rm BS}$ and the domain\n$\\widetilde{\\mathcal A}=\\{P\\in\\operatorname{Conv}\\{\\mathrm{PR},\\mathrm{SR},\\mathrm I\\}:\\operatorname{CHSH}(P)\\ge3/4\\}$.\nThe orbit includes every full binary parenthesization of $k$ copies of $P$.\nThe right product multiplies each fresh copy on the right, so its parentheses\nare left-nested. Botteron's thesis, arXiv:2510.09457, repeats the question as\nConjecture 6.11. The source definitions and their locators are in\n[the Library note](../Library/QuantumBounds/botteron2024nonlocalboxorbit.md).","title":"Right multiplication maximizes CHSH over the nonlocal-box orbit","url":"https://the-omega-institute.github.io/trureturing-pages/research/botteron-et-al-2024-nonlocal-box-orbit-right-product/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e562f3d601b46aaf1130488552a862327938fb754e85239d5e5294782b0f0cd3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:boudabsa-simon-2021-kilbas-saigo-hyperbolic-bounds","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Boudabsa and Simon, *Some properties of the Kilbas–Saigo function*,\narXiv:2012.05666v1, §4.1, Conjecture 4.5 (Conjecture 4 in the journal):\n\n> For every α ∈ (0, 1], m > 0, l > m − 1/α and x ≥ 0, one has 1/(1 + [Γ(1+α(l−m))/Γ(1+α(l−m+1))] x) ≤ E_{α,m,l}(−x) ≤ 1/(1 + [Γ(1+αl)/Γ(1+α(1+l))] x).\n\nHere\n\n$$E_{\\alpha,m,l}(z)=\\sum_{n=0}^{\\infty}c_nz^n,\\qquad\nc_n=\\prod_{k=0}^{n-1}\n\\frac{\\Gamma(1+\\alpha(km+l))}{\\Gamma(1+\\alpha(km+l+1))},\\quad c_0=1.$$\n\nThe source records the cases $l=m-1$ and $m=1$ as known.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.KilbasSaigo`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $c_n$ | `Defs.ksCoeff` | The product over $0\\le k0$ and positivity of every Gamma argument used. |\n| $f(x)=E_{\\alpha,m,l}(-x)$ | `Series.g` at order zero | `Series.kilbasSaigo_neg_eq_g0` identifies the two series. |\n| $g_k=(-1)^kf^{(k)}$ | `Series.g` | `Series.hasDerivAt_g` gives $g_k'=-g_{k+1}$ for every natural $k$. |\n| Both bounds and their quantifiers | `Defs.claim`, `Result.result` | All real $\\alpha,m,l,x$ with exactly the five displayed hypotheses; both inequalities are non-strict. |","title":"Hyperbolic bounds for the Kilbas–Saigo function","url":"https://the-omega-institute.github.io/trureturing-pages/research/boudabsa-simon-2021-kilbas-saigo-hyperbolic-bounds/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b79e837730fc15360e3451fe9e03f453fff3c0320e53411e55d4fe5e528a63b2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bouras","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Bouras, OEIS A363347, COMMENTS, Conjecture 3:\n\n> Conjecture 3: Except for 5, the primes all appear exactly twice.\n\nM. Bouras, OEIS A363102, COMMENTS, Conjecture 2:\n\n> Conjecture 2: All prime numbers appear either twice (same as A356247 and A357127) or three times.\n\nFor $n\\ge3$, let $K(n,x)$ be the reciprocal of the right-folded fraction\n$2-3/(3-4/(\\cdots- n/x))$. Define\n$a_4(n)=\\operatorname{den}(K(n,-4))$ and\n$a_2(n)=\\operatorname{den}(K(n,-2))$, using the positive reduced rational denominator.\nThe first claim counts exactly two indices for every occurring prime $p\\ne5$;\nthe second counts exactly two or three for every occurring prime $p$.\nThis dossier anchors A363347 Conjecture 3; the sibling settlement has its own\n[dossier](bouras-a363102.md). Neither statement asserts that every prime occurs. This is the binding reading\nin preregistrations #18626 and #18627, both tier 1 external OEIS problems.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Continued fraction, terminal value $-4$ | `BourasA363347.a`, using `BourasA356360.K` and `nested` | Literal reciprocal and right fold; `Rat.den` |\n| Continued fraction, terminal value $-2$ | `BourasA363102.a`, using the same `K` | Literal reciprocal and right fold; `Rat.den` |\n| Prime values | `Nat.Prime` and an existential occurrence guard | Includes precisely the primes taken by the sequence |\n| Except for $5$ | `BourasA363347.claim` has the guard $p\\ne5$ | No exclusion is added to the second claim |\n| Indices start at three | Both occurrence sets include $3\\le n$ | Matches the OEIS offsets |\n| Exactly twice | `BourasA363347.claim` uses `Set.ncard = 2` | Its proof constructs a finite pair |\n| Twice or three times | `BourasA363102.claim` uses the disjunction of cardinalities two and three | Its proof constructs a pair or triple |\n\nThe natural polynomial `BourasPolynomials.U` is $U(n)=n^2-2$, with truncated\nnatural subtraction. At $n\\ge3$ it agrees with the integer polynomial.\nThe source's rational fold expressions are represented by the existing frozen\n`BourasA356360.nested` and `K` declarations in this module. The public\n`BourasA372763OnceRefutation.nested` and `K` owners are textually equal; reusing\nthem here would require changing or removing those frozen declarations, so\nboth copies remain. No duplicate-retention authorization is claimed.","title":"Prime multiplicities in Bouras's continued fractions","url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5cd34b641918fe83ff5a662bafc15e164db35414401e63d9124b1f54044e589e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bouras-a363102","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Bouras, OEIS A363102, COMMENTS, Conjecture 2:\n\n> Conjecture 2: All prime numbers appear either twice (same as A356247 and A357127) or three times.\n\nFor $n\\ge3$, $a(n)$ is the positive reduced denominator of the rational\ncontinued fraction $1/(2-3/(3-4/(\\cdots-n/(-2))))$.\nThe claim quantifies over primes that occur, as specified in preregistration\n#18627. It does not assert that every prime occurs. This dossier anchors only\nConjecture 2; the $-4$ sequence has its own [dossier](bouras.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Rational continued fraction with tail $-2$ | `BourasA363102.a` uses `BourasA356360.K n (-2)` and `Rat.den` | Literal reciprocal right fold |\n| Prime values | `Nat.Prime` and the existential occurrence hypothesis in `claim` | Excludes primes absent from the sequence |\n| OEIS offset $n=3$ | $3\\le n$ in the occurrence guard and counted set | All larger indices are included |\n| Twice or three times | The disjunction of `Set.ncard = 2` and `Set.ncard = 3` | The proof exhibits a pair or a triple |","title":"Two or three prime occurrences in Bouras A363102","url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras-a363102/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e83fa5e3710de79361a864b85fbd5ac28fc57092489cdb444135ae8047aca69.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bouras372763-once","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A372763, COMMENTS (unpaginated), states: \"Conjecture 2: Except for 2 and 3, all primes appear in the sequence once.\"\nIts NAME is \"Denominator of the continued fraction 1/(2-3/(3-4/(4-5/(...(n-1)-n/(n+5))))).\"\nThe source index range is $n \\ge 3$. The source and statement are preregistered in [#18628](https://github.com/the-omega-institute/trureturing/issues/18628).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Inner fraction from $2$ through $n-1$, terminal value $n+5$ | `nested`, `K`, `a` | Right fold with rational division and the literal terminal value. |\n| Denominator | `a` | Positive denominator of the reduced rational number. |\n| All primes except $2$ and $3$ | `claim` | Natural prime binder and the two explicit exclusions. |\n| Appear once in the sequence | `claim` | Unique natural index with $3 \\le n$ and $a(n)=p$. |\n| Conjecture is false | `result` | Negation of the complete claim. |\n\nThe parameter $x$ in `nested` and `K` exposes the source's terminal argument; `a` specializes it to $n+5$. Total rational division and truncated natural subtraction extend the definitions to all natural indices; `claim` restricts the assertion to the source range. There are no additional hypotheses.","title":"bouras372763 once","url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras372763-once/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83df0b39e1f02ee0cce1355b61cb931930e98d9fe83765986a24bba4bbbf9e38.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bourin-2008-matrix-subadditivity-question-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jean-Christophe Bourin, *Matrix subadditivity inequalities and block-matrices*, arXiv:0805.1954v1, p. 3, Question 1; published as *Matrix subadditivity inequalities and block-matrices*, International Journal of Mathematics 20 (2009), 679–691, DOI 10.1142/S0129167X09005509. The source note is [bourin2008matrixsubadditivity](../Library/QuantumContext/bourin2008matrixsubadditivity.md).\n\n> Let Φ : M_n → M_k be a positive linear map between the algebras of matrices of size n and k and let F_n and F_k be the sets of non-expansive, positive operators in M_n and M_k respectively. Does Φ⁻¹(F_k) ⊂ F_n imply ‖f ∘ Φ(M)‖ ≤ ‖Φ ∘ f(M)‖ for all M ≥ 0, all symmetric norms and all non-negative concave functions on the positive half line?\n\nThe matrix algebras are over $\\mathbb C$. Write $F_j=\\{X:0\\preceq X\\preceq I_j\\}$. The inclusion means non-strict set inclusion, and the inverse image is taken in the entire domain algebra. A symmetric norm is invariant under left and right multiplication by unitaries. The preregistration is [#18573](https://github.com/the-omega-institute/trureturing/issues/18573).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Matrix sizes $n,k$ and positive linear map $\\Phi:M_n\\to M_k$ | `claim` quantifies over natural sizes and complex `LinearMap`; its first premise is positivity | All sizes and all positive maps |\n| $\\Phi^{-1}(F_k)\\subseteq F_n$ | Full-domain preimage of the literal sets $\\{X\\mid 0\\le X\\land X\\le1\\}$ | No extra predicate or domain restriction |\n| Every $M\\ge0$ | Universal matrix binder and nonnegativity premise | Loewner order |\n| Every symmetric norm | Frozen `D5.S3.Quantum.Schatten.UINorm k` and its field `N` | The six preregistered norm and two-sided unitary-invariance laws |\n| Every nonnegative concave function on the positive half line | Real function, `ConcaveOn` on `Set.Ici 0`, pointwise nonnegativity there | Both function premises retained |\n| $\\|f(\\Phi(M))\\|\\le\\|\\Phi(f(M))\\|$ | Norm evaluations of the two `cfc` expressions in `claim` | Same inequality direction |\n\nThere is no additional public hypothesis. The positive-contraction condition is inlined literally; the existing `UINorm` supplies the preregistered norm structure.","title":"Bourin's matrix subadditivity Question 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/bourin-2008-matrix-subadditivity-question-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a591b1ce7254f8a872f0e2826ddc3e3e8b190c3d84680591f97c5065b443bcfd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bourin-lee-2021-essentially-hermitian-block-norm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J.-C. Bourin and E.-Y. Lee, *Eigenvalue inequalities for positive block matrices with the inradius of the numerical range*, arXiv:2111.15180v1, printed page 6, Conjecture 3.3, states:\n\n> Let $X\\in\\mathbb{M}_n$. If the inequality\n>\n> $$\\left\\|\\begin{bmatrix}A&X\\\\X^*&B\\end{bmatrix}\\right\\|_\\infty\\le\\|A+B\\|_\\infty$$\n>\n> holds for all positive block-matrix with $X$ as off-diagonal block, then $X$ is essentially Hermitian.\n\nThe source says: “If W(T) is line segment, then T is a so-called essentially Hermitian matrix.” The binding convention of [#13752](https://github.com/the-omega-institute/trureturing/issues/13752) is the literal affine expression $X=\\alpha H+\\beta I$, with $H=H^*$ and $\\alpha,\\beta\\in\\mathbb C$, including points. The norm is the Euclidean operator norm, and positive means positive semidefinite. The same question is Conjecture 7.3.3 of Bourin's *A journey into Matrix Analysis*, arXiv:2307.03064v1, printed page 114.","title":"Essentially Hermitian off-diagonal blocks","url":"https://the-omega-institute.github.io/trureturing-pages/research/bourin-lee-2021-essentially-hermitian-block-norm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/240cb00eae47119bfe9a416dede26d360af848cd30d74643b8debd846cbbd5ac.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bourin-lee-2026-normal-modulus-constant-n2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J.-C. Bourin and E.-Y. Lee, *Triangle inequalities for the operator symmetric modulus*, arXiv:2602.19607v1, §4, Corollary 4.2, state verbatim:\n\n> Corollary 4.2. Let A, B ∈ Mn be normal. Then, for some unitary V ∈ Mn, |A + B| ≤ |A| + |B| + (1/4) V(|A| + |B|)V*. If n ≥ 3, the constant 1/4 is the smallest possible one.\n\nThe following paragraph asks verbatim:\n\n> However, contrarily to the Schur version (4.1), it seems necessary to request n ≥ 3 in Corollary 4.2 to get sharpness of 1/4 (what is the best constant for n = 2 ?).\n\nHere $|X|=(X^*X)^{1/2}$ is the positive modulus and the inequality uses positive-semidefinite order. The article is in press in *Proceedings of the American Mathematical Society*, doi:10.1090/proc/17861; the quotations and locators refer to the arXiv version.","title":"The normal-modulus triangle constant in dimension two","url":"https://the-omega-institute.github.io/trureturing-pages/research/bourin-lee-2026-normal-modulus-constant-n2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/64ea54b3fa5da8316f95faba2486aa9d5d93010b8c2165460d78b0b8f8d21c70.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bouthat-chavez-sheng-2026-path-chs-remark43","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bouthat, Chávez and Sheng, *A general framework for inequalities on simple graphs*, arXiv:2606.17356v1, Remark 4.3, state the two path CHS identities as unproved and then assert a one-term simplification under $n+1\\ge d/2$.\n\nFor every $n\\ge1$ and even $d\\ge2$, the symmetric value $h_d(\\lambda(P_n))$ equals both the trigonometric sum and the bilateral zero-extended binomial sum printed in Remark 4.3.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $\\|P_n\\|_{X,d}^d=h_d(\\lambda(P_n))$ | `PathCHSNormClosedForm.chs` | Same complete homogeneous symmetric value. |\n| All $n\\ge1$ and even $d\\ge2$ | Arithmetic and parity hypotheses in `claim` | No degenerate path witness is used. |\n| Trigonometric sum, $1\\le k\\le n$ | `trigForm`, with `Finset.Icc 1 n` | Every factor and exponent is literal. |\n| Sum over all integers | `bilateralForm`, using `finsum` | Zero-extended binomials have finite support. |\n| Both equalities | The two conjuncts in `claim` | Both are proved by `result`. |","title":"The two identities of Remark 4.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/bouthat-chavez-sheng-2026-path-chs-remark43/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f447c8dfc270dc6b17c721e9fa2f8e6bd3f9b564eb57af3ac98d2f6a31e72bbb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bouthat-chavez-sheng-2026-path-chs-remark43-closed-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bouthat, Chávez and Sheng, *A general framework for inequalities on simple graphs*, arXiv:2606.17356v1, Remark 4.3, state the two path CHS identities as unproved and then assert a one-term simplification under $n+1\\ge d/2$.\n\nThe closing assertion is false. At $n=2,d=6$, $h_6(\\lambda(P_2))=1$, whereas $((n^2+n-d)/((n+d)(n+d+1)))\\binom{{n+d+1}}{{d/2}}=0$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $\\|P_n\\|_{X,d}^d=h_d(\\lambda(P_n))$ | `PathCHSNormClosedForm.chs` | Same complete homogeneous symmetric value. |\n| All $n\\ge1$ and even $d\\ge2$ | Arithmetic and parity hypotheses in `claim` | No degenerate path witness is used. |\n| $d/2\\le n+1$ | Natural quotient guard in `claim` | Exact for even $d$; equality is included. |\n| One-term rational-binomial expression | Literal right side of `claim` | Real arithmetic and `Nat.choose`. |\n| Failure of the universal formula | `result : ¬ claim` | Witness $n=2,d=6$ satisfies every guard. |","title":"The one-term formula at the boundary","url":"https://the-omega-institute.github.io/trureturing-pages/research/bouthat-chavez-sheng-2026-path-chs-remark43-closed-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/072376e5fd9741f3b8acadfc8fef58fe091f8aa22824cfe6aa74e39338b19a74.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bozovic-2026-sierpinski-limited-domination","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Božović, *On k-limited domination: complexity and Sierpiński graphs*,\narXiv:2610.01584v1, Section 6, Conjecture 12, states:\n\n> Let n, m, and k be integers such that n ≥ 1 and 1 ≤ k ≤ m − 1. Then γ_k^L(S(n, m)) = (m − k) m^(n−1).\n\nSection 2 defines S(n,m) for n ≥ 1 and m ≥ 2. Its vertices are words of\nlength n over {0,…,m−1}. Adjacency means that for some coordinate h the\npreceding coordinates agree, the h-coordinates differ, and each later\ncoordinate equals the other word's h-coordinate. A dominating set D is\nk-limited when every selected vertex has at most k neighbors outside D;\nγ_k^L is its minimum cardinality.","title":"Božović Conjecture 12: k-limited domination of Sierpiński graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/bozovic-2026-sierpinski-limited-domination/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b067d5c26c455c1d1836597cdb05d645cbbb6774afd11c7a35a0b10d3b111d30.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bradshaw-arithmetic-derivative-collatz-squarefree-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z. P. Bradshaw, *On a Family of Solutions to Arithmetic Differential\nEquations Involving the Collatz Map*, Journal of Integer Sequences\n28 (2025), Article 25.1.8, Conjecture 20, printed page 16:\n\n> The solutions to the commutation problem D(C_{a,b}(n)) = C_{a,b}(D(n)) are squarefree for every a ≡ b (mod 2).\n\nThe same page defines `C(a,b,n) = (a*n+b)/2` for odd n and `n/2` for\neven n. Printed page 3 defines `D : Nat -> Nat` by `D(0)=D(1)=0`,\n`D(p)=1` at each prime, and `D(m*n)=D(m)*n+m*D(n)` for all m,n.\nThe closed formal claim quantifies over such D and all natural a,b,n,\nwith equal parity of a,b and `1 <= n`, and concludes `Squarefree n`\nfrom the displayed commutation equality.","title":"Bradshaw's Conjecture 20","url":"https://the-omega-institute.github.io/trureturing-pages/research/bradshaw-arithmetic-derivative-collatz-squarefree-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7583456ffcf4fc5321ab9d4e2c063dc5ba6663190fafee13afb9be877f1160ce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:brask-chaves-2017-outcome-communication-chained-inequality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. B. Brask and R. Chaves, *Bell scenarios with communication*, J. Phys. A: Math. Theor. 50(9), 094001 (2017), arXiv:1607.08182v1, subsection “Generalization to more inputs”, conjecture following Eq. (25). The source member `Bell_comm_arXiv_v1.tex`, line 325, states verbatim:\n\n> For the inequality to hold for the COD model, Alice's outputs must be relabelled for every second input. That is $a=0 \\leftrightarrow a=1$ for even $x$. With this relabelling, we conjecture that \\eqref{eq.validchained} holds for any odd number of inputs.\n\nThe source label `eq.validchained` denotes Eq. (25), at TeX lines 318–324:\n\n$$[a_1-b_1]+[b_1-a_2]+[a_2-b_2]+\\cdots+[a_n-b_n]+[b_n-a_1-1]\\geq1.$$\n\nAt line 325, the bracket is defined by $[a_x-b_y]=\\sum_{j=0}^1j\\,p(a-b=j\\bmod2\\mid x,y)$. For binary outputs, the ordinary brackets are unequal-output probabilities and the closing bracket is an equal-output probability.\n\nIssue [#24183](https://github.com/the-omega-institute/trureturing/issues/24183) fixes the finite hidden-variable reading of the source's causal outcome-dependent (COD) model, Eq. (7), TeX lines 150–155:\n\n$$p(a,b\\mid x,y)=\\sum_{\\lambda\\in\\Lambda}w_\\lambda A_x(a\\mid\\lambda)B_y(b\\mid a,\\lambda),$$\n\nwhere $\\Lambda$ is finite, $w_\\lambda\\geq0$, $\\sum_\\lambda w_\\lambda=1$, and both response functions are nonnegative and normalized over their binary output, including Bob's normalization for each raw Alice output. Let $r_x=1$ for even one-based inputs $x$, and $r_x=0$ otherwise. The inequality reads Alice's relabelled output $c=a\\oplus r_x$, while Bob receives the raw $a$. Writing $q(c,b\\mid x,y)=p(c\\oplus r_x,b\\mid x,y)$, the statement is\n\n$$\\sum_{i=1}^nq(c\\ne b\\mid i,i)+\\sum_{i=1}^{n-1}q(c\\ne b\\mid i+1,i)+q(c=b\\mid1,n)\\geq1$$\n\nfor every odd natural $n$ and every such model. This is proved by `result : claim`. The definitions and source locator are also in [the literature note](../Library/Observer/braskchaves2017bellcommunication.md).","title":"The relabelled chained inequality holds for every odd-input binary COD model","url":"https://the-omega-institute.github.io/trureturing-pages/research/brask-chaves-2017-outcome-communication-chained-inequality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d14c1f9a712fe9bc9bad590b30c8ce8fb7af76b974603a6f41f3e39933a3193e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:braun-bruegge-2022-symmetric-edge-polytope-facet-sum-conjecture-5-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Braun and K. Bruegge, *Facets of Symmetric Edge Polytopes for Graphs with Few Edges*, arXiv:2201.13303v4, Conjecture 5.1 (Conjecture 3.15 in v1), asks for two endpoint maxima and two monotone transfers of the facet sum from Proposition 4.4. The verbatim statement is recorded in #19695 and the cited Library note.\n\nFor sorted positive natural numbers $a\\ge b\\ge c$ of common parity, the literal source specialization is\n\n$$\nF(a,b,c)=\\sum_{j=0}^{c}\\binom cj\\binom b{(b-c)/2+j}\\binom a{(a-c)/2+j}.\n$$\n\nThe natural divisions are exact on this domain. Outside it, the Lean definition remains a total natural-valued expression; the conjecture asserts nothing there.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Proposition 4.4, three entries, ordered and of common parity | `F`, `Admissible` | The sum and the five domain conjuncts are literal binding conventions. |\n| $n=2k$, $k\\ge2$, sum $n+1$, maximum at $(n-1,1,1)$ | `result.1` | Every admissible triple with that sum is bounded by the endpoint. |\n| $n=2k-1$, $k\\ge3$, sum $n+1$, maximum at $(n-3,2,2)$ | `result.2.1` | Natural subtraction has the source value under the stated bound on $k$. |\n| Transfer from the third coordinate | `result.2.2.1` | $c\\ge3$ makes $(a+2,b,c-2)$ positive and sorted. |\n| Transfer from the second coordinate, preserving order | `result.2.2.2` | The additional condition $c\\le b-2$ is exactly preservation of the lower order inequality. |\n\nThe endpoint triples themselves satisfy the domain and sum conditions. There are no additional hypotheses on the four assertions.","title":"Braun–Bruegge Conjecture 5.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/braun-bruegge-2022-symmetric-edge-polytope-facet-sum-conjecture-5-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16e20517f05532c902c16c2127938e1eb514dcf201b4d074f85090123df579d6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:braunstein-ghosh-severini-2006-star-formation-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Braunstein, Ghosh and Severini, *The Laplacian of a graph as a density matrix: a basic combinatorial approach to separability of mixed states*, Annals of Combinatorics 10 (2006), Conjecture 6.7, states:\n\n> Let $\\mathcal G_n^c$ be the set of all connected graphs on $n$ vertices. Let $G\\in\\mathcal G_n^c$ with $|V|=pq$. Then\n> $$\\max_{G\\in\\mathcal G_n^c}E_F(\\sigma(G))=E_F(\\sigma(K_{1,n-1})).$$\n\nHere $\\sigma(G)=L(G)/d_G$ is the graph Laplacian density matrix and $E_F$ is the convex-roof entanglement of formation over finite pure-state ensembles.","title":"Refutation of Braunstein–Ghosh–Severini Conjecture 6.7","url":"https://the-omega-institute.github.io/trureturing-pages/research/braunstein-ghosh-severini-2006-star-formation-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1d81d3bf227067917a9c0862cfbcdc37d14c05f29fda497eeebef41040e62ae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bresar-2024-direct-product-cycle-path-percolation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"In `r`-neighbour bootstrap percolation a vertex becomes infected once it has\nat least `r` infected neighbours, and `m(G, r)` is the least size of a nonempty\ninitial set that eventually infects every vertex. For the direct product\n`C_n × P_m` of a cycle and a path, Brešar, Hedžet and Herrman\n(arXiv:2403.10957, Discuss. Math. Graph Theory 46 (2026)) ask:\n\n> Problem 5. Determine m(C_n × P_m, 2).\n\nTheir Proposition 3 gives `m(C_n × P_m, 2) ≤ n`, and they note that equality\nappears to hold for small examples.","title":"Problem 5 of Brešar, Hedžet and Herrman: m(C_n × P_m, 2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/bresar-2024-direct-product-cycle-path-percolation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b35392e3edc78a33ff7f42ad4159da432e979abbaba7ec9ed495c8ac9fb488e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bresar-ferme-hu-infinite-path-six-reader-question","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Brešar, Ferme and Hu, *Partitioning an S-packing coloring into broadcast dominating sets*,\narXiv:2610.03477v1, Section 3.2, immediately after Theorem 3.6, ask:\n\n> For k=6, the problem is reduced to determining which of the two possible values in {4,5} is attained by χ_{ρ,6}(P∞), a question we leave for the reader.\n\nThe graph P∞ has vertices Z and distance |u-v|. For t in Nat, Q(t) asserts the existence of\nf:Z→Nat and one common c:Z→Fin 6 such that 1≤f(x)≤t for every x, distinct vertices of the\nsame colour j are at distance greater than j, and for every x and every class i there is a\nvertex a with c(a)=i and |x-a|≤f(a). Neither map is assumed periodic or regular.\nThe exact answer is 5: Q(5) holds, and Q(t) implies 5≤t for every natural t.","title":"Six broadcast-dominating packing classes on the two-way infinite path","url":"https://the-omega-institute.github.io/trureturing-pages/research/bresar-ferme-hu-infinite-path-six-reader-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff236d50225d42e5a34cf1928dfa9b512e91364d863b33b16a140b10253dbb36.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bresar-ferme-hu-packing-domatic-paths","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Boštjan Brešar, Jasmina Ferme and Wenjie Hu, *Partitioning an S-packing coloring into broadcast dominating sets*,\narXiv:2610.03477v1, Section 5: \"Problem 2. Is it true that χ_{ρ,k}(P_n) ≤ k + 1 for any k ≥ 3 and any n ≥ 2k?\"\nA packing k-domatic colouring f with colours in [t] keeps two vertices of colour j at distance greater than j and\nadmits a partition of the vertices into k classes, each broadcast-dominating every vertex x through some a in the\nclass with d(x, a) ≤ f(a); χ_{ρ,k}(G) is the least such t.","title":"Packing k-Domatic Colourings of Paths Need More Than k + 1 Colours","url":"https://the-omega-institute.github.io/trureturing-pages/research/bresar-ferme-hu-packing-domatic-paths/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2395ce843f1b805ed445b59a03595613e1f94e60fdefc789bd0d89f9054f929b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:briet-harremoes-2009-trace-exp-exponential-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jop Briët and Peter Harremoës, *Properties of classical and quantum Jensen-Shannon divergence*, Physical Review A 79, 052311 (2009), arXiv:0806.4472v4, Definition 4 (Exponential convexity):\n\n> Let X be a convex set and φ : X → R a mapping. Then φ is said to be exponentially convex if the function X × X → R given by (x, y) → φ((x+y)/2) is positive definite.\n\n§VI (Conclusions and open problems):\n\n> Whether (QJD1)^{1/2} is a metric for all mixed states remains unknown. However, based on a large amount of numerical evidence, we conjecture the function A → Tr(e^A) to be exponentially convex for density matrices A. Proving this would imply that QJDα is negative definite for α ∈ (0, 2], and hence the square of a metric that can be embedded in a real Hilbert space.","title":"The trace exponential is not exponentially convex on density matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/briet-harremoes-2009-trace-exp-exponential-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9202d113c09216571b65b33ca3980831bdca3c07902e243bf3d26e53bb6cfb15.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:brinkley-fickus-jasper-mixon-2026-suntf-two-five-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"W. J. Brinkley, M. Fickus, J. Jasper and D. G. Mixon, *Incoherent filter banks*,\narXiv:2610.05302v1, Section 4, Example 4.4, asks whether a SUNTF_R(2, 5, 3)\nexists. The last paragraph states:\n\n> We do not yet know whether a SUNTF_R(2, 5, 3) exists, and we leave this for future research.\n\nDefinition 3.2 requires each of the R layers to be a unit norm tight frame in\ndimension D and requires the averaged squared inner products between every two\ndistinct columns to be constant.","title":"A SUNTF_R(2, 5, 3) exists","url":"https://the-omega-institute.github.io/trureturing-pages/research/brinkley-fickus-jasper-mixon-2026-suntf-two-five-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8ad0902b9d947a933756a3d46223d4aa8fff3394d590ab23e355aa9123166ca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:brody-graefe-melanathuru-2026-damped-kernel-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. C. Brody, E.-M. Graefe and R. Melanathuru, *Phase-space measurements and decoherence for angular momentum systems*, arXiv:2605.02696v1, Section VI, after equation (52):\n\n> For J > 1/2 we do not have an exact result, but it seems reasonable to conjecture that positivity is ensured provided that the damped σ kernel in (44), due to decoherence, becomes no sharper than the Husimi (σ = −1) kernel. That is, if e^{−½γL(L+1)} ( C(2J,L)/C(2J+L+1,L) )^{−σ/2} ≤ ( C(2J,L)/C(2J+L+1,L) )^{1/2} (53) for all L, then we restore positivity.\n\nEquation (44) includes the complex conjugate of the harmonic:\n\n$$\nF^\\sigma(\\theta,\\phi,t)=a_J\\sum_{L,k}e^{-\\gamma L(L+1)t/2}\n\\left(\\frac{\\binom{2J}{L}}{\\binom{2J+L+1}{L}}\\right)^{-\\sigma/2}\n\\rho_{Lk}(0)\\overline{Y_L^k(\\theta,\\phi)},\\qquad a_J=(2J+1)^{-1/2}.\n$$\n\nThe tensor entries and multipoles follow equations (12) and (13); the harmonics have the Condon–Shortley phase and the paper's factor $\\sqrt{4\\pi}$ (footnote 1). Equation (53) omits $t$ in print. The refuting example has $t=1$, where the printed and time-dependent readings coincide.","title":"The damped angular-momentum kernel positivity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/brody-graefe-melanathuru-2026-damped-kernel-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7de119de4ea8a44105e679f2d4df669cd5d97ba25f787a8b9934356d7202207e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:brown-george-2025-degree-polynomial-root-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. I. Brown and I. C. George, *On the roots of degree polynomials*,\nAustralasian Journal of Combinatorics 92(2), 164–183, Conjecture 4.1, p. 180,\nstates:\n\n> If z is a nonzero degree root of a graph of order n with argument arg(z) ∈ (−π, π], then |z| ≤ (n − 1)^{|arg(z)/π|}.\n\nSection 1 defines $D(G;x)=\\sum_{v\\in V(G)}x^{\\deg(v)}$.\nThe arXiv version is *The Degree Polynomial*, arXiv:2505.04882v1.\nThe settled proposition is\n\n$$\n\\forall n\\ge1\\;\\forall G\\text{ simple on }\\operatorname{Fin}(n)\\;\\forall z\\in\\mathbb C,\n\\quad z\\ne0\\land D(G;z)=0\\Longrightarrow\n|z|\\le(n-1)^{|\\arg z|/\\pi}.\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Graph of positive order $n$ | Natural `n`, `1 ≤ n`, `SimpleGraph (Fin n)` in `claim` | The preregistered simple-graph convention; empty graphs are excluded |\n| Degree polynomial | `degreePolynomial`, the sum of `Polynomial.X ^ G.degree v` over every vertex | Includes degree-zero vertices |\n| Nonzero degree root | Complex `z`, `z ≠ 0`, polynomial evaluation equal to zero | Both hypotheses retained |\n| $\\arg z\\in(-\\pi,\\pi]$ | `Complex.arg z` | Pinned Mathlib branch convention |\n| Modulus and power | Complex norm and real power | $\\pi>0$ gives $|\\arg(z)/\\pi|=|\\arg z|/\\pi$ |\n| All nonzero roots | `result : claim` | Universal inequality; no extra root hypothesis |\n\nThe public definitions `degreePolynomial` and `claim` reproduce the two\npreregistered expressions literally. Positive order is explicit: at order\nzero the vertex sum is the zero polynomial, so a universal root bound\nwould have a different meaning.","title":"Brown–George's argument-dependent degree-root bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/brown-george-2025-degree-polynomial-root-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f3012bfaf752d70095603a2ac327a1d0bdb48807f5127cd91d9ae2ed521009a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bu-gu-jaffe-2025-mixed-state-inverse-sumset","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Bu, W. Gu and A. Jaffe, *Quantum Ruzsa Divergence to Quantify Magic*, IEEE Transactions on Information Theory 71(4) (2025) 2726–2740, DOI 10.1109/TIT.2025.3543276; arXiv:2401.14385v3, Remark 45, equation (58). The v3 source `IEEE_Final_BGJ.tex`, line 2106, states verbatim:\n\n> In addition, we also expect that the property (2) in Theorem \\ref{thm:QIST} holds, which has the following form\n\nThe reference is to Theorem 44. Its proposed mixed-state inequality, at lines 2107–2109, is\n\n$D(\\rho\\|\\mathcal M(\\rho))\\leq\\dfrac{\\log R(\\rho)}{\\log R(\\rho)-\\log[1+\\lambda(R(\\rho)\\operatorname{Tr}[\\rho^2]-1)]}\\log C.\\qquad(58)$\n\nHere $R(\\rho)=\\operatorname{rank}\\mathcal M(\\rho)$ and $\\lambda=(1-MG(\\rho))^2$. The characteristic function is $\\Xi_\\rho(p,q)=\\operatorname{Tr}[\\rho w(-p,-q)]$; the mean state retains exactly its coefficients of modulus one. The magic gap $MG$ is one minus the largest modulus among the nonzero coefficients whose modulus is not one, with $MG=0$ if that finite set is empty. The source definitions and their locations are in [the literature note](../Library/Divergence/bugujaffe2025ruzsa.md).\n\nIssue [#24984](https://github.com/the-omega-institute/trureturing/issues/24984) fixes the single-qudit reading encoded by `claim`: for every nonzero prime dimension $d$, every nonzero $s,t\\in\\mathbb Z_d$ with $s^2+t^2=1$, every three density matrices $\\rho,\\sigma,\\tau$ with $\\sigma=\\mathcal M(\\rho)$ and $\\tau=\\rho\\boxtimes_{s,t}\\rho$, and every real $C$, require (58) whenever\n\n$1<\\delta_q[\\rho]=\\exp(S(\\tau)-S(\\rho))\\leq C,\\qquad A=1+\\lambda(R(\\rho)\\operatorname{Tr}[\\rho^2]-1)>0,\\qquad\\log R(\\rho)-\\log A>0.$\n\nDensity matrices are positive semidefinite complex matrices of trace one. Entropy and relative entropy are the natural-logarithm trace expressions $S(\\rho)=-\\operatorname{Tr}(\\rho\\log\\rho)$ and $D(\\rho\\|\\sigma)=\\operatorname{Tr}[\\rho(\\log\\rho-\\log\\sigma)]$. The two positivity assumptions restrict to a positive logarithm argument and denominator. This restriction of the expected universal bound is refuted.","title":"The Bu–Gu–Jaffe mixed-state quantum inverse sumset bound is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/bu-gu-jaffe-2025-mixed-state-inverse-sumset/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/efdd3dce55118056982b02dcf630333d061d9509100368cc9c9690bc4fc12d00.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bubboloni-caceres-neighbourhood-upsets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Daniela Bubboloni and José Cáceres, *The neighbourhood convexity*,\narXiv:2608.25912v1, “Conclusions and future lines of work,” paragraph\nbeginning “By exploring the connection with the neighbourhood preorder P(G)”:\n\n> We conjecture that, given a convex geometry (G,n), any upset in P(G) is an n-convex set, having checked it computationally for graphs up to 9 vertices.\n\nFor every finite nonempty set $V$ and simple undirected graph $G=(V,E)$,\nput $B(x)=\\{x\\}\\cup\\{y:xy\\in E\\}$ and\n$N(X)=\\bigcap_{x\\in X}B(x)$, with $N(\\varnothing)=V$.\nLet $C=\\{N(Y):Y\\subseteq V\\}\\cup\\{\\varnothing\\}$, let\n$h(\\varnothing)=\\varnothing$ and $h(X)=N(N(X))$ for nonempty $X$, and put\n$\\operatorname{ex}(K)=\\{x\\in K:K\\setminus\\{x\\}\\in C\\}$.\nAssume $h(\\operatorname{ex}(K))=K$ for every $K\\in C$.\nThen, for every $U\\subseteq V$,\n\n$$\n\\left(\\forall x\\in U,\\ \\forall y\\in V,\\ B(x)\\subseteq B(y)\\Rightarrow y\\in U\\right)\n\\Rightarrow U\\in C.\n$$\n\nThe source anchors are Sections 2.1 and 2.2, Definitions 3, 5, 9 and 28,\nand Proposition 12. The quantifiers include all positive graph orders,\ndisconnected graphs, universal vertices, equal closed neighbourhoods and the\nempty upset. No connected, star-free or twin-free hypothesis is substituted.","title":"Every upset in a neighbourhood convex geometry is convex","url":"https://the-omega-institute.github.io/trureturing-pages/research/bubboloni-caceres-neighbourhood-upsets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0583ec2c9471def0a1418c329853bd2e14b1f83748a9d909b735251ba85ce492.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:buhrman-grinko-verduyn-lunel-weggemans-2026-g-test-threshold-prior","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Buhrman, D. Grinko, P. Verduyn Lunel and J. Weggemans, *Permutation tests for quantum state identity*, arXiv:2405.09626v2, distinguish the twirled unequal-case state ρ̃^µ_≠ (Eqs. (11)–(12)) from the equal-case state ρ^n_= (Eq. (13)).\n- **The G-test.** For a subgroup G of S_n, the G-test measures Π_= = Π_G = (1/|G|) Σ_{π∈G} ψ(π) (Eqs. (61)–(62)). Its completeness is 1 (Eq. (63)).\n- **The Bayesian comparison.** With prior {p, 1 − p}, the success probability is P_succ = p P_c + (1 − p) P_s. The trivial test always answers \"not equal\" and succeeds with probability 1 − p. For G = S_n, Theorem 5 gives the threshold p∗(µ) = 1/(1 + (n choose µ)).\n- **The question.** §6: \"A further open question is to determine, for a given subgroup G and partition µ, the threshold prior p∗(G, µ) below which the trivial test outperforms the G-test, analogous to the threshold p∗(µ) established for the permutation test in Theorem 5.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/buhrmangrinkoverduynlunelweggemans2026permutation.md).\n\nIssue [#17307](https://github.com/the-omega-institute/trureturing/issues/17307) fixes the reading:\n- W_µ is the set of words of length n over {1, …, d} with µ_i letters i, of size M = n!/∏_i µ_i!;\n- O is the number of orbits of G on W_µ.\n\nThe answer is p∗(G, µ) = O/(M + O):\n- the G-test has strictly larger success probability than the trivial test exactly when p > O/(M + O);\n- the two probabilities are equal exactly when p = O/(M + O);\n- the soundness error is Tr[Π_G ρ̃^µ_≠] = O/M.","title":"The threshold prior of the G-test is O/(M + O)","url":"https://the-omega-institute.github.io/trureturing-pages/research/buhrman-grinko-verduyn-lunel-weggemans-2026-g-test-threshold-prior/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82c65ecbea4cd47e0d32b0ee12fb190957c02251aa70df1cffc16d8dbabf9cd6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bulchandani-burnell-sondhi-2023-polygon-game-classical-value","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"V. B. Bulchandani, F. J. Burnell and S. L. Sondhi, *Playing nonlocal games with phases of quantum matter*, Phys. Rev. B 107, 045412 (2023), arXiv:2206.11252v2, Section 4.1, \"Classical winning probabilities\", state in `ArxivResub.tex`, lines 714–717, the equation labelled `eq:classwinbound` and its conjecture:\n\n> $p_{\\mathrm{cl}}^*(P) \\geq \\frac{2^P - F_{P-1}}{2^P} = 1 - \\frac{F_{P-1}}{2^P}, \\quad P \\,\\mathrm{odd}.$\n>\n> We conjecture that this bound is tight.\n\nHere $P\\ge3$ is the number of players and $F_1=F_2=1$. The game's rules are in the same section, `ArxivResub.tex`, lines 629–655; see also [the literature note](../Library/Resource/bulchandaniburnellsondhi2023polygon.md).\n\nIndex players and oriented edges by $\\mathbb Z/P$, with edge $i$ joining $i$ to $i+1$. A deterministic strategy assigns triples $(a_i,b_i^0,c_i)$ on input $0$ and $(d_i,b_i^1,e_i)$ on input $1$. The module allows the middle bit to depend on the input. For a matching $M$, let $x_M$ be the input whose ones are exactly its covered vertices. On input $x$, the strategy wins when every returned triple has even parity, the returned middle bits have even total parity, and, for every matching with $x_M=x$,\n\n$$\\sum_{i\\in M}(d_i+e_{i+1})+\\sum_{j\\text{ uncovered by }M}a_j\\equiv |M|\\pmod2.$$\n\nThe empty matching gives Condition 3; matchings with one, two and more edges give Conditions 4–6. Inputs are uniform over all $2^P$ Boolean strings, including those arising from no matching. Write $w_P(s)$ for the fraction of these inputs won by a deterministic strategy $s$, and $W_P(\\mu)=\\sum_s\\mu(s)w_P(s)$ for a nonnegative weight function of total mass one on all deterministic strategies.\n\nIssue [#26289](https://github.com/the-omega-institute/trureturing/issues/26289) fixes the reading: for every odd $P\\ge3$, every such $\\mu$ satisfies $W_P(\\mu)\\le1-F_{P-1}/2^P$, and some deterministic $s$ attains equality. This is proved.","title":"The classical value of the odd polygon game is its Fibonacci bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/bulchandani-burnell-sondhi-2023-polygon-game-classical-value/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8f466759d2862dcf75b9a3e61be8daac2157e9666c680b8687135c054117ed2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:buniy-feger-kephart-2024-tripartite-mec-length-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. V. Buniy, R. P. Feger and T. W. Kephart, *Decoherence and the classes of maximally entangled states*, Int. J. Quantum Inf. 22, 2450035 (2024), arXiv:2210.07618v1, classify tripartite qudit states by algebraic entanglement invariants.\n- **The invariants.** For a state v of ℂ^{d₁} ⊗ ℂ^{d₂} ⊗ ℂ^{d₃} with coordinates v_{ijk}, they are (n₁, n₂, n₃, n₁,₂,₃) (Appendix A):\n - n_i, the kernel dimensions of the three flattenings;\n - n₁,₂,₃, the dimension of the space of arrays z with Σ_{j,k} v_{ajk}z_{bjk} = Σ_{i,k} v_{iak}z_{ibk} = Σ_{i,j} v_{ija}z_{ijb} = 0 for all a, b.\n- **The class.** A state is maximally entangled when every invariant takes its least value, and these states form the maximally entangled class C_D (Definition II.1).\n- **The length.** It is the number of nonzero coordinates, and L_D is the least length in C_D (Eq. (2)).\n- **The conjecture.** From exact values of L_D for dimensions up to 5 (Table XII), Conjecture III.4 states \"L_(d₁,d₂,d₃) = Σ_{i=1}^{3} d_i − 2 for min(d₁, d₂, d₃) ≥ 3\", together with two clauses for d₁ = 2.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/buniy2024maximallyentangledclasses.md).\n\nIssue [#16992](https://github.com/the-omega-institute/trureturing/issues/16992) fixes the reading: the lower-bound half of the third clause, i.e. every v ∈ C_D has L(v) ≥ d₁ + d₂ + d₃ − 2 when all d_i ≥ 3. Refuting it refutes the conjecture. It is refuted at D = (3, 3, 7).","title":"A maximally entangled (3, 3, 7) state of length 9 refutes the length formula Σ d_i − 2","url":"https://the-omega-institute.github.io/trureturing-pages/research/buniy-feger-kephart-2024-tripartite-mec-length-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/995ec3ec74855ce1dc02a02dce59e4c6c968bf02abd8356bd9125b8ca4b7d059.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bunth-pitrik-titkos-virosztek-2024-wasserstein-complexity-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Bunth, J. Pitrik, T. Titkos and D. Virosztek, *On the metric property of quantum Wasserstein divergences*, Phys. Rev. A **110** (2024), 022211, [arXiv:2402.13150v4](https://arxiv.org/abs/2402.13150v4), Section 5.1, asks:\n\n> We note that whether the Wasserstein complexity (37) is convex or not is an open question.\n\nFor a finite Hermitian observable family $\\mathcal A$, the complexity is $C_W(\\Phi)=\\max_\\rho d_{\\mathcal A}(\\rho,\\Phi(\\rho))$. The density matrices are positive semidefinite and have trace one. The couplings have marginals $\\omega$ and $\\rho^T$, and the quadratic cost and divergence are those of equations (eq:2-Wass-quant-def) and (eq:mod-qw-dist-def). [The literature note](../Library/QuantumChannels/bunthpitriktitkosvirosztek2024wasserstein.md) gives the definitions and locators. [Preregistration #15149](https://github.com/the-omega-institute/trureturing/issues/15149) specifies the universal convexity statement, its conventions and the qubit counterexample.","title":"A qubit dephasing midpoint refutes convexity of Wasserstein complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/bunth-pitrik-titkos-virosztek-2024-wasserstein-complexity-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/704439e603f79637699797114cedace85ebcc180955409fabdb78b63769364db.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:burke-haris-madhavendra-increasing-function-jets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Maxim R. Burke, Maleeha Haris and Madhavendra, *Repeated integrals of increasing\nfunctions*, arXiv:2512.02151v1, Conjecture 1.2, p. 2:\n\n> $(P_n)$ is true for all nonnegative integers $n$.\n\nFor each $n\\geq 0$, let $W_n\\subseteq\\mathbb R^{n+1}$ consist of vectors\n$b=(b_0,\\ldots,b_n)$ for which some $f\\in C^n[0,1]$ has nondecreasing,\nnonconstant $D^n f$ and satisfies $D^j f(0)=0$ and $D^j f(1)=b_j$ for every\n$0\\leq j\\leq n$. Here $D^0f=f$ and endpoint derivatives are one-sided.\nNondecreasing does not mean strictly increasing, and absolute continuity is\nnot an input hypothesis.\n\nThe assertion $(P_n)$ requires both openness of $W_n$ and, for every $b\\in W_n$,\none function $F\\in C^\\infty[0,1]$ satisfying all of the following:\n\n- $D^jF(0)=0$ and $D^jF(1)=b_j$ for every $0\\leq j\\leq n$;\n- $D^{n+1}F(x)>0$ for every $0n+1$.\n\nThe sole problem anchor is this original Conjecture 1.2, including $n=0$.\nThe follow-up arXiv:2512.23949v1 calls the same interval property $Q_n$ and\nretains its all-order assertion as Conjecture 2.5.","title":"The all-order increasing-function endpoint-jet assertion","url":"https://the-omega-institute.github.io/trureturing-pages/research/burke-haris-madhavendra-increasing-function-jets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2fe6a6f45ead0cb30031bd979987b3840cbdfd71ac2d6fe71ca0a60488390db8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:burmester-kuhn-2025-conjecture-4-29-ari-coproduct-filtration","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Annika Burmester and Ulf Kühn, *On post-Lie structures for free Lie\nalgebras*, arXiv:2504.19661v1, Conjecture 4.29, state:\n\n> For k ≥ 0 we define the k-level of a word v_{k_1}⋯v_{k_d} by\n> d − #{j | k_j=k}, and extend this to an increasing filtration on\n> Q⟨V⟩. For w∈Q⟨V⟩, write as usual Δ_a(w)=w_(1)⊗w_(2). Then,\n> we always have k-level(w_(2)) ≤ k-level(w).\n\nLet $H=\\mathbb Q\\langle v_0,v_1,\\ldots\\rangle$ be the algebra of finite\nordinary rational word polynomials, including its unit. For a word $w$,\nlet $\\lambda_k(w)$ count letters different from $v_k$ and put\n\n$$\nF_r^k=\\operatorname{span}_{\\mathbb Q}\n \\{w:\\lambda_k(w)\\le r\\}.\n$$\n\nThe precise filtration assertion is, for every $k,r\\in\\mathbb N$ and\nevery $P\\in F_r^k$,\n\n$$\n\\Delta_a(P)\\in\\operatorname{range}\n \\bigl(\\operatorname{id}_H\\otimes\\iota_r^k:\n H\\otimes_{\\mathbb Q}F_r^k\\longrightarrow H\\otimes_{\\mathbb Q}H\\bigr),\n$$\n\nwhere $\\iota_r^k$ is the natural submodule inclusion. All tensor products\nare ordinary algebraic tensor products. A Sweedler expression is read\nthrough this subspace assertion, rather than through a chosen tensor\ndecomposition. The resolution is **Proved** by\n`D5/S1/Words/PostLie/AriCoproductFiltration.result`, with\n`claim : Prop` and `result : claim`.\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Original rational word algebra on all nonnegative indices | `AriWordOperations.H = RationalWordPolynomial Nat`, with `Word = FreeMonoid Nat` and the existing ordinary monoid-algebra basis |\n| Every natural $k$, including zero | The unrestricted `k : Nat` in `claim`; the generator $v_0$ and mixed-zero multiplicities retain their source conventions |\n| Increasing level filtration at every $r$ | `level k w = w.countP' (· ≠ k)` and `filtration k r`, the rational span of all words of level at most $r$ |\n| Every finite rational polynomial in that span | `P : H` with membership in `filtration k r`; no length, weight, homogeneity, coefficient-sign or cancellation restriction |\n| Original ari product and dual coproduct | `letterAction`, `primitiveUnshuffle`, `star`, `transposeProperty`, and the proved uniquely characterized `delta` |\n| Full ordinary pairing | `pairing`, `tensorPairing`, `delta_transpose` and `delta_basis_coefficient`, including empty words and both unit tensor terms |\n| Membership in the actual right-factor tensor image | `rightFiltrationImage k r = LinearMap.range (TensorProduct.map id (filtration k r).subtype)` and `result` |\n\nThe source operations and locators are documented in\n[the source note](../Library/LieTheory/burmesterkuhn2025postlie.md).\nThe claim is the entire target preregistered in\n[#23147](https://github.com/the-omega-institute/trureturing/issues/23147).","title":"The original ari coproduct preserves every right-factor level filtration","url":"https://the-omega-institute.github.io/trureturing-pages/research/burmester-kuhn-2025-conjecture-4-29-ari-coproduct-filtration/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be0419d3eda0ea97b16c2c9f41ac40b055a48c4a398b0b2641de091b8ed1f204.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:burri-2025-sw-petz-decoder-comparison","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Laura Burri, *Entanglement fidelity of Petz decoder for one-shot entanglement\ntransmission*, arXiv:2502.17411v1, Section 3.3, Question 1 asks:\n\n> Which performs better: the SW decoder or the Petz decoder?\n\nThe source domain is every density state and CPTP noise channel (Section 3.1,\nTheorem 1). The question concerns the actual Appendix A SW decoder and ordinary\nPetz decoder, measured by canonical entanglement fidelity. It is separate from\nQuestion 2's comparison with a lower bound.","title":"Burri's SW–Petz decoder comparison","url":"https://the-omega-institute.github.io/trureturing-pages/research/burri-2025-sw-petz-decoder-comparison/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/26acdf1ac613537c1c579681ed75f2d3f0c22bd608f5eb2dff19253ec8ca8688.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:butera-2015-dimer-virial-bounds-regular-biconnected","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Butera, P. Federbush, M. Pernici, *Positivity of the virial coefficients\nin lattice dimer models and upper bounds on the number of matchings on\ngraphs*, Physica A 437 (2015) 278–294, arXiv:1502.06734v2, Section IV B:\n\n> For all the graphs examined in this section, Eq. (\\ref{Delta0}) is\n> satisfied for $k \\le 4$. It would be interesting to know whether these\n> bounds, Eq. (\\ref{Delta0}) for $k \\le 4$, are always satisfied for\n> regular biconnected graphs.\n\nEq. (1) is $\\Delta^k\\ln(i!\\,N(i))\\le0$ for $k=2,\\dots,\\nu$ and\n$i=0,\\dots,\\nu-k$, where $N(i)$ is the number of configurations of $i$\ndimers on the graph ($i$-edge matchings), $\\Delta$ is the forward difference\nin $i$, and $\\nu$ is the matching number.","title":"The dimer bounds for k at most 4 on regular biconnected graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/butera-2015-dimer-virial-bounds-regular-biconnected/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c6529d5430be7cb22fe8796852fb08ae518421ac0878955e530dda14d32b037.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:byrapuram-pell-partial-sum-max-index","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Byrapuram et al., *Fibonacci Partial Sums Tricks*, arXiv:2409.01296v1,\nSection 4.5, states verbatim:\n\n> **Conjecture 18.** For the Pell sequence, we have\n> m^P_{4k+1}=m^P_{4k+2}=1, m^P_{4k-1}=2k, and m^P_{4k}=2k+1.\n\nHere `P(0)=0`, `P(1)=1`, `P(n+2)=2P(n+1)+P(n)`, and\n`S(n)=P(1)+...+P(n)`. For positive `n`, `m_n^P` is the greatest\npositive `j` satisfying `P(j)|S(n)`. The first two equalities hold for\nevery natural `k`; the latter two hold for every `k>=1`.\nThe formal predicate is `IsGreatest {j | 1<=j and P(j)|S(n)} m`, so\ndivisibility, positivity of the selected index, and maximality are all\npart of each clause.","title":"Greatest Pell indices dividing initial partial sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/byrapuram-pell-partial-sum-max-index/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe0f4335b072abb789b8c41199a1ce3422038a2fba16fd2168c6ded4fd7664f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:bytsko-2009-positive-coupling-real-spectrum-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. G. Bytsko, *Non-Hermitian spin chains with inhomogeneous coupling*, arXiv:0911.4476v3 (St. Petersburg Math. J. 22, 393–410, 2011), Conjecture 3:\n\n> For $|\\gamma| < \\frac{\\pi}{2S}$, the Hamiltonian (22) of a spin chain with inhomogeneous coupling has a real spectrum if all $a_n >0$.\n\nHere:\n- **The chain:** $N$ sites, each carrying the spin-$S$ representation of $U_q(sl_2)$ with $q=e^{i\\gamma}$, in the source's normalization.\n- **The Hamiltonian (22):** $H=\\sum_{n=1}^{N-1}a_nP^{S,s}_{n,n+1}$, where $P^{S,s}$ is the non-orthogonal projector onto the spin-$s$ summand of $V^S\\otimes V^S$, given by the source's Casimir polynomial.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/bytsko2011nonhermitian.md).\n\nIssue [#20040](https://github.com/the-omega-institute/trureturing/issues/20040) fixes the reading: for every $S$, $s\\in\\{0,\\dots,2S\\}$, $N\\ge2$, $0<|\\gamma|<\\pi/(2S)$ away from projector poles, and positive couplings, every eigenvalue of $H$ is real. This is refuted.","title":"Positive couplings do not force a real spectrum for U_q(sl_2)-invariant spin chains","url":"https://the-omega-institute.github.io/trureturing-pages/research/bytsko-2009-positive-coupling-real-spectrum-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f11d4f270e16e81a80d879ffe7cab76888e14864b803faf4d75ebd656c13fd5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:calderon-2026-quadratic-order-reciprocal-moments","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kevin Calderon, *Ljunggren–Jacobsthal and Bailey-Type Congruences for Rectangular\nGaussian Binomial Coefficients*, arXiv:2608.00347v1, Conjecture 6.3, PDF page 19:\n\n> Let $\\omega$ satisfy (6.1), and let $p$ satisfy (6.3). Then, for every $k\\geq1$,\n> $H_{1,\\omega}(k)\\in p^{2k}\\mathcal O_{\\omega,p}$,\n> $H_{2,\\omega}(k)\\in p^k\\mathcal O_{\\omega,p}$.\n\nEquation (6.1) is $\\omega^2-T\\omega+N=0$, with $T,N\\in\\mathbb Z$ and\n$\\Delta=T^2-4N<0$. Equation (6.3) requires a rational prime $p>5$,\n$p\\nmid\\Delta$, and $(\\Delta/p)=-1$. The local order is\n$\\mathcal O_{\\omega,p}:=\\mathbb Z_p[\\omega]$. Its positive representatives are\n\n$$\nU_{\\omega,k}:=\\{x+y\\omega:1\\leq x,y\\leq p^k,\\ p\\nmid(x,y)\\},\n\\qquad H_{r,\\omega}(k):=\\sum_{z\\in U_{\\omega,k}}z^{-r}.\n$$\n\nThe coordinate condition excludes simultaneous divisibility by $p$; both\nintervals include $p^k$. The Lean ring `R` is the adjoin-root of $X^2-TX+N$\nover `PadicInt p`, and `U` is the finite image of exactly this filtered\npositive rectangle. All its elements are units in the admissible setting.\nConsequently `Ring.inverse` in `H1` and `H2` is the underlying inverse unit\nvalue, without adding an assumption or a term to the source's sums.\n\nThe fully quantified statement is: for every natural prime $p$, every pair\nof integers $T,N$ satisfying all four admissibility clauses, and every\nnatural $k\\geq1$, both displayed memberships hold. The public `claim`\nencodes this statement; `result : claim` has no additional hypotheses.","title":"Calderon's reciprocal moments in imaginary quadratic orders","url":"https://the-omega-institute.github.io/trureturing-pages/research/calderon-2026-quadratic-order-reciprocal-moments/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a37c1d4643d32d3403d34793f872dc164436f57da58078f75827970a561752a2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:caldwell-2024-super-catalan-action-graph-ntable","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Caldwell, A. Cochran, N. Glisson, B. Jennings, K. McDicken, L. Proctor,\nS. Klanderman and A. Tebbe, *Catalan Number Sequences and Generalized Action\nGraphs*, Ball State Undergraduate Mathematics Exchange **18**(1), Fall 2024,\n88–106, [Conjecture 5](https://openjournals.bsu.edu/mathexchange/article/view/5831);\n[arXiv:2507.22719v1](https://arxiv.org/abs/2507.22719v1), Conjecture 5.6:\n\n> The subsequent super Catalan number can be computed from the n-table of its previous action graph via S(0, n + 1) = Σ_{ℓ=0}^{n} (2/2^ℓ) Σ_{v=0}^{n} Kℓ,v,n.\n\nFor every $n\\ge0$, with precisely the graphs of Definition 5.2 and the path\ncounts of Definition 5.4, the named assertion is\n\n$$S(0,n+1)=\\sum_{\\ell=0}^{n}\\frac{2}{2^\\ell}\\sum_{v=0}^{n}K_{\\ell,v,n}.$$\n\nHere Definition 5.1 is used literally:\n$S(m,n)=(2m)!(2n)!/(m!n!(m+n)!)$, in $\\mathbb Q$.\nPreregistration: [#14721](https://github.com/the-omega-institute/trureturing/issues/14721), CALD-1.","title":"Caldwell and coauthors' super Catalan action-graph identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/caldwell-2024-super-catalan-action-graph-ntable/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/45103d1546819680e7f18799e0d00c998a0ec67e655b5c596fe90ea2ee0e2596.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:callan-mansour-inversion-class-152","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David Callan and Toufik Mansour, *Inversion Sequences Avoiding Quadruple Length-3 Patterns*,\nIntegers 23 (2023), #A78, Section 1, Conjecture 1:\n\n> Class 152: {010, 100, 102, 210} ∼_I {011, 201, 210}\n\nHere ∼_I means that, for every n, equally many inversion sequences of length n avoid each set. The authors\nchecked the conjecture up to n = 13 and list the comparison as \"still open\" in Table 1.","title":"Inversion Sequences Avoiding {010, 100, 102, 210} and {011, 201, 210}","url":"https://the-omega-institute.github.io/trureturing-pages/research/callan-mansour-inversion-class-152/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b01c2f1e2996d924b874102c0074c33ab0a9a6e3ef39e0a4a10ee8b020adc6a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:callan-mansour-weak-ascent-class-215","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David Callan and Toufik Mansour, *Ascent Sequences and Weak Ascent Sequences Avoiding a Quadruple of\nLength-3 Patterns*, Integers 25 (2025), #A80, Section 1 and Table 2:\n\n> Theorem 2. The number WAS4 of WA-Wilf-equivalence classes among quadruples of length-3 patterns is\n> either 228 or 229. The slight ambiguity in Theorem 2 is because the status of Class 215 is left open in\n> Table 2 below.\n\nTable 2 lists Class 215 as the pair {100, 101, 110, 201} and {021, 101, 201, 210}, marked \"Open\".","title":"Weak Ascent Sequences Avoiding {100, 101, 110, 201} and {021, 101, 201, 210}","url":"https://the-omega-institute.github.io/trureturing-pages/research/callan-mansour-weak-ascent-class-215/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/10070ca11cbd7820d29d9933f6c12b1981e0a472479710956bc79ed003a87d14.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:camargo-nishida-2026-gaussian-multientropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. A. Camargo and M. Nishida, *Genuine Multi-Entropy of Fully Symmetric Gaussian States*,\narXiv:2609.30754v1, compute the tripartite genuine Rényi multi-entropy\n$\\mathrm{GM}^{(3)}_n(A:B:C)$ of the fully symmetric pure Gaussian state of $N$ bosonic modes\n(diagonal $a\\ge1$, off-diagonal $e^-$ of their Eq. (3.1)) for $N\\le8$ and $n\\le4$, observe\n$\\mathrm{GM}^{(3)}_n\\sim\\frac{2-n}{2n}\\log a$ as $a\\to\\infty$ (Eq. (3.8)), and write \"we\n\\emph{conjecture} that the asymptotic behavior \\eqref{AsymB} is valid for the fully symmetric\nGaussian states with any $n$ and $N$.\" The verbatim definitions are in\n[the literature note](../Library/QuantumStates/camargo2026gaussianmultientropy.md).\n\nIssue [#13990](https://github.com/the-omega-institute/trureturing/issues/13990) reads the\nconjecture for every replica order $n\\ge2$ and every tripartition with positive party sizes\n$N_A,N_B,N_C$, with the replica partition functions as the literal Lebesgue integrals of products\nof the real Gaussian density kernels under the twists $g_A,g_B,g_C$, and asymptotic equivalence in\nthe sense of `Asymptotics.IsEquivalent` (for $n=2$ both sides are the zero function).","title":"The large-squeezing law of the genuine multi-entropy of fully symmetric Gaussian states","url":"https://the-omega-institute.github.io/trureturing-pages/research/camargo-nishida-2026-gaussian-multientropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6c53e9e83387948fb80f6b09f99cc8f49bd77f1d4d387b50a0d0c38be1065907.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:camilo-landi-eliens-2019-bosonic-gaussian-renyi-strong-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Camilo, G. T. Landi and S. Eliëns, *On the Strong Subadditivity of the Rényi entropies for bosonic and fermionic Gaussian states*, arXiv:1810.07070v2 (Phys. Rev. B 99, 045155, 2019), Section IV.A, `RenyiGaussian_v2.tex`, line 463:\n\n> It provides clear evidence that SSA is violated for all $\\alpha>2$, while no violation is found for $0\\le\\alpha\\le2$. We conjecture that this result holds true in general.\n\nSection V, line 532, requests:\n\n> An obvious continuation of this work is to prove the conjectured domains of validity of the SSA for the R\\'enyi-$\\alpha$ entropies.\n\nFor $n$ modes, put $\\Omega=\\bigoplus_{j=1}^n\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}$. The covariance formulation takes a real symmetric $2n\\times2n$ matrix $V$ with $V+i\\Omega\\succeq0$. For a mode set $X$, retain both quadratures to obtain $V_X$ and $\\Omega_X$. The module defines\n\n$$s_\\alpha(\\nu)=\\frac{\\log\\left[((\\nu+1)/2)^\\alpha-((\\nu-1)/2)^\\alpha\\right]}{\\alpha-1},\\qquad\nS_\\alpha(V_X)=\\frac12\\sum_{\\lambda\\in\\operatorname{spec}(V_X^{1/2}(i\\Omega_X)V_X^{1/2})}s_\\alpha(|\\lambda|),$$\n\nwith eigenvalues counted with multiplicity. This is the covariance entropy formula of Eq. `renyi_bosonic_1` (TeX lines 216–220), using both signs of the symplectic spectrum. The empty sum is zero.\n\nIssue [#26322](https://github.com/the-omega-institute/trureturing/issues/26322) fixes the statement: for every $n$, every such physical $V$, every pairwise disjoint mode sets $A,B,C$, and every $0<\\alpha\\le2$ with $\\alpha\\ne1$,\n\n$$S_\\alpha(V_{A\\cup B})+S_\\alpha(V_{B\\cup C})\\ge S_\\alpha(V_{A\\cup B\\cup C})+S_\\alpha(V_B).$$\n\nThis statement is proved. The settlement excludes order zero, order one, the existence of violations at every order above two, fermions and non-Gaussian states.","title":"Strong subadditivity of bosonic Gaussian Rényi entropy for $0<\\alpha\\le2$, $\\alpha\\ne1$","url":"https://the-omega-institute.github.io/trureturing-pages/research/camilo-landi-eliens-2019-bosonic-gaussian-renyi-strong-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea7b76c8326353c5bb19db2d3057412439fec77bb847b474d57b930aff2a45f8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:campbell-2010-log-negativity-resource-independence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. T. Campbell, *Optimal entangling capacity of dynamical processes*, Phys. Rev. A **82**, 042314 (2010), arXiv:1007.1445v2, Section VII, p. 7, asks:\n\n> The most obvious open question is whether log-negativity EC is resource independent for all unitaries, or indeed all CPTP maps, or whether this is a singular feature of 2-qubit gates.\n\nThe all-CPTP clause is refuted. The unitary clause and the source's two-qutrit unitary conjecture remain open. The statement and source scope are fixed in [#17047](https://github.com/the-omega-institute/trureturing/issues/17047); the source quotations are in the [Campbell note](../Library/QuantumContext/campbell2010optimalentangling.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Every bipartite CPTP map | `claim` quantifies local dimensions and finite Kraus families with completeness | Finite-dimensional square input/output spaces; the Choi–Kraus representation supplies this interpretation. |\n| Free local ancillary systems | `extend`, and the suprema in `capSep` and `capAssisted` | Every pair of natural-number ancillary dimensions; the cut is $AA':BB'$. |\n| Eq. (1), separable inputs | `capSep` | Density matrices that are finite convex mixtures of products of density matrices. |\n| Eq. (2), increase of entanglement | `capAssisted` | All density inputs, subtracting the input's own log-negativity. |\n| Eqs. (4)–(5), logarithmic rescaling | `Real.logb 2` of `traceNorm (partialTranspose ...)` | Base two; the partial transpose acts on $BB'$. |\n| Resource independence | Equality of the two capacities in `claim` | Suprema are in `EReal`, preserving unbounded suprema. |\n| A negative answer to the all-CPTP clause | `result` | A fixed two-qutrit channel with 18 Kraus operators and $t=1$. |\n\nThe generalized `IptPptSquaredEquivalence.IsSeparable` retains the finite-convex-mixture condition. Its local factors are positive semidefinite with trace one, the body of the reused `IsDensity` predicate. The auxiliary density condition in `capSep` is consistent with those mixtures; it does not exclude a source separable state.","title":"Log-negativity entangling capacity depends on the input resource for CPTP maps","url":"https://the-omega-institute.github.io/trureturing-pages/research/campbell-2010-log-negativity-resource-independence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ba5da94e7035dc429e15d24233813b10530a226fd38d7f70dc4e297ce087210.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:campbell-2020-group-graph-reciprocal-pairs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kirsty Campbell, *Group-Graph Reciprocal Pairs*, arXiv:2005.13566v1, Conjecture on page 3:\n\n> Conjecture: Let (Γ, G) be an irreducible reciprocal pair. Then one of the following hold:\n> 1. (Γ, G) is one of the two trivial cases: (a) Γ is a null graph and G is any group with no odd permutations, (b) Γ is the complete graph with n vertices and G = Sn.\n> 2. Γ is a 4-cycle and G ≅ Aut(Γ).\n> 3. Γ is a k-star and G is defined in Theorem 1.\n\nThe source definitions and locators are in [Campbell’s Library note](../Library/GraphInvariants/campbell2020reciprocal.md). [Cameron–Semeraro](../Library/GraphInvariants/cameronsemeraro2018cycle.md), *The cycle polynomial of a permutation group*, DOI 10.37236/7299, supplies the literature background. The quantified conventions and external Tier-1 settlement are preregistered in [#16260](https://github.com/the-omega-institute/trureturing/issues/16260).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope and fidelity |\n| --- | --- | --- |\n| Finite simple graph and automorphism group | `claim` quantifies a finite vertex type, a simple graph and a finite permutation subgroup; `IsAut` preserves adjacency in both directions | The decidability instances support finite counting and do not restrict classical finite graphs. |\n| Cycles include fixed points | `cyclePoly` uses the cardinality of the SameCycle quotient | The literal cycle-count convention is inlined. Its proof correspondence `cycleConvention` uses the existing `NikolovSegal.Equation47TypeII.actualCycleCount`. |\n| Proper colourings fixed by a permutation | `fixedColourings` filters all maps to the finite colour set by properness and fixedness | Both conditions and every vertex are quantified. |\n| Reciprocal polynomial identity | `IsReciprocal` equates the colouring total with the signed evaluation of the cycle polynomial for every positive integer | These infinitely many evaluations determine the polynomial identity. |\n| No nontrivial direct-product decomposition into reciprocal factors | First branch of `Reducible`: at least two nonempty parts, no cross-part edges, generation by subgroups supported on individual parts | Omitting the reciprocity requirements enlarges reducibility. The corresponding irreducibility premise is stronger. |\n| No nontrivial wreath-product decomposition with isomorphic part graphs | Second branch of `Reducible`: all induced part graphs are pairwise isomorphic | Omitting all group conditions enlarges reducibility. |\n| Null, complete, four-cycle, or k-star with the specified groups | Four alternatives of `Listed`, retaining the graph conditions | Omitting group restrictions enlarges the conclusion. The k-star centre is nonempty and proper, has the stated cardinality, and gives exactly the source adjacency rule. |\n| Every irreducible reciprocal pair belongs to the list | `claim` | A stronger premise and a weaker conclusion make the formal claim no stronger than the source conjecture. |\n| Refutation | `result` negates `claim` | A counterexample to this conservative consequence refutes the source conjecture. No source clause is dropped without the stated conservative direction. |","title":"Campbell’s classification of irreducible reciprocal pairs is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/campbell-2020-group-graph-reciprocal-pairs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a486e93cb14a537c9cf1e9eadfd8113eca5d599a9a37350004b81cd0134c4802.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:campbell-currie-rampersad-eq-11","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Write `rho(n)` for the reduced abelian complexity of the Thue-Morse word: the\nnumber of classes of length-`n` factors, where two factors are equivalent when\ncollapsing each maximal constant run to one letter leaves words of equal length\nthat are rearrangements of one another. Campbell, Currie and Rampersad display\n\n |rho(4n+2) - rho(4n)| = 0 if t(n+1) = t(3n+1), and 1 otherwise,\n\nas equation (11) in the Conclusion of arXiv:2509.16034v1, with the sentence \"It\nappears that (11) holds\", and state a few lines later that they leave proving it\nas an open problem. The verified journal version retains equation (11) and\nthis question in Section 3, printed page 14; its citation is recorded in\n`Library/Words/campbell2025reduced.md`. The paper's letter indices start at one,\nso the zero-indexed reading is `t(n) = t(3n)`. Factor lengths do not shift.","title":"Equation (11) for the reduced abelian complexity of the Thue-Morse word","url":"https://the-omega-institute.github.io/trureturing-pages/research/campbell-currie-rampersad-eq-11/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8a54170a65c91d776824748f8e33271bcde0c2f7bbc6d0be66b52bac8ed59065.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:campoamor-stursberg-et-al-2025-zernike-ladder-factorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Campoamor-Stursberg, Herranz, Latini, Marquette and Blasco, Nonlinearity 39, 055008, arXiv:2502.02491v2, Conjecture 1(iii), asks for number and ladder operators constructed from the quantum symmetries (2.8), satisfying (2.9)–(2.10) with the factorization (5.1) for every $N\\geq1$. Preregistration: [#19957](https://github.com/the-omega-institute/trureturing/issues/19957).\n\nThe binding polynomial formulation is proved by `D5.S3.Quantum.CampoamorStursbergZernikeLadder.result`. Operators are complex linear endomorphisms of $\\mathbb C[q_1,q_2]$. For every $N\\geq1$ and every complex coefficient sequence there are two symmetries with the kinetic terms and correction span of (2.6), and number and ladder operators in the unital algebra generated by $K_1,K_2,K_3$, satisfying all four identities of the claim.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Coordinate multiplication and $-i$ times partial differentiation, (2.1) | Core `q`, `p` | `Fin 2` indices 0 and 1 encode source coordinates 1 and 2. |\n| Hamiltonian and angular momentum, (2.3), (2.5) | Core `dot`, `hamiltonian`, `angularMomentum` | Composition of polynomial endomorphisms; arbitrary complex coefficients. |\n| Normal-ordered correction monomials of positive degree at most $2N$ and grading zero, (2.6) | Core `monomial`, `corrections` | The defining span uses exactly these conditions. |\n| Two commuting symmetries with prescribed kinetic terms, (2.6)–(2.7) | Core `IsSymmetry`; both first conjuncts of `claim` | Momentum-order upper bound; no exact-order assertion at degenerate coefficients. |\n| $K_1=C$, $K_2=(I'-I)/2$, $K_3=[K_1,K_2]$, (2.8) | Core `k1`, `k2`, `k3` | Literal binding definitions. |\n| Operators constructed from these three generators | Three `Algebra.adjoin` memberships in `claim` | The unital generated algebra; no extra hypotheses. |\n| Raising and lowering commutators, (2.9) | First two final conjuncts of `claim` | Equalities of endomorphisms. |\n| $K_+K_-=\\Phi(H,K)$ and the shifted commutator, (2.9)–(2.10) | Last two conjuncts of `claim` | Core `phi1`, `phi2`, `phi` give exactly (5.1). |\n| Every order and every coefficient choice | Universal binders of `claim` | $N\\geq1$; only indices 1 through $N$ are used. |\n\nThe divided-difference definitions are auxiliary polynomial expressions. The claim has no localization, invertibility, nondegeneracy or analytic-domain hypothesis. No clause of the binding claim is dropped.","title":"Factorized ladders for generalized quantum Zernike Hamiltonians","url":"https://the-omega-institute.github.io/trureturing-pages/research/campoamor-stursberg-et-al-2025-zernike-ladder-factorization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e7a076ff5a1af0a4cbf7265a04bcb3397499ebb3260af8bbe81b309cc7fea51.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cao-chen-miller-egg-drop-conjecture-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cao, Chen, and Miller, *Egg Drop Problems: They Are All They Are Cracked Up\nTo Be!*, arXiv:2511.18330v2, printed page 10, Conjecture 1:\n\n> In a d-dimensional setting with sides N_1, N_2, ..., N_d and k eggs, the minimum\n> number of drops required in the worst-case scenario under the same strategy satisfies\n\n$$P_d(k) \\le \\left\\lceil (k-d+1)(N_1+\\cdots+N_d)^{1/(k-d+1)}\\right\\rceil,$$\n\n> for k >= d.\n\nThe subscripted names and inequality above transcribe the printed notation.\nPreregistration issue #8448 writes the positive integer domains explicitly.\nThe hidden point ranges over the product of the integer intervals\n`{1, ..., N_i}`. A drop at `a` returns intact exactly when `a_i < x_i` for\nevery coordinate; otherwise it breaks one egg. The objective is exact recovery.\n\nThe formal `claim` weakens success of the paper's named strategy to existence\nof any deterministic adaptive strategy with those outcomes and egg semantics:\n\n```text\nforall (d k : Nat) (N : Fin d -> Nat),\n 1 <= d -> d <= k -> (forall i, 0 < N i) ->\n exists s : EggStrategy N k (paperBound d k N), Correct s\n```\n\nRefuting this weaker existence assertion refutes the named-strategy assertion.","title":"Cao-Chen-Miller Conjecture 1: the multidimensional egg-drop bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/cao-chen-miller-egg-drop-conjecture-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d16c2d8b2e13c5737ba6ee670c1144e5ede11e610247f2426ffd4a59e685152.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cao-fitzsimmons-mann-pereira-laflamme-2023-semi-nonnegative-backflow-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Cao, M. Fitzsimmons, Z. Mann, R. Pereira and R. Laflamme, *Quantum Maps Between CPTP and HPTP*, arXiv:2308.01894, classify the Hermiticity-preserving trace-preserving (HPTP) maps that can describe the evolution of an open system.\n- A map is semi-nonnegative (SN) if it sends some density matrix to a density matrix (Definition 2), equivalently if Ψ ∘ Φ = Ξ for some CPTP Φ and Ξ (Theorem 2).\n- It is semi-positive (SP) if it sends some invertible density matrix to an invertible one (Definition 1). SP maps are the intermediate maps of CP-indivisible, non-Markovian evolutions.\n- For qubits, the non-SP SN maps send only one pure state to a state, so they \"signal no information backflow\".\n- The Outlook conjectures: \"We conjecture that non-SP SNTP maps still signal no information backflow and are not uniquely defined even for higher dimensional H.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/caofitzsimmonsmannpereiralaflamme2023cptphptp.md).\n\nIssue [#18048](https://github.com/the-omega-institute/trureturing/issues/18048) fixes the reading of the first conjunct: in every context Ψ ∘ Φ = Ξ of a non-SP SNTP map, the intermediate map does not increase the trace distance of two states. This conjunct, and with it the conjecture, is refuted.","title":"A non-semi-positive semi-nonnegative qutrit map can signal information backflow","url":"https://the-omega-institute.github.io/trureturing-pages/research/cao-fitzsimmons-mann-pereira-laflamme-2023-semi-nonnegative-backflow-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4bcf5e5281e80eb2d896cc214e28d5669471867b570f222ea4a113db82b08f34.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cao-zhang-shi-zhao-2026-mabk-optimal-robustness-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Shen Cao, Xingjian Zhang, Fei Shi and Qi Zhao, *Size-Independent Robustness\nin Multipartite Bell Self-Testing*, arXiv:2608.30851v1, Supplemental Material,\n“Efficient Verification of Optimal Lower Bound”, state:\n\n> For $n \\le 5$ this bound is already established analytically [20, 23]; for\n> $n \\geq 6$ the two remaining cases ($h = 1, 2$) are an open conjecture\n> supported by the numerical evidence above.\n\nThe conjecture is that the five-term expression $\\lambda_{\\vec a}$ on\nPDF p. 37 is nonnegative at every point of $[0,\\kappa]^n$, where\n$\\kappa=1-1/\\sqrt2$, for every $n\\ge6$ and $h\\in\\{1,2\\}$.\nThe source has complementary index sets $T$ and $J$, of cardinalities\n$h$ and $d=n-h$. Its convention $h\\le d$ follows from these hypotheses.\nIssue #11561 preregisters the literal expression, full quantifiers,\nTier 1 classification and literature check.\n\nThe definition `lambdaA` preserves all five terms, their coefficients,\nboth complementary products in each paired term, and the real square roots.\nThe bijection from `Fin n` to the source indices is $i\\mapsto i+1$.\nThe definitions `kappa` and `claim` express the endpoint and complete\nconjecture. The theorem `result : claim` proves it.","title":"Cao–Zhang–Shi–Zhao MABK positivity for the two remaining blocks","url":"https://the-omega-institute.github.io/trureturing-pages/research/cao-zhang-shi-zhao-2026-mabk-optimal-robustness-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d632b52adacda0b2582654d7b4a3493c8be028fe20c744f1ed44118d626d9a5a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:capela-celeri-chaves-modi-2022-classical-markov-monogamy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Capela, L. C. Céleri, R. Chaves and K. Modi, *Quantum Markov monogamy inequalities*, Phys. Rev. A 106, 022218 (2022), arXiv:2108.11533v4, Section II, Conjecture 1 (`PRA-v1.tex`, lines 153–159):\n\n> Consider the Markovian process $X_n \\rightarrow \\cdots \\rightarrow X_1 \\rightarrow Y_1 \\rightarrow \\cdots \\rightarrow Y_n$. The variables $X_1$ and $Y_1$ are to be interpreted as input and output of a given channel, respectively. The variables $X_i$ and $Y_i$, with $i=2,\\dots,n$, are interpreted as pre-processed and post-processed variables, respectively. Then for any bijective function $f \\colon \\{1,\\dots,n\\} \\rightarrow \\{1,\\dots,n\\}$, it holds that\n>\n> $\\sum_{i=1}^{n} I(X_i:Y_i) \\geq \\sum_{i=1}^{n} I(X_i:Y_{f(i)}),$\n>\n> where $I(X:Y)$ denotes the mutual information of random variables $X$ and $Y$.\n\nThe source reports at line 161:\n\n> The conjecture above has been checked for $n$ up to 4, that is, Markov chains with 8 random variables\n\nHere $I(X:Y)=H(X)+H(Y)-H(X,Y)$, with Shannon entropy. The definitions and source locators are in [the literature note](../Library/Entropy/capelacelerichavesmodi2022markov.md).\n\nIssue [#22399](https://github.com/the-omega-institute/trureturing/issues/22399) fixes the finite-alphabet reading. The module proves it for every $n\\in\\mathbb N$, including $n=0$ with empty sums: choose finite alphabets $\\alpha_t$, a nonnegative initial mass $p:\\alpha_0\\to\\mathbb R$ with $\\sum_xp(x)=1$, and matrices $k_t:\\alpha_t\\to\\alpha_{t+1}\\to\\mathbb R$ satisfying $k_t(x,y)\\ge0$ and $\\sum_y k_t(x,y)=1$ whenever $t+1<2n$. For every permutation $f$ of $\\mathrm{Fin}\\,n$, the sum paired by $f$ is at most the diagonal sum.\n\nPrecisely, `composedTransition` is the identity at equal positions and the product of successive matrices for ordered positions $s\\le t$; write it as $K_{s,t}$. The definitions `mixture` and `mixtureJoint` give $p_s(x)=\\sum_zp(z)K_{0,s}(z,x)$ and $q_{s,t}(x,y)=p_s(x)K_{s,t}(x,y)$. For $i\\in\\mathrm{Fin}\\,n$, position $n-1-i.val$ represents $X_{i.val+1}$ and position $n+i.val$ represents $Y_{i.val+1}$. Each summand in `claim` is `mutualInformation` of the corresponding $q_{s,t}$, defined as its KL divergence from the product of its marginals, in nats.","title":"Classical Markov monogamy for every finite alphabet and every permutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/capela-celeri-chaves-modi-2022-classical-markov-monogamy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/449054b7d99ccdb86a6345f42ec3177c163b7fff9f1095674732179f7dc480fd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:capitelli-somma-2024-all-odd-nonconvergence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nicolás Capitelli and Francisco Somma, “On sequences arising from randomizing subtraction games”, arXiv:2405.19593v1, Section 3, p. 7, state:\n\n> However, we do not show that an all-odd subtraction set necesarily do not converge (although we conjeture it) nor we provide the exact limits of the even and odd subsequences for the (conjectured) non-convergent case.\n\nThe section assumes a finite nonempty subtraction set of positive integers with gcd one. A player selects uniformly among legal subtractions, and loses when none is available. Writing $A_n=\\{s\\in S:0 At present, however, we are unable to establish the sign of $\\left(b_n^{(A)}\\right)^2$ in full generality. Based on all the examples discussed in this manuscript, we conjecture that $\\left(b_n^{(A)}\\right)^2>0$ for every $n$, and hence that all the coefficients $b_n^{(A)}$ are real.\n\nThe source evolves a normalized pure state by $U(t)=e^{-\\mathrm{i}Ht}$,\nwith a time-independent Hermitian Hamiltonian on a bipartite Hilbert\nspace. Its reduced density and fixed-normalization return amplitude are\n\n$$\n\\rho_A(t)=\\operatorname{Tr}_B(U(t)|\\psi\\rangle\\langle\\psi|U(t)^\\dagger),\n\\qquad\nR_A(t)=\\frac{\\operatorname{Tr}(\\rho_A(t)\\rho_A(0))}\n {\\operatorname{Tr}(\\rho_A(0)^2)}.\n$$\n\nEquations (3.3) and (3.5) define\n$\\mu_n^{(A)}=\\partial_t^nR_A(t)|_{t=0}$ and\n$(b_1^{(A)})^2=(\\mu_1^{(A)})^2-\\mu_2^{(A)}$.\nThe finite-dimensional $n=1$ clause quantifies over both subsystem\ndimensions, every normalized pure state and every Hermitian Hamiltonian.\nA negative first square disproves the full conjecture, including a\nnonnegative reading that allows terminating zero coefficients.","title":"Subsystem Lanczos positivity for general initial states","url":"https://the-omega-institute.github.io/trureturing-pages/research/caputa-di-giulio-loc-subsystem-lanczos-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db978a2d66f8a7ced24dc4530353e1ff78e3a36e6da257a95462a740a1b56d7a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:carli-silva-2026-theta-representation-convexity-clause","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Marcel K Carli Silva, Gabriel Coutinho, Thiago Oliveira and Levent Tunçel,\n*On some k-fold generalizations of Lovász theta and their sandwich theorems*,\narXiv:2609.06312v1, §7, Open questions, item (iv), ask:\n\n> Is the orthonormal representation set $\\texttt{M}_k(G)$ convex? If not, can one characterize its closed convex lower hull, i.e., the convex corner whose support function is $\\varphi_k(G,\\cdot)$?\n\nThis dossier identifies **only the first convexity clause** of that version\nand item. The closed convex lower hull characterization is a separate,\nunresolved question. The preregistration is\n[issue #15881](https://github.com/the-omega-institute/trureturing/issues/15881).\n\nIn §5.3, the proof of Theorem 16 defines, for a finite ordinary simple graph\n$G=(V,E)$ and $n=|V|$,\n\n$$\n\\texttt{M}_k(G)=\\left\\{x\\in\\mathbb R_+^V:\\begin{array}{l}\n\\exists M\\in\\mathbb S_+^V,\\quad I-M\\in\\mathbb S_+^V,\\quad \\operatorname{tr}(M)=k,\\\\\n\\exists u:V\\to\\mathbb R^V,\\quad \\forall i\\in V,\\ \\langle u_i,u_i\\rangle=1,\\\\\n\\forall i,j\\in V,\\ ij\\in E\\Rightarrow\\langle u_i,u_j\\rangle=0,\\\\\n\\forall i\\in V,\\ x_i=u_i^{\\mathsf T}Mu_i\n\\end{array}\\right\\}.\n$$\n\nHere $\\mathbb S_+^V$ consists of real symmetric positive semidefinite\n$V\\times V$ matrices. The source's order $M\\preccurlyeq I$ means\n$I-M\\succeq0$, and its matrix inner product is $\\langle A,B\\rangle\n=\\operatorname{tr}(AB)$, so $\\langle I,M\\rangle=k$ is exact trace equality.\nOne common matrix and one common vector family realize every coordinate.\nThere is no rank or projector restriction. The ambient real dimension is\nexactly $|V|$. In the source's convention, $u$ is an orthonormal\nrepresentation of $\\overline G$: orthogonality is imposed on edges of $G$,\nwith no condition on its nonedges. Vectors at nonadjacent vertices may\ncoincide; the family need not be a mutually orthogonal basis.","title":"The theta representation set: Question (iv), first convexity clause","url":"https://the-omega-institute.github.io/trureturing-pages/research/carli-silva-2026-theta-representation-convexity-clause/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/006a94b362acdea613079820a4cd5cf7e9480ce6c454ab4ff3f575fd40bc1eb7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:carmeli-heinosaari-toigo-2025-upper-boundary-instrument-uniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Carmeli, T. Heinosaari and A. Toigo, *Postponing the choice: advantage of deferred measurements in quantum information processing*, arXiv:2502.19871v1 (New J. Phys. 27, 094501, 2025), after the proof of the compatibility theorem for a sharp meter and the identity channel:\n\n> We conclude that, when the point $(s,t)$ attains the second equality in \\eqref{eq:QI_comp_neg}, the instrument $\\tilde{\\ii}^{(t)}$ is actually the unique $W$-covariant joint instrument for $\\Qo_s$ and $I_t$. It is an open question whether in this case there exist other (necessarily not $W$-covariant) joint instruments.\n\nHere:\n- **The meter:** $\\mathsf Q_s(x)=s|\\varphi_x\\rangle\\langle\\varphi_x|+(1-s)I/d$.\n- **The channel:** $I_t=t\\,\\mathrm{id}+(1-t)\\,\\mathrm{tr}(\\cdot)\\,I/d$.\n- **Joint instrument:** completely positive maps $J_x$ with $\\sum_xJ_x=I_t$ and $\\mathrm{tr}\\,J_x(\\rho)=\\mathrm{tr}(\\rho\\,\\mathsf Q_s(x))$.\n- **The upper boundary:** $d^2s=(d^2-2)(1-t)+2\\sqrt{(1-d^2)t^2+(d^2-2)t+1}$, for $t\\in[-1/(d^2-1),1]$.\n\nThe definitions and the instrument $\\tilde{\\mathcal J}^{(t)}$ are quoted in [the literature note](../Library/QuantumContext/carmeliheinosaaritoigo2025postponing.md).\n\nIssue [#20028](https://github.com/the-omega-institute/trureturing/issues/20028) fixes the reading: on the upper boundary, every joint instrument equals $\\tilde{\\mathcal J}^{(t)}$, for every $d\\ge2$ and every orthonormal basis. This is proved.","title":"The upper-boundary joint instrument is unique","url":"https://the-omega-institute.github.io/trureturing-pages/research/carmeli-heinosaari-toigo-2025-upper-boundary-instrument-uniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/249b8c656929cc6cb312e14b86f514a873a92a07f99a01837c14c80b4c4d4674.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:carnero-bravo-2025-hereditary-lexicographic-products","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Andrés Carnero Bravo, *On independence complexes of graph products*,\narXiv:2505.06457v2, Section 4, Question 22 asks whether, under the hypotheses\nof Theorem 21, $G\\circ H$ belongs to $\\mathcal W$. Both graphs are finite,\nhave at least two vertices, and belong to $\\mathcal{SW}$.\nSection 2 defines $\\mathcal{SP}$ as contractibility or the homotopy type of\nan ordinary wedge of spheres, and $\\mathcal P$ additionally permits a\ndisjoint union of such wedges. The hereditary classes require the respective\nproperty on induced subgraphs. The requested conclusion is unsuspended.\n\nThe paper does not separately specify the vertexless convention. This\nsettlement states it explicitly: sourceSW requires SP on every nonempty\ninduced vertex set; the output sourceW includes every induced vertex set,\nincluding the empty one. The native realization of a vertexless graph is\nempty. Its output P witness is an empty outer coproduct, rather than a\ncontractible point or a nonempty ordinary wedge.\nThe initial preregistration sketch represented the vertexless input by an\naugmented empty-sphere model. The retained statement makes that boundary\nexplicit through the nonempty-induced input convention and the independent\nempty-output proof; it does not assert ordinary SP for an empty realization.","title":"Carnero Bravo's hereditary lexicographic product question","url":"https://the-omega-institute.github.io/trureturing-pages/research/carnero-bravo-2025-hereditary-lexicographic-products/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c43e2c1d19b4fcda12932d5fedae0df47bb16880a5a8c35627e7180b14c6e6d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:caro-graswald-2021-markovian-divisibility-maximal-index","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. C. Caro and B. R. Graswald, *Necessary criteria for Markovian divisibility of linear maps*, Journal of Mathematical Physics 62, 042203 (2021), arXiv:2009.06666v3, Conjecture 4.19 (`CnjAMGMMatrices`), `main.tex`, lines 729–737. The following quotation is checked against the v3 TeX; its macro `\\IR` is expanded as defined in `commands.tex`:\n\n> Let $p\\in\\mathbb{R}^d_{\\geq 0}$ with $\\sum\\limits_{i=1}^d p_i = 1$. Define the matrices $a,g\\in\\mathbb{R}^{d\\times d}$ via\n>\n> $a_{ij} = \\frac{p_i + p_j}{2},\\quad g_{ij}=\\sqrt{p_i p_j}.$\n>\n> Denote by $a_k^\\downarrow$ and $g_k^\\downarrow$ the $k^{th}$ largest entry of $a$ and $g$, respectively. Define\n>\n> $A = \\sum\\limits_{k=1}^{h(d)} a_k^\\downarrow,\\quad G = \\sum\\limits_{k=1}^{h(d)} g_k^\\downarrow.$\n>\n> We conjecture that the maximal integer $h(d)$ such that $A+G\\leq d$ holds for any probability vector $p$ is given by $h(d) = 2d-5$.\n\nFor a natural index $m$, write $A_m(p)$ and $G_m(p)$ for the sums of the largest $\\min(m,d^2)$ entries of the two matrices, sorted independently, counting both ordered positions and retaining equal values at distinct positions. Let $S_d=\\{m\\in\\mathbb{N}:\\forall p\\in\\mathbb{R}^d_{\\ge0},\\ \\sum_i p_i=1\\Rightarrow A_m(p)+G_m(p)\\le d\\}$. The module's `admissibleIndices d` is $S_d$.\n\nIssue [#22748](https://github.com/the-omega-institute/trureturing/issues/22748) fixes the settlement as two greatest-element assertions: $\\max S_d=2d-5$ for every $d\\ge5$, and $\\max S_3=2$. Thus the conjectured formula is proved in the specified range, while the three-dimensional value differs from the printed expression $2\\cdot3-5=1$. The source states no dimension restriction; the settlement does not assert its formula in every dimension.","title":"The Caro–Graswald maximal index is $2d-5$ for $d\\ge5$, and $2$ for $d=3$","url":"https://the-omega-institute.github.io/trureturing-pages/research/caro-graswald-2021-markovian-divisibility-maximal-index/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84cc0814cc32bf492908eb853ca932e05dbe8b6bfd2b4196a43fa4d2ebadaa14.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:caro-petrusevski-skrekovski-tuza-grid-three-eighths","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Caro, M. Petruševski, R. Škrekovski and Zs. Tuza, *The odd independence number of graphs, II:\nFinite and infinite grids and chessboard graphs*, arXiv:2510.01897v1, Problem 29: does there exist a\nsequence ε_n with ε_n → 0 such that any (3/8 + ε_n)n² independent vertices of P_n □ P_n contain all\nfour neighbours of some vertex?","title":"Dense Independent Sets of the Square Grid Contain a Full Cross","url":"https://the-omega-institute.github.io/trureturing-pages/research/caro-petrusevski-skrekovski-tuza-grid-three-eighths/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/761c684f2a4f54220419552d5b48da06938fcb17a433194bb05364f0cb287893.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cc2025","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Costa and Catarino, *On Matrices with Bidimensional Fibonacci Numbers*, Annales Mathematicae Silesianae 40(1), 59–76 (2026), DOI 10.2478/amsil-2025-0014, state an unnumbered conjecture at the end of §4 (p. 73): every order-two bidimensional Fibonacci matrix has nonzero determinant for all nonnegative indices. The source definitions are Definition 2.1 (p. 63), Lemma 2.5 (p. 64), and Definition 3.1 (p. 67). Preregistration and the verbatim statement are in issue [#22448](https://github.com/the-omega-institute/trureturing/issues/22448).\n\n### Statement echo\n\nThe Lean declaration `D5.S1.Recurrence.CostaCatarinoBidimensionalFibonacci.claim` keeps the source recurrences with guards $2 \\le m$ and $2 \\le n$, all four complex corner values, and the matrix with both negative entries. The definition `sourceClosedForm` separately retains the literal closed form. The claim adds an existence conjunct so the universal determinant assertion cannot be vacuous, and quantifies over every source sequence, every $m$, and every $n$; axes and the diagonal are included.","title":"Costa–Catarino bidimensional Fibonacci matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/cc2025/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ccf74c236d58208fb811ef604927f70793102fa84f4329f43a3e1a97ade8d01c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cenni-lami-acin-mehboudi-2022-gaussian-thermometry-local-global-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. F. B. Cenni, L. Lami, A. Acín and M. Mehboudi, “Thermometry of Gaussian\nquantum systems using Gaussian measurements”, arXiv:2110.02098v4, §3.2,\nEq. (47), ask whether the optimal joint Gaussian Fisher information equals\nthe optimal local Gaussian Fisher information for every product thermal state:\n\n> \"${\\cal F}^{\\rm C}(\\oplus_k {\\bm \\sigma}_{k}; {\\bm \\sigma}^M_{\\max}) \\overset{?}{=} {\\cal F}^{\\rm C}(\\oplus_k {\\bm \\sigma}_{k}; \\oplus_k{\\bm \\sigma}_{k,\\max}^M)$\"\n\n> Based on these observations we conjecture this is generally true, however, a rigorous proof is missing currently.\n\nIssue [#11666](https://github.com/the-omega-institute/trureturing/issues/11666)\npreregisters this Tier 1 conjecture, the quantified version 2 and the two-mode\ncounterexample. The formal claim compares each physical joint value with the\nsupremum over all physical block-diagonal measurements.","title":"Local versus joint Gaussian thermometry for unequal-frequency modes","url":"https://the-omega-institute.github.io/trureturing-pages/research/cenni-lami-acin-mehboudi-2022-gaussian-thermometry-local-global-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da0284b9340dada0d2d3cc724f14cc3c239ccdbd71a0a498c7feb825b0f2f054.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:centered-reduced-residue-progressions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let `n` be even and squarefree with at least three distinct prime factors. Let\n`p` be its greatest prime factor and put `d=n/p`. Under `d<2p`, determine the\nlongest positive-step arithmetic progression in the fixed centered reduced\nresidue system from `-n/2+1` through `n/2`.\n\nThe unnumbered conjecture on printed page 14 of the cited paper states that the\nmaximum length is `floor(p-2p/d)`. The formal target expresses the maximum as\nan `IsGreatest` statement, so it requires an attaining progression and a bound\nfor every positive step, together with the exact rational-floor equality.","title":"Centered reduced residue progression maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/centered-reduced-residue-progressions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18e94983fadc6125883a45e21a2fde1a8df5a2ed00d20ce61b2f35c51e9db48e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:central-edges","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dovgoshey and Rovenska, *On the Graph Corresponding to Center of Distances of\nMetric Space*, arXiv:2610.11596v1, Conjecture 4.2, p. 13, assert an equivalence\nfor every finite ultrametric space $X$ with $|X|\\ge4$ and three central distances.\nThe center is\n$C(X)=\\{t\\ge0:\\forall p\\in X\\;\\exists y\\in X,\\ d(y,p)=t\\}$.\nThe central graph joins distinct vertices whose distance belongs to $C(X)$;\nthe diametrical graph joins vertices at the diameter.\n\nClause (i) says that the diametrical graph is complete bipartite on a partition\n$X=A\\sqcup B$ with $\\min(|A|,|B|)=2$, and that the central graph is its union\nwith one star on each part. Clause (ii) says that its central graph has at most\nas many edges as every metric space $Y$ with $|Y|=|X|$ and $|C(Y)|=3$.\nThe comparator is unrestricted: it need not be ultrametric.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Definition 1.1, nonnegative center including zero | `center` | The realizing point may depend on the base point. |\n| Definition 1.2, simple central graph | `centralGraph` | `SimpleGraph.fromRel` excludes loops; metric symmetry gives the stated unordered adjacency. |\n| Definition 2.14, diameter graph | `diametricalGraph` | `Metric.diam` on the whole carrier. |\n| Finite ultrametric, order at least four, three central values | `claim` quantifiers and hypotheses on $X$ | No restriction omitted. |\n| Complete bipartite partition, smaller part of size two, two stars | `conditionI` | Disjoint sets cover the carrier; subtype star centers ensure nonempty parts. Graph maps preserve the original carrier. |\n| Every equal-order three-central metric comparator | `claim` quantifiers on $Y$ and edge inequality | No ultrametric assumption on $Y$. |\n| Equivalence of (i) and (ii) | `claim` biconditional | `result` proves its literal negation. |\n\n`Fintype` supplies the source finiteness. `Nat.card` counts actual center values\nand unordered edges. No binder adds a condition absent from the source on the\nnonempty finite carriers in question. The five public definitions retain the\npreregistered expressions; the module uses the existing metric and graph\nconstructions rather than redefining those constructions.","title":"Three-central edge extremality against all metrics","url":"https://the-omega-institute.github.io/trureturing-pages/research/central-edges/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ee9fca441794100c94872eabb5ed702c87416db6bf46cdc1726286d6aed1322e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cepollaro-2025-power-of-two-zero-magic-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Cepollaro, G. Cuffaro, M. B. Weiss, S. Cusumano, A. Hamma and S. Lloyd, *Stabilizer\nEntropy of Subspaces*, arXiv:2512.23013v1, define the average stabilizer entropy gap of an\nisometry $\\mathcal E:\\mathbb C^{d_S}\\to\\mathbb C^{d_B}$ as the Haar average over pure states of\nthe difference between the linear stabilizer entropy of $\\mathcal E(\\psi)$ for the single-qudit\nWeyl–Heisenberg group of $\\mathbb C^{d_B}$ and that of $\\psi$ for the group of $\\mathbb C^{d_S}$,\nand write it in closed form as\n$\\Delta M(\\mathcal E)=d_S\\,\\mathrm{tr}(Q_S\\mathcal A_4^S)-d_B\\,\\mathrm{tr}(Q_B\\mathcal E^{\\otimes4}(\\mathcal A_4^S))$.\nIn §V.A they report from numerical optimization that \"when $d_B$ is a power of 2, a zero ASE\nsubspace of dimension three is always achievable\", and in their conclusion that \"it is unclear\nwhy\" this happens. The verbatim definitions and statements are in\n[the literature note](../Library/QuantumStates/cepollaro2025stabilizer.md).\n\nIssue [#13822](https://github.com/the-omega-institute/trureturing/issues/13822) reads a\nthree-dimensional subspace as the image of an isometry $V:\\mathbb C^3\\to\\mathbb C^{d_B}$ and the\nHaar fourth moment in the paper's closed form $\\binom{6}{4}^{-1}\\frac1{24}\\sum_{\\sigma\\in S_4}T_\\sigma$.","title":"Zero-magic-gap three-dimensional subspaces in every power-of-two dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/cepollaro-2025-power-of-two-zero-magic-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bcb3b84fa0878550b4960a28f0d7a156ea4e3fedd4b70f70eba6394c847c5721.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cereceda-2017-three-qubit-cabello-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. L. Cereceda, *Cabello's nonlocality for generalized three-qubit GHZ\nstates*, arXiv:1609.04763v2, Section 2, and Quantum Stud. Math. Found. 4(3)\n(2017) 205–215, conjectures:\n\n> we conjecture that, for the case in which $Q >0$, the maximum quantum violation of inequality (ineq2) (subject to the fulfillment of conditions (hc2)-(hc4)) is achieved, over all possible states and choice of observables, for the GHZ state $|\\psi \\rangle =\\frac{1}{\\sqrt{2}}(|v_1, v_2,v_3 \\rangle + |w_1, w_2, w_3 \\rangle)$, with the maximum violation being equal to $C_{\\text{max}} =\\frac{9}{64}$.\n\nHere $P = P(U_1,U_2,U_3|+++)$, $Q = P(D_1,D_2,D_3|---)$, the conditions\n(hc2)–(hc4) are $P(D_1,U_2,U_3|+++) = P(U_1,D_2,U_3|+++) =\nP(U_1,U_2,D_3|+++) = 0$, and (ineq2) is $C = P - Q \\le 0$; each $U_k$, $D_k$\nis a $\\pm1$-valued projective qubit observable. The conclusions restate the\nconjecture for all entangled states of three qubits.","title":"The maximal three-qubit violation of Cabello's nonlocality argument","url":"https://the-omega-institute.github.io/trureturing-pages/research/cereceda-2017-three-qubit-cabello-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/924cf559e06fa787def6c17f5616282259bacb0d5863efdc5e2d96af7b0ba167.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cha-2026-equiprobable-pgm-active-set","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Cha and J. Lee, *Structural perspectives from quantum states and\nmeasurements in optimal state discrimination*, Quantum Information Processing\n25, 310 (2026), arXiv:2507.05778v2. For an ensemble\n$\\tilde\\sigma_i=p_i\\sigma_i$, $I_+$ is the set of labels whose optimal\nminimum-error effect is nonzero, and\n\n> $P_\\mathcal{E}^{\\text{PGM}+} = \\sum_{i \\in \\text{I}_+(\\mathcal{E})} \\text{tr}(\\Tilde{\\sigma}_i E_i)$, where $S \\equiv \\sum_{i \\in \\text{I}_+(\\mathcal{E})} \\Tilde{\\sigma}_i$ and $E_i \\equiv S^{-1 / 2} \\Tilde{\\sigma}_i S^{-1 / 2}$.\n\nFor equal priors the paper conjectures (journal Eq. (18)):\n\n> $(|\\text{I}_+(\\mathcal{E})| - 1) \\left( P_\\mathcal{E}^{\\text{PGM}+} - \\frac{1}{N} \\right) \\leq (N - 1) \\left( P_\\mathcal{E}^{\\text{PGM}} - \\frac{1}{N} \\right).$ […] Consequently, we conjecture that it holds universally for equiprobable states.","title":"The equiprobable pretty-good-measurement comparison of Cha and Lee","url":"https://the-omega-institute.github.io/trureturing-pages/research/cha-2026-equiprobable-pgm-active-set/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d04592feda385fff8294f98601d06dacb4f7c0ff9f7a2308b517febd1a6d9c01.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chaffin-sloane-gijswijt-merge-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chaffin, Linderman, Sloane and Wilks, *Journal of Integer Sequences* 16 (2013),\nArticle 13.4.3 = arXiv:1212.6102v3, Section 5, Theorem 23, state: \"Assume the\ncurling number conjecture is true. Let S be an initial sequence not containing a\n1, let S^(e) be its 'extension' (defined in §1), and let S^(inf) be its infinite\ncontinuation. Then S^(inf) = S^(e) G.\" Here `G` is Gijswijt's sequence A090822,\n`cn(S)` is the largest `k` with `S = X Y^k` for nonempty `Y`, the continuation\nappends `cn` repeatedly, and the extension is the first prefix whose curling\nnumber equals one.\n\nThe section closes: \"We do not know if the theorem is still true if S is allowed\nto contain a 1 but does not end with 1.\" Section 6 lists this as open question\n12: \"The question implicit in the last sentence of §5.\"","title":"Chaffin-Linderman-Sloane-Wilks Gijswijt Merge Refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d84ae892ed866ebd3f1dfcf0a52699bf0f16be2770b98f09f582b002ac01877b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:champagne-johnston-macdonald-pipes-2022-symmetric-witness-spectrum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Champagne, N. Johnston, M. MacDonald and L. Pipes, *Spectral properties of symmetric quantum states and symmetric entanglement witnesses*, Linear Algebra Appl. 649, 273–300 (2022), arXiv:2108.10405v1, study the eigenvalues of symmetric entanglement witnesses.\n- **Witnesses.** A symmetric entanglement witness is a Hermitian W on C^d ∨ C^d with (v ⊗ v)*W(v ⊗ v) ≥ 0 for all v (§2.3); positive semidefinite W are included (footnote 1).\n- **Theorem 3.** For d = 2 and real µ1 ≥ µ2 ≥ µ3, the inequalities (a) µ2 ≥ 0 and (b) µ3 ≥ −√(µ1µ2) are sufficient for the existence of a symmetric witness with these eigenvalues, and necessary for decomposable ones. Every symmetric witness satisfies (a) and a weaker inequality (c).\n- **The conjecture.** After Theorem 3 and in §6(c) (page 20): \"We conjecture that inequalities (a) and (b) from that theorem provide the correct characterization.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/champagnejohnstonmacdonaldpipes2022spectral.md).\n\nIssue [#17536](https://github.com/the-omega-institute/trureturing/issues/17536) fixes the reading:\n- C²∨C² is identified with C³ through the orthonormal Dicke basis e₀₀, (e₀₁ + e₁₀)/√2, e₁₁, in which v ⊗ v has coordinates (v₀², √2 v₀v₁, v₁²).\n- \"Eigenvalues µ1 ≥ µ2 ≥ µ3\" means that the characteristic polynomial is (X − µ1)(X − µ2)(X − µ3).\n- The conjecture: for all real µ1 ≥ µ2 ≥ µ3, a symmetric witness with these eigenvalues exists if and only if µ2 ≥ 0 and µ3 ≥ −√(µ1µ2).\n\nThe conjecture is proved in this reading.","title":"The eigenvalues of symmetric two-qubit entanglement witnesses satisfy µ3 ≥ −√(µ1µ2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/champagne-johnston-macdonald-pipes-2022-symmetric-witness-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/000830d24518e1854e436254318885601cdfdda2fd49209a0207326209920304.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chan-lopez-martin-ruiz-2026-rule30-sign-pattern","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Chan-López and A. Martín-Ruiz, *Symmetric Nonlinear Cellular Automata as Algebraic References for Rule 30*, arXiv:2604.00165v3, Remark 3, states:\n\n> A direct computation for m ≤ 256 shows that ϵ(m) ≤ 0 precisely at the Mersenne indices m = 2^k − 1, with ϵ = 0 for k ≤ 3 and ϵ < 0 for 4 ≤ k ≤ 8.\n\nSection 9 asks:\n\n> Is the sign pattern of Remark 3 exact for all k?\n\nThe quantified sign-pattern interpretation is\n\n```text\nclaim := ∀ m : ℕ, 1 ≤ m → (eps m ≤ 0 ↔ ∃ k : ℕ, 1 ≤ k ∧ m = 2^k − 1).\n```\n\nHere `row g 0 r = decide (r = 0)` and `row g (m+1) r = g (row g m (r−1)) (row g m r) (row g m (r+1))` on the full integer lattice. The Boolean algebraic normal forms are `g30 a b c = a xor b xor c xor (b and c)` (Proposition 1) and `g22 a b c = a xor b xor c xor (a and b and c)` (Eq. (1)). Definition 2 counts the full-row set `{r : ℤ | row g m r = true}` with `Set.ncard`. Equation (11) defines `eps m` as the integer difference of these cardinalities. This interprets the word “precisely” as a biconditional for every positive row; it does not identify that biconditional with the separate strict-negativity assertion at every Mersenne row.","title":"A non-Mersenne exception to the Rule 30 / Rule 22 sign pattern","url":"https://the-omega-institute.github.io/trureturing-pages/research/chan-lopez-martin-ruiz-2026-rule30-sign-pattern/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8525cc4fd6e51c5d60ee1aa9885e17d01a129e4abdbb0ed6ea86d890d16a6e50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:charlier-harju-puzynina-zamboni-question-two","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Émilie Charlier, Tero Harju, Svetlana Puzynina and Luca Q. Zamboni, *Abelian bordered factors and periodicity*,\narXiv:1501.07464v1, Section 7, Question 2: let w be an infinite bounded weakly abelian periodic word over a k-letter\nalphabet such that its graph G_w belongs to a k-dimensional cylinder with axis with rational coefficients, and each\ntangential line to G_w has points of G_w on it with bounded gaps. Does it follow that w has only finitely many weakly\nabelian unbordered factors?","title":"Weak Abelian Periodicity Does Not Force Finitely Many Weakly Abelian Unbordered Factors","url":"https://the-omega-institute.github.io/trureturing-pages/research/charlier-harju-puzynina-zamboni-question-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/521b94db4c88239a7b81a0979dc1fff32c7e4a367700105ba9092f3836f394a4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chauve-zhang-condensed-neighborhood-unary-minimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chauve and Zhang, *On the size of the neighborhoods of a word*,\narXiv:2505.13796v2, Section 6, printed page 7, ask:\n\n> It was also shown in [6] that unary words have the smallest neighborhoods\n> among all words of the same length over a given alphabet, thus leading to\n> lower bounds for the size of neighborhoods. It is thus natural to ask if a\n> similar property holds for condensed and super condensed neighborhoods,\n> namely that unary words have the smallest condensed or super condensed\n> neighborhoods.\n\nThis dossier tracks the condensed assertion. Equations (2.1)--(2.2) define\n`N(w,d)` as the words at Levenshtein distance at most `d` and `CN(w,d)` as the\nmembers of `N(w,d)` with no proper prefix in `N(w,d)`. The assertion asks\nwhether, for every finite alphabet, length `n`, radius `d`, letter `a`, and\nword `w` of length `n`,\n\n`|CN(a^n,d)| <= |CN(w,d)|`.","title":"Chauve--Zhang condensed-neighborhood unary minimality","url":"https://the-omega-institute.github.io/trureturing-pages/research/chauve-zhang-condensed-neighborhood-unary-minimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3745e9adff060f083ac483d418ba80109b2916f458e1f5ef7c79e33999d0d72.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chauve-zhang-super-condensed-neighborhood-unary-minimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chauve and Zhang, *On the size of the neighborhoods of a word*,\narXiv:2505.13796v2, Section 6, printed page 7, ask:\n\n> It was also shown in [6] that unary words have the smallest neighborhoods\n> among all words of the same length over a given alphabet, thus leading to\n> lower bounds for the size of neighborhoods. It is thus natural to ask if a\n> similar property holds for condensed and super condensed neighborhoods,\n> namely that unary words have the smallest condensed or super condensed\n> neighborhoods.\n\nThis dossier tracks the super-condensed assertion. Equations (2.1) and (2.3)\ndefine `N(w,d)` as the words at Levenshtein distance at most `d` and\n`SCN(w,d)` as the members of `N(w,d)` with no proper contiguous subword in\n`N(w,d)`. The assertion asks whether, for every finite alphabet, length `n`,\nradius `d`, letter `a`, and word `w` of length `n`,\n\n`|SCN(a^n,d)| <= |SCN(w,d)|`.","title":"Chauve--Zhang super-condensed-neighborhood unary minimality","url":"https://the-omega-institute.github.io/trureturing-pages/research/chauve-zhang-super-condensed-neighborhood-unary-minimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4fca2019169955685d5dec36e1c6f8cf26ae466243c5774071d1c11f7d6fa75.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chavez-2023-multipartite-chs-norm-extrema","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chávez et al., *Extremal polynomial norms of graphs*, arXiv:2311.04689v3, Section 8, Problem 3: “Can one find the extrema of the CHS norms over other families of graphs such as regular, unicyclic and complete multipartite graphs?”\n\nFor a complete $r$-partite graph with positive part sizes $a_1,\\ldots,a_r$ and total order $n$, the cited article defines, for even $d=2m$, the CHS norm by\n\n$$\\|G\\|_{2m}=h_{2m}(\\lambda_1,\\ldots,\\lambda_n)^{1/(2m)},$$\n\nwhere the $\\lambda_i$ are the adjacency eigenvalues. This dossier settles the complete multipartite branch only, for each fixed number of parts. The regular and unicyclic branches remain open.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Complete multipartite graph | `Vtx a` and `adj a` | Literal vertices and adjacency matrix for the part sizes. |\n| Complete homogeneous symmetric polynomial | `chs` | Evaluation of `MvPolynomial.hsymm` at the real adjacency eigenvalues. |\n| Even-degree norm | `chsNorm` | Real power with exponent `1 / d`, specialized to `d = 2 * m`. |\n| Minimum part multiset | `minParts r n` | One part of size `n-r+1` and `r-1` singleton parts. |\n| Maximum part multiset | `balancedParts r n` | Parts differ by at most one, with the prescribed quotient and remainder. |\n| Unique extrema | `claim` and `Result.result` | Both inequalities and both equality classifications up to permutation. |","title":"Complete homogeneous symmetric norms of complete multipartite graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/chavez-2023-multipartite-chs-norm-extrema/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9684de1c2f1d63b7c3ddb2c1aeee4ddb09384d442f296a91348ca7d5c7d73153.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chavez-2023-path-chs-norm-closed-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chávez et al., *Extremal polynomial norms of graphs*, arXiv:2311.04689v3, Section 5, Problem 1: “Compute ‖P_n‖_d for d ≥ 4.” The surrounding norm definition requires even $d$.\n\nFor $d=2m$, set $L=2m+n-1$ and interpret $\\binom Lr$ as zero outside $0\\le r\\le L$. The answer is\n\n$$h_{2m}(\\lambda(P_n))=\\sum_{k=0}^{\\lfloor m/(n+1)\\rfloor}\\left[\\binom L{m-k(n+1)}-2\\binom L{m-k(n+1)-1}+\\binom L{m-k(n+1)-2}\\right].$$\n\nThe norm is the real $2m$-th root of this expression, for every $n\\ge1,m\\ge2$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Path adjacency | Literal matrix with entry one exactly at consecutive natural indices | `pathAdjHermitian` identifies a Hermitian path matrix. |\n| Complete homogeneous symmetric polynomial | `chs`, using Mathlib `hsymm` at Hermitian eigenvalues | Exact source object; eigenvalue ordering is immaterial. |\n| Norm from Theorem 1.4 | `chsNorm`, real power with reciprocal degree | Exact real-root convention. |\n| Every even degree at least four | `claim`: all natural $n,m$, $1\\le n$, $2\\le m$ | $d=2m$. |\n| Compute the value | Both equalities in `claim`, proved by `result` | Gives the power and the norm; no source clause is dropped. |\n| Zero-extended binomial | Frozen `IntegerIndexBinomial.binom` | Its value is zero outside the natural binomial range. |","title":"Complete homogeneous symmetric norms of paths","url":"https://the-omega-institute.github.io/trureturing-pages/research/chavez-2023-path-chs-norm-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5bc8045394ac2985230b3d9389d793e72b0b8c151b148c074d4e2c3705e16d65.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chavez-2023-regular-chs-norm-extrema","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chávez et al., *Extremal polynomial norms of graphs*, arXiv:2311.04689v3, Section 8, Problem 3: “Can one find the extrema of the CHS norms over other families of graphs such as regular, unicyclic and complete multipartite graphs?”\n\nFor even degree $d=2m\\ge2$, Theorem 1.4 defines\n\n$$\\|G\\|_{2m}=h_{2m}(\\lambda_1,\\ldots,\\lambda_n)^{1/(2m)},$$\n\nwhere $\\lambda_1,\\ldots,\\lambda_n$ are the real adjacency eigenvalues. Among connected regular simple graphs of order $n\\ge3$, the cycle $C_n$ attains the minimum and the complete graph $K_n$ attains the maximum:\n\n$$\\|C_n\\|_{2m}\\le\\|G\\|_{2m}\\le\\|K_n\\|_{2m}.$$\n\nHere connected means that every two vertices are joined by a walk. Section 2 of the source uses the word connected for graphs without isolated vertices; under that reading the minimum changes for even orders, since a perfect matching is regular without isolated vertices and has smaller coefficients, for example $h_2(2K_2)=2<4=h_2(C_4)$. The formal statement uses ordinary connectedness. In fact the lower coefficient bound $h_{2m}(C_n)\\le h_{2m}(G)$ holds for every $r$-regular simple graph of order $n\\ge3$ with $r\\ge2$, without connectedness. This dossier concerns the regular family. The unicyclic branch is not addressed; the complete multipartite branch is settled separately.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Simple graph of order n | `SimpleGraph (Fin n)` | Literal finite simple graph. |\n| Adjacency eigenvalues and complete homogeneous polynomial | `Defs.chs` | `MvPolynomial.hsymm` evaluated at the real Hermitian adjacency eigenvalues. |\n| Positive even-degree norm | `Defs.chsNorm`, `2 * m`, `1 ≤ m` | Real power with exponent one divided by the degree. |\n| Connected regular family | `G.Connected` and `∃ r, G.IsRegularOfDegree r` | Usual graph connectedness; no isolated-vertex convention is substituted. |\n| Every n at least three | `3 ≤ n` in `Defs.claim` | Excludes the smaller orders. |\n| Minimum and maximum with attainment | `Result.result : Defs.claim` | Both inequalities and connected regular endpoint graphs. |\n| Lower bound without connectedness | Private `Result.chs_cycle_le_regular` | Every regular degree at least two, at every even coefficient degree. |","title":"Complete homogeneous symmetric norms of regular graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/chavez-2023-regular-chs-norm-extrema/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a27f92dc0548cce7c694cd7e6f1c5779835257810c06a1da84fcee862ba9f613.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-2017-exclusion-moving-particle-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. P. Chen, *The moving particle lemma for the exclusion process on a weighted graph*, Electron. Commun. Probab. 22 (2017), paper 47, arXiv:1606.01577v4, §1, `OctopusMoving_v3.tex`, lines 177–179, states:\n\n> However, it is unclear in general when \\eqref{Reffinf} becomes an equality.\n\n> That being said, we conjecture that Theorem \\ref{thm:dirichletex} is optimal on any finite weighted graph, and more generally, on resistance spaces \\cite{KigamiRF}, in light of the connection between the moving particle lemma and the spectral gap of an interacting particle system \\cite{Morris, Jara, Quastel}.\n\nThese are verbatim TeX excerpts; the labels refer to equation (1.9) and Theorem 1. The definitions and source locators are also recorded in [the literature note](../Library/StatisticalMechanics/chen2017movingparticle.md).\n\nFor a finite vertex set $V$, let $c:V\\times V\\to\\mathbb R$ be nonnegative and symmetric, with zero weights representing absent edges. A configuration is $\\zeta\\in\\{0,1\\}^V$, $\\zeta^{xy}$ exchanges its occupations at $x,y$, and $\\nabla_{xy}f(\\zeta)=f(\\zeta^{xy})-f(\\zeta)$. Write $\\nu_\\alpha$ for the product Bernoulli measure of constant density $\\alpha$, and put\n\n$\\mathcal E^{\\rm el}(h)=\\tfrac12\\sum_{u\\in V}\\sum_{w\\in V}c_{uw}(h(u)-h(w))^2$,\n\n$\\mathcal E^{\\rm EX}_{\\nu_\\alpha}(f)=\\tfrac14\\sum_{u\\in V}\\sum_{w\\in V}c_{uw}\\nu_\\alpha[(\\nabla_{uw}f)^2]$,\n\n$C(x,y)=\\inf\\{\\mathcal E^{\\rm el}(h):h:V\\to\\mathbb R,\\ h(x)=1,\\ h(y)=0\\}$.\n\nThe factors remove ordered-pair double counting and retain the source's additional Dirichlet-energy factor $\\tfrac12$ for exclusion. For positive $C(x,y)$, the source's effective resistance is $R_{\\rm eff}(x,y)=1/C(x,y)$.\n\nIssue [#24189](https://github.com/the-omega-institute/trureturing/issues/24189) fixes the finite-graph reading: for every such $V,c$, every distinct $x,y\\in V$, every $0<\\alpha<1$ and every real $k$, if\n\n$\\tfrac12\\nu_\\alpha[(\\nabla_{xy}f)^2]\\le k\\mathcal E^{\\rm EX}_{\\nu_\\alpha}(f)$ for every $f:\\{0,1\\}^V\\to\\mathbb R$,\n\nthen $1\\le kC(x,y)$. This is the proposition `claim` proved by `result`.","title":"Effective resistance is the optimal finite-graph moving particle constant","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-2017-exclusion-moving-particle-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aecdb0b1aae345ac22fdcc590129bbace40a6c38673cce6b89ceaeba32524c91.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-chen-step-m-eulerian-parity-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For each m=-3,-4,-5, let T map pairs of natural numbers to integers, with\nT(n,1)=T(n,n)=1 for n>=1 and\n\n`T(n,k)=(m*(n-k)+1)*T(n-1,k-1)+(m*(k-1)+1)*T(n-1,k)`\n\nfor 1=0 for every positive even n and every 1<=k<=n. The exact Lean\n`claim` conjoins, for each parameter, existence of a solution and this\nassertion for every solution. Row zero and entries outside the triangle\nare unconstrained, and no uniqueness of the total extension is assumed.","title":"Chen and Chen's step-m Eulerian parity positivity","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chen-step-m-eulerian-parity-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/761baa7ec364e39ae40ca98f24fb6d82d4f140511cc14a878fd74c489f034369.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-chu-qian-shen-2019-multi-qutrit-completely-symmetric-separability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Chen, D. Chu, L. Qian and Y. Shen, *Separability of completely symmetric states in a multipartite system*, Phys. Rev. A 99, 032312 (2019), arXiv:1812.04182v4, Conclusion, item 2 (v4 TeX, line 1857):\n\n> Are multi-qutrit CS states separable?\n\nThe source's Definition 1 (TeX lines 360–391) requires invariance of matrix entries under every permutation of all $2d$ ket and bra index positions, including permutations mixing ket and bra. Its multipartite convention is explicit (lines 449–452):\n\n> For simplicity, a separable state will indicate the fully separable one in the multipartite case.\n\nIssue [#23788](https://github.com/the-omega-institute/trureturing/issues/23788) fixes the assertion as follows. For every natural $d\\ge3$, every positive semidefinite complex matrix $\\rho$ of trace one on $(\\mathbb C^3)^{\\otimes d}$ that is completely symmetric is fully separable. Here full separability means a finite convex combination, with strictly positive real weights summing to one, of projectors onto products of unit local complex vectors. This assertion is `claim`; it is refuted.\n\nThe source definitions and locator are also recorded in [the literature note](../Library/Entropy/chenchuqianshen2019cs.md).","title":"Completely symmetric three-qutrit states need not be fully separable","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chu-qian-shen-2019-multi-qutrit-completely-symmetric-separability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b91ea64eba5793a1594ef389facb165df723c91c5a863cd6d297f8e67a4e5d01.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-dey-2020-orthostochastic-naive-equations-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Chen and P. Dey, *The 4 × 4 orthostochastic variety*, arXiv:2001.10691v1, Experimental Mathematics 32(4) (2023), 692–700, DOI 10.1080/10586458.2021.1982427, Question 5 (cf. Chterental–Đoković, Remark 3.4), asks:\n\n> Do the naive equations define the orthostochastic variety inside B_n? In other words, if a doubly stochastic matrix satisfies C_{i,j}, R_{i,j} = 0 for all i < j, must it be orthostochastic?\n\nSection 5 states:\n\n> Next, for n = 4, 5 it remains open whether the naive equations cut out the orthostochastic variety inside the Birkhoff polytope set-theoretically.\n\nThe order-five clause is **Refuted** by $B=(4J_5-3I_5)/17$. [Preregistration #17606](https://github.com/the-omega-institute/trureturing/issues/17606) fixes the source, quantifiers and conventions. The [Library note](../Library/QuantumBounds/chendey2023orthostochastic.md) contains the source locators and quotations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A real orthostochastic matrix is the entrywise square of an orthogonal matrix | `IsOrthostochastic` quantifies over every real orthogonal matrix and uses `RHLinalg.normSqMatrix` | For real entries, the squared norm is the square |\n| The equivalent sign condition after Definition 4 for $C_{i,j}=0$ | `ColumnNaive` quantifies over all real signs equal to $1$ or $-1$ and requires the signed radical sum to vanish | A sign on every term is equivalent to fixing the first sign to $1$, by reversing all signs together |\n| Row equations are obtained “by transposing indices” | `RowNaive` exchanges the row and column positions in that sum | No symmetry assumption on the quantified matrix |\n| Every pair $i State $e_i$ is the characteristic vector of $i$. It is easy to see that there is perfect state transfer from state $e_1$ to $e_6$ at step $k=1$. But up to $k=300000$ steps, there is no perfect state transfer observed from $e_6$ to $e_1$. We suspect that there is no perfect state transfer from $e_6$ to $e_1$. We would like to find a condition on graph $G$ that determines whether or not perfect state transfer is symmetric.\n\nThe checked v1 TeX has md5 `2e3f9a1a91cb695794df49b0123dfc37`. Its evolution convention is $U^kz$ (lines 135–142), and its bipartite-walk definition is $U=(2P-I)(2Q-I)$ (lines 154–163). The projections for the seven-edge example are printed at lines 320–338, and its displayed transition matrix at lines 340–351 and 1278–1286. These definitions are documented in [the literature note](../Library/QuantumContext/chenetal2024bipartitewalkhamiltonians.md).\n\nUse the source's one-based coordinate labels throughout this dossier. Let $U_{\\rm def}=(2P-I)(2Q-I)$ for those explicit projections and let $U_{\\rm disp}$ be the printed matrix. Exact rational comparison gives one discrepancy: their entries at row 3, column 5 are respectively $-1/3$ and $+1/3$; all other entries agree.\n\nIssue [#21532](https://github.com/the-omega-institute/trureturing/issues/21532) fixes a reading covering both matrices. In Lean, `PerfectStateTransfer U a b` means that there exist $k\\in\\mathbb N$ and $\\gamma\\in\\mathbb C$ with $\\|\\gamma\\|=1$ and $U^ke_a=\\gamma e_b$. The predicate imposes no unitarity assumption and includes step zero. The declaration `claim` says that neither $U_{\\rm def}$ nor $U_{\\rm disp}$ has such transfer from $e_6$ to $e_1$. This claim is proved by `result`.","title":"No reverse perfect state transfer in the seven-state bipartite walk","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-godsil-sobchuk-zhan-2024-bipartite-walk-reverse-state-transfer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cfaca4d94856ee813b16804718709e8e2e2b9dee3844d6dcb278617bc9c70eb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-he-lai-li-2026-local-average-dominating-stem","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Tingyun Chen, Weihua He, Hong-Jian Lai and Jianping Li, “On the local average\norder of dominating sets”, arXiv:2610.11446v1, Conjecture 6.1:\n\n> Let G be a graph of order n ≥ 2 without isolated vertices, and let v be an\n> l-stem of G, where l ≥ 1. Then avd_v(G) ≤ (4n + 1)/6, with equality if and\n> only if l = 1 and G is star-like.\n\nA dominating set contains or neighbours every vertex. The local average is\nthe mean cardinality over the dominating sets containing v. A leaf has\ndegree one; an l-stem has exactly l leaf neighbours. Literally following\nIain Beaton and Ben Cameron, “A Tight Upper Bound on the Average Order of\nDominating Sets of a Graph”, arXiv:2208.10475v2, a graph is star-like when\nevery vertex is a leaf or a stem with one or two leaf neighbours.\n\nThe closed Lean `claim` quantifies over `V : Type`, its finite and decidable\nequality structures, a finite simple graph with decidable adjacency, v and\nl. It retains the order, positive-degree and leaf-count hypotheses and both\nthe upper bound and the equivalence characterizing equality. All averages\nare exact rational numbers.","title":"Chen–He–Lai–Li Conjecture 6.1: the local average at a stem","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-he-lai-li-2026-local-average-dominating-stem/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1865c1a9922f6503e4c36a8dd5fd1bc35b827826d96b195d6b0bacdb2dbc050a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-kato-brandao-2020-completely-contractive-cmi","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão,\n*Matrix Product Density Operators: when do they have a local parent Hamiltonian?*,\narXiv:2010.14682v3, Section III.C, Conjecture III.1:\n\n> For any channel $\\mathcal E:C\\rightarrow C'$ with trivial correctable algebra\n> $\\mathcal A(\\mathcal E)=\\mathbb C I$, there exists a constant $\\eta<1$ such that\n> for any tripartite system $ABC$ and any state $\\rho_{ABC}$, it holds that\n> $I(A:C'|B)_{\\mathcal E(\\rho)}\\le\\eta I(A:C|B)_\\rho$.\n\nEquation (22) defines the whole-space algebra as\n$\\mathcal A(\\mathcal E)=\\mathrm{Alg}\\{O:[O,K_i^\\dagger K_j]=0\\ \\forall i,j\\}$.\nEquation (1) defines\n$I(A:C|B)=S(AB)+S(BC)-S(B)-S(ABC)$, with von Neumann entropy in bits.\nThe Lean expression uses the same eigenvalue entropy in nats; all terms are\nmultiplied by the common positive factor $\\log 2$. Singular density matrices\nare part of the source's domain.","title":"Chen–Kato–Brandão completely contractive CMI data processing","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-kato-brandao-2020-completely-contractive-cmi/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/668659145059ba33eb0528fcced6d649224534eabdecfaee00f31ae38b8e5bfd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-kato-brandao-2020-trace-norm-cmi-contraction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C.-F. Chen, K. Kato and F. G. S. L. Brandão, *Matrix Product Density Operators: when do they have\na local parent Hamiltonian?*, arXiv:2010.14682v3, Section III.C (PDF p. 17), Conjecture III.2:\n\n> For any channel $\\mathcal E: C \\rightarrow C'$ with local contraction ratio\n> $\\eta_{1,C}:=\\sup_{\\rho_C,\\rho'_C}\\frac{\\lVert\\mathcal E_C[\\rho_{C}]-\\mathcal E_C[\\rho'_{C}]\\rVert_1 }{\\lVert\\rho_{C}-\\rho'_{C}\\rVert_1} <1$\n> There exists a global constant $\\eta < 1$ such that for any tripartite system $ABC$ and any\n> state $\\rho_{ABC}$, it holds that $I_1(A:C'|B)_{\\mathcal E(\\rho)} \\le \\eta I_1(A:C|B)_\\rho$.\n\nHere $I_1(A:C|B)=\\lVert\\rho_{ABC}-\\rho_A\\otimes\\rho_{BC}\\rVert_1-\\lVert\\rho_{AB}-\\rho_A\\otimes\\rho_B\\rVert_1$\n(Definition 2, PDF p. 14). The verbatim texts are in\n[the literature note](../Library/QuantumChannels/chen2020mpdoparent.md).\nIssue [#14728](https://github.com/the-omega-institute/trureturing/issues/14728) preregisters the\nreading over finite-dimensional Kraus channels and density matrices, and the refutation.","title":"A completely contractive data-processing inequality for the trace-norm CMI","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-kato-brandao-2020-trace-norm-cmi-contraction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ee40d49e31446a97e52bfadbbb6421a6d8a23710b22eaca7e9df6c711cab549.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chen-klco-2026-hildebrand-negative-index-persistence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Chen and N. Klco, *Susceptibility of qudit-qudit entanglement to quantum noise: insights from the negativity rank*, arXiv:2610.10820v1, derive the maximum purity of a qutrit-qutrit APPT state from Hildebrand's spectral criterion.\n- **The criterion.** For the ordered eigenvalues λ1 ≥ … ≥ λ9 ≥ 0, ρ is APPT iff the two matrices L1, L2 of Eq. (5) are positive semidefinite (Theorem II.1, citing Hildebrand).\n- **The rank bound.** \"If λ⃗ is an ordered vector of qutrit-qutrit density matrix eigenvalues, then L1 and L2 of Eq. (5) have at most one negative eigenvalue\" (Theorem II.2, p. 5).\n- **The question.** \"it may be prudent to explore whether components of the proof, e.g., the limited rank available for inequality violation established in Theorem II.2, persists for the increasing number of corresponding inequalities in qudit-qudit systems with d > 3\" (§V, p. 16).\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/chenklco2026negativityrank.md). The corresponding inequalities are Hildebrand's ([note](../Library/QuantumStates/hildebrand2007pptspectra.md)): for a pair of orderings (σ₊, σ₋) in Σ±, Λ(σ₊, σ₋) + Λ(σ₊, σ₋)ᵀ ⪰ 0.\n\nIssue [#17687](https://github.com/the-omega-institute/trureturing/issues/17687) fixes the reading:\n- S₊ = {(k, l) : k ≤ l} and S₋ = {(k, l) : k < l}, with orderings as bijections onto ranks.\n- Σ± is the set of pairs compatible, in the sense of Hildebrand's Definition 1, with some x₁ > … > x_d > 0.\n- L = Λ + Λᵀ has diagonal entries 2λ_{d²+1−σ₊(k,k)} and off-diagonal entries λ_{d²+1−σ₊(k,l)} − λ_{σ₋(k,l)}. For d = 3 the two pairs of Σ± give L1 and L2.\n- The property: for every d > 3, every ordered spectrum (λ_k ≥ 0, Σλ_k = 1, no APPT condition, as in Theorem II.2) and every pair in Σ±, L has at most one negative eigenvalue counted with multiplicity.\n\nThe property is refuted in this reading.","title":"The one-negative-eigenvalue bound for Hildebrand's APPT matrices fails in dimension four","url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-klco-2026-hildebrand-negative-index-persistence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c717ea1032730852ec7c13f42f4d365d067158d698cebd80702811d394431d0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cheng-milne-hall-wiseman-2016-four-qubit-correlation-monogamy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Cheng, A. Milne, M. J. W. Hall and H. M. Wiseman, *Volume monogamy of quantum steering ellipsoids for multi-qubit systems*, Phys. Rev. A 94, 042105 (2016), arXiv:1606.02390v2, Sec. III C, p. 6, Eq. (36) (`finalversion.tex`, lines 395–399):\n\n> Finally, we conjecture that inequality (35) generalises to\n>\n> $\\operatorname{Tr}[T_{AB}T_{AB}^{\\top}]+\\operatorname{Tr}[T_{AC}T_{AC}^{\\top}]+\\operatorname{Tr}[T_{AD}T_{AD}^{\\top}]\\leq 3.$\n>\n> for all pure 4-qubit states.\n\nThe source defines $T_{jk}=\\operatorname{Tr}[\\rho_{AB}\\sigma_j\\otimes\\sigma_k]$ for $j,k=1,2,3$ in Sec. II, Eq. (2) (`finalversion.tex`, lines 129–133). Thus each term in Eq. (36) is the sum of the nine squared real entries of a two-qubit correlation matrix.\n\nIssue [#26136](https://github.com/the-omega-institute/trureturing/issues/26136) fixes the reading. In the module, a vector $\\psi:(\\mathrm{Fin}\\,4\\to\\mathrm{Fin}\\,2)\\to\\mathbb C$ has its sixteen computational-basis amplitudes indexed by four bits. The qubits are ordered $A,B,C,D$. The indices $X=0,1,2$ denote targets $B,C,D$, and $i,j=0,1,2$ denote Pauli directions $x,y,z$. Let $P_{Xij}$ be `wordOp (twoBodyWord X i j)`: Pauli $i$ on $A$, Pauli $j$ on target $X$, and identities elsewhere. The definition `rankOneDensity` is the outer product $\\psi\\psi^{\\dagger}$, and `correlation` is\n\n$$c_{Xij}(\\psi)=\\operatorname{Re}\\operatorname{Tr}[\\psi\\psi^{\\dagger}P_{Xij}].$$\n\nThe proposition `claim` states, exactly,\n\n$$\\forall\\psi,\\quad \\sum_q\\overline{\\psi(q)}\\psi(q)=1\\quad\\Longrightarrow\\quad\n\\sum_{X=0}^{2}\\sum_{i=0}^{2}\\sum_{j=0}^{2}c_{Xij}(\\psi)^2\\leq3.$$\n\nThe module proves `result : claim`. The correspondence with reduced-state correlation matrices is explained below.","title":"Four-qubit pair-correlation monogamy","url":"https://the-omega-institute.github.io/trureturing-pages/research/cheng-milne-hall-wiseman-2016-four-qubit-correlation-monogamy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3fa1eac38c4dfad93c1f1759f03720ca8cec96392c4898011f44b2b0d09e365f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chenkitaevzhang-2024-separable-record-peak-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chen, Kitaev and Zhang, *Distributions of statistics on separable\npermutations*, arXiv:2404.18517v1, Section 3, Conjecture 2, page 17, asks:\n\n> The distribution of rmax or lmin (resp., lmax or rmin) on irreducible\n> (resp., reducible) separable permutations is unimodal with the peak\n> being at k=3 for n>=5 (see Table 4).\n\nThe journal reference is *Discrete Applied Mathematics* 355 (2024),\n169–179. The carrier is the actual classical $2413/3142$-avoiding\npermutations, rather than a recursive replacement class. A proper direct\ncut separates a nonempty prefix whose values are all below those of the\nnonempty suffix. Irreducibility means positive length and no proper direct\ncut; reducibility means a proper direct cut. The empty permutation belongs\nto neither positive class. The singleton is irreducible and has one record\nof each type; the reducible singleton class is empty.\n\nThe four strict record statistics are right maxima, left minima, left\nmaxima and right minima. A maximum exceeds every value on its indicated\nside, and a minimum is below every value on its indicated side. These\ncounts are unshifted. For any of the pairs\n$(\\mathrm{irreducible},\\mathrm{rmax})$,\n$(\\mathrm{irreducible},\\mathrm{lmin})$,\n$(\\mathrm{reducible},\\mathrm{lmax})$ and\n$(\\mathrm{reducible},\\mathrm{rmin})$, let $a(n,k)$ be the corresponding\nactual record-fiber cardinality. The complete target is, for every\n$n\\ge5$ and every natural $k$,\n\n$$\n\\begin{aligned}\nk<3&\\implies a(n,k)\\le a(n,k+1),\\\\\nk\\ge3&\\implies a(n,k+1)\\le a(n,k),\\\\\na(n,k)&\\le a(n,3).\n\\end{aligned}\n$$\n\nA maximum at three is interpreted weakly. Strict inequalities on zero\ntails and uniqueness are not required. [Issue 12664](https://github.com/the-omega-institute/trureturing/issues/12664)\nregisters this full target. A partial rising or transport result does not\nsettle the conjecture.","title":"CKZ Conjecture 2: the four actual record distributions","url":"https://the-omega-institute.github.io/trureturing-pages/research/chenkitaevzhang-2024-separable-record-peak-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4993d79c6d68af80a6fa302eb22acc623318e08460da018d5611de184e106fe9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chern-fu-improper-partition-matrix-fixed-image","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Shane Chern and Shishuo Fu, *Signed counting of partition matrices*,\narXiv:2508.21318v2, Section 5, Question 5.5, page 26:\n\n> Find a characterization (possibly in terms of pattern avoidance) of the inversion sequences produced by improper partition matrices under the action of the Claesson–Dukes–Kubitzke bijection.\n\nFor each integer $n\\geq1$, write $[n]=\\{1,\\ldots,n\\}$ and let\n$\\mathbf I_n=\\{e:[n]\\to\\mathbb N:0\\leq e_j\\leq j-1\\text{ for all }j\\in[n]\\}$.\nA native partition matrix on $[n]$ is an upper triangular square matrix $P$ of\nsets, of any permitted positive dimension $k$, whose nonempty cells partition\n$[n]$, every row and column is nonempty, and\n$\\operatorname{col}_P(a)\\leq\\operatorname{col}_P(b)$ whenever $a Conjecture 11. Case k = 4 from n = 6:\n>\n> $n \\equiv 0,1 \\pmod{3}: D_4(n)=\\frac{n(n-1)}{6}-1,$\n>\n> $n \\equiv 2 \\pmod{3}: D_4(n)=\\frac{n(n-1)}{6}+\\frac{2}{3}.$\n\nSection 8.1 specifies the cycles `(i,i+1,i+2,i+3)`, for\n`i = 0,...,n-4`, with no wrapping. Section 8.5 specifies Cayley graphs,\nnot Schreier coset graphs, and inverse-closed generators. The vertices are\nall permutations of `{0,...,n-1}` and distance is inverse-closed word\ndistance. This dossier concerns the complete clause for every `n >= 6`.","title":"CayleyPy-4 Conjecture 11: consecutive four-cycle diameter","url":"https://the-omega-institute.github.io/trureturing-pages/research/chervov-2026-cayleypy4-conjecture-eleven-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/095d18f939936cc9eb49c382a470f3ea85373da7c98a0b04f507cd5ec2a68213.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chervov-2026-cayleypy4-conjecture-fifteen-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"CayleyPy-4 (arXiv:2603.22195v1) studies the Schreier coset graph\n`S_N / (S_L × S_{N−L})` on binary words with `L` zeros and `N − L` ones,\ngenerated by the consecutive `k`-cycles `(i, …, i + k − 1)`. Conjecture 15\ntreats generators without their inverses and gives the diameters as\nquasipolynomials in `n` and `t = n − (l + 1)`; its `k = 5` clause reads:\n\n> For $k=5$ with period~$4$ (again $t = n - (l+1)$, for all $L \\ge 4$, starting from $t+1 \\ge 9$ that is, $N \\ge L + 9$): $D(L,N) = L \\left\\lfloor \\frac{t+1}{4} \\right\\rfloor + \\mathbf{1}_{(t+1) \\equiv 0 \\; (\\mathrm{mod}\\ 4)}$.\n\nIssue #10253 fixes the readings: vertices and the central state as in\n#10230; a directed move rotates a window of five consecutive letters one\nplace, to the left in one convention for the action of a cycle on a word and\nto the right in the other; the diameter is read both as the largest directed\ndistance from the central state and as the largest directed distance between\ntwo vertices. The formal claim is that the clause holds for some convention\nand some reading.","title":"CayleyPy-4 Conjecture 15 fails at k = 5, L = 4, N = 13","url":"https://the-omega-institute.github.io/trureturing-pages/research/chervov-2026-cayleypy4-conjecture-fifteen-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da9bf1390618d58d5be734c3c2a19295b255861c495cd5bec2b842292410124b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chervov-2026-cayleypy4-conjecture-sixteen-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"CayleyPy-4 (arXiv:2603.22195v1) studies the Schreier coset graph\n`S_N / (S_L × S_{N−L})` on binary words with `L` zeros and `N − L` ones,\ngenerated by the consecutive `k`-cycles `(i, …, i + k − 1)` for `N > k`.\nConjecture 16 treats the inverse-closed case, in which a move rotates `k`\nconsecutive letters one place in either direction:\n\n> For $k=3$ from $L\\ge 2$(?): $D(L, N) = \\left\\lceil \\frac{L (N - L)}{2} \\right\\rceil$\n> For $k=4$ from $L\\ge 2$: $D(L, N) = \\frac{L}{3}(N - L)$ if $3 \\mid L$, $\\left\\lfloor \\frac{L(N - L) + 2}{3} \\right\\rfloor$ if $3 \\nmid L$.\n> For $k=5$ for $N,L,N-L$ large enough: $D(L, N) = \\left\\lfloor \\frac{LN + 2}{4} \\right\\rfloor - 2\\left\\lfloor \\frac{L^2}{8} \\right\\rfloor - \\mathbf{1}_{L \\equiv 2 \\pmod{4}} \\, \\mathbf{1}_{N \\equiv 2 \\pmod{4}}$\n\nIssue #10252 fixes the readings as in #10230: words are `List Bool`, and the\ndiameter is read both as the largest distance from the central state\n`[0]^L + [1]^{N−L}` and as the largest distance between two vertices; the\n`k = 5` range reads \"for some constant `C`, all `L ≥ C` and `N − L ≥ C`\".","title":"CayleyPy-4 Conjecture 16 fails at k = 4, L = 2, N = 5","url":"https://the-omega-institute.github.io/trureturing-pages/research/chervov-2026-cayleypy4-conjecture-sixteen-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/077614c67b72d9c645b853588e4e34c493323c7cc0309e5b8537e37f8febbfd9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chervov-2026-cayleypy4-three-cycle-grassmannian-diameter","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"CayleyPy-4 (arXiv:2603.22195v1) studies the Schreier coset graph\n`S_N / (S_L × S_{N−L})` whose vertices are the binary words with `L` zeros and\n`N − L` ones and whose generators are the consecutive `k`-cycles\n`(i, i + 1, …, i + k − 1)`, `0 ≤ i ≤ N − k`, for `N > k`. In the\ninverse-closed case a move rotates `k` consecutive letters one place in either\ndirection. The `k = 3` clause of Conjecture 16 states:\n\n> For $k=3$ from $L\\ge 2$(?): $D(L, N) = \\left\\lceil \\frac{L (N - L)}{2} \\right\\rceil$\n\nIssue #10230 fixes the readings: words are `List Bool` with `false` for `0`;\n\"diameter\" is read both as CayleyPy computes it, the largest distance from the\ncentral state `[0]^L + [1]^{N−L}`, and as the largest distance between two\nvertices; the range is `2 ≤ L ≤ N` and `N > 3`.","title":"The diameter of the inverse-closed consecutive 3-cycle Schreier coset graph","url":"https://the-omega-institute.github.io/trureturing-pages/research/chervov-2026-cayleypy4-three-cycle-grassmannian-diameter/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74de0053f84834f0e1f41e3ade922ba5197dd25b71f2ad4ac9c56e601e0fc775.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chiribella-aurell-zyczkowski-2021-channel-time-reversal","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chiribella, Aurell and Życzkowski, “Symmetries of quantum evolutions”,\narXiv:2101.04962v5, Section V.B, conjecture that the no-go theorem for time\nreversal continues to hold for symmetries of the full set of quantum channels:\n\n> We conjecture that this assumption can be lifted, and that the no-go theorem\n> to time reversal applies also to symmetries of the set of quantum channels.\n\nFor every natural dimension $d \\ge 2$, `IsChannelSymmetry S` is the\npredicate saying that `S` is a bijection of the completely positive\ntrace-preserving maps on $d\\times d$ complex matrices and preserves every binary\nrandomisation. The claim requires that no such `S` sends every unitary channel\n`Ad_U` to `Ad_{U†}`, and that none sends every `Ad_U` to `Ad_{Uᵀ}`.","title":"No randomisation-consistent time reversal on the full channel set","url":"https://the-omega-institute.github.io/trureturing-pages/research/chiribella-aurell-zyczkowski-2021-channel-time-reversal/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7d318ea405b0e06e26a8e4552b46401ab1b1f96e94da50943b99d4f8dd9007a4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chiribella-davidson-paulsen-rahaman-2023-ppt-ipt-squared-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Chiribella, K. R. Davidson, V. I. Paulsen and M. Rahaman, *Positive maps and entanglement in real Hilbert spaces*, arXiv:2207.02510v2, Ann. Henri Poincaré 24 (2023), DOI 10.1007/s00023-023-01325-x, Section 9.2, asks:\n\n> We do not know if these conjectures are equivalent.\n\nThe source defines PPT-squared over complex matrices and IPT-squared over real and complex matrices. The source implications are $\\mathrm{PPT}^2(\\mathbb C) \\Rightarrow \\mathrm{IPT}^2(\\mathbb C) \\Rightarrow \\mathrm{IPT}^2(\\mathbb R)$.\n\n### Statement echo\n\nThe source conventions are represented by the public definitions `SEP`, `IsEB`, `ptFirst`, `IsPPT`, `IsIPT`, `PPT2C`, `IPT2` and `claim` in `D5/S3/Quantum/QuantumChannels/IptPptSquaredEquivalence.lean`. Amplification and complete positivity use `MatrixMap.kron` and `MatrixMap.IsCompletelyPositive` from `FiniteKrausChannel`; the consumed `kron_id_apply` proves the source coordinate formula. The input-first Choi expression is `MatrixMap.choi_matrix` from `FiniteKrausRepresentation`, with `submatrix Prod.swap Prod.swap`; its entries are exactly the source formula. Thus the claim remains $(\\mathrm{PPT2C}\\leftrightarrow\\mathrm{IPT2}(\\mathbb C))\\land(\\mathrm{IPT2}(\\mathbb C)\\leftrightarrow\\mathrm{IPT2}(\\mathbb R))$. The settling declaration is `result : claim`. The matrix indices use `Fin`, scalars are generic over `[RCLike 𝕜]`, and the flag identification is `(finCongr (Nat.mul_comm 2 n)).trans finProdFinEquiv.symm`.","title":"PPT-squared and IPT-squared equivalence","url":"https://the-omega-institute.github.io/trureturing-pages/research/chiribella-davidson-paulsen-rahaman-2023-ppt-ipt-squared-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd0ea4d925a82558d977ad822e23953992c0887476de8301689c3a83096f9c17.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chirvasitu-cusick-2024-rules-matrix-order-conjecture-6-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Chirvasitu and T. W. Cusick, *Quadratic rotation symmetric Boolean functions*, arXiv:2304.12734v1, Section 6, Conjecture 6.1:\n\n> For i > j ≥ 1 the eigenvalues of the matrix R(i, j) of (6-3) are all √2-scaled roots of unity. Moreover, the multiplicative order of R(i,j)/√2 is precisely 2 · lcm(4, i − j, i + j).\n\nThe source and its definitions are recorded in [the Library note](../Library/Fourier/chirvasitucusick2024quadraticrotation.md). The preregistration is [#19397](https://github.com/the-omega-institute/trureturing/issues/19397). The settlement refutes the conjunction through the order conjunct at $(i,j)=(8,4)$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $H_0=[[1,1],[1,-1]]$, $H_1=[[1,1],[-1,1]]$ | `H0`, `H1` | Integer entries in source order. |\n| Append $k$ zero columns after each column | `pad` | Column $b$ reads column $b/(k+1)$ precisely when $b\\bmod(k+1)=0$. |\n| Rotate every row rightward $k$ times | `rot` | Column $b$ reads column $b-k$ modulo the width; `[NeZero n]` expresses a nonempty cyclic index set. |\n| Alternate the blocks in batches of $2^{j-1}$ | `blockH` | Starts with $H_0$, then alternates with $H_1$. |\n| Stack the blocks of Eq. (6-3) | `R` | Row $2k+c$ reads row $c$ of block $k$. |\n| The same rules matrix over the real numbers | `Rsq` | Row-bound reindexing and the integer-to-real entry map; no altered entries. |\n| For all $i>j\\ge1$ | `claim`, natural binders and the two inequalities | The source range is retained. The dimension is $2\\cdot2^{i-1}=2^i$ throughout this range. |\n| Eigenvalues are $\\sqrt2$-scaled roots of unity | First conjunct of `claim` | Complex spectrum; a positive exponent may depend on the eigenvalue. |\n| Order is precisely $2\\operatorname{lcm}(4,i-j,i+j)$ | Second conjunct of `claim` | `orderOf` in the monoid of real square matrices. |\n| Refutation of the complete assertion | `result` | The proposition is exactly the negation of `claim`. |\n\nThe public definitions agree with the binding definitions in #19397. Integer entries, zero-based finite indices and real scalar multiplication specify the source matrix without extra hypotheses on the conjecture.","title":"A sign obstruction to Chirvasitu–Cusick Conjecture 6.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/chirvasitu-cusick-2024-rules-matrix-order-conjecture-6-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d189edc5f875c0801f9144cf82cfe19d9b294bb1c9bb8e4eb9bdc8778c13741.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chitambar-et-al-2023-entanglement-assisted-communication-value-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Chitambar, I. George, B. Doolittle and M. Junge, *The Communication Value of a Quantum Channel*, arXiv:2109.11144v1 (IEEE Trans. Inf. Theory 69, 1660–1679, 2023), section on the entanglement-assisted communication value:\n\n> \\noindent\\textit{Conjecture}: For any channel $\\mc{N}\\in\\cptp(A\\to B)$, $\\cv^*(\\mc{N})\\leq d_B\\cdot \\cv(\\mc{N})$.\n\nHere:\n- **cv(N):** the supremum, over all finite families of input states $\\rho_x$ and output POVMs $\\{\\Pi_y\\}$, of $\\sum_y\\max_xP(y|x)$ with $P(y|x)=\\mathrm{tr}[\\Pi_y\\mathcal N(\\rho_x)]$.\n- **cv\\*(N):** the same supremum when sender and receiver share a state $\\varphi^{A'B'}$ of any dimension, the sender applies channels $\\mathcal E_x^{A'\\to A}$ and the receiver measures $BB'$.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/chitambargeorgedoolittlejunge2023communication.md).\n\nIssue [#20033](https://github.com/the-omega-institute/trureturing/issues/20033) fixes the reading with the operational definitions of the source. This is proved.","title":"Entanglement assistance raises the communication value by at most the output dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/chitambar-et-al-2023-entanglement-assisted-communication-value-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e29fdfc8fffaf22478020a6175aff96607ef2b3737ce5a6c7e1f7ebaa709f459.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:choi-kiem-kye-2020-maximal-birank-edge-states","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Choi, Y.-H. Kiem and S.-H. Kye, *Entangled edge states of corank one\nwith positive partial transposes*, J. Math. Phys. 61, 062202 (2020),\narXiv:1903.10745v2, Conjecture 6.4 ([Library note](../Library/QuantumBounds/choikiemkye2020maximalbirank.md)):\n\n> For any $n\\ge 3$, the open set of parameters\n> $(\\alpha_1,\\ldots,\\alpha_n,\\beta_1,\\ldots,\\beta_n)$ which give PPT entangled\n> edge states of bi-rank $(n^2-1,n^2-2n+3)$ is nonempty.\n\nThe binding statement in issue #13859 asks for the Section 6 construction\nin every dimension $n\\ge3$, with unit phases, $\\alpha_1=\\beta_n=1$,\n$\\alpha_i^{-1}\\alpha_j\\ne\\beta_i^{-1}\\beta_j$ for $i We conjecture that the maximum of the success probability appearing in the temporal version of Hardy's\n> argument is $25\\%$ for any spin-$s$ system.\n\nThe conclusion of the source states the same conjecture for spin measurements. The source proves that the success probability is at most 1/4 for every system and every projective measurement, and attains it with spin components for s = 1 and s = 3/2. The experiment and the four probabilities are quoted in [the literature note](../Library/QuantumContext/choudhary2013temporalhardy.md).\n\nIssue [#19643](https://github.com/the-omega-institute/trureturing/issues/19643) fixes the reading. For every spin s = n/2 with n ≥ 1, some state and four spin-component observables u·S, with Lüders spectral projectors for chosen outcomes, satisfy the three zero conditions and have success probability 1/4. The conjecture is proved.","title":"The temporal Hardy maximum 1/4 is attained with spin measurements for every spin","url":"https://the-omega-institute.github.io/trureturing-pages/research/choudhary-goyal-konrad-ghosh-2013-temporal-hardy-spin-attainment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2c01852d62ae7f412c21183eea408a6ec63b14f6e189dadc873052da9789c934.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:christiana-2025-homflypt-yang-baxter-kunneth-splitting-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Christiana, Clingenpeel, Guo, Oh, Przytycki, Wang and Yun,\n*Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial*,\narXiv:2502.20659v1, §5.1, Conjecture 5.6:\n\n> The short exact sequence of chain complexes\n> $0 \\to C_\\bullet^{(m,A,B)} \\to C_\\bullet^m \\to C_n^m/C_\\bullet^{(m,A,B)} \\to 0,$\n> splits. Thus homology $H_n(C_\\bullet^m)$ splits.\n\nDefinition 5.4 includes every cutoff from zero to the word length. The\nhypotheses are $A\\cup B=X_{(m)}$ and every letter of $A$ at least every\nletter of $B$. Splitting means a chain-map retraction of the inclusion.","title":"The HOMFLYPT Yang–Baxter Künneth subcomplex need not split","url":"https://the-omega-institute.github.io/trureturing-pages/research/christiana-2025-homflypt-yang-baxter-kunneth-splitting-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9738d7f441f6a2c3c7af494f0ddb092b864fa4536dc30ea8c486dbeb4e0d571f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:christiansen-2025-fermionic-correlational-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Martin Ravn Christiansen, *A Correlational Bound for Eigenvalues of Fermionic\n2-Body Operators*, arXiv:2505.21167v1, Section 1.1, Conjecture 3, asks for a\nconstant $C>0$, independent of the normalized canonical pair state\n$\\Phi=\\sum_k\\lambda_k u_k\\wedge v_k$, such that every even $N\\ge2$ with\n$N\\lambda_{\\max}^2\\le1$ and every normalized $N$-particle state satisfy\n\n$$\n\\langle\\Phi,\\gamma_2^\\Psi\\Phi\\rangle\\le\nN\\left(1-\\frac{N-2}{2}\\sum_k\\lambda_k^4+C(N\\lambda_{\\max}^2)^2\\right).\n$$\n\nThe [source note](../Library/QuantumBounds/christiansen2025correlational.md)\nquotes the conjecture and the RDM definition. [Preregistration #14135](https://github.com/the-omega-institute/trureturing/issues/14135)\nfixes CHR-3 with $C=5/8$ over every separable complex Hilbert space, including\ninfinite-dimensional spaces. The source conventions are $\\lambda_k\\ge0$,\n$\\sum_k\\lambda_k^2=1$, mutual orthonormality of the pair modes,\n$u\\wedge v=(u\\otimes v-v\\otimes u)/\\sqrt2$, and RDM trace $N(N-1)$.","title":"Christiansen's fermionic correlational bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/christiansen-2025-fermionic-correlational-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11927f6eb0581b43cf5f5115b29eb787e4f9816b72c1ce6976e816eb3e463b0c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chruscinski-2022-qdeformed-commutator-negative-q-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Chruściński, G. Kimura, H. Ohno and T. Singal, *Bounding the Frobenius norm of a q-deformed\ncommutator*, arXiv:2202.11520v2, state Conjecture 2: \"For any $q\\le 0$, if $A$ or $B$ is traceless,\nthe sharp bound is $\\|[A,B]_q\\|_F^2\\le\\max[g(n)(1-q)^2,1+q^2]\\,\\|A\\|_F^2\\|B\\|_F^2$\", with\n$[A,B]_q=AB-qBA$ and $g(n)=(n^2-3n+3)/(n(n-1))$. Their Proposition 3 proves the case $n=2$ with\n$\\mathrm{tr}A=0$ for every real $q$. The verbatim statements are in\n[the literature note](../Library/QuantumBounds/chruscinski2022qdeformed.md).\n\nIssue [#14632](https://github.com/the-omega-institute/trureturing/issues/14632) reads the inequality\nclause for every $n\\ge2$, every $q\\le0$ and all $A,B\\in M_n(\\mathbb C)$ with $\\mathrm{tr}A=0$ or\n$\\mathrm{tr}B=0$.","title":"A rank-one partner breaks the negative-q bound for a traceless matrix","url":"https://the-omega-institute.github.io/trureturing-pages/research/chruscinski-2022-qdeformed-commutator-negative-q-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1d5b95b74237cf406e4f7451e5da45a4d146f69cbc0a72e79e8daa1f92a874e7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chruscinski-2022-qdeformed-commutator-traceless-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Chruściński, G. Kimura, H. Ohno and T. Singal, *Bounding the Frobenius norm of a q-deformed\ncommutator*, arXiv:2202.11520v2, prove $\\|AB-qBA\\|_F^2\\le(1+q^2)\\|A\\|_F^2\\|B\\|_F^2$ for $q>0$ when\n$A$ or $B$ is normal, note that it fails in general, and state Conjecture 1: \"For any $q > 0$, if\n$A$ or $B$ is traceless, the inequality ... holds and is sharp.\" They prove it for $n=2$. The\nverbatim statements are in [the literature note](../Library/QuantumBounds/chruscinski2022qdeformed.md).\n\nIssue [#13930](https://github.com/the-omega-institute/trureturing/issues/13930) reads the inequality\nclause for every $n$, every $q>0$ and all $A,B\\in M_n(\\mathbb C)$ with $\\mathrm{tr}A=0$ or\n$\\mathrm{tr}B=0$.","title":"A traceless counterexample to the q-deformed Böttcher–Wenzel bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/chruscinski-2022-qdeformed-commutator-traceless-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be1e3fa9881d64896739af5335f5ff6e958dfc81ce640e1e8b6c3f530531b3c4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chruscinski-denisov-tarnowski-zyczkowski-2025-dissipator-spectral-disk-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Chruściński, S. Denisov, W. Tarnowski and K. Życzkowski, *Spectral delineation of Markov generators: classical vs quantum*, arXiv:2504.07903v1 (2025), Section “Beyond complete positivity”, Conjecture 3. The source is `bounding_6.tex`, lines 1044–1050, md5 `31cb6e7a93cfb4dac99f83105a784d57`. The wording below is verbatim; the source's bra–ket macros are expanded for display:\n\n> If $\\mathcal{L}$ is a generator of positive trace-preserving dynamics and\n>\n> $\\nu_{\\text{q},3} := \\max_{\\|\\psi\\|=1} | \\langle \\psi|\\mathcal{L}(|\\psi\\rangle\\langle\\psi|)|\\psi\\rangle|$,\n>\n> then the spectrum of $\\frac{1}{ \\nu_{\\text{q},3}} \\mathcal{L}$ is confined in the unit disk $\\mathbb{D}(-1,1)$.\n\nThe disk is closed, with centre $-1$ and radius $1$. The paper sets its purely dissipative scope at lines 227 and 702. Its GKLS formula is at lines 205–209; the canonical traceless, Hilbert–Schmidt orthonormal jumps appear at lines 794–799. The source and publication identifiers are collected in [the literature note](../Library/QuantumContext/chruscinskidenisovtarnowskizyczkowski2025spectral.md).\n\nIssue [#20152](https://github.com/the-omega-institute/trureturing/issues/20152) fixes the restricted reading: for every $N,m\\in\\mathbb N$, every finite family of traceless complex $N\\times N$ jumps $L_k$ and nonnegative real rates $\\gamma_k$, put $\\mathcal L=\\sum_{k\\in\\mathrm{Fin}\\,m}\\gamma_k\\mathcal D_{L_k}$, where $\\mathcal D_L(\\rho)=L\\rho L^\\dagger-\\tfrac12(L^\\dagger L\\rho+\\rho L^\\dagger L)$. Define $\\nu$ as the supremum of $|\\langle\\psi|\\mathcal L(|\\psi\\rangle\\langle\\psi|)|\\psi\\rangle|$ over $\\sum_i|\\psi_i|^2=1$. If $\\nu>0$, every nonzero eigenmatrix $X$ with $\\mathcal L(X)=\\lambda X$ would satisfy $|1+\\lambda/\\nu|\\le1$. This is `claim`; `result` refutes it.","title":"The pure-state scale does not confine purely dissipative spectra to the unit disk","url":"https://the-omega-institute.github.io/trureturing-pages/research/chruscinski-denisov-tarnowski-zyczkowski-2025-dissipator-spectral-disk-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/989ac310e0b1e0ca3c9ef1bc308fb5c71d1bebb4ea60313355be8c545a2ace94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chu-2026-schreier-multiset-recurrence-q-two","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chu, Geng, King, Miller, Tresch, and Vasseur define on printed page 19:\n\n> A^{(s)}_{p,q,n} := {F ⊂ {1, …, 1, …, n − 1, …, n − 1, n} : n ∈ F and q min F ≥ p|F|}\n\nIn the printed display, an underbrace labelled `s` spans the copies of each\nentry from `1` through `n − 1`: there are `s` copies of every one of those\nentries, followed by one copy of `n`. The page then says:\n\n> Below are the data and conjectured recurrences we gather:\n\nIts first item is:\n\n> (|A^{(2)}_{1,2,n}|)^∞_{n=1}: 1, 2, 4, 9, 19, 41, 88, 189, 406, 872, 1873, 4023, 8641, . . . with a_n = a_{n−1} + 2a_{n−2} + a_{n−3};\n\nThe formal reading makes three decisions. First, `s = 2` means two copies of\neach of `1,...,n−1` and one copy of `n`. Second, `(p,q) = (1,2)` means\n`2 · min F ≥ |F|`, with `min F = F.toFinset.min'` on the witness `n ∈ F`\nand `|F| = Multiset.card`. Third, the recurrence starts at `4 ≤ n`, the least\nindex for which all four sequence indices are at least one; at the boundary,\n`a_4 = 9 = a_3 + 2a_2 + a_1`.","title":"The first q = 2 Schreier-multiset recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/chu-2026-schreier-multiset-recurrence-q-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55953182c7fce5284dcea786550ca96d5dc43ba9285b22c167a74459fa64d4c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chu-miller-tresch-2025-gamma-split-linear-growth","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hùng Việt Chu, Steven J. Miller and Garrett Tresch, *Problems Regarding a Pair of Diophantine Equations*, arXiv:2512.12681v1, Section 3.4, p. 11, ask:\n\n> As noted in Remark 2.7, the sequence grows exponentially fast (faster than the Fibonacci sequence). Is there a construction involving a strictly increasing sequence that grows slower than the Fibonacci sequence, or perhaps even like n^k for some k?\n\nThe question extends Corollary 2.6 to every prescribed split $p\\in[0,1]$. The answer is affirmative. The construction has $-9\\le a_n-8n\\le15$, so $a_n=8n+O(1)$, and its zero-label count $C_N$ satisfies $0\\le pN-C_N<5$.\n\nPreregistration: [#20691](https://github.com/the-omega-institute/trureturing/issues/20691). Source note: `Library/ArithUnits/chumillertresch2025diophantine.md`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Section 1: the gcd-reduced first equation has a nonnegative integral solution exactly when the label is zero | `Gamma` | Literal natural-number equation, integer division, truncated natural subtraction; complementary branch returns one. The source's exactly-one theorem supplies the interpretation for positive inputs. |\n| Corollary 2.6: count the first $N$ pairs of a sequence indexed from zero | `zeroCount` | Inclusive indices $1\\le n\\le N$ and pair $(a_{n-1},a_n)$. |\n| Every split $p\\in[0,1]$ | `claim`, binders `p : ℝ`, `0 ≤ p`, `p ≤ 1` | Both endpoints included. |\n| A strictly increasing positive integer sequence | `claim`, existential `a : ℕ → ℕ`, strict monotonicity and positivity | Index zero and natural-valued entries. |\n| Growth like $n^k$ for some $k$ | Polynomial-bound conjunct of `claim`; `result` chooses exponent one and coefficient 23 | Upper bound for every $n\\ge1$. The sharper two-sided linear estimate is private. |\n| Limiting zero proportion $p$ | Final conjunct of `claim` and `result` | Real division, natural index tending to infinity. |\n\nThe definitions `Gamma`, `zeroCount` and `claim` retain the literal preregistered bodies. The proof uses the existing `D5.S1.Words.Mechanical.lowerMechanicalLetter p 0` schedule; no second definition of that schedule is introduced. The exponent, coefficient and positivity clauses specify the polynomial-growth question without adding a hypothesis on $p$.","title":"Linear growth for every Gamma-label split","url":"https://the-omega-institute.github.io/trureturing-pages/research/chu-miller-tresch-2025-gamma-split-linear-growth/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/48a63cfe9ddd1004867a3f0a39fc71c0373f81588c2964e02ffe50a1cec42779.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:chuysurichay-2026-tribonacci-four-bonacci-local-dold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chuysurichay, Jaidee, Panraksa and Ward, *The local Dold congruence for\nFibonacci and Lucas sequences*, arXiv:2609.24544v1, section 4, PDF page 11:\n\n> Another natural problem from the results here is to try and characterize these sets of primes for linear recurrence sequences in general, starting with the intriguing observation that the classical tribonacci sequence (1,1,2,4,...) and 4-bonacci sequence (1,1,1,2,5,...) seem (computationally) to have G_D = ∅.\n\nThe quotation normalizes mathematical notation and whitespace. It is an\nempirical open observation, without a numbered conjecture. Its positive\nsequences are interpreted by their preceding-three and preceding-four\nrecurrences: T1=T2=1, T3=2, and B1=B2=B3=1, B4=2. Thus\nB=(1,1,1,2,5,9,17,33,...).\n\nFor a positive natural sequence a and prime p, put b(n)=p^valuation_p(a(n)).\nThe assertion is that neither sequence has any prime p for which every\npositive n divides the integer sum of mu(n/d)b(d) over positive divisors d\nof n. Both sequences and every prime are included.","title":"The tribonacci and four-Bonacci local Dold observations","url":"https://the-omega-institute.github.io/trureturing-pages/research/chuysurichay-2026-tribonacci-four-bonacci-local-dold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a54d01593379d2e5033bb8949c3bcbd8a907da2601a0ba1a449308cb508396b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-boundary-motzkin-hankel","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin\npaths*, arXiv:2204.09910v4, Section 2, Conjecture 2.1, equation (2.3). M_{n,k}(t,s) is the weighted number of Motzkin\npaths from (0,0) to (n,k) that never go below the axis, with up and down steps of weight 1, horizontal steps of weight s\nat height 0 and t above; d_m(n,t,s) = det(M_{m+i+j,0}(t,s))_{0≤i,j 0, the conjecture states\n\nΣ_{n≥0} d_m(n,t,s) xⁿ = R_m(x,t,s) / ∏_{j=0}^{⌊m/2⌋} A_{0,m−2j}(x,t)^{1+j(m−j)},\n\nwith R_m an integer polynomial in x, s, t of x-degree binom(m+1,3) + 1.","title":"Reduced Denominators for Hankel Determinants of Boundary-Weighted Motzkin Paths","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-boundary-motzkin-hankel/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a90ef9674823f54203cebeb45516cf8c81d1813072e003a35f5940643ec1a73a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-catalan-power-shifted-hankel","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some experimental observations about Hankel determinants of convolution powers of Catalan numbers*,\narXiv:2308.07642v2, Section 2.2, Conjecture 11: with C_{r,j} = r/(2j+r) · binom(2j+r, j) for j ≥ 0, C_{r,j} = 0 for\nj < 0, and D_{r,s}(N) = det(C_{r,i+j+s})_{0 ≤ i,j < N}, D_{r,s}(0) = 1, for every k ≥ 1, 0 ≤ m ≤ k+1 and n ≥ 0,\nD_{2k+1,m−k+1}((2k+1)n+k) = (−1)^{kn+binom(k,2)} (2k+1)^m (n+1)^m. The shift m − k + 1 may be negative.","title":"Shifted Hankel Determinants of Odd Catalan Powers","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-catalan-power-shifted-hankel/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8894200314cf609185456ca84d52efdf741d8a9c8603692caab77cfac3523640.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-motzkin-column-hankel","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some remarks and conjectures about Hankel determinants of polynomials which are related to Motzkin\npaths*, arXiv:2204.09910v4, Section 1, Conjecture 1.3, equation (1.30). M_{n,k}(t) is the weighted number of Motzkin\npaths from (0,0) to (n,k) that never go below the axis, with up and down steps of weight 1 and horizontal steps of\nweight t; d_m^{(k)}(n,t) = det(M_{m+i+j,k}(t))_{0≤i,j 0 (equation\n(1.28)), the conjecture states\n\nD_m^{(k)}(x,t) = r_m^{(k)}(x,t) / ∏_{j=0}^{⌊m/2⌋} A_{k,(k+1)(m−2j)}(x,t)^{1+j(m−j)},\n\nwith deg_x r_m^{(k)} = binom(m+1,3) + k (binom(m,1) + binom(m,2) + binom(m,3)).","title":"Reduced Denominators for Hankel Determinants of Motzkin-Triangle Columns","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-motzkin-column-hankel/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e762afbcd5a086311b9d01a3e57af4710886f09f48b1621ec52afe1a823a3fc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-narayana-strip-expansion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana\npolynomials for q=-1*, arXiv:2608.03363v2, Section 4, Conjecture 3 (Conjecture 2 in v1): for m ≥ 1 and n ≥ 0,\n\nC_{n+1}^{(2m)}(t) = Σ_{j=0}^{⌊n/2⌋} C_j^{(m−1)} binom(n, 2j) t^j (1 + t)^{n−2j},\n\nc_{n+1}^{(2m)}(t) = Σ_{j=0}^{⌊n/2⌋} (−1)^j C_j^{(m−1)} binom(⌊n/2⌋, j) t^j (1 + t)^{n−2j}.\n\nHere C_n^{(h)}(t) and c_n^{(h)}(t) are the weighted counts of Dyck paths of semilength n in the strip\n0 ≤ y ≤ h, a down-step to height k having weight τ_k with τ = (1, t, 1, t, …) and τ = (1, t, −1, −t, …)\nrespectively, and C_j^{(m−1)} is the number of Dyck paths of semilength j in the strip of height m − 1.","title":"Expansions of Narayana-Weighted Dyck Paths in an Even Strip","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-narayana-strip-expansion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/706c9512868320ef88819c5fbc5b8bccab0a80256c98296d3269e3fa2a4ba54a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-narayana-strip-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana\npolynomials for q=-1*, arXiv:2608.03363v2, Section 4, Conjecture 2, equation (79) (Conjecture 1, equation (62), in\nv1): for m ≥ 1,\n\nC^{(4m)}(t², z²) = c^{(4m)}(t, z) c^{(4m)}(−t, −z), C^{(4m+1)}(t², z²) = c^{(4m+1)}(t, z) c^{(4m+1)}(−t, −z).\n\nHere C^{(h)}(t, z) = Σ_n C_n^{(h)}(t) zⁿ and c^{(h)}(t, z) = Σ_n c_n^{(h)}(t) zⁿ, where C_n^{(h)}(t) and c_n^{(h)}(t) are\nthe weighted counts of Dyck paths of semilength n in the strip 0 ≤ y ≤ h, a down-step to height k having weight\nτ_k with τ = (1, t, 1, t, …) and τ = (1, t, −1, −t, …) respectively. The identity is the bounded analogue of the\nunrestricted identity c(t, z) c(−t, −z) = C(t², z²), equation (6) of the source.","title":"A Product Formula for Narayana Strip Series of Heights 4m and 4m + 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-narayana-strip-product/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e797dfac0e3bc444b05f71425c5b76a413a3aafe9bade2ad3511dc62a2a7ff1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-partial-theta-hankel","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Hankel determinants of backward shifts of the coefficients of a partial theta function*,\narXiv:2407.05768v2, Section 1, the Conjecture after equation (2): with a(s, q) = q^{binom(s,2)} for s ≥ 0 and\na(s, q) = 0 for s < 0, D_{−m,N}(q) = det(a(−m + i + j, q))_{0≤i,j The functions r_{m,n}(q) are monic polynomials with integer coefficients with deg r_{m,n}(q) = mn(n + m + 2)/2 which\n> satisfy r_{m,n}(1) = 1 and r_{m,n}(0) = (−1)^{mn}.\n\nThe paper proves the cases m = 1 and m = 2.","title":"Hankel Determinants of Backward Shifts of Partial Theta Coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-partial-theta-hankel/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70035706a021f1253aaa98763a29cc28ec0bef7d617035bce08dcd97cfc92813.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-signed-strip-cycle-walks","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some sequences and number triangles which are related to Narayana polynomials and to q-Narayana\npolynomials for q=-1*, arXiv:2608.03363v2, Section 3, Conjecture 1, equation (74) (absent from v1): for k ≥ 1, with\nU_k the adjacency matrix of the cycle on 4k vertices,\n\nc_{2n+1}^{(4k−2)} = U_k^{2n+1}(0, 1), c_{2n+2}^{(4k−2)} = U_k^{2n+2}(0, 0) = 2 c_{2n+1}^{(4k−2)}.\n\nHere c_r^{(H)} = c_r^{(H)}(1) is the weighted count of Dyck paths of semilength r in the strip 0 ≤ y ≤ H, a down-step to\nheight j having weight (−1)^{⌊j/2⌋}, that is the q = −1 weights (1, t, −1, −t, …) at t = 1.","title":"Signed Dyck Paths in the Strip of Height 4k − 2 Count Walks on the 4k-Cycle","url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-signed-strip-cycle-walks/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fca9927370235bf1d8519cb7e968c466ea9a793e345f0d154dfe2da6ac87937.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cigler-sixteen-literal-numerator-degree","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johann Cigler, *Some experimental observations about Hankel determinants of\nconvolution powers of Catalan numbers*, arXiv:2308.07642v2, Section 3,\nConjecture 16, equation (30), printed page 10, asserts the following whole\nconjunction. For every integer $k\\ge1$ and every integer $m\\ge1-k$, there is\nan actual polynomial $Q=Q_{2k-1,m}$ such that\n\n$$\n\\left(1+(-1)^k x^{2k-1}\\right)^{\\binom{k+m}{2}+1}\n\\sum_{N\\ge0}D_{2k-1,m}(N)x^N=Q(x),\n\\qquad \\deg Q=\\pi_{2k+1}(m+k-1)+k.\n$$\n\nHere $C(x)=\\sum_{n\\ge0}C_nx^n$, where $C_n=\\binom{2n}{n}/(n+1)$ is the\nCatalan number; $C_{r,t}=[x^t]C(x)^r$ for $t\\ge0$, and $C_{r,t}=0$ for\n$t<0$. The actual determinant is\n\n$$\nD_{r,m}(N)=\\det(C_{r,i+j+m})_{0\\le i,j Let a_n be the number of simple permutations of size n that can be sorted by a machine consisting of two\n> pop stacks in parallel where entries are allowed to bypass the pop stacks. Then a_0 = a_1 = 1, a_2 = 2,\n> a_n = F_{2n−5} − 1 if n ≥ 3 is odd, and a_n = F_{2n−5} if n > 3 is even.\n\nThe sortable permutations are the class C = Av(2341, 25314, 42513, 42531, 45213, 45231, 52314, 642135,\n642153); a permutation is simple when it has no interval of length strictly between 1 and n.","title":"Simple Permutations Sortable by Two Parallel Pop Stacks with Bypass","url":"https://the-omega-institute.github.io/trureturing-pages/research/cioni-ferrari-smith-pop-stack-simple/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba677b4c2d3212038a36a50ae21b67d7e19c329a8a74f9d0343f71167d7bdf93.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:circular-two-choice-parking-fixed-fiber-bijection","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Open Problem 1 of the linked paper asks:\n\n> Give a bijective proof of the factorization of Corollary 1 for d=1,\n> refining Theorem 2 to a canonical bijection between each anchor class\n> {E=j} intersect C_kappa and the set of classical parking functions.\n\nThe relevant specialization has `n>=1`, `m=n+1`, `s=2`, and `d=1`. Every car\nhas an ordered pair consisting of an anchor and a distinct second choice. The\nanchor is tried first. If it is occupied, the car begins a clockwise scan at\nthe second choice itself, so offset zero must be inspected before later spots.\nThe matrix `kappa` fixes the positive clockwise increment from anchor to second\nfor every car, and `j` fixes the actual unique vacancy.\n\nThe exact formal target is `result n hn increments j`: for every natural `n`\nwith `1<=n`, every per-car increment matrix with entries in `1..n`, and every\ncircular spot `j`, it certifies `Function.Bijective (fixedFiberEquiv n increments\nj)`. The referenced equivalence remains the explicit forward map and inverse\nbetween the corresponding literal actual fiber and classical length-`n` parking\nfunctions; `result` is its sole source-level settlement theorem.","title":"A canonical fixed-fiber bijection for circular two-choice parking","url":"https://the-omega-institute.github.io/trureturing-pages/research/circular-two-choice-parking-fixed-fiber-bijection/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51c683c5e9e1870c3f9912d8e1f0ed6699f68e15b8c4b3e2189d109a71f8db10.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:clavier-foissy-lopez-paycha-2022-locality-tensor-product-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. J. Clavier, L. Foissy, D. A. López and S. Paycha, *Tensor products and the Milnor-Moore theorem in the locality setup*, arXiv:2205.14616v1 (European Journal of Mathematics 11, 41, 2025), Conjectural statement 6.16, states:\n\n> Given a locality vector space $( {E},\\top)$ and $V_1$, $V_2$ two of its subspaces, the subspace $I_{\\rm bil} {(V_1, V_2)} {\\cap\\mathbb{K}(V_1\\times_\\top V_2)}\\subset \\mathbb{K}(V_1\\times_\\top V_2)$ is locality compatible with $\\top_{V_1\\times_\\top V_2}$.\n\nThe locator is the downloaded TeX `Clavier_Foissy_Lopez_Paycha_arxiv.tex`, lines 1984–1987, label `conj:main1`, md5 `1f6041a82cd80114573aa776694ccf20`. Quotations expand typesetting macros without changing the wording.\n\nThe paper's Notations, `Clavier_Foissy_Lopez_Paycha_arxiv.tex`, line 321, specifies that $\\mathbb K$ is a commutative field of characteristic zero. Both conjectures below, under both generator-relation readings, quantify over fields of characteristic zero. The thesis states no characteristic restriction; refuting its assertion even on characteristic-zero fields also refutes the unrestricted assertion.\n\nThe tensor-product conjecture is stated explicitly by P. J. Clavier, *TRAPs, Generalisations of MZVs, Locality and Resurgence for Quantum Field Theories*, arXiv:2506.09493v1 (2025), chapter `LOC.tex`, lines 1797–1799, label `conj:loc_tensor`, md5 `41f5e772e40d5eaf8583e8903293dbbb`:\n\n> Let $(E,\\top)$ be any locality vector space, $V$ and $W$ any two locality vector subspaces of $E$. Then $(V\\otimes_\\top W,\\overline{\\top})$ is a locality vector space.\n\nFor a field $\\mathbb K$, a locality vector space is a $\\mathbb K$-vector space $E$ with a symmetric relation $r$, with $r(0,0)$, such that the polar $U^r=\\{x:\\forall u\\in U,\\ r(u,x)\\}$ of every subset $U\\subseteq E$ is a linear subspace. For subspaces $V,W\\subseteq E$, put $F=\\mathbb K(V\\times_\\top W)$, the free span of pairs $(v,w)\\in V\\times W$ satisfying $r(v,w)$. In the ambient free span on all pairs, let $I_{\\rm bil}(V,W)$ be the span of the two additivity and two scalar-multiplication relations, and put $I=F\\cap I_{\\rm bil}(V,W)$.\n\nThe module uses two generator relations on pairs $(v,w)$ and $(v',w')$:\n\n- `Prod.RProd r r`: $r(v,v')\\wedge r(w,w')$, as in the 2022 paper, Eq. `eq:locreltoptimes`, TeX lines 447–449;\n- `fourWay`: $r(v,w)\\wedge r(v,v')\\wedge r(v,w')\\wedge r(w,v')\\wedge r(w,w')\\wedge r(v',w')$, all six pairs among the four entries, as in the thesis's locality cartesian power, Eq. `eq:lprod`, `LOC.tex` lines 219–221, and its description of tensor locality through $E^{\\times_\\top4}$, lines 743–747.\n\nThe sources disagree: the paper's Remark at lines 1972–1975 identifies its componentwise relation, when $V=W$, with $V^{\\times_\\top4}$, whereas the thesis's cartesian-power definition requires all six pairs to be related. The two generator relations differ in general, even on local pairs; the computed example in Triage distinguishes them.\n\nFor either generator relation $g$, write $s_g(a,b)$ when every pair in the support of $a\\in F$ is $g$-related to every pair in the support of $b\\in F$. The relation on $F/I=V\\otimes_\\top W$ is existential: two classes are related precisely when they have $s_g$-related representatives. Locality compatibility of $I$ means that for all $x,y,z\\in F$ and $w\\in I$, if $s_g(x,y)$ and $s_g(x+w,z)$, there is $w'\\in I$ with both $s_g(x+w',y)$ and $s_g(x+w',z)$.\n\nThe problem is tracked in issue [#21552](https://github.com/the-omega-institute/trureturing/issues/21552), including [comment 6396883196](https://github.com/the-omega-institute/trureturing/issues/21552#issuecomment-6396883196). The module defines `TensorConjecture g` as the tensor-product assertion and `PairConjecture g` as the compatibility assertion for the chosen generator relation, each universally quantified over every field $\\mathbb K$ of characteristic zero, every $\\mathbb K$-vector space $E$ in `Type`, every locality relation and every pair of subspaces, without a finite-dimensionality hypothesis. The module defines `claim := TensorConjecture (fun _ r => Prod.RProd r r) ∨ TensorConjecture fourWay ∨ PairConjecture (fun _ r => Prod.RProd r r) ∨ PairConjecture fourWay` and proves `result : ¬ claim`, refuting both universal assertions for each generator relation.\n\nThe definitions and source locators are recorded in [the paper note](../Library/HomologicalAlgebra/clavieretal2022localitytensor.md) and [the thesis note](../Library/HomologicalAlgebra/clavier2025localityhdr.md).","title":"Locality tensor products need not be locality vector spaces","url":"https://the-omega-institute.github.io/trureturing-pages/research/clavier-foissy-lopez-paycha-2022-locality-tensor-product-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12700d9b533c5620391a532f75819e73fa72359ace101456343464288c2cadeb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cobucci-tavakoli-2024-odd-ghz-ten-setting-witness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Cobucci and A. Tavakoli, “Detecting the dimensionality of genuine\nmulti-particle entanglement”, Science Advances 10, eadq4467,\nDOI 10.1126/sciadv.adq4467, arXiv:2402.06234v4, Supplementary\nMaterial V, printed page 22, states:\n\n> We have shown this explicitly for odd d ≤ 17 and conjecture it to hold for odd d in general.\n\nFor the nine labelled triples in (V1), the three-party witness operator\n$\\mathcal W_d$ in (V2), and every odd natural local dimension $d$, the\nconjectured operator comparison is\n\n$$\n\\mathcal W_d\\preceq9|\\mathrm{ghz}_{3,d}\\rangle\n\\langle\\mathrm{ghz}_{3,d}|+3\\mathbf1. \\tag{V5}\n$$\n\nConsequently, every density state $\\rho$ satisfies\n\n$$\nF_{\\mathrm{ghz}}(\\rho)\\geq\\frac{W_d(\\rho)-3}{9},\n\\qquad\nW_d(\\rho)\\leq3+\\frac{9D_{\\mathrm{GME}}(\\rho)}{d}. \\tag{V4,V3}\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| All odd local dimensions | `claim`, binder `d` and hypothesis `Odd d` | All odd natural dimensions, including composite dimensions and 1 |\n| Nine triples (V1) | `tuples` | A list retains all nine labelled multiplicities |\n| Witness (V2) | `WindowRegister.windowRoot`, `basisVec`, `PureStateHandshake.rankOneDensity`, `witness`, `GHZMeasureBiseparableBound.expect` | The source exponential phase, coordinate vectors, rank-one projectors and labelled witness sum |\n| GHZ vector, Eq. (5) and fidelity | `ghz`, `RHLinalg.hermForm` | The normalized three-party vector and real expectation |\n| Schmidt rank and Schmidt number | `cutRank`, `SchmidtNumberLE` | Literal coefficient matrices indexed by `Fin d` and `Fin d × Fin d`, and finite convex decompositions across each cut |\n| Definition 1, Eq. (2) | `GMEDecomposable`, `gmeDim` | Mixtures across all three cuts; the natural infimum is attained |\n| (V5) | `witness_quadratic_bound`, `witness_density_bound` | Quadratic-form and density-state comparisons |\n| (V4) and (V3) | `result : claim` | Both inequalities for every density state |\n\nExcept for the named phase, projector, real trace expectation and quadratic-form owners, names in the table belong to\n`D5.S3.Quantum.Entanglement.GHZ.OddGhzTenSettingWitness`.\nThe state carrier is the canonical `FiniteStateChannel.DensityState`.\nThe witness expressions follow the formulas fixed by\n[preregistration #16991](https://github.com/the-omega-institute/trureturing/issues/16991).","title":"The Cobucci–Tavakoli ten-setting GHZ witness in odd dimensions","url":"https://the-omega-institute.github.io/trureturing-pages/research/cobucci-tavakoli-2024-odd-ghz-ten-setting-witness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06d843df9f8e2e5f5ad4489af99b1aa31d6743a32f9df8632313f42f5020e611.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cohen-consecutive-cube-cousin-prime-threshold-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cohen, Journal of Integer Sequences 28 (2025), Article 25.4.7, Section 2.5,\nConjecture 29, asserts that there are at least two cousin-prime pairs between\nconsecutive cubes for every natural `n > 1` and that the count eventually\nexceeds every positive integer. Its printed threshold table further states\n`N(8) = N(9) = N(10) = 12`: every natural `n >= 12` has at least eight,\nnine, and ten such pairs, respectively.\n\nThe formal `claim29` retains both general assertions and all ten printed\nthreshold clauses. The theorem `result29` proves its negation.","title":"Cohen Conjecture 29","url":"https://the-omega-institute.github.io/trureturing-pages/research/cohen-consecutive-cube-cousin-prime-threshold-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f8982a4d3f163f23dc2bb4a9d547e3b215f824cd7c7f202abb9b74ea88bfd40.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cohen-consecutive-cube-twin-prime-threshold-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cohen, Journal of Integer Sequences 28 (2025), Article 25.4.7, Section 2.5,\nConjecture 28, asserts that there are at least two twin-prime pairs between\nevery pair of consecutive positive cubes and that the count eventually\nexceeds every positive integer. Its printed threshold table further states\n`N(10) = 11`: for every natural `n >= 11`, there are at least ten such pairs\nstrictly between `n^3` and `(n+1)^3`.\n\nThe formal `claim28` retains both general assertions and all seventeen printed\nthreshold clauses. The theorem `result28` proves its negation.","title":"Cohen Conjecture 28","url":"https://the-omega-institute.github.io/trureturing-pages/research/cohen-consecutive-cube-twin-prime-threshold-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb05fa4c892432185242b883e23e82513638a445fbde3f1abf87e1a71b83e540.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:condon-dugan-goldman-williams-2026-l-polyomino-skipped-sequence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. M. Condon, E. B. Dugan, L. M. Goldman and E. R. Williams,\n*Polyomino Density*, arXiv:2608.29231v1, Section 6.3, p. 28:\n\n> We believe S(5, 1, 2) is the same as the instance sequence for [L pentomino], and we suspect that the instance sequence for the L n-omino is S(n, 1, 2) in general.\n\nThe bracketed label represents the inline L pentomino diagram. For every\ninteger n >= 3, the L n-omino has n-1 left-aligned bottom-row cells and\none top-row cell. Instances are translated copies, as in Section 2.1.\nFor every N >= 1, its instance minimum is the least cardinality of a\nnonempty edge-connected finite board containing at least N instances.\nThe sequence S(n,1,2) starts at n, then increases by one at position k\nif k appears before position k, and by two otherwise.","title":"The L n-omino instance sequence","url":"https://the-omega-institute.github.io/trureturing-pages/research/condon-dugan-goldman-williams-2026-l-polyomino-skipped-sequence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0c0e3c48a1387c491c65e898caadd4d217c36a9fa818f4fa32bb89b517df087.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:connelly-2017-property-t-universal-state-transfer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Connelly, N. Grammel, M. Kraut, L. Serazo and C. Tamon (arXiv:1701.04145,\nquant-ph and math.CO; Linear Algebra Appl. 531 (2017) 516–532) study the\ncontinuous-time quantum walk `U(t) = exp(−i t A)` of a graph with Hermitian\nadjacency matrix `A`. The walk has universal perfect state transfer if for all\nvertices `u, v` some time gives `|U(t)_{v,u}| = 1`. A graph has property `𝕋`\nif every nonzero entry of `A` has modulus `1`. The paper states:\n\n```latex\n\\begin{conjecture} $\\Circ(0,-\\ii,\\ii)$ is the only circulant with property\n$\\TT$ which has universal perfect state transfer. \\end{conjecture}\n```\n\nand, in its introduction,\n\n```latex\nThe only known examples of complex unit gain graphs\nwith the universal property are the circulants $K_{2}$ and $\\Circ(0,-\\ii,\\ii)$.\nWe conjecture that this set is unique.\n```\n\nIssue #11528 fixes the reading. `Circ(a)` has entries `C_{jk} = a_{k−j}`\nover `ℤ/nℤ`, \"the only\" is read up to the paper's switching equivalence\n(`MA = BM` with `M` a permutation matrix times an invertible diagonal matrix)\nand together with `K₂`, the order is `n ≥ 2`, and the circulant is Hermitian\nwithout loops. The formal `claim` is: every such circulant with property `𝕋`\nand universal perfect state transfer is switching equivalent to `K₂` or to\n`Circ(0, −i, i)`. The result refutes it.","title":"A unit-gain circulant on three vertices with universal perfect state transfer","url":"https://the-omega-institute.github.io/trureturing-pages/research/connelly-2017-property-t-universal-state-transfer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce13f7331b3543bc65f0feab6f85804080ee7d82a892d9a9c64154fda7b1b721.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:consiglio-apollaro-wiesniak-2022-xmems-css-shape","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Consiglio, T. J. G. Apollaro and M. Wieśniak, *A Variational Approach to the Quantum Separability Problem*, arXiv:2209.01430v2 (Phys. Rev. A 106, 062413, 2022), Appendix A, subsection “$n$-qubit $X$-MEMS”, state in `PRA_v2.tex`, line 459 (acronym macros expanded):\n\n> While we have not explicitly derived the CSS of $n$-qubit X-MEMS in general, we conjecture that it is of the form\n\nThe following matrix is Eq. (A11), lines 460–473. Put $N=2^{n-1}$ and order computational basis words with qubit 1 most significant. Its diagonal entries are $a/2$ at the all-zero and all-one words, $b/(N-1)$ at every other first-bit-zero word, and $(1-a-b)/(N-1)$ at every other first-bit-one word. Its opposite corners are $d,d^*$ and all other entries vanish. Line 474 restricts $a,b\\in[0,1]$, $a+b\\le1$, and $|d|\\le a/2$.\n\nThe source defines the CSS as a minimizer over fully separable states of $D_{HS}(\\rho,\\sigma)=\\operatorname{Tr}((\\rho-\\sigma)^2)$ (Section II, lines 70–93). The definitions and source locator are in [the literature note](../Library/Divergence/consiglioapollarowiesniak2022separability.md).\n\nIssue [#25134](https://github.com/the-omega-institute/trureturing/issues/25134) fixes the low-branch reading: for every integer $n\\ge2$ and every $\\gamma\\in\\mathbb C$ with $|\\gamma|\\le1/(N+1)$, there exists a fully separable $\\sigma$ of Eq. (A11) such that\n\n$\\operatorname{hsSq}(\\rho,\\sigma)\\le\\operatorname{hsSq}(\\rho,\\tau)$ for every fully separable $\\tau$.\n\nHere $\\rho=\\operatorname{xmems}(n,\\gamma)$ has diagonal $1/(N+1)$ at all first-bit-zero words and the all-one word, zero at the remaining words, and opposite corners $\\gamma,\\gamma^*$; this is Eq. (A1) on the branch where $f=g=1/(N+1)$ (lines 341–373). The module defines $\\operatorname{hsSq}(\\rho,\\sigma)=\\operatorname{Re}\\operatorname{Tr}((\\rho-\\sigma)^\\dagger(\\rho-\\sigma))$. The statement asserts attainment as well as comparison with every fully separable competitor. It is refuted.","title":"A closest fully separable state of an X-MEMS need not have the conjectured shape","url":"https://the-omega-institute.github.io/trureturing-pages/research/consiglio-apollaro-wiesniak-2022-xmems-css-shape/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39017fa5126735f2c94b3190b799d748d5c46898cf9f3262a3f9207b21ac1163.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cope-2021-binary-detection-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Cope, *The Binary-Outcome Detection Loophole*, New Journal of Physics 23 (2021) 073032, DOI 10.1088/1367-2630/ac10a5, arXiv:2005.03344v3, Section 4.3, printed p. 6, states:\n\n> In order to prove that our conjecture of $\\eta^*=4/(m+4)$ for $m_{\\mathrm{A}}=m_{\\mathrm{B}}=m$ is correct, one would need to provide a extremal $\\mathcal{NS}$ distribution $p^{\\mathrm{NS}}$, and corresponding Bell inequality $s_{a'b'}^{xy}$, such that $\\sum_{a',b',x,y}s_{a'b'}^{xy}p^{\\mathrm{NS}}_{\\eta}(a'b'|xy) \\not \\leq k,\\; \\forall \\eta > \\eta^*$.\n\nThe named conjecture asserts $\\eta^*(m)=4/(m+4)$ for every $m\\ge2$. Its definitions and the refutation criterion are fixed in [#15572](https://github.com/the-omega-institute/trureturing/issues/15572). The source definitions and locator are in [the Library note](../Library/QuantumBounds/cope2021binarydetection.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Two parties, $m$ inputs each, binary outputs | Two `Fin m` inputs and two `Bool` outputs in `IsNSBox` | Equal input counts; failure is an additional output only after loss. |\n| Conditional probabilities and no signalling (Section 2) | Four conjuncts of `IsNSBox`: nonnegativity, normalization, and both input-independent marginals | Every input and output is quantified. |\n| Independent equal-efficiency loss, equation (3) | All four branches of `lossy` | `some` denotes a detected bit and `none` denotes $F$; the two single-failure terms use the original marginals. |\n| Convex hull of deterministic local responses | `IsLocal` | A finite weight for every pair of response functions, nonnegative weights, sum one, and pointwise reconstruction. |\n| No-signalling threshold, minimum of extremal-box thresholds | `etaStar m`, the supremum of $\\eta\\in[0,1]$ making every no-signalling box local after loss | The all-box formulation below is the convention of #15572 and expresses the same threshold. |\n| Symmetric conjecture $\\eta^*=4/(m+4)$ | `claim` | Quantifies every natural $m\\ge2$; division is in $\\mathbb R$. |\n| Proposed extremal-box Bell-inequality certificate above the conjectured threshold | No additional Lean hypothesis | This is the source's proposed way to establish tightness, not an assumption of its equality conjecture. The local construction above that conjectured value disproves the equality instead. |\n| A counterexample to the universal equality | `result : ¬ claim` | The private `lower_bound` proves $6/25\\le\\eta^*(13)$, whereas the equality gives $4/17$. |\n\nThe threshold formulation follows from convexity. Every no-signalling box is a convex combination of extremal boxes. At a fixed efficiency, loss preserves normalized convex combinations and the local polytope is convex. Thus every box is local after loss if and only if every extremal box is. Further independent erasure preserves locality, so the admissible efficiencies of each extremal box form a downward interval. There are finitely many extremal boxes in this finite polytope; the supremum of efficiencies shared by all of them is the minimum of their thresholds. This is the mathematical interpretation of `etaStar`, not an additional Lean theorem in this module.\n\nThe efficiency interval $[0,1]$ is physical, and $m\\ge2$ restricts the conjecture to its nontrivial input domain. Neither restricts the refuting instance $m=13$. Nonnegativity and normalization express a conditional probability distribution and are not extra assumptions about the conjecture.","title":"Refutation of the symmetric binary-outcome detection-threshold conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/cope-2021-binary-detection-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d22bfc545b71dd02af341eb24facc5cdf0d6702a538e9d7ed53db4493a766a5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:coprime-divisor-tuples","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A062319, \"Number of divisors of n^n, or of A000312(n)\", carries a comment\nby Gus Wiseman dated May 02 2021 and still standing at revision #60 of\nAug 30 2026:\n\n> Conjecture: The number of divisors of n^n equals the number of pairwise\n> coprime ordered n-tuples of divisors of n. Confirmed up to n = 30.\n\nThe comment goes on to print the tuples counted for `n = 1` through `n = 5`.\nThose lists fix two readings the words leave open: the value one may occur and\nmay repeat, so the constant tuple of ones counts, and the order of the\ncoordinates matters, so `(1,2)` and `(2,1)` count separately.","title":"Pairwise Coprime Ordered Tuples of Divisors","url":"https://the-omega-institute.github.io/trureturing-pages/research/coprime-divisor-tuples/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71294f305add4c1115234aff9814b093d7b9be6f575e5f2f111a31080614a66d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cork-disney-hogg-2025-jnr-remark13-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cork and Disney-Hogg, *Locations of JNR skyrmions*, Appendix A, Remark 13,\nconjecture:\n\n> The above proof shows that this is positive so long as Σᵢ (λᵢ²/|yᵢ|²) yᵢ = 0, however we conjecture that the sum in (18) is positive for any choice of yᵢ ∈ ℝ³∖{aᵢ}.\n\nThe source and equation are recorded in the\n[Library note](../Library/Geometry/corkdisneyhogg2025jnr.md).\nThe exact statement is preregistered in\n[#19039](https://github.com/the-omega-institute/trureturing/issues/19039).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Interpretation |\n| --- | --- | --- |\n| $N+1$ poles, at least two | `N : ℕ`, `1 ≤ N`, `Fin (N + 1)` | Every finite size in the stated range |\n| Pure imaginary quaternion positions in $\\mathbb R^3$ | `EuclideanSpace ℝ (Fin 3)`, `a`, `x` | The source's Euclidean norm and real inner product |\n| $\\lambda_i>0$, $\\mu>0$ | `∀ i, 0 < lam i`, `0 < mu` | Strict positivity retained |\n| Distinct poles | `Function.Injective a` | The source configuration assumption |\n| $y_i=a_i-x_*$ and the printed exclusion | `fun i => a i - x`, `∀ i, x ≠ a i` | The off-pole interpretation fixed by #19039 |\n| Equation (3) at zero | `rho0` | The full weighted reciprocal-square sum |\n| Equation (18) | `S` | All ordered triples, denominator factors and normalization retained |\n| Positive without condition (15) | `result : claim` | No balance premise |\n\nNo further binder or hypothesis restricts the claim. Division in the definitions\nis total real division; the theorem's hypotheses make every relevant denominator\nstrictly positive.","title":"Positivity of the JNR triple sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/cork-disney-hogg-2025-jnr-remark13-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8bbf09934429401fa75eb347b35c6b0bb7dd1c3e13ec43249c9ba3c281a5c282.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:coskun-eden-2025-vidinli-octonion-pushforward","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Remark 5.9 of Coşkun–Eden, arXiv:2511.09395v1, asks whether the exceptional\nVidinli algebra $V_7$ has a push-forward description using octonion multiplication.\nThe affirmative assertion is the existence of real-linear maps\n$\\nu:V_7\\to\\mathbb O$ and $\\pi:\\mathbb O\\to V_7$ such that\n$a\\mathbin{\\veebar_7}b=\\pi(\\nu(a)\\nu(b))$ for every $a,b$.\nThe preregistration is [#20661](https://github.com/the-omega-institute/trureturing/issues/20661).\n\n### Statement echo\n\n`vidinliMul` is the coordinate expression of Definition 5.1, with zero-based\nindices in `Fin 7 → ℝ`; index $i$ denotes the paper's coordinate $i+1$.\n`claim` quantifies over both arbitrary real-linear maps and every pair of vectors.\nNeither map is assumed multiplicative or unital, and no retraction identity is\nassumed. `result` proves `¬ claim`.\nThe octonion carrier, multiplication and multiplicative square norm are reused\nfrom the closed source module `D5/S3/VertexAlgebra/MonsterTwistedOctonionNorm`.\nThat owner has no frozen state record on the searched dev revision.\nThe source's higher Cayley–Dickson clause is a separate question.","title":"Coşkun–Eden Remark 5.9: the octonion clause","url":"https://the-omega-institute.github.io/trureturing-pages/research/coskun-eden-2025-vidinli-octonion-pushforward/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7369646b3afad73f6422f6ab9e8b3464680ce76e0c9eaac368a99fc8eb707389.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cotfas-2026-uncertainty-condition-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nicolae Cotfas, *Frame-dependent coherence of Gaussian quantum states*, arXiv:2610.05623v1, asks after Eq. (22): “As concern ρ̃_σ, we do not know if (11) is sufficient in order to have ρ̃_σ ≥ 0. Based on our numerical investigations, we conjecture that this is true.”\n\n[Preregistration #18946](https://github.com/the-omega-institute/trureturing/issues/18946) fixes the source formulas and quantifiers. [The source note](../Library/QuantumContext/cotfas2026dvgs.md) gives the locators and verbatim statements. The settlement is **Proved** by `D5.S3.Quantum.CotfasDVGS.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Eq. (10), continuous Wigner Gaussian | `W`, real inverse covariance and determinant | Literal formula |\n| Odd dimension $d=2s+1\\ge3$ | Natural $s$ with $1\\le s$ in `claim` | Same range |\n| Functions on $\\{-s,\\ldots,s\\}$; addition modulo $d$ | Integer `Finset.Icc` subtype; `ZMod.valMinAbs` | Centered modular representative |\n| Eq. (18), displaced parity | `parity`, coefficient in row $m$, column $j$ | Same phase and selected column |\n| Eq. (22), signed double lattice sum | `wSum`, `tsum` over $\\mathbb Z^2$ | Real division in both coordinates; scalar normalization separated |\n| Eq. (21), weighted finite parity sum | `unnormalized` | Both sums use the centered indices |\n| $c'_\\sigma$ is a normalizing constant | `rhoTilde`, inverse complex trace times `unnormalized` | The proof establishes a nonzero trace |\n| Positive Planck parameter | $h\\in\\mathbb R$, $0 For all $\\lambda \\vdash k$,\n> $\\mathrm{ch}^{(b)}((\\mathcal J_1+\\cdots+\\mathcal J_k)\\cdot\\mathbf p_\\lambda)\n> =D(b)\\cdot\\mathrm{ch}^{(b)}(\\mathbf p_\\lambda)$.\n\nThe original source uses the macros `\\J` and `\\p` for the two symbols above.\nHere every positive integer k and every partition lambda of k is in scope,\nthe scalar field is C(b), and b is an indeterminate. The class vector has\ncoefficient one at precisely the matchings of coset type lambda. Membership\nof the operator image in the class-constant space is part of the assertion,\nsince the characteristic map has that space as its domain. Explicitly,\none coefficient function c on all partitions of k must satisfy both\n\n1. `S_X(indicator lambda)(p) = c(ct(p))` for every actual matching p;\n2. `sum_mu (c(mu)/A_mu) p_mu = D_X((A_lambda)^(-1) p_lambda)`.\n\nThe normalization is literally\n`A_mu=(1+b)^(number of parts of mu) product_r r^(m_r(mu)) m_r(mu)!`.\nThe differential operator acts on C(b)[p_1,p_2,...], with all positive\nderivative indices and the original ordered cut and join sums.","title":"Coulter–Do Conjecture 5.8","url":"https://the-omega-institute.github.io/trureturing-pages/research/coulter-do-2025-conjecture-5-8/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12dc7da691033c0e0109ca3abfe2a300ad2a492c0ba914c972d732f545ad7f29.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:coulter-do-2025-deformed-jucys-murphy-noncommutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Xavier Coulter and Norman Do, arXiv:2506.04002v1, Conjecture 5.4(a), page 34:\n\n> The 𝒥-operators commute when restricted to 𝒳(k) — that is, 𝒥ₘ 𝒥ₙ (v) = 𝒥ₙ 𝒥ₘ (v) for 1 ⩽ m ⩽ n ⩽ k and for all v ∈ 𝒳(k).\n\nHere k is a positive integer, the scalars are ℂ(b) with b an indeterminate,\nand 𝒳(k) is the orbit of the consecutive identity pair partition under the\nunital algebra generated by the operators in Definition 5.1.\nThe theorem `result` proves `¬ claim`, with all these quantifiers retained.","title":"Coulter–Do Conjecture 5.4(a)","url":"https://the-omega-institute.github.io/trureturing-pages/research/coulter-do-2025-deformed-jucys-murphy-noncommutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dad713c645920d389deada5b336330e377d4799419059e367b8370fe6df0da9d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:coutinho-baptista-2023-normalized-laplacian-tree-state-transfer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Coutinho and P. F. Baptista, *Quantum Walks in the Normalized Laplacian*, arXiv:2209.04668v1 (2022; journal publication online 2023, DOI 10.1080/03081087.2023.2236282), Conclusion, `norm-laplacian-v2.tex`, line 427:\n\n> For trees, we were not able to show a complete analogous result to the combinatorial Laplacian case --- that is, that perfect state transfer does not occur. We were however able to significantly restrict the possible cases, and therefore we conjecture that apart from the paths on $2$ and $3$ vertices, no tree admits perfect state transfer according to the normalized Laplacian.\n\nThis quotation was checked against the v1 TeX source, md5 `b274bca76e7a9e881ea5b092556075a9`. The definitions preceding equation (1), lines 89–98, and the transfer definition, lines 161–164, use $\\mathcal L=D^{-1/2}(D-A)D^{-1/2}$, $U(t)=\\exp(it\\mathcal L)$ and $U(\\tau)e_a=\\lambda e_b$ for a complex unit $\\lambda$. See also [the literature note](../Library/QuantumContext/coutinhobaptista2023normalized.md).\n\nIssue [#20154](https://github.com/the-omega-institute/trureturing/issues/20154) fixes the statement: for every natural number $n\\ge4$, every simple graph $G$ on $\\operatorname{Fin}(n)$ with decidable adjacency, and every pair of distinct vertices $a,b$, if $G$ is a tree then there are no $\\tau\\in\\mathbb R$ and $c\\in\\mathbb C$ with $|c|=1$ and $U(\\tau)e_a=c e_b$. This is `claim`, and it is refuted. The two- and three-vertex trees are precisely the excluded paths.","title":"A seventeen-vertex tree admits normalized-Laplacian perfect state transfer","url":"https://the-omega-institute.github.io/trureturing-pages/research/coutinho-baptista-2023-normalized-laplacian-tree-state-transfer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db13367490e7073f67709732df01875562e684d9df0e05f1c7cab4316fc16ac3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:coutinho-guo-schmeits-2025-laplacian-peak-transfer-trees","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gabriel Coutinho, Krystal Guo and Vincent Schmeits, *Peak state transfer in\ncontinuous quantum walks*, arXiv:2505.11986v4, Section 7, Open Problem 7.1\n(p. 21), ask:\n\n> Determine whether infinitely many such trees, not isomorphic to the star graph, admit Laplacian peak state transfer.\n\nFigure 10 (p. 22) includes a root with three hubs and two leaves per hub.\nFor the real Laplacian $L=D-A=\\sum_\\theta\\theta E_\\theta$, Section 3 defines\n$U(t)=e^{itL}$ and $B(L)=\\sum_\\theta|E_\\theta|$, summing over distinct\neigenvalues and taking absolute values entrywise. Peak transfer between\ndistinct vertices $r,h$ means $|U(\\tau)_{h,r}|=B(L)_{h,r}$ at some real time.\nThe source does not require that value to equal one.","title":"Infinitely many non-star trees admit Laplacian peak state transfer","url":"https://the-omega-institute.github.io/trureturing-pages/research/coutinho-guo-schmeits-2025-laplacian-peak-transfer-trees/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c56be450206c65c8a52be5e9847ab5c866591bb53859efd0a88b27cff6ced1ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cowtan-burton-2024-css-monic-pushout-ldpc","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alexander Cowtan and Simon Burton, *CSS code surgery as a universal construction*, Quantum 8 (2024), 1344, arXiv:2301.13738v6, Appendix D, Conjecture D.1:\n\n> Let C• ←f• A• →g• D• be a basis-preserving monic span in Ch(MatF2), and let Q• be the pushout chain complex of this monic span. Further, let the monic span be a representative of a family of monic spans which are parameterised by some n ∈ ℕ, and let A•, C• and D• be the Z-type complexes of quantum LDPC codes. Then (Q•, Q•) is also LDPC. Formulating this conjecture properly requires specifying what it means for a monic span to be parameterised.\n\nThe settlement is Proved for the proposition in #19488, interpreting parameterisation as an arbitrary natural-indexed family.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Based length-two binary complexes | `BasedComplex`: bases `Fin (dim i)`, two `ZMod 2` matrices and their zero composite |\n| Basis-preserving chain maps, Definition 3.4 | `basisMatrix`, `BPChainMap`, including both commutation identities |\n| Monic span | `BPChainMap.Monic`: injectivity of each component matrix map |\n| Basis-preserving pushout, Definition 5.1 and Lemma 5.4 | `glue`, `PBasis`, `sumD2`, `sumD1`, `pushoutD2`, `pushoutD1`; coefficients are fiber sums of a chosen representative column |\n| Natural-indexed family | `claim` quantifies over arbitrary `A`, `C`, `D`, `φ`, `ψ` |\n| Apex and feet are LDPC | The three `IsLDPC` hypotheses in `claim`; `weights2` and `weights1` count all four weights |\n| Pushout family is LDPC | `result : claim`, with uniform bound $w_C+w_D$ |\n\nWrite $E_i=\\operatorname{Fin}(C.\\mathrm{dim}(i))\\sqcup\\operatorname{Fin}(D.\\mathrm{dim}(i))$ and $q_i:E_i\\to\\mathrm{PBasis}(\\varphi,\\psi,i)$ for the quotient map. Under both monicity hypotheses, `D5.S3.Quantum.Information.CssMonicPushoutLdpc.pushoutD2_representative_independent` proves, for every $Y:\\mathrm{PBasis}(\\varphi,\\psi,2)$, $y:E_2$ with $q_2(y)=Y$, and $X:\\mathrm{PBasis}(\\varphi,\\psi,1)$,\n\n$$\\mathrm{pushoutD2}(\\varphi,\\psi)(X,Y)=\\sum_{x\\in E_1,\\ q_1(x)=X}\\mathrm{sumD2}(C,D)(x,y).$$\n\n`D5.S3.Quantum.Information.CssMonicPushoutLdpc.pushoutD1_representative_independent` proves the analogous formula for $Y:\\mathrm{PBasis}(\\varphi,\\psi,1)$, $y:E_1$ with $q_1(y)=Y$, and $X:\\mathrm{PBasis}(\\varphi,\\psi,0)$:\n\n$$\\mathrm{pushoutD1}(\\varphi,\\psi)(X,Y)=\\sum_{x\\in E_0,\\ q_0(x)=X}\\mathrm{sumD1}(C,D)(x,y).$$\n\nThus the two matrices represent the induced maps $[y]\\mapsto[\\partial y]$, independently of the selected representative. The degree-two and degree-one formulas use `BPChainMap.comm2` and `BPChainMap.comm1`, respectively. The identity $\\partial^Q_1\\partial^Q_2=0$ is checked by the pinned experiment below; it is not delivered as a Lean theorem.\n\nThe categorical universal property is proved in prose here, rather than by a Lean categorical theorem. In degree $i$, take the free binary vector space on $\\mathrm{PBasis}(\\varphi,\\psi,i)$. Its maps from the two feet send a basis vector $x$ to $[x]$ and agree on every apex basis vector. Any two linear maps from the feet agreeing on the apex assign equal values to each generating glued pair, so extend uniquely to a linear map on the quotient basis. The representative formulas above identify its differentials with $[y]\\mapsto[\\partial y]$. If the two maps from the feet are chain maps, that extension commutes with the differential on every quotient generator, hence everywhere by linearity. Uniqueness follows because those generators span. This identifies the construction with the basis-preserving pushout of Lemma 5.4.","title":"Cowtan–Burton Conjecture D.1: LDPC monic CSS pushouts","url":"https://the-omega-institute.github.io/trureturing-pages/research/cowtan-burton-2024-css-monic-pushout-ldpc/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/48d01f813d8ee08835cd391d3131f35ecac671e03aadb11356fe8c2da3073a13.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:crim-rectair-square-conjecture-two-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bašić, Gottlieb, and Krnc, *CRIM: A Natural Game on Integer Partitions*,\narXiv:2606.16828v1, printed page 18, Conjecture 2 states a three-branch\nSprague-Grundy formula for every square rectair `R^k_{r,r}` with\n`0 <= k < r`.\n\nThe formal claim quantifies natural numbers `r` and `k` with `k < r` and\nuses the published three branches: value zero when `r` is even or\n`k < r - 1`, value one for `r` equal to 3 or 5, and value two otherwise.","title":"Refutation of the CRIM square-rectair formula at its boundary","url":"https://the-omega-institute.github.io/trureturing-pages/research/crim-rectair-square-conjecture-two-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50c66bf7341daba8ec8d7a58593bec1a983938083c37d4549bf99a2d6ddab6d1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:critch-morton-2014-open-boundary-mps-marginal-birationality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Critch and J. Morton, *Algebraic geometry of matrix product states*, SIGMA 10 (2014), 095, state Conjecture 8 in the source TeX `sigma14-095.tex`, lines 430–434:\n\n> For $N\\geq 4$, a~generic~$N$-qubit state can be recovered from its improper margina\\-li\\-zation to any three consecutive states.\n> That is, each improper marginalization map $\\overline\\OB(2,2,N)\\to\\overline\\OB(2,2,3)$ is\n> a~birational equivalence of varieties.\n\nThe paper defines an open-boundary state by $\\Psi_{i_1\\dots i_N}=b_0^TA_{i_1}\\cdots A_{i_N}b_1$ and defines $\\overline\\OB(2,2,N)$ as the Zariski closure of these states (source lines 349–363). Its improper marginalization sums amplitudes over all indices outside a chosen consecutive triple (lines 419–426). The Lean reading uses `closure n` for the affine zero locus of the vanishing ideal and `improperMarginal n a h` for this coherent amplitude sum, where $h:a+3\\le n$.","title":"Every three-site improper marginalization is birational for open-boundary binary bond-two MPS","url":"https://the-omega-institute.github.io/trureturing-pages/research/critch-morton-2014-open-boundary-mps-marginal-birationality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eefeae3038bc9e39afee9cd456a2a6c3d5abbf60944e619e0d04799ffa9b7063.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:crown-order-polytope-log-concavity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 3.7 on PDF page 12 of the pinned v3 source states:\n\n> For any positive integer n, the entries of the f-vector of O(C₂ₙ) form a log-concave sequence.\n\nThe calligraphic symbols are transcribed here as `O` and `C`.\nThe crown has source cover relations `1<2>3<...>2n-1<2n>1`.\nIts order polytope consists of real coordinates between zero and one\nrespecting those comparisons. Its full f-vector includes the empty face\nand the whole `2n`-dimensional polytope. Precisely, for all `n,k : Nat`\nwith `00`. The full vector has `2n+2` entries.","title":"Crown order-polytope f-vector log-concavity","url":"https://the-omega-institute.github.io/trureturing-pages/research/crown-order-polytope-log-concavity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2b53e55e1648f9d26b43c4e7a9276e31fa3dd3a5353bc439e43abfddcde913f7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:crux-problem-2623-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zejnulahi and Arslanagic, *Problem 2623*, Crux Mathematicorum with\nMathematical Mayhem 27, no. 2 (2001), displayed page 139, define positive\ncyclic variables by `x_(n+r) = x_r` and\n\n`S_k = sum_(j=1)^n (sum_(i=0)^k x_(j+i)) / (sum_(i=0)^k x_(j+i+1))`.\n\nThey ask verbatim:\n\n> Prove or disprove that S_k >= S_(k+1).\n\nThe formal claim quantifies over every `n >= 2`, every positive real-valued\ncyclic function on `ZMod n`, and every interior index `k` with `k+2 <= n`.\nIt asserts `S_(k+1) <= S_k`.","title":"Refutation of Crux Problem 2623","url":"https://the-omega-institute.github.io/trureturing-pages/research/crux-problem-2623-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af7fd1f8d0ed834767208bc50c484dc7102c45a82b952dc38ed2fafc02f2ac2c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cruz-menendez-2023-fiedler-resistance-invariant-weights","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cruz Ortega and Menéndez-Conde, *Conectividad algebraica máxima para caminos con resistencia total fija*, arXiv:2306.08187v1 and *Pesquimat* 26(2) (2023), Section 5, define\n\n$$\n\\breve a(G)=\\max\\{\\lambda_2(G;\\omega)\\mid R(G;\\omega)=R(G)\\}\n$$\n\nand conjecture that the maximum is always attained by edge weights invariant under graph automorphisms. Issue #15242 records the literal source statement, the $K_{2,m}$ route and the Lean conventions.","title":"Refutation of the fixed-resistance Fiedler invariance conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/cruz-menendez-2023-fiedler-resistance-invariant-weights/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e8156b06cd1972d62ecf980d7a0e29614bc4feb513797814245660919c053f9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cui-mehta-mousavi-nezhadi-2020-chsh-group-decomposition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Cui, Mehta, Mousavi and Nezhadi, *A generalization of CHSH and the algebraic structure of optimal strategies*, Quantum 4, 346 (2020), arXiv:1911.01593v5, Section 1.4 (Further work), item 2, ask:\n\n> In Section 5, we give efficient explicit presentations for G_n and its multiplication table. Can we go further and characterize these groups in terms of direct and semidirect products of small well-known groups? The first few cases are as follows\n\nFor every natural $n\\ge2$, the answer is\n\n$$G_n\\cong\\left(C_{2n}\\times C_2^{\\,n-2}\\right)\\rtimes_{\\alpha_n}C_n.$$\n\nThe group $G_n$ has the source's Section 5 presentation on $P_0,P_1,J$: the relators are $P_0^n,P_1^n,J^n,[J,P_0],[J,P_1]$ and $J^i(P_0^iP_1^{-i})^2$ for $1\\le i\\le\\lfloor n/2\\rfloor$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The Section 5 presentation | `P0`, `P1`, `J`, `rels`, `G` | The relators are words equal to the identity; the negative exponent uses the integer cast of the natural index. |\n| Every $n\\ge2$ | `claim` quantifies over natural `n` with `2 ≤ n` | No further assumption on $n$. |\n| Small cyclic factors | `L` is `Multiplicative (ZMod (2*n))` times the functions from `Fin (n-2)` to `Multiplicative (ZMod 2)` | The independent involutions have zero-based indices. |\n| An explicit semidirect product | `alpha` and the `MulEquiv` in `claim` | The action is the binding action in [the definition comment on #16204](https://github.com/the-omega-institute/trureturing/issues/16204#issuecomment-6146781824). |\n| A product characterization | `result : claim` | An actual multiplicative equivalence, rather than an order comparison. |\n\nThe binding comment fixes `cyclicHom`, `s`, `t`, `liftL`, `ts`, `u`, `e`, `step`, `stepAut` and `alpha`. Here `t n i` represents the source's $t_{i+1}$, and `u n k` uses one-based indices. In particular, $u_{n-1}=s^n\\prod_{i=1}^{n-2}t_i$, while $u_k=1$ outside the indicated range. For $n=2$, $u_1=s^2$. The action below two is trivial; the claim has no conclusion for those values.","title":"A product decomposition of the generalized CHSH groups","url":"https://the-omega-institute.github.io/trureturing-pages/research/cui-mehta-mousavi-nezhadi-2020-chsh-group-decomposition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad38912d356d24258204b51b10af1cabda9c54c9c9135ccf34e636c1a7dc65a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cui-sun-2026-legendre-odd-power-telescoper","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Li-Li Cui and Zhi-Hong Sun, *Curious identities involving Legendre polynomials and Apéry-like numbers*, arXiv:2607.12330v1, page 10, state:\n\n> Suppose that m, p ∈ Z^+. Then there are integral polynomials f_i(t) with degree i (i = 0, 1, …, m − 1) such that (1 − x)^{m+1} Σ_{n=0}^{p−1} (2n + 1)^{2m+1} P_n(x) = pL_m(p, 1 − x)P_{p−1}(x) − pL_m(−p, 1 − x)P_p(x), where L_m(p, t) = (2p + 1)^{2m} t^m + f_{m−1}((2p + 1)^2) t^{m−1} + · · · + f_1((2p + 1)^2) t + f_0.\n\nEquation (1.1), page 1, specifies:\n\n> The famous Legendre polynomials {P_n(x)} are given by P_0(x) = 1, P_1(x) = x and (n + 1)P_{n+1}(x) = (2n + 1)xP_n(x) − nP_{n−1}(x) (n ≥ 1).\n\nThe coefficient family is chosen once for each positive m and works for every positive p. The model is an identity in ℚ[X], with integral coefficient polynomials in ℤ[s]; the endpoint parameter of L is an integer.","title":"Cui–Sun Conjecture 2.1: integral Legendre telescopers","url":"https://the-omega-institute.github.io/trureturing-pages/research/cui-sun-2026-legendre-odd-power-telescoper/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1337d82a392b8411d21261c140c630fa6d9d75c5c70b0d515917c217e53df4df.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cullinan-young-2026-krawtchouk-sign-leading-coefficient-remark-8","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Cullinan and E. Young, *The Gibbs phenomenon for the Krawtchouk polynomials*,\narXiv:2603.05408v1, Remark 8, state that numerical experimentation suggests\nthat for even $N$ the leading coefficient of the sign interpolant $I_N$ is\n\n$$(-1)^{(N-2)/2} C((N-2)/2)/(N-1)!,$$\n\nwhere $C(n)$ is the $n$th Catalan number. Section 2 defines\n\n$$I_N(x)=\\sum_{i=0}^{N}\\operatorname{sgn}(i-N/2)\\ell_i(x),qquad\n\\ell_i(x)=\\prod_{0\\le j\\le N,\\,j\\ne i}\\frac{x-(j-N/2)}{i-j}.$$\n\nThe settling declaration is `D5.S1.Recurrence.Interpolation.CullinanYoungKrawtchouk.result`.\n\n### Statement echo\n\nThe source nodes $j-N/2$, signs, products, sums, and the even case are literal\nin `ell`, `I`, and `claim`. The Lean proposition quantifies $N\\in\\mathbb N$,\nassumes $2\\le N$ and `Even N`, and equates\n`Polynomial.leadingCoeff (I N)` with the rational Catalan expression using\n`Nat.div` and `Nat.factorial`. The source's Proposition 6 identification\n$F_N=I_N$ is outside this claim.","title":"Cullinan–Young Remark 8 leading coefficient","url":"https://the-omega-institute.github.io/trureturing-pages/research/cullinan-young-2026-krawtchouk-sign-leading-coefficient-remark-8/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5446eabcd2b93a9eb600b551ab0079fa8d3508a7d57f46eca59ad7424333bc29.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cuntz2009","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Michael Cuntz, *Integral modular data and congruences*, Section 7, Question (5), arXiv:math/0611233v2 p. 21, journal p. 380, asks:\n\n> Does an integral Fourier matrix $s$ exist with a column $k$ such that $\\forall i$ $s_{ik}\\ne0$ and $\\exists i$ $s_{ik}\\ne\\pm1$?\n\nThe answer is negative for the exact existential assertion. Preregistration: [#22614](https://github.com/the-omega-institute/trureturing/issues/22614). The source note is [Cuntz](../Library/Fourier/cuntz2009integralmodulardata.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive finite size and an integer square matrix | `claim` quantifies `n`, `hn`, `s` | All sizes; source indices $1,\\ldots,n$ correspond to `Fin n`, with source index one represented by zero. |\n| Definition 4.1(1): first column one and nonzero determinant | First two conjuncts of `FourierAxioms` | Integer determinant, unchanged. |\n| Definition 3.1: positive square roots | Third conjunct of `FourierAxioms` | Every square root is positive. |\n| Definition 4.1(2): $ss^T=\\operatorname{diag}(d)$ | Fourth conjunct of `FourierAxioms` | Exact integer-to-real cast; $d$ is existentially quantified by `IsIntegralFourierMatrix`. |\n| Definition 4.1(3): $\\sqrt{d_j}s_{ij}=\\sqrt{d_i}s_{ji}$ | Fifth conjunct of `FourierAxioms` | Every pair of indices. |\n| Definition 4.1(4): integral structure constants | Sixth conjunct of `FourierAxioms` | Every triple; each real sum equals an integer witness. |\n| A zero-free column containing a nonunit | Final witnesses of `claim` | Every entry nonzero and some entry different from both $1$ and $-1$. |\n| Negative answer | `result` | The exact negation of `claim`. |\n\nDefinition 4.1 names integral Fourier matrices before adding congruence conditions (5)–(6); neither these conditions nor a $T$-matrix is required. The matrix equation fixes row norms. The reversed indices in Definition 3.1's explanatory norm sum are not an additional hypothesis. No extra binder or conjunct weakens the source question.","title":"Cuntz's integral Fourier zero-free column question","url":"https://the-omega-institute.github.io/trureturing-pages/research/cuntz2009/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b36b435e5a2be27819cd62afc6ba965b1be02b9cfd3e00844e85b1ed3497e86.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:cycle-star-irregularity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Salma–Nagesh–Narahari, *On edge irregularity strength of cycle-star graphs*,\nProyecciones 43(3), 725–741, arXiv:2405.12263v1, Conjecture 3.1, states:\n\n> Let $G=CS_{k,n-k}$ be a cycle-star graph. For $k\\geq8$ and $n-k\\geq1$,\n> $$es(G):=\\begin{cases}\\lceil(n+1)/2\\rceil&\\text{for }k+1\\leq n\\leq2k-4,\\\\n-k+2&\\text{for }n\\geq2k-3.\\end{cases}$$\n\nThe graph consists of a cycle of length $k$ with $n-k$ leaves appended to\none vertex. A vertex labeling takes values in $\\{1,\\ldots,K\\}$ and may repeat\nlabels; different unordered edges must have different endpoint sums.\nThe strength is the least positive feasible bound and includes attainment.\nThe source and binding conventions are in [#22868](https://github.com/the-omega-institute/trureturing/issues/22868).\nThe source note is [Salma–Nagesh–Narahari](../Library/FiniteGeometry/salmanageshnarahari2024cyclestar.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every $k\\geq8$ and $n-k\\geq1$ | `claim` quantifies natural $k,n$ with both hypotheses | Exact source domain |\n| Vertices $v,v_i$, $1\\leq i\\leq n-1$ | `cycleStar` has carrier `Fin n`; zero represents $v$ | Exact indexing |\n| Leaf edges, both cycle-closing edges, consecutive internal edges | `cycleStar` uses the three source index ranges inside `SimpleGraph.fromRel` | Undirected, loopless edge set |\n| Labels $1$ through $K$, repetitions allowed | `IsKLabeling` | No vertex-injectivity assumption |\n| Edge weights are endpoint sums; different edges have different weights | `edgeWeight`, `IsEdgeIrregularLabeling`, unordered equality in `Sym2` | Reversing orientation preserves the edge |\n| Minimum feasible label bound | `IsEdgeIrregularityStrength` uses `IsLeast` on positive feasible bounds | Attainment and comparison with every feasible bound |\n| $k+1\\leq n\\leq2k-4$: $\\lceil(n+1)/2\\rceil$ | First conjunct of `claim`, natural division $(n+2)/2$ | Same ceiling |\n| $n\\geq2k-3$: $n-k+2$ | Second conjunct of `claim` | Exact source value |\n\nNo hypothesis assumes a labeling exists. Positivity of feasible bounds is\nthe source's label-bound convention, rather than an additional graph\nhypothesis. Under the source domain, the two ranges exhaust all parameters.","title":"Edge irregularity strength of cycle-star graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/cycle-star-irregularity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16fe600f51cd555a2c307929032f004a93fdbe7ba3bfacf8ceef2c161f0815e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:czaplinski-michalek-seynnaeve-2023-umps-expected-dimension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Czapliński, M. Michałek and T. Seynnaeve, *Uniform matrix product states from an algebraic geometer's point of view*, arXiv:1904.07563v1 (2019; Adv. Appl. Math. 142 (2023), 102417), Section 2, Conjecture 2.14:\n\n> For every choice of $(D,d,N)$, $\\IMPS(D,d,N)$ has expected dimension.\n\nThis is the verbatim text of `uMPS-final.tex`, line 312, inside the conjecture at lines 311–313; the source's macro `\\IMPS` denotes uMPS. Definition 2.1 fixes $D,d,N\\ge1$. For $d$ complex $D\\times D$ matrices, the coefficient indexed by a word $(i_1,\\ldots,i_N)$ is $\\operatorname{tr}(M_{i_1}\\cdots M_{i_N})$, and $\\mathrm{uMPS}(D,d,N)$ is the image of this parametrization (lines 180–193).\n\nThe expected-dimension equality in lines 299–305 is\n\n$\\dim\\overline{\\mathrm{uMPS}(D,d,N)}=\\min\\{(d-1)D^2+1,\\dim\\mathrm{Cyc}^N(\\mathbb C^d)\\}$,\n\nwhere $\\mathrm{Cyc}^N(\\mathbb C^d)$ is the space of cyclically invariant tensors (line 209). Dimension is interpreted through the coordinate ring of the Zariski closure, as in the preregistration and the module's `varietyDim`. The source introduces this closure at line 211 and its dimension bound at lines 272–274.\n\nIssue [#23659](https://github.com/the-omega-institute/trureturing/issues/23659) fixes this universal reading. It is refuted at $(D,d,N)=(2,5,2)$. The definitions and source locators are also recorded in [the literature note](../Library/Geometry/czaplinskimichalekseynnaeve2023umps.md).","title":"Two-site uniform matrix product states need not have expected dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/czaplinski-michalek-seynnaeve-2023-umps-expected-dimension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/647989c865ce968c3bf7aa7662fd39e0f52500b1e3a824bfa698bdf8630575c4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:czartowski-braun-zyczkowski-2019-qutrit-complementary-entropy-frontier","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Czartowski, D. Braun and K. Życzkowski, *Trade-off relations for operation entropy of complementary quantum channels*, Int. J. Quantum Inf. 17, 1950046 (2019), arXiv:1908.03492v1, study the pairs (S_map(Φ), S_map(Φ̃)) of the map entropy of a channel and of its complementary channel.\n- **The map entropy.** S_map(Φ) is the von Neumann entropy of the normalized Choi state. By Proposition 1, S_map(Φ) = S(Φ̃(I/N)) and S_map(Φ̃) = S(Φ(I/N)).\n- **The conjecture.** Conjecture 1 (page 5): \"For qutrit channels with qutrit environment the lower boundary of the allowed set A_3 is given by the parametric curve p(a) = (−a log a − (1 − a) log(1 − a), log 3/3 − (a + 1/3) log(a + 1/3) − (a − 1/3) log(a − 1/3)) (20) with a ∈ (0, 1/3), together with reflection through line S = S̃\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/czartowski2019tradeoff.md).\n\nIssue [#17038](https://github.com/the-omega-institute/trureturing/issues/17038) fixes the reading:\n- The last term of Eq. (20) is read as −(1/3 − a) log(1/3 − a): the printed term is not real for a < 1/3, and the source's attaining channels give the corrected one.\n- Put h(a) = H(1 − a, a) and g(a) = H(1/3 + a, 1/3, 1/3 − a). The lower boundary means two things for a ∈ [0, 1/3]:\n - for every isometry V : ℂ³ → ℂ³ ⊗ ℂ³, S_map(Φ) ≤ h(a) implies S_map(Φ̃) ≥ g(a), and the same with Φ and Φ̃ exchanged;\n - the curve and its reflection are attained.\n\nThe conjecture is proved in this reading.","title":"The lower boundary of the entropy region of qutrit channels and their complements is the conjectured curve","url":"https://the-omega-institute.github.io/trureturing-pages/research/czartowski-braun-zyczkowski-2019-qutrit-complementary-entropy-frontier/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1fa82ec0d3fe9205528b16b78bad9f97f0005b6bc4751d0d423d8927d83a49f4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:czartowski-deoliveirajunior-korzekwa-2023-thermal-recall-truncation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Czartowski, A. de Oliveira Junior and K. Korzekwa, *Thermal Recall: Memory-Assisted Markovian Thermal Processes*, PRX Quantum 4, 040304 (2023), arXiv:2303.12840v2, introduce a memory-assisted protocol for approximating permutations by sequential two-level thermalisations. Conjecture 1, “Improved convergence,” is stated in the source TeX `TR.tex`, lines 417–427:\n\n> In the infinite temperature limit, $\\beta=0$, and for an $N$-dimensional memory, $\\widetilde{\\mathcal{P}}^{{\\Pi}}$ gives a better approximation of a permutation $\\Pi$ than $\\mathcal{P}^{\\Pi}$:\n> $$\\delta\\left( \\Pi \\mathbf p,\\tilde{\\mathbf q}\\right) \\leq \\delta\\left( \\Pi \\mathbf p, \\mathbf{q}\\right),$$\n> where $\\tilde{\\mathbf{q}}$ and $\\mathbf{q}$ are defined via $\\widetilde{\\mathcal{P}}^{{\\Pi}}(\\mathbf{p}\\otimes \\boldsymbol\\eta_M) = \\tilde{\\mathbf{q}}\\otimes \\boldsymbol\\eta_M$ and $\\mathcal{P}^{{\\Pi}}(\\mathbf{p}\\otimes \\boldsymbol\\eta_M) = \\mathbf{q}\\otimes \\boldsymbol\\eta_M$.\n\nThe repository proposition `claim` is the following exact finite reading. For every $d,n\\in\\mathbb N$, $p:\\mathrm{Fin}\\,d\\to\\mathbb R$, and word $w:\\mathrm{List}(\\mathrm{Fin}\\,d\\times\\mathrm{Fin}\\,d)$, if $0 Let f : (0, ∞) → R. Then f is radially convex of order n (with respect to the Frobenius norm) if and only if f is matrix monotone of degree n.\n\narXiv:2610.17393v1, §4, directly after Conjecture 4.10, verbatim:\n\n> By Proposition 3.9, matrix monotonicity of degree n implies radial convexity of order n. Thus, it remains to prove the converse implication. The next theorem establishes the conjecture for n = 2.\n\nThe matrices are complex Hermitian positive definite, and the order is Loewner order. Radial convexity requires both endpoint distance inequalities for every ordered triple $A\\preceq C\\preceq B$. Matrix monotonicity allows either of the two uniform order directions. The universal assertion quantifies over every $n\\ge1$ and every real function on the positive half-line.","title":"Czerwińska–Franco–Merino: radial convexity and matrix monotonicity","url":"https://the-omega-institute.github.io/trureturing-pages/research/czerwinska-franco-merino-2026-radial-convexity-matrix-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50c199953a026461e4c28000dfbc343f3baea692911e9e172441eb19d2f7be9f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dai-2026-cross-polytope-face-counts","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dai, Hou, Liu, Thawinrak and Wang, *Counting Lattice Points in Minkowski\nSums of Cross Polytopes*, arXiv:2608.16037v2, Problem 5.4, asks:\n\n> Given their close connection to Minkowski sums of simplices, can one apply similar tools and techniques in [15] to study the faces of Minkowski sums of cross polytopes? In particular, can one give a formula for their f- and h-vectors (for those simple polytopes)?\n\nEquation (3) specifies $Q=\\sum_i\\lambda_i\\diamondsuit_{I_i}$ with\nnonempty coordinate supports $I_i$ and nonnegative integer coefficients.\nThe face-count statement below extends the coefficients to nonnegative\nreals. For arbitrary natural $n,m$, set\n$A=\\{i:\\lambda_i>0\\}$, $U=\\bigcup_{i\\in A}I_i$ and $d=|U|$.\nFor $0\\le r\\le d$, $f_r(Q)$ counts actual nonempty exposed faces of\nactual affine dimension $r$, including the whole polytope. The empty-face\nentry is separately $f_{-1}=1$.","title":"Face counts for coordinate cross-polytope Minkowski sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/dai-2026-cross-polytope-face-counts/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19cc0bab36de240ba408c923d5331bcb4c3074ce4b39f066ad13a165a1d15fe1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dalfo-fiol-reyes-three-quarter-conjecture-24","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 2.4 of Dalfó, Fiol, and Reyes, *A note on three-quarters circulant\ndigraphs*, section 2, quantifies over every integer `k >= 1`. It specifies\n`N=k^2+4k-2`, steps `a=1` and `b=-k-4`, and lattice columns `(k+2,1-k)` and\n`(2,k)`. It says these data give a tile of a three-quarters digraph of sector\ndiameter `k`. The paper defines the sector distance using exactly the three\ncongruences `m*a+n*b`, `-m*a+n*b`, and `m*a-n*b` for `m,n >= 0`.","title":"Dalfó-Fiol-Reyes Conjecture 2.4","url":"https://the-omega-institute.github.io/trureturing-pages/research/dalfo-fiol-reyes-three-quarter-conjecture-24/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/030ba557f39a9f39e8ad13f3c5aa1434ebe160f5777c7d0aa6945d64a4aebe81.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:daly-2022-superchannel-minimal-extension-extremality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. C. Daly, *Axiomatic Approach to Quantum Superchannels*,\narXiv:2210.00370v1, Remark 4.1, asks:\n\n> The set of possible superchannel extensions of a QSC is a convex set. In the example given the extensions with minimal e are extreme points. A natural question to ask is whether it is generally true that the extensions which give minimal dimensions e are extreme points.\n\nThe [Library note](../Library/QuantumContext/daly2022axiomatic.md) quotes\nthe definitions and locators. Here $e$ is exactly the source's rank of the\nChoi matrix after tracing out both output factors. Convexity and extremality\nuse real scalars. This question is the universal assertion about **every**\nminimizer; existence of **some** extreme minimizer is a weaker assertion.\n\n### Statement echo\n\nAll names in this table belong to\n`D5.S3.QuantumChannels.DalySuperchannelMinimalExtension`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Definition 3.1: complex span of CPTP maps | `SCPTP` | `Submodule.span` of completely positive maps satisfying trace preservation on every matrix |\n| Remark 3.3: tensor product of matrix maps | `kronBilin`, `kronLift` | Bilinear `MatrixMap.kron`, lifted to all tensors |\n| Definition 3.4, conditions (1) and (2) | `IsQSC` | CPTP preservation and complete CP preservation on channel spans, for every auxiliary dimension and tensor |\n| Remark 3.7, all three bullets | `IsQuantumSuperchannel` | CP preservation, complete CP preservation and TP preservation on the full map spaces |\n| Full superchannels agreeing with the QSC | `extensions` | Equality of matrix maps on every element of the channel span |\n| Section 2: $C_L=(L(E_{ij}))$ | `choi`, `choiEquiv` | Input-first block entries and the double-sum inverse |\n| Induced map $\\widetilde S=R_2SR_1^{-1}$ | `induced` | Composition with the two Choi equivalences |\n| Section 4: $e=\\operatorname{rank}(\\operatorname{Tr}_{r_1}\\operatorname{Tr}_{r_2}C_{\\widetilde S})$ | `e` | Reindex to input-pair/output-pair order and trace out the output pair |\n| All positive dimensions, all QSCs, every minimizing extension is extreme | `claim` | Four lower bounds, QSC hypothesis, membership, comparison with every extension, real extreme points |\n| Negative answer | `result` | Exact negation of `claim` |\n\n`Fin n` uses zero-based indices. Complex-linear spaces and real convex\ncombinations express the two different scalar roles in the source; neither\nadds a hypothesis to the question. The input-first Choi owner used in `e`\nhas the same matrix-unit expression as `choi`.","title":"A non-extreme minimum-rank superchannel extension","url":"https://the-omega-institute.github.io/trureturing-pages/research/daly-2022-superchannel-minimal-extension-extremality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6862963356d67c6ae24dad30269aa1080ebe73f7c0d18ca71f6d472f9c6ecaa6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:das-2026-whole-cyclic-sgb-extension-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Das–Erfanian–Nath, *Certain topological indices and spectral properties of SGB-graphs of finite\ncyclic groups*, arXiv:2602.07587v1, closes Section 5 with an unlabelled invitation to extend\nits preceding complete list to any cyclic group. The source's ambient groups are finite.\nThe list was established there for orders $p^n,pq,p^2q,p^2q^2$, with distinct primes $p,q$\nand $n\\geq1$; those families are known source results.\n\nFor every finite cyclic group $G$, including order one, let $B(G)$ have vertex type\n$(G\\times G)\\sqcup\\operatorname{Subgroup}(G)$. All ordered and repeated pairs, the identity pair,\nand all subgroups including the trivial one remain present. Cross adjacency is exactly\n$\\langle a,b\\rangle=H$; same-part vertices are never adjacent.\n\nThe whole extension asserts the conjunction of these ten clauses:\n\n1. The adjacency spectrum is not integral.\n2. The Laplacian spectrum is integral.\n3. The signless Laplacian spectrum is integral.\n4. The common-neighbour spectrum is integral.\n5. $E(B(G))<|V(B(G))|$.\n6. $E(B(G))\\leq E(K_{|V(B(G))|})$.\n7. $LE(B(G))\\leq LE(K_{|V(B(G))|})$.\n8. $LE^+(B(G))\\leq LE^+(K_{|V(B(G))|})$.\n9. $E_{CN}(B(G))\\leq E_{CN}(K_{|V(B(G))|})$.\n10. $E(B(G))\\leq LE(B(G))$.\n\nHere $A$ is literal adjacency, $D$ is diagonal degree, $L=D-A$, $Q=D+A$, and $CN$ counts\ncommon neighbours off the diagonal and is zero on it. Spectra are actual symmetric-matrix\nspectra with algebraic multiplicity. Integral means every real eigenvalue is the real image\nof an integer. Energies sum absolute eigenvalues; only $LE,LE^+$ subtract the whole graph's\n$2m/N$. Each benchmark uses its respective energy of an actual same-carrier complete graph.\nThus no separate formula or indexing transport is assumed for a benchmark.","title":"The whole finite-cyclic SGB extension","url":"https://the-omega-institute.github.io/trureturing-pages/research/das-2026-whole-cyclic-sgb-extension-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01a943165338389828e4eb8ae8a5fa2a6d7078a33f3dd893decffed6a2a6ab1f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:das-et-al-2025-onb-tomography-holevo-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Das, L. O. Conlon, J. Suzuki, S. K. Yung, P. K. Lam and S. M. Assad, *Holevo Cramér-Rao bound: How close can we get without entangling measurements?*, Quantum 9, 1867 (2025), DOI 10.22331/q-2025-09-30-1867; arXiv:2405.09622v2, Section 3.2.1, after Theorem 3, `DraftQuantumSubmission.tex`, lines 1147–1155:\n\n> Although we prove an upper bound of~$d+2$, based on numerical evidence shown in Fig.~\\ref{fig:estimateallarbitstate}, we expect the attainable bound to be~$d+1$ and expect~$\\rat[\\{\\rho_\\theta\\}]$ to be maximised at~$\\theta=0$ over~$\\Theta$. Thus for the linear GMM model, we propose~$\\rat[\\{\\rho_\\theta\\}]\\leq d+1$.\n\nThe quotation preserves the TeX macros: `\\rat` denotes $\\mathcal R^{\\mathrm{NH}}$. The source defines $\\mathcal R^{\\mathrm{NH}}[\\{\\rho_\\theta\\}]=\\max_{\\theta\\in\\Theta}\\mathrm C_{\\mathrm{NHCRB}}[\\rho_\\theta]/\\mathrm C_{\\mathrm{HCRB}}[\\rho_\\theta]$ in Eq. `ratiodef`, lines 798–801.\n\nFor $d\\ge2$, put $n=d^2-1$. Let $b_1,\\ldots,b_n$ be traceless Hermitian $d\\times d$ complex matrices with $\\operatorname{tr}(b_jb_k)=\\delta_{jk}$, and let $\\theta\\in\\mathbb R^n$. The model is $\\rho_\\theta=I_d/d+\\sum_j\\theta_jb_j$, with $\\Theta$ consisting of the parameters making $\\rho_\\theta$ positive semidefinite. Its trace is one. This includes the source's generalized Gell-Mann model, whose normalization is specified in Section 5.2, lines 2071–2074.\n\nThe bounds use identity weight and the source's uncentered locally unbiased operators. Precisely:\n\n- `IsLUB` requires every $X_k$ to be Hermitian, $\\operatorname{tr}(\\rho_\\theta X_k)=\\theta_k$, and $\\operatorname{tr}(b_jX_k)=\\delta_{jk}$.\n- `zMatrix` has entries $Z_{jk}=\\operatorname{tr}(\\rho_\\theta X_jX_k)$; `imPart` takes the entrywise imaginary part and embeds it back into complex matrices. The trace norm is $\\lVert A\\rVert_1=\\operatorname{Re}\\operatorname{tr}\\sqrt{A^\\dagger A}$.\n- `IsAdmissible` requires Hermitian blocks $L_{jk}=L_{kj}$ and positive semidefiniteness of the matrix with blocks $L_{jk}-X_jX_k$, indexed by parameter–quantum pairs.\n- `hcrb` is $\\inf_X\\{\\operatorname{Re}\\operatorname{tr}Z+\\lVert\\operatorname{Im}Z\\rVert_1\\}-\\sum_j\\theta_j^2$, over `IsLUB` families; `nhcrb` is $\\inf_{X,L}\\operatorname{Re}\\sum_j\\operatorname{tr}(\\rho_\\theta L_{jj})-\\sum_j\\theta_j^2$, over `IsLUB` families and `IsAdmissible` blocks.\n\nThese are the infimum forms of the source's Eqs. `HNCRBdefn`, `origunbiaseq` and `Holevodefn1`, lines 588–602 and 658–666. The definitions are also described in [the literature note](../Library/Estimation/dasetal2025holevo.md).\n\nIssue [#25805](https://github.com/the-omega-institute/trureturing/issues/25805) fixes the settlement as the following multiplicative inequality: for every such $d,b,\\theta$ with $\\rho_\\theta$ positive semidefinite, $\\mathrm C_{\\mathrm{NHCRB}}[\\rho_\\theta]\\le(d+1)\\mathrm C_{\\mathrm{HCRB}}[\\rho_\\theta]$. This is `claim`, proved by `result`.","title":"The NHCRB/HCRB ratio for full orthonormal qudit tomography is at most $d+1$","url":"https://the-omega-institute.github.io/trureturing-pages/research/das-et-al-2025-onb-tomography-holevo-ratio/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c4520a61319df40fb162a846768555777238443e6dadda600324ad12123a0b9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:das-mohanty-singular-squared-distance-laplacian","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The single target is the complete two-clause concluding conjecture identified by `D5/L/QuadraticForms/dasmohanty2024inverse`. The Library note owns the bibliography, equation (3.6), omitted-list locators and source qualifications. The exact proposition is `DasMohantySingularSquaredDistance.completeConjecture`; the theorem `DasMohantySingularSquaredDistance.result` has that proposition as its type. Issue https://github.com/the-omega-institute/trureturing/issues/25619 and immutable preregistration comment 6574192179 identify the external event.\n\nFor every finite number t of parts with t at least two, and every positive integer part-size function n on Fin t, let N be the sum of the sizes, s the number of parts of size at least two and h the number of singleton parts. Vertices are pairs consisting of a part and a vertex in that part. The real squared-distance matrix delta has diagonal zero, same-part off-diagonal four and between-part one. Assume delta is singular and the sum of all its cofactors is nonzero. The latter is the sum of all adjugate entries, not a principal cofactor.\n\nThe functions phi, psi and theta, their single and double omissions, and all three entry cases of sourceL are exactly the author-defined omitted-list functions and equation (3.6). Empty products are one and empty sums zero. No expanded hatted-Psi display is substituted for the omitted-list definition.\n\nThe matrix sourceL is Hermitian. Its entire positive real characteristic-root multiset equals N-t copies of 1/4 together with h-1 copies of one, with algebraic multiplicity and no other positive roots. Its positive, zero and negative indices are N-s-1, one and s, respectively. These are a conjunction on the same actual matrix and domain. All finite part counts, repeated sizes and zero multiplicities are included; no additional matrix identity, genericity, classification or spectral-transfer premise is assumed.","title":"The complete singular squared-distance Laplacian conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/das-mohanty-singular-squared-distance-laplacian/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9069259375290a2e38a0aea85cc61235464f91834694d8979e281b9b28ca0fad.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:das-mori-2026-purification-lower-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. N. Das and T. Mori, *Krylov complexity of purification*, Phys. Rev. Lett. 136, 030201 (2026), arXiv:2408.00826v4, compare the Krylov operator complexity C_K(ρ(t)) of a mixed state ρ(t) = U_t ρ U_t† (U_t = e^{−iHt}) with complexities of its purifications.\n- **The time-dependent purification.** |Ψ^{U*}_ρ(t)⟩ = (U_t ⊗ U_t^*)|Ψ_ρ⟩ (Eq. (2)), generated by H ⊗ I − I ⊗ H^*.\n- **The complexities.** C_S is the spread complexity Σ_n n|ψ_n(t)|² of a state in its Lanczos basis (Eq. (S.4)), and C_K is its operator analogue for the Liouvillian L = [H, ·] with the Hilbert–Schmidt inner product (Eqs. (S.5)–(S.7)).\n- **The conjecture.** \"Krylov operator complexity of the mixed state is upper bounded by the operator complexity of the time-independent purification and is lower bounded by the state complexity of the time-dependent purification, C_S(|Ψ^{U*}_ρ(t)⟩) ≤ C_K(ρ(t)) ≤ C_K(|Ψ^I_ρ(t)⟩) (4)\". The authors confirm it analytically for one-qubit states and numerically for Werner states.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/das2026krylovpurification.md).\n\nIssue [#16717](https://github.com/the-omega-institute/trureturing/issues/16717) fixes the reading.\n- The claim is the left inequality of Eq. (4), for every full-rank density matrix, Hermitian H and t ≥ 0. Full rank is a special case of the source's statement, so its failure refutes the source's.\n- For full-rank ρ, |Ψ_ρ⟩ is the canonical purification Σ_{ij}(√ρ)_{ij}|i⟩|j⟩.\n- The left inequality is refuted.","title":"A full-rank qutrit refutes the purification lower bound on Krylov operator complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/das-mori-2026-purification-lower-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ed3c89b4c0a700e9157b5c1af725d1f00bccfa725bd328dc62571971362271a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:datta-dorlas-jozsa-benatti-2014-conditional-subentropy-conjecture-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Datta, Dorlas, Jozsa and Benatti, *Properties of subentropy*, arXiv:1310.1312v1, §2.5.1, Conjecture 2:\n\n> For a c-q state ρ_XB, the conditional subentropy Q(X|B) satisfies the following bound: Q(X|B) ≤ H_min(X|B) = − ln p_guess(X|B).\n\nFor every finite family of positive semidefinite complex matrices $A_x$ of total trace one, the joint state is $\\bigoplus_x A_x$ and its marginal is $\\sum_x A_x$. The claim uses equation (4), with normalized simplex measure from equation (3), for subentropy. Natural logarithms are used. Zero blocks, zero eigenvalues and repeated eigenvalues remain in scope.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (2), $C_n=1/2+\\cdots+1/n$ | `C` | Inclusive sum from 2 to n. |\n| Equation (3), normalized simplex measure | `simplexCoords`, `subentropySpectrum` | Nonnegative first n−1 coordinates of sum at most one; factorial times Lebesgue measure. |\n| Equation (4), $Q=n\\int\\eta(\\lambda\\cdot x)\\,dx-C_n$ | `subentropySpectrum`, `subentropy` | `Real.negMulLog`, the existing simplex `chart`, and all Hermitian eigenvalues. |\n| Finite c-q state | `claim` | Arbitrary finite X and B; positive semidefinite blocks, total trace one. |\n| Equation (13), $Q(X\\mid B)=Q(\\rho_{XB})-Q(\\rho_B)$ | `claim` | Block diagonal state and summed marginal. |\n| All POVMs; guessing min-entropy | `pGuess`, `result` | Supremum of the exact finite POVM scores; negative natural logarithm. |\n| Typeclass and Hermitian binders | `claim` | Finite indexing and matrix well-formedness; Hermiticity follows from positivity. |","title":"Conditional subentropy and guessing min-entropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/datta-dorlas-jozsa-benatti-2014-conditional-subentropy-conjecture-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/276f08c7b306681885d671c7dc985e49e4735a3e31bee5dc866bc254bfd4a606.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:datta-dorlas-jozsa-benatti-2014-conditional-subentropy-guessing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture 2. For a c-q state ρXB, the conditional subentropy Q(X|B) satisfies the following bound: Q(X|B) ≤ Hmin(X|B) = − ln pguess(X|B).\n\nThe supplied locator is arXiv:1310.1312v1, §2.5.1, Conjecture 2.\nLet p be any finite probability distribution and rho_x any finite-dimensional\ndensity states. Conditional subentropy is\nQ(recordState p rho) minus Q(mixtureState p rho), using the canonical states.\nThe strengthened target is\n\n$$\nQ(X|B)\\le1-p_{\\rm guess}(X|B)\\le-\\log p_{\\rm guess}(X|B).\n$$","title":"Datta–Dorlas–Jozsa–Benatti: conditional subentropy and guessing","url":"https://the-omega-institute.github.io/trureturing-pages/research/datta-dorlas-jozsa-benatti-2014-conditional-subentropy-guessing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/401d1cc75b0250cbfadd3b3bfd5af05dae6af8a62785c235bf222de2981ed36c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:datta-dorlas-jozsa-benatti-2014-subentropy-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture 1. Subentropy is subadditive: Q(ρAB) ≤ Q(ρA) + Q(ρB).\n\nThe supplied locator is arXiv:1310.1312v1, §2.3.1, Conjecture 1.\nThe related restatement in arXiv:1510.08859v2, §7.1 is:\n\n> At present, we do not know wether the subadditivity inequality of subentropy is true or not\n\nThe assertion concerns every finite-dimensional bipartite density state, with\nno commutation, rank or separability restriction. In the canonical partial-trace\ninterface, `marginalRight` retains A and `marginalLeft` retains B.","title":"Datta–Dorlas–Jozsa–Benatti: subadditivity of subentropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/datta-dorlas-jozsa-benatti-2014-subentropy-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8136e90eeef163ae94e4badd253116a7ce963cf31107e8d7655ec6aa9a3d781b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:daugherty-latin-tableaux-conjecture63","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every integer k at least five, let lambda=(k,k-1,...,5,4,3,3).\nConjecture 6.3 of arXiv:2610.29291v1 states LT(lambda)=9*4^(k-3).\nThe source count is the number of all fillings of the fixed left-aligned\ncells by actual integer symbols, with no repetition in any row or column,\nand type lambda-prime. It is not the reduced count. Both bottom rows\nremain distinct positions. There is no upper cutoff or additional\nwide-partition hypothesis.\n\nThe exact Lean statement is `Counting.conjecture63_integer`:\nfor `k : Int` with `5 <= k`,\n`Source.LT k.toNat = 9 * 4^(k.toNat - 3)`.\nOn this domain conversion to Nat is exact. `Source.shape k.toNat` is\nprecisely the source diagram. `Source.SourceLatin` imposes the conjugate\nmultiplicity equation for every integer, including symbols outside the\npositive row intervals.","title":"Daugherty et al., Latin tableaux Conjecture 6.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/daugherty-latin-tableaux-conjecture63/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81f8289fda162eef23aac7b46eef03ded3aab865bbf5f09a32c54b3748d1c22c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:davalos-ziman-pineda-2019-qutrit-divisibility-entanglement-breaking","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Davalos, M. Ziman and C. Pineda, *Divisibility of qubit channels and dynamical maps*, Quantum 3, 144 (2019), arXiv:1812.11437v4, ask in the abstract:\n\n> Finally, we show that every divisible but not infinitesimal divisible qubit channel (in positive maps) is entanglement breaking, and open the question if something similar occurs for higher dimensions.\n\nThis sentence is in the source TeX `paper.tex`, lines 162–164. In Section 2.1 the authors define a divisible channel by the negation of indivisibility, where indivisibility means that a concatenation into two channels forces one factor to be unitary (lines 256–261). In Section 2.2 they write:\n\n> we say $\\mathcal E\\in\\mathsf C^P$ if there exists a P-divisible dynamical map $\\mathcal E_t$ such that $\\mathcal E=\\mathcal E_t$ (with arbitrary precision) for some value of $t$ (including $t=\\infty$).\n\nThis is lines 337–339. Theorem 6 states:\n\n> Consider a qubit channel $\\mathcal E$. If $\\det\\hat\\mathcal E<0$, then $\\mathcal E$ is entanglement breaking, i.e. all qubit channels outside $\\mathsf C^P$ are entanglement breaking.\n\nThis is lines 1237–1241. The settled qutrit reading is the assertion that every completely positive trace-preserving qutrit map that is divisible and outside the formal `pDivisibleSet` is entanglement breaking. The module names this assertion `claim`.","title":"Divisibility does not force entanglement breaking for qutrits","url":"https://the-omega-institute.github.io/trureturing-pages/research/davalos-ziman-pineda-2019-qutrit-divisibility-entanglement-breaking/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/27c926f00fd8f7ff56e6846d2d274c587af3f071538ec5630b48aa27e7745141.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:davies-marigliano-2021-circulant-orbit-eigenvalue-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Isobel Davies and Orlando Marigliano, *Coloured Graphical Models and their Symmetries*, arXiv:2012.01905v5, Section 4, Conjecture 4.4, PDF page 11:\n\n> If $G$ is an uncoloured circulant graph, then the number of orbits of the action of $\\operatorname{Aut}(G)$ on the set $V*V$ of unordered pairs of vertices of $G$ coincides with the number of distinct eigenvalues of its adjacency matrix.\n\nThe preceding definition says: “By definition, these are graphs that have a symmetry which is a cyclic permutation of its vertices.” Corollary 3.3, page 7, uses the entrywise action of the full automorphism group; its count $\\binom{n+1}{2}$ includes diagonal pairs. The source's adjacency matrix is the real symmetric zero-one matrix of the simple undirected graph. Preregistration: [#21260](https://github.com/the-omega-institute/trureturing/issues/21260).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every uncoloured simple undirected graph of positive order | `claim` quantifies over every natural number and every `SimpleGraph (Fin n)`; positivity is part of `IsCirculant` | No connectivity assumption is added |\n| A symmetry which is a cyclic permutation of all vertices | `IsCirculant` uses a graph self-isomorphism whose permutation is `Equiv.Perm.IsCycleOn` on `Set.univ` | The whole vertex set is used |\n| Unordered pairs, including repetitions | `pairOrbits` uses `Sym2 (Fin n)` | Diagonal pairs remain |\n| Entrywise action of the full automorphism group | Each orbit is the range of `Sym2.map` over every graph self-isomorphism; the outer range collects distinct orbit sets | No subgroup substitution |\n| Number of distinct adjacency eigenvalues | `eigenvalueCount` counts `Module.End.Eigenvalues` of the real adjacency endomorphism with `Nat.card` | Scalar values are counted without multiplicity |\n| Equality for every circulant | `claim`; `result` proves its negation | No source clause is dropped |\n\nThe public definitions retain the literal preregistered expressions. Proofs use Mathlib's common-neighbour and circulant-matrix constructions directly, with the finite adjacency correspondence established in `intMatrix_adj`.","title":"A circulant counterexample to Davies–Marigliano Conjecture 4.4","url":"https://the-omega-institute.github.io/trureturing-pages/research/davies-marigliano-2021-circulant-orbit-eigenvalue-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f520cc452c835c29e32b1037cd40662276165b1fc2faf00aa9e46db5491ff1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:davies-sandhu-seo-tan-2026-multivariate-occupancy-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Davies, Sandhu, Seo and Tan, arXiv:2605.05149v1, Section 1 after Theorem 1,\nstate:\n\n> Strengthening the conjecture, we believe that the multivariate version should hold for any λ in the positive orthant.\n\n> The bound in Theorem 1 is tight by the example of a disjoint union of complete graphs (such that λ is constant on each component), though we believe that the upper bound on the entries of λ can be removed.\n\nThe proposed bound is\n\n$$\n\\mathbb E_{G,\\lambda}|I|\\geq\n\\sum_{v\\in V(G)}\\frac{\\lambda_v}{1+(d_v+1)\\lambda_v},\n\\qquad \\lambda_v>0.\n$$\n\nLee and Seo, arXiv:2602.02450v2, Section 5, write:\n\n> Having seen the Davies–Kang conjecture, it seems plausible to look for a strengthening of Theorem 1.1 in terms of occupancy fractions.\n\nTheir inequality (5.2), at t = 1, is the same bound after multiplication\nby the vertex count and identification of the sum of vertex marginals with\nexpected independent-set cardinality. Its nonnegative orthant contains\nthe strictly positive witness used here. The formal `claim` quantifies all\nnatural n, all simple graphs on Fin n, all adjacency-decision instances,\nand every strictly positive real fugacity vector. Classical decidability\nsupplies such an instance for every graph. The empty graph gives 0 ≤ 0.","title":"The unrestricted multivariate hard-core occupancy bound is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/davies-sandhu-seo-tan-2026-multivariate-occupancy-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3f6f2c23fdc282fe4fcc3e5ad6e0d68c27fa9f5785c4d0ed6d711cb981c9b9c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:davino-et-al-2025-energy-constrained-deterministic-violation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. D'Avino, G. Senno, M. Alimuddin and A. Acín, *Entanglement in the energy-constrained prepare-and-measure scenario: applications to randomness certification and channel discrimination*, arXiv:2510.27559v3 (2025 preprint; v3, 2026), Section IV.A, state:\n\n> We conjecture that this holds for all $\\omega > 0$.\n\nThis is verbatim from `paper_Quantum.tex`, line 284 (MD5 `012a72f4d8cfd30e14dc3813eda8f546`). The preceding sentence's assertion is the existence of behaviors allowing entanglement that are deterministic for input $x=0$ but violate the separable deterministic-input inequality. The global restriction is $\\omega\\in[0,1/2)$ (line 154), so the conjecture concerns every $0<\\omega<1/2$.\n\nThe paper considers a shared state $\\sigma_{PM}$, local channels $\\Lambda_x:P\\to S$ for $x\\in\\{0,1\\}$, and a binary POVM $\\{\\Pi^0,\\Pi^1\\}$ on $S\\otimes M$. Its behavior formula is $p(b|x)=\\operatorname{Tr}[\\Pi^b(\\Lambda_x\\otimes\\mathrm{id}_M)(\\sigma_{PM})]$ (lines 137–148), and $E_x=p(0|x)-p(1|x)$ (lines 165–166). For separable devices with $p(0|0)=1$, its displayed inequality is:\n\n> $E_1\\geq 2(1-2\\omega)^2-1=:I_\\mathrm{det}^{\\omega,\\mathrm{sep}}$\n\nThis is line 268, equation labelled `eq:deterministic-ineq-for-fixed-input-0`. The source defines $\\mathcal Q_\\omega^{\\mathrm{ent}}$ by allowing entanglement, rather than by requiring an additional entanglement test (line 158).\n\nIssue [#22022](https://github.com/the-omega-institute/trureturing/issues/22022) fixes the following finite-dimensional reading, which is proved. For every real $0<\\omega<1/2$, there exist natural dimensions $d_p,d_s,d_m$, a complex matrix $\\sigma$ on $P\\otimes M$, two complex-linear matrix maps $\\Lambda_x:P\\to S$, a matrix $H$ and vector $g$ on $S$, and two effects $\\Pi^b$ on $S\\otimes M$, such that:\n\n- $\\sigma\\succeq0$ and $\\operatorname{Tr}\\sigma=1$;\n- each $\\Lambda_x$ is completely positive and preserves the trace of every complex matrix;\n- $H=H^\\dagger$, $g^\\dagger g=1$, $Hg=0$, and $H-(I-|g\\rangle\\langle g|)\\succeq0$;\n- $\\Pi^b\\succeq0$ for both outcomes and $\\Pi^0+\\Pi^1=I$;\n- $\\operatorname{Re}\\operatorname{Tr}[H\\Lambda_x(\\operatorname{Tr}_M\\sigma)]\\leq\\omega$ for both inputs;\n- $p(0|0)=1$ and $p(0|1)-p(1|1)<2(1-2\\omega)^2-1$.\n\nHere the Lean definition `probability` takes the real part of the Born trace displayed above. The claim has no separate separability or entanglement predicate. The source permits arbitrary, including infinite, dimensions; finite witnesses are sufficient for its existential question. See also [the literature note](../Library/Entropy/davinoetal2025energyconstrained.md).","title":"Deterministic-input violations exist at every positive energy budget below one half","url":"https://the-omega-institute.github.io/trureturing-pages/research/davino-et-al-2025-energy-constrained-deterministic-violation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec987fff7cb3cb091c05684237c7853c751b5213c4b6d359a9613145baaabb3c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:davis-width-k-descent-difference-coprime-formula","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Robert Davis, *Width-k Generalizations of Classical Permutation Statistics*,\nJIS 20 (2017), gives the following descent-set line, generating function, and\nMacMahon identification together at the top of printed page 2. Printed page\n1 ends with the preceding descent-count formula:\n\n> where Des σ = {i ∈ [n − 1] | a_i > a_{i+1}}.\n> Given any statistic st, one may form the generating function\n> F_n^{st}(q) = Σ_{σ∈S_n} q^{st σ}.\n> A famous result due to MacMahon [6] states that F_n^{des}(q) =\n> F_n^{exc}(q), and that both are equal to the Eulerian polynomial A_n(q).\n> The Eulerian polynomials themselves may be defined via the identity\n> Σ_{j≥0} (1 + j)^n q^j = A_n(q)/(1 − q)^{n+1}.\n\nPrinted page 2 defines width-k descents:\n\n> For each of the following definitions, we assume n ∈ Z_{>0}, k ∈ [n − 1],\n> ∅ ≠ K ⊆ [n − 1], and σ = a_1 a_2 ··· a_n ∈ S_n. We define a width-k\n> descent of σ to be an index i ∈ [n − k] for which a_i > a_{i+k}. Thus the\n> width-1 descents are the usual descents of a permutation. Let\n> Des_k(σ) = {i ∈ [n − k] | a_i > a_{i+k}} denote the set of all width-k\n> descents of σ, and set des_k(σ) = |Des_k(σ)|.\n\nPrinted page 6 introduces the Laurent polynomial and the named conjecture:\n\n> We now show that interesting behavior occurs when considering the function\n> G_{n,k}(q) = Σ_{σ∈S_n} q^{des_k(σ)−des_{n−k}(σ)}.\n> According to computational data, the following conjecture holds for all\n> n ≤ 9 and 1 ≤ k < n for which gcd(k, n) = 1.\n>\n> Conjecture 9. If gcd(k, n) = 1, then\n> G_{n,k}(q) = n q^{1−k} A_{n−1}(q).\n\nThe formal theorem makes the surrounding range explicit: for all natural\n`n` and `k`, if `1 <= k`, `k < n`, and `Nat.Coprime k n`, then\n\n```text\nG(n,k) = n * q^(1-k) * eulerian(n-1).\n```\n\nHere `eulerian(m)` is the descent generating function that the paper\nidentifies with `A_m` by MacMahon's theorem. The paper's separate\ninfinite-series definition of the Eulerian polynomials is not formalized.\nOne-based positions are transported to `Fin n`, and signed exponents live in\nthe Laurent polynomial ring over the integers.","title":"Davis's coprime width-descent difference formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/davis-width-k-descent-difference-coprime-formula/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7107d8aedf40614285701f19520aec42264b435a40da3b47351a0f5038871736.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dawsey-russell-urban-partition-polynomial-derivative-question-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dawsey, Russell, and Urban define the partition polynomial on printed page 2:\n\n> To define this new polynomial, let λ = ⟨1^{m_1}, 2^{m_2}, . . . , k^{m_k}⟩ be a partition written in frequency notation. We define the partition polynomial f_λ by f_λ(x) = Σ_{i=1}^{k} m_i x^i. (1)\n\nPrinted page 9 gives the range convention and asks Question 9:\n\n> For the following question, let lg(λ) denote the largest part of the partition λ, and recall that f_λ^{(d)}(x) = 0 for all d > lg(λ).\n>\n> **Question 9.** If λ, λ′ are any two unequal partitions, is it true that f_λ^{(d)}(1) ≠ f_{λ′}^{(d)}(1) for some positive integer d ≤ min{lg(λ), lg(λ′)}?\n\nThe formal `claim` follows these printed quantifiers: the two partitions may\nhave different sizes and lengths.","title":"Refutation of the printed partition-polynomial derivative question","url":"https://the-omega-institute.github.io/trureturing-pages/research/dawsey-russell-urban-partition-polynomial-derivative-question-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3089fdec3ffcaad3e7f2a2f8a982353f413ae9e2b4580f091a23f60062e3ec91.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:day-sarkar-2021-t-optimizer-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. N. Day and A. Sarkar, *On a Conjecture of Nagy on Extremal Densities*, SIAM Journal on Discrete Mathematics 35 (2021), 294–306, DOI 10.1137/19M1296525, arXiv:1910.13465v1, Section 5, p. 14, state verbatim:\n\n> **Conjecture 2.** For each graph $G_l$, we have that $q(G_l,\\beta)$ is an increasing function of $\\beta$.\n\nThe source takes a maximiser of $q\\mapsto t(G_l,T_n^e(q))$ on $[0,1]$. The public `claim` uses the weakest reading: for every finite labelled simple graph, there exists a nondecreasing selection of maximisers at all densities in $[0,1]$. Its negation also excludes a strictly increasing selection.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every labelled finite graph $G_l$ | `v : Nat`, `G : SimpleGraph (Fin v)` in `claim` | All finite orders, including zero; labels introduce no restriction. |\n| Class sizes $\\sqrt\\beta qn$, $(1-\\sqrt{1-\\beta(1-q^2)})n$, $(\\sqrt{1-\\beta(1-q^2)}-\\sqrt\\beta q)n$ | `yW`, `rW`, `bW`, `vertexClass` | Yellow and red sizes use natural floors; blue is the remainder. Floors do not change the proved limit. |\n| Yellow and red cliques; red joined to both other classes; blue independent; no yellow–blue edges | `classAllowed`, `tGraph` | Adjacency requires unequal endpoints. |\n| Adjacency-preserving functions, not necessarily injective | `homCount` | Literal cardinality of the source homomorphisms. |\n| Homomorphism density as a limit | `tDens` | Literal `limUnder` along natural graph orders; `tDens_eq_weightingSum` proves the limit formula. |\n| Parameter attaining the supremum on $[0,1]$ | Membership and `IsMaxOn` conjuncts in `claim` | Every chosen parameter is a maximiser. |\n| Increasing with density | `MonotoneOn` in `claim` | Nondecreasing selection is the weaker interpretation. The witness densities lie strictly inside $(0,1)$. |\n| Fractional independence weightings | `independentWeight`, `admissible`, `weightingSum` | Classes red, yellow and blue have indices $0,1,2$, representing weights $0,1/2,1$. |","title":"The optimal T-parameter is not monotone in edge density","url":"https://the-omega-institute.github.io/trureturing-pages/research/day-sarkar-2021-t-optimizer-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bea4cbc3e23b531d6bb31d66eba6134bae413cbfc3c5cfe79d7053bd3a3d997b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:de-bievre-2022-noncommuting-not-coinc-all-dimensions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stephan De Bièvre, arXiv:2207.07451v1, §5, conjecture following Proposition 7:\n\n> We conjecture it is true that (ii) does not imply (i) in all dimensions\n> $d\\geq4$, but we have not produced such examples in other dimensions than\n> $4$ and $6$.\n\nThe space is $\\mathbb C^d$ and the bases are arbitrary orthonormal bases.\nCondition (ii) says that every pair of coordinate orthogonal projectors\nindexed by nonempty proper sets fails to commute. Condition (i), complete\nincompatibility, says that their images have zero intersection whenever\nthe two index-set cardinalities sum to at most $d$; empty sets are included.\nIssue #11785 preregisters the exact quantified statement before any Lean.","title":"Strong noncommutativity without complete incompatibility in every dimension at least four","url":"https://the-omega-institute.github.io/trureturing-pages/research/de-bievre-2022-noncommuting-not-coinc-all-dimensions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60929ee6bf764b814e4df036e7be08394b08ec283ef6a2c38a563528e4cbbca5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:de-boer-di-giulio-keski-vakkuri-tonni-2024-wigner-majorization-chains","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"De Boer, Di Giulio, Keski-Vakkuri and Tonni, *Continuous majorization in quantum phase space for Wigner-positive states and proposals for Wigner-negative states*, arXiv:2412.19698v3, Section V.A.1, ask for an analytic understanding of ordered subsets of the vacuum/one-photon mixture family. The v1 question is: “It would be interesting to identify subsets of W̃_u in which all elements exhibit a Wigner majorization relationship with one another. We leave this question for future works.” The v3 conjecture concerns the branch above one half. Preregistration #18571 specifies DBKT-1: both closed intervals form chains under Proposal 1.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Fock Wigner function, Eq. (99) | `fockWigner n x p`, with the reused `genLaguerre 0 n` | Literal expression over real phase-space coordinates |\n| Mixture, Eq. (100) | `mixtureWigner u x p` | $(1-u)W_0+uW_1$ |\n| Positive-part functional, Eq. (97) | `levelFunctional W t` | Product Lebesgue integral on $\\mathbb R^2$ of $\\max(|W|-t,0)$ |\n| Proposal 1, Eq. (98) | `WignerMajorizes W₁ W₂` | Every real threshold $t\\geq0$ |\n| DBKT-1(A) | First conjunct of `claim` and `result` | $1/2\\leq u\\leq v\\leq1\\Rightarrow\\widetilde W_v\\succ_W\\widetilde W_u$ |\n| DBKT-1(B) | Second conjunct of `claim` and `result` | $0\\leq u\\leq v\\leq1/2\\Rightarrow\\widetilde W_u\\succ_W\\widetilde W_v$ |\n\nDefinitions of the functions are meaningful for every real parameter; the theorem imposes the source's physical parameter intervals. Endpoints and all nonnegative thresholds are included. No clause is dropped.","title":"Vacuum and one-photon Wigner-majorization chains","url":"https://the-omega-institute.github.io/trureturing-pages/research/de-boer-di-giulio-keski-vakkuri-tonni-2024-wigner-majorization-chains/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19b399a5af622d607433a74cda29774468a42d1117e788fbfb37f8902d5f8344.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:de-lazzari-motwani-seynnaeve-2022-umps-two-by-two-linear-span","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. De Lazzari, H. J. Motwani and T. Seynnaeve, *The Linear Span of Uniform Matrix Product States*, SIGMA 18 (2022), 099, DOI 10.3842/SIGMA.2022.099; arXiv:2204.10363v3, Section 3.2, PDF page 9 (`sigma22-099.tex`, lines 540–547):\n\n> **Conjecture 3.8 (Corollary of Conjecture 3.7).**\n>\n> $\\dim \\langle \\mathrm{uMPS}(2,2,d)\\rangle = \\begin{cases}\\frac{1}{192}\\big(d^4-4d^2+192d+192\\big) & \\text{for }d\\text{ even,}\\\\ \\frac{1}{192}\\big(d^4-10d^2+192d+201\\big) & \\text{for }d\\text{ odd.}\\end{cases}$\n\nFor two complex $2\\times2$ matrices $X,Y$, let $A_0=X$, $A_1=Y$, and let the tensor $\\phi_d(X,Y)$ have coefficient $\\operatorname{tr}(A_{i_1}\\cdots A_{i_d})$ at the binary word $(i_1,\\ldots,i_d)$. The source defines $\\mathrm{uMPS}(2,2,d)$ as the closure of the image of $\\phi_d$ (the parametrization and following paragraph; TeX lines 215–225). Its linear span is taken over $\\mathbb C$.\n\nIssue [#23810](https://github.com/the-omega-institute/trureturing/issues/23810) fixes the reading for every positive integer $d$. The Lean `claim` uses `UniformMPSExpectedDimensionRefutation.uMPS 2 2 d`, the image of this ordered trace parametrization, and asserts the displayed equality after casting the natural-number dimension to $\\mathbb Q$. The image/closure bridge is explained below. This reading of Conjecture 3.8 is proved.","title":"The exact linear-span dimension of binary bond-two uniform matrix product states","url":"https://the-omega-institute.github.io/trureturing-pages/research/de-lazzari-motwani-seynnaeve-2022-umps-two-by-two-linear-span/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/607d53dd7429e8e4059b1fea365f90c0ebe26752ef3eefd8117745aa67260dc4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:de-oliveira-barbosa-galvao-2024-symmetric-ghz-resource","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. de Oliveira, L. S. Barbosa and E. F. Galvão, *Quantum advantage in temporally flat measurement-based quantum computation*, arXiv:2212.03668v4, Quantum 8, 1312 (2024), Section 4.2, Conjecture 2:\n\n> The number of qubits in a GHZ state necessary for the deterministic evaluation of a complete symmetric function Cᵏ, of constant degree k, with an n bit input string and a symmetric measurement assignment, scales as Ω(n^{⌊k/2⌋−1}).\n\nThe [source note](../Library/QuantumBounds/deoliveira2024symmetricghz.md) quotes Definitions 5, 15 and 16, Equations (70)–(72), and Corollary 1. Issue [#16724](https://github.com/the-omega-institute/trureturing/issues/16724) specifies the mathematical statement and source correspondence.\n\n### Statement echo\n\nAll Lean names in this dossier belong to `D5.S3.Quantum.Contextuality.SymmetricGHZResourceLowerBound`.\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| $n$ Boolean inputs | `Fin n → Bool` | Every input, including the all-false input |\n| Parity basis on $S$, Eq. (71) | Literal `Odd`-cardinality expression in `layer` | Integer zero or one according to odd subset weight |\n| All parity bases of each chosen size, Definition 16 | `layer`, `T : Finset ℕ` | Each selected size satisfies $1\\le i\\le n$ |\n| Equal coefficients within a size, Eq. (72) | `a : ℕ → ℝ`, `ComputesSym` | One real coefficient per size; no constant term |\n| Complete symmetric function, Definition 5 | `csf` | Degree-$k$ squarefree monomials counted modulo two |\n| Integer lift congruent modulo two, Definition 15 | `ComputesSym` | For every input there is an integer with the required real value and parity |\n| One qubit per selected parity basis | `qubits` | $\\sum_{i\\in T}\\binom ni$, including selected zero-coefficient layers |\n| Constant degree and asymptotic lower bound | `claim`, `result` | For every $k\\ge1$, positive $c$ and threshold $N$ depend only on $k$, uniformly in $n,T,a$ |\n\nThere are no additional public hypotheses. The integer exponent in `claim` is $\\lfloor k/2\\rfloor-1$, including $-1$ for $k=1$. Selected zero-coefficient layers can only increase the counted resource. The precise conventions are the literal expressions in #16724.","title":"Symmetric GHZ resources for complete symmetric Boolean functions","url":"https://the-omega-institute.github.io/trureturing-pages/research/de-oliveira-barbosa-galvao-2024-symmetric-ghz-resource/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60b0b8b91576682a0c05fd53416493d6d760b578707a58f935e578f360e90ced.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:de-vos-de-baerdemacker-2014-unitary-scaling-landscape-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. De Vos and S. De Baerdemacker, *Scaling a unitary matrix*, arXiv:1401.7883v2, Open Syst. Inf. Dyn. 21 (2014) 1450013, Appendix B:\n\n> We conjecture that all these stationary points are either maxima or saddle points or global minima. In other words: we conjecture that no local minima exist. Moreover, we conjecture that the global minima satisfy~$\\Psi=0$.\n\nThe landscape Ψ(λ, ρ) = n² − |sum(B)|² with B_{jk} = e^{i(λ_j+ρ_k)} A_{jk} is quoted in [the literature note](../Library/QuantumContext/devos2014scalingunitary.md).\n\nIssue [#19521](https://github.com/the-omega-institute/trureturing/issues/19521) fixes the reading: for every unitary A, every local minimum of Ψ_A is a global minimum. This first sentence of the conjecture is refuted. The second sentence, that global minima have Ψ = 0, is the existence of a scaling, attributed in the v2 comment to Idel and Wolf.","title":"The unitary scaling landscape has a nonglobal local minimum","url":"https://the-omega-institute.github.io/trureturing-pages/research/de-vos-de-baerdemacker-2014-unitary-scaling-landscape-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b7809a7de7f142d1bc6f1ef69576b419d84c5492a3e4080a71ccd25cba2388be.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debrota-2017-hoggar-sic-sum-negativity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"John B. DeBrota and Christopher A. Fuchs, *Negativity Bounds for\nWeyl–Heisenberg Quasiprobability Representations*, arXiv:1703.08272v2,\nSection 6, page 17, write:\n\n> Although dimension $5$ was the last in which we were able to explicitly calculate the sum negativity for the SIC Q-reps by exhaustive combinatorial searching, we suspect that we have found the correct sum negativity for $\\{Q_j^-\\}$ constructed with the Hoggar SIC in dimension $8$. Rather than calculating the eigenvalues of every partial sum matrix (since this is infeasible for $2^{64}$, $8\\times8$ matrices), we used a numerical local maximization procedure and around $10^6$ random pure state seeds. The overall maximum value we found, $7/8$, occurred frequently in our data and is significantly larger than all of the smaller local maxima. Of course, we could still be falling short of the global maximum value if it occurs at very hard to access positions. The states whose quasiprobability representations achieve the sum negativity of $7/8$ consist of $28$ copies of value $-1/32$ and $36$ copies of value $5/96$.\n\nThe precise question, preregistered in issue #11512, is whether $7/8$ is\nthe attained greatest value of sum negativity over all density matrices.\nWith the three-qubit Pauli orbit of\n$v=(-1+2i,1,1,1,1,1,1,1)^T$, write\n$\\Pi_j=u_ju_j^*/12$, $Q_j^-=3\\Pi_j-I/4$ and\n$q_\\rho(j)=\\operatorname{Re}\\operatorname{Tr}(\\rho Q_j^-)/8$.\nEquations (8)–(11) define\n$N^1(\\rho)=\\sum_j(|q_\\rho(j)|-q_\\rho(j))/2$ and maximize it over\npositive-semidefinite complex $8\\times8$ matrices of trace one.","title":"The sharp Hoggar-SIC sum negativity is seven eighths","url":"https://the-omega-institute.github.io/trureturing-pages/research/debrota-2017-hoggar-sic-sum-negativity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/56057c1418801b30e1ece6d722078e6e910bcae9b78626931d043aeded3de796.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debrota-2020-orthocross-gram-inverse-half-integer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. B. DeBrota, C. A. Fuchs and B. C. Stacey (arXiv:1812.08762, quant-ph;\nInt. J. Quantum Inf. 19 (2021) 2040005) study minimal informationally complete\nmeasurements (MICs) and their Gram matrices `G_{ij} = tr(E_i E_j)`. For an\northonormal basis `{|j⟩}` of `ℂ^d`, the orthocross MIC has elements\n`E_α = Ω^{−1/2} Π_α Ω^{−1/2}`, where `Π_α` runs over the `d²` projectors onto\n`|j⟩`, `(|j⟩ + |k⟩)/√2` and `(|j⟩ + i|k⟩)/√2` (`j < k`) and `Ω = Σ_α Π_α`. The\npaper states:\n\n> Conjecture. For any orthocross MIC, the entries in G^{−1} are integers or\n> half-integers.\n\nIssue #11385 fixes the reading: every dimension `d` and every orthonormal\nbasis (the columns of a unitary `U`), `Ω^{−1/2}` the positive square root of\n`Ω⁻¹`, and \"integers or half-integers\" as `2 (G⁻¹)_{αβ} ∈ ℤ`, with `G`\ninvertible.","title":"The inverse Gram matrix of an orthocross MIC is half-integral","url":"https://the-omega-institute.github.io/trureturing-pages/research/debrota-2020-orthocross-gram-inverse-half-integer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dcae979bc9a9c3d87635fce3e87c1037bd5d9cc1197c84b9df673cd663e93252.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debrota-2020-orthocross-nonorthogonality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. B. DeBrota, C. A. Fuchs and B. C. Stacey (arXiv:1812.08762v5, quant-ph; Int. J. Quantum Inf. 19\n(2021) 2040005) define the orthocross MIC of an orthonormal basis $\\{|j\\rangle\\}$ of $\\mathbb C^d$:\n- $\\Pi_\\alpha$ runs over the $d^2$ projectors onto $|j\\rangle$, $(|j\\rangle+|k\\rangle)/\\sqrt2$ and\n $(|j\\rangle+i|k\\rangle)/\\sqrt2$ ($j The entries in $G$ for orthocross MICs can become arbitrarily small with increasing $d$, but no two\n> elements of an orthocross MIC can be exactly orthogonal.\n\nThe verbatim statements are in [the literature note](../Library/QuantumStates/debrota2020varieties.md).\nIssue [#14213](https://github.com/the-omega-institute/trureturing/issues/14213) reads the conjecture\nas two clauses, both for every $d$ and every orthonormal basis:\n- (A) every off-diagonal Gram entry is a strictly positive real number;\n- (B) every Gram entry tends to zero uniformly as $d\\to\\infty$.","title":"Orthocross MIC elements are never orthogonal, and their Gram entries tend to zero","url":"https://the-omega-institute.github.io/trureturing-pages/research/debrota-2020-orthocross-nonorthogonality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1905a94b0a7a8b5a9176462dffeb161a1d97334b408f354005a7cd1a165367fb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debrota-2020-rank-one-mic-seven-orthogonal-pairs-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. B. DeBrota, C. A. Fuchs and B. C. Stacey, “The Varieties of Minimal\nTomographically Complete Measurements”, arXiv:1812.08762v5 (quant-ph),\nConjecture 1 on printed page 6 states:\n\n> A rank-1 MIC in dimension 3 can have no more than 7 pairs of orthogonal elements.\n\nA MIC is a POVM of exactly d² positive semidefinite effects summing to the\nidentity whose real span is the Hermitian operators. Orthogonality is\n`tr(E_a E_b) = 0`; distinct unordered pairs are counted once. Issue #11474\npreregisters the literal reading for d = 3 before Lean implementation.\nThe conjecture imposes no unbiasedness condition, although its preceding\nseven-pair example is explicitly called unbiased.","title":"A rank-one qutrit MIC with nine orthogonal pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/debrota-2020-rank-one-mic-seven-orthogonal-pairs-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d59642179e0c16707cea6bac7140d16c44dfa14fd42ec9c6d81ab36b8b0e6bc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debrota-stacey-2019-luders-entropy-saturation-sic","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. B. DeBrota and B. C. Stacey, *Lüders channels and the existence of symmetric-informationally-complete measurements*, Phys. Rev. A 100, 062327 (2019), arXiv:1907.10999v2, Section III, “Entropic Optimality”, Conjecture 1 (`lmc.tex`, lines 417–420):\n\n> Equality is achievable in the statement of Theorem\n> \\ref{avgentropybnd} only if a SIC exists in dimension $d$.\n\nThe reference is to Theorem 2 (lines 385–390): every Lüders MIC channel has Haar-averaged pure-input output entropy at most\n$B_d=\\log(d+1)-\\frac{2}{d+1}\\log2$, with equality achievable if a SIC exists in dimension $d$.\n\nA MIC is a family $E=(E_i)_{i=1}^{d^2}$ of positive semidefinite complex $d\\times d$ matrices, linearly independent over $\\mathbb C$, with $\\sum_iE_i=I$. Its Lüders channel is\n$\\mathcal E_E(X)=\\sum_i\\sqrt{E_i}\\,X\\sqrt{E_i}$, using the positive square roots. A SIC is a rank-one equiangular MIC. These are the definitions in the source, lines 80–90 and 104–114, also recorded in [the literature note](../Library/Entropy/debrotastacey2019luders.md).\n\nIssue [#25617](https://github.com/the-omega-institute/trureturing/issues/25617) fixes the statement for every positive natural dimension $d$: if a MIC $E$ has Haar-averaged pure-input output entropy $B_d$, then there exists a SIC in dimension $d$. This is the proposition `claim`, proved by `result`.","title":"Entropy saturation of a Lüders MIC channel forces SIC existence","url":"https://the-omega-institute.github.io/trureturing-pages/research/debrota-stacey-2019-luders-entropy-saturation-sic/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/054c42e514225f0cdf559865017582606de1a46a9361f03a57586477bbb4382f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:debski-barrycade-omitted-element-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Michał Dębski, Jarosław Grytczuk, Paweł Naroski, Bartłomiej Pawlik,\nJakub Przybyło, and Małgorzata Śleszyńska-Nowak, *Finite and infinite\nbarrycades*, arXiv:2609.18476v1, Section 3.1, Conjecture 2 (2), print:\n\n> A2(i) = A1(i) + 1 for every i >= 3.\n\nThe paper defines `A1(i)` as the least positive integer omitted from the\nquasi-permutation `rho_i` and `A2(i)` as its first entry `rho_i(1)`.","title":"Refutation of the printed barrycade omitted-element relation","url":"https://the-omega-institute.github.io/trureturing-pages/research/debski-barrycade-omitted-element-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1ea63e82e2be1c12bfd6d7470c58f1de02212b50f90e37cf197129e296fe425.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dedania-patel-2024-spatial-numerical-range-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. V. Dedania and A. B. Patel, *Spatial numerical range in non-unital, normed algebras and their unitizations*, arXiv:2306.16172v1, *Sarajevo Journal of Mathematics* 20(2) (2024), §1, p. 2, write:\n\n> Let $S(A)=\\{x\\in A:\\|x\\|=1\\}$ be the unit sphere in $A$. Let $A^*$ be the Banach space dual of $A$. Let $D_A(x)=\\{\\varphi\\in A^*:\\|\\varphi\\|=1=\\varphi(x)\\}$ for each $x\\in S(A)$. Further, let $V_A(a;x)=\\{\\varphi(ax):\\varphi\\in D_A(x)\\}$ for $a\\in A$ and $x\\in S(A)$. Then $V_A(a)=\\cup\\{V_A(a;x):x\\in S(A)\\}$ is the spatial numerical range (SNR) … It is still open problem wether $V_A(a)$ is always convex? This problem is discussed in [5].\n\nThe question is whether $V_A(a)$ is convex as a subset of the real vector space underlying $\\mathbb C$, for every element of every complex non-unital normed algebra.","title":"Convexity of the spatial numerical range in non-unital normed algebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/dedania-patel-2024-spatial-numerical-range-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8920cc22c54e8b8849b6f00cdf576a0561df6be506083ce01930146fd6016819.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dekking-keane-2022-two-block-thue-morse-frequency","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dekking and Keane, *Two-block substitutions and morphic words*, arXiv:2202.13548v1,\nSection 4, define\n$\\kappa_{\\rm TM}(00)=001$, $\\kappa_{\\rm TM}(01)=010$,\n$\\kappa_{\\rm TM}(10)=101$, and $\\kappa_{\\rm TM}(11)=110$.\nTheir fixed point $x^{(00)}$ starts with $00$ and satisfies\n$x_{3n}=x_{2n}$, $x_{3n+1}=x_{2n+1}$, and $x_{3n+2}=1-x_{2n+1}$.\n\n> **Conjecture 4.** The frequency of 1 in $x^{(00)}$ exists and equals $\\frac12$.\n\nThe source is v1, Section 4, pp. 5–6. ArXiv v2 and the journal version change\nthe substitution; this conclusion concerns the displayed v1 table.","title":"The Thue–Morse two-block fixed point has frequency one half","url":"https://the-omega-institute.github.io/trureturing-pages/research/dekking-keane-2022-two-block-thue-morse-frequency/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2de84f80ec12293255b82f73cff79e38dec0386554296232236c8efee8abf335.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:delascuevas-klingler-netzer-2024-nonfactorizable-doubled-triangle","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. De las Cuevas, A. Klingler and T. Netzer, *Polynomial decompositions with invariance and positivity inspired by tensors*, arXiv:2109.06680v2; Linear Algebra and its Applications 698 (2024), 537–588, DOI 10.1016/j.laa.2024.05.025, ask in Question 38:\n\n> Are there non-factorizable $(\\Omega,G)$ structures?\n\nThis is the verbatim question at lines 1546–1549 of the v2 TeX source `Manuscript_final.tex` (md5 `e601e0a3202f352cba4bf6d4d13e7db4`). Question 38 is also on PDF page 28; the conclusion repeats the existence question at TeX line 2035. Definitions and further quotations are in [the literature note](../Library/QuantumContext/delascuevasetal2024invariantpolynomialdecompositions.md).\n\nThe source's definitions at lines 318–332, 425–444 and 1515–1522 give the following reading. A weighted simplicial complex on $[n]=\\{0,\\ldots,n\\}$ assigns natural weights $\\Omega(S)$ to vertex subsets, with $\\Omega(S)\\mid\\Omega(T)$ whenever $S\\subseteq T$, and nonzero singleton weights. Each maximal nonzero-weight subset $F$ is a facet with $\\Omega(F)$ distinct copies. Write $\\widetilde{\\mathcal F}$ for these multifacets and $\\widetilde{\\mathcal F}_i$ for those containing $i$. A finite group $G$ acts on vertices preserving weights and on multifacets compatibly with collapse to their underlying facets. Its action on labelings is inverse pullback, ${}^g\\gamma(x)=\\gamma(g^{-1}x)$.\n\nFor each finite index set $\\mathcal I$ and labeling $\\alpha:\\widetilde{\\mathcal F}\\to\\mathcal I$, define\n$K_\\alpha=\\left|\\left\\{\\gamma:\\widetilde{\\mathcal F}\\to\\mathcal I:\\ \\forall i\\in[n],\\ \\exists g_i\\in G,\\ g_i i=i\\ \\text{and}\\ ({}^{g_i}\\gamma)|_i=\\alpha|_i\\right\\}\\right|$.\nFactorizability requires, for every finite $\\mathcal I$, strictly positive real factors $C^{[i]}_\\beta$, equivariant under $C^{[gi]}_{{}^g\\beta}=C^{[i]}_\\beta$, satisfying $\\prod_{i\\in[n]}C^{[i]}_{\\alpha|_i}=K_\\alpha^{-1}$ for every global labeling $\\alpha$.\n\nIssue [#21655](https://github.com/the-omega-institute/trureturing/issues/21655) fixes the formal `claim` as universal factorizability: for every natural $n$, every such weight function, every finite group and every compatible pair of actions, the structure is factorizable. `result : ¬ claim` refutes that universal assertion and answers the quoted existence question affirmatively.","title":"A nonfactorizable doubled triangle","url":"https://the-omega-institute.github.io/trureturing-pages/research/delascuevas-klingler-netzer-2024-nonfactorizable-doubled-triangle/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e1b9c91f34512edc92fca08b0d16f117cdf51e09a57b3d538f8b5ff000896ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dement-2005-a113249-odd-index-squares","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The Name field of [OEIS A113249](https://oeis.org/A113249) defines the integer family by\n$a_0=-1$, $a_1=4$, $a_2=-13+6(m-1)+3(m-1)^2$,\n$a_3=(m^2-8)^2$, and\n$a_{n+4}=m^4a_n+4m^2a_{n+1}-4a_{n+3}$.\nIts Comments field states:\n\n> Conjecture: a(m, 2*n+1) is a perfect square for all m, n.\n\nThe parameter and index range over natural numbers including zero, and the values\nare integers. Subtraction in $m-1$ is integer subtraction. Preregistration\n[#19316](https://github.com/the-omega-institute/trureturing/issues/19316) fixes these conventions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Four stated initial values | `a` at indices 0, 1, 2, 3 | Literal integer expressions, including all casts of the natural parameter. |\n| Fourth-order recurrence | `a` at index `n+4` | The source recurrence with all indices shifted by four. |\n| All family parameters and odd indices | `claim` | Both quantifiers range over the naturals, including zero. |\n| Perfect square | Mathlib `IsSquare` over the integers | Existence of an integer whose product with itself is the term. |\n| Entire conjecture | `result` | No additional hypothesis. |","title":"Dement's odd-index square conjecture for OEIS A113249","url":"https://the-omega-institute.github.io/trureturing-pages/research/dement-2005-a113249-odd-index-squares/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c17e4c724e1fd258f3064e02eefcdfe35de3fbe710bd9f28e1e0d40c5ac451ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:demo-2026-strong-period-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gregory Demo, David Lund and Oleksiy Al-saadi, *On the Periodicity of\nk-distance Graphs*, [arXiv:2606.25493v1](https://arxiv.org/abs/2606.25493v1),\nSection 4, Conjecture 1, printed PDF page 11:\n\n> Conjecture 1. Any connected graph G=(V,E) has ρ²(G)≤|V|−3.\n\nThe exact target and integer interpretation are registered in\n[#23003](https://github.com/the-omega-institute/trureturing/issues/23003).\nThe [source note](../Library/Geometry/demo2026periodicity.md) records the\nsource qualification; the\n[earlier metamour note](../Library/Geometry/erickson2024metamour.md) records\nthe known fixed-point example.\n\n### Definitions and statement echo\n\nLet $V$ be a finite vertex set and $G$ a simple undirected graph on $V$.\nDefine $T(G)$ on the same vertices by\n$u\\sim_{T(G)}v\\iff\\operatorname{edist}_G(u,v)=2$.\nThe extended distance is infinity for unreachable pairs, so $T$ is defined\nalso on disconnected graphs and every iterate remains in its domain.\nWrite $T^0(G)=G$. A positive return is an integer iteration count $t>0$\nwith an ordinary graph isomorphism $T^t(G)\\cong G$. Define\n$\\rho_2(G)=\\min\\{t\\in\\mathbb N: t>0,\\ T^t(G)\\cong G\\}$ when that set is\nnonempty, and $\\rho_2(G)=0$ otherwise. The source starts at $G_1^2=G$,\nso its return index is $f=t+1$ and its period $f-1=t$.\n\nThe full closed assertion is\n\n$$\\forall V\\text{ finite},\\ \\forall G\\text{ simple on }V,\\quad\nG\\text{ connected}\\ \\Longrightarrow\n\\operatorname{Int.ofNat}(\\rho_2(G))\\leq\n\\operatorname{Int.ofNat}(|V|)-3.$$\n\nBoth casts precede integer subtraction. There is no size lower bound,\nnontriviality requirement, edge requirement, periodicity hypothesis or\nrestriction on intermediate graphs. Isomorphism need not preserve labels;\neventual periodicity is a different assertion.\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Finite simple undirected graph | `V : Type`, `Fintype V`, `SimpleGraph V` in `claim` | All finite orders admitted by connectedness |\n| Exact geodesic distance two on the same vertices | `distanceTwo` | Extended distance also handles disconnected iterates |\n| Positive isomorphism return to the initial graph | `PositiveReturn` | Strictly positive count and ordinary graph isomorphism |\n| Least return, or zero if absent | `strongPeriod` | `Nat.find` on precisely that predicate, otherwise zero |\n| Only connectedness is assumed | `claim` | No added size hypothesis |\n| Integer upper bound | `claim` | Period and cardinality cast before subtracting three |\n| Negation of the entire assertion | `result` | Literally `Not claim`, with no free term or universe parameters |\n\nAll declaration names in this table belong to\n`D5.S3.Combinatorics.Graph.Walks.DemoStrongPeriodBoundRefutation`.","title":"Refutation of the unrestricted strong distance-two period bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/demo-2026-strong-period-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1264d708aedda146e29498e410b9bff473f2ba0d10fcbb9e9e266a556be0d7dc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:denis-lacaille-martin-serrano-ensastiga-2026-anticoherence-trace-distance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Denis, T. Lacaille, J. Martin and E. Serrano-Ensástiga, *Total, quantum, and classical measures of anticoherence for mixed spin states* (2026), arXiv:2605.29436v1, leave the trace-distance case of axiom (T4) open. The downloaded TeX member `arxiv.tex`, line 363, immediately following Proposition 2, states verbatim:\n\n> We also find numerical evidence that axiom~\\eqref{ax:T4} remains valid for the trace-distance case $p=1$; a proof is left open.\n\nThe TeX label `ax:T4` denotes monotonicity under every completely positive trace-preserving channel covariant under the spin representation of $\\mathrm{SU}(2)$ (Definition 2, lines 180–200). Identify a spin-$j$ state $\\rho$ with a symmetric state $\\rho_S$ of $N=2j$ qubits and put $\\rho_t=\\operatorname{Tr}_{N-t}(\\rho_S)$. In the normalized Dicke basis the reference state is $\\rho_0^{(t)}=I_{t+1}/(t+1)$. The source's candidate (lines 333–358) is\n\n$\\mathcal A_t^{T,d_1}(\\rho)=1-\\|\\rho_t-\\rho_0^{(t)}\\|_1/K_t$, with $K_t=\\max_{\\sigma\\in\\mathcal D(\\mathcal H_{t/2})}\\|\\sigma-\\rho_0^{(t)}\\|_1$.\n\nThe source uses the full Schatten one-norm. Its mixed-state domain includes every $t\\in\\{1,\\ldots,N\\}$; the footnote at line 178 says:\n\n> Whenever we consider pure states, we usually assume $t\\le N/2$.\n\nIssue [#24441](https://github.com/the-omega-institute/trureturing/issues/24441) fixes the universal reading: for every integer $N\\ge1$, every integer $1\\le t\\le N$, every such covariant channel $\\Phi$, and every positive semidefinite spin-$N/2$ matrix $\\rho$ of trace one, $\\mathcal A_t^{T,d_1}(\\Phi(\\rho))\\ge\\mathcal A_t^{T,d_1}(\\rho)$. This assertion is refuted.\n\nThe source definitions and locators are collected in [the literature note](../Library/Observer/denisetal2026anticoherencemixed.md).","title":"Trace-distance anticoherence can decrease under covariant spin noise","url":"https://the-omega-institute.github.io/trureturing-pages/research/denis-lacaille-martin-serrano-ensastiga-2026-anticoherence-trace-distance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/87554cfb258c5fea6f593dcb6e4e65e9ac95b9ea0cde81841f6cbe476e2ba0b7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:denker-imai-guhne-2025-flip-conjugate-geometric-measure","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Denker, S. Imai and O. Gühne, *Chiral Symmetries and Multiparticle Entanglement*, arXiv:2506.15609v1, Appendix H.2:\n\n> **Conjecture:** The geometric measure of entanglement for any state in the flip-conjugate symmetric subspace $\\mathcal{H_I}$ (and respectively $\\mathcal{H}_{\\bar{\\mathcal{I}}}$) is given by $G(\\ket{\\psi}) = 1-\\frac{d^2}{(d^2-1)(d+1)}$, where $d$ denotes the local dimension, which coincides with the dimension $\\dim(\\mathcal{H}_\\mathcal{I})$ of the subspace.\n\nHere:\n- **The subspace:** $\\mathcal H_{\\mathcal I}\\subset(\\mathbb C^d)^{\\otimes3}$ is spanned by $\\phi_n=(\\bar z_d\\sum_i|ini\\rangle+z_d\\sum_i|iin\\rangle)/\\sqrt{d+1}$ with $z_d=\\tfrac12(1+i\\sqrt{(d+1)/(d-1)})$.\n- **The geometric measure:** $G(\\psi)=1-\\max|\\langle abc|\\psi\\rangle|^2$ over unit product vectors.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/denkerimaiguhne2025chiral.md).\n\nIssue [#20037](https://github.com/the-omega-institute/trureturing/issues/20037) fixes the reading: for every $d\\ge2$ and every unit vector of either subspace, $G=1-d^2/((d^2-1)(d+1))$. This is proved.","title":"The geometric measure of the flip-conjugate symmetric subspace","url":"https://the-omega-institute.github.io/trureturing-pages/research/denker-imai-guhne-2025-flip-conjugate-geometric-measure/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/868b8a790a200a6b2c0006b1912f0d65f1cada15d24a25cf777c81593a0b76de.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:derksen-friedland-lim-wang-2017-density-tensor-nuclear-norm-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Derksen, S. Friedland, L.-H. Lim and L. Wang, *Theoretical and computational aspects of entanglement* (2017), arXiv:1705.07160v1, §6, immediately following Lemma 6.2, conjecture:\n\n> We conjecture that equality in (6.4) implies that $\\mathcal A$ is a pure density state corresponding to maximum entangled state.\n\nThis is the wording on PDF page 13. The corresponding TeX is `dflw19May17.tex`, line 878, with the equation label `maxinsepin`. The source TeX has md5 `1a948baafea109a303b169459eff9371`. See also [the literature note](../Library/Entropy/derksenfriedlandlimwang2017entanglement.md).\n\nFor $\\mathbf n=(n_1,\\ldots,n_d)$, the source defines $\\|\\mathcal T\\|_1$ as the complex tensor nuclear norm, the minimum of $\\sum_a\\prod_j\\|x_{a,j}\\|_2$ over finite product decompositions $\\mathcal T=\\sum_a\\bigotimes_j x_{a,j}$ (TeX lines 494–497), and defines $\\alpha(\\mathbf n,\\mathbb C)=\\max_{\\|\\mathcal T\\|_2=1}\\|\\mathcal T\\|_1$ (lines 525–526). A density tensor is the coefficient tensor of a positive semidefinite trace-one matrix, with its row and column modes kept separate (lines 680–695). Equation (6.4) is $\\log_2\\|\\mathcal A\\|_1\\le 2\\log_2\\alpha(\\mathbf n,\\mathbb C)$.\n\nIssue [#23962](https://github.com/the-omega-institute/trureturing/issues/23962) fixes the reading: for every mode count and mode-size function, every positive semidefinite trace-one complex matrix $A$ satisfying $\\|\\operatorname{densityTensor}(A)\\|_1=\\alpha(\\mathbf n,\\mathbb C)^2$ is $tt^*$ for some $t$ with $\\|t\\|_2=1$ and $\\|t\\|_1=\\alpha(\\mathbf n,\\mathbb C)$. This universal implication is refuted at $d=2$, $\\mathbf n=(2,4)$.","title":"A mixed density tensor attains the nuclear-norm maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/derksen-friedland-lim-wang-2017-density-tensor-nuclear-norm-equality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7665cfaddcfac26db7028d9aa2568a820dd409006eda9db6ce7fb5af347b6102.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:descamps-2024-binary-stabilizer-local-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"É. Descamps and B. Dakić (arXiv:2309.09815, quant-ph; J. Phys. A 57, 455301\n(2024)) call a state of `N` qubits stabilized by a stabilizing set `𝒜` when it\nis, up to a complex factor, the unique common `+1` eigenvector of finitely many\ntensor products of elements of `𝒜`. Standard stabilizer states are those\nstabilized by `𝒫 = {±1, ±i}·{𝟙, X, Y, Z}`. After an exhaustive analysis of two\nand three qubits the paper states:\n\n```latex\n\\begin{conj} All states stabilized by the set $\\mathcal A$ composed of binary\noperators and the identity {\\it i.e.}, $\\mathcal{A}=\\{A_{\\theta,\\phi},\\1_2\\}$,\nare locally equivalent to standard stabilizer states (\\textit{i.e.},\nstabilized by the Pauli group\n$\\mathcal{P}=\\{\\pm 1,\\pm i\\}\\cdot\\{\\1,\\sigma_X,\\sigma_Y,\\sigma_Z\\}$).\n\\end{conj}\n```\n\nIssue #11565 fixes the reading. `A_{θ,φ} = cos θ Z + sin θ (cos φ X + sin φ Y)`,\nstabilization needs no commutativity, and local equivalence is\n`ψ = (U_1 ⊗ ⋯ ⊗ U_N) φ` with `U_i ∈ U(2)`. The formal `claim` is the conjecture\nfor every `N`; the result refutes it.","title":"A binary-stabilized six-qubit state that is not locally a stabilizer state","url":"https://the-omega-institute.github.io/trureturing-pages/research/descamps-2024-binary-stabilizer-local-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9581317f00f2a163192671b35d6a3de7c34d152a31231ce73cff8f34c43a7761.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:designolle-vertesi-pokutta-2023-symmetric-bell-cross-polytope","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Designolle, T. Vértesi and S. Pokutta, *Symmetric multipartite Bell\ninequalities via Frank-Wolfe algorithms*, arXiv:2310.20677v3 and\n*Physical Review A* 109, 022205 (2024), DOI\n10.1103/PhysRevA.109.022205, conjecture in Section V that the symmetrised\nlocal polytope is affinely equivalent to the cross-polytope in dimension\n$\\lceil m/2\\rceil$ for the stated multipartite range. The source defines\ndeterministic strategies and their local polytope in Eqs. (1)--(2), and the\nsymmetry generators in Eq. (25). The literal claim is encoded by\n`D5/S3/Quantum/Entanglement/SymmetricBellPolytopeNotCrossPolytope.claim`.","title":"Refutation of the symmetric Bell cross-polytope conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/designolle-vertesi-pokutta-2023-symmetric-bell-cross-polytope/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/087c155fddc99c99e5c7e09eae9682057cb69305942c5bbc366a67d2518de42c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dhamapurkar-deng-2023-lattice-walk-cross-integral-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Dhamapurkar and X.-H. Deng, *Quantum walk mixing is faster than classical on periodic lattices*, Physica A 630, 129252 (2023), arXiv:2309.16352v1, Conjecture 1 in its case d = 2 (Eq. (15)):\n\n> $\\left| \\int_{0}^{T} n_1(t) n_2(t) dt \\right|\\leq 32 n_1(n_2 \\log(n_2))^2+ 32 n_2(n_1 \\log(n_1))^2$\n\nfor odd, relatively prime cycle orders $n_1 > n_2$. Here $n_i(t)$ is the time-dependent part of the transition amplitude product on $\\mathbb{Z}_{n_i}$, Eq. (B8): $n(t) = \\sum_{j \\neq k,\\, j+k \\neq n} e^{-it\\sin(\\pi(j+k)/n)\\sin(\\pi(j-k)/n)} w^{l(j-k)}$ with $w = e^{2\\pi i/n}$ and displacement $l$. The verbatim statements are in [the literature note](../Library/QuantumChannels/dhamapurkar2023latticemixing.md).\n\nIssue [#18528](https://github.com/the-omega-institute/trureturing/issues/18528) fixes the reading: the bound for all odd coprime $n_1 > n_2$, all displacements and every $T \\ge 0$. This is refuted.","title":"The cross-integral bound for continuous-time walks on two cycles","url":"https://the-omega-institute.github.io/trureturing-pages/research/dhamapurkar-deng-2023-lattice-walk-cross-integral-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f32ad730b9ce8dfbe95c80d67611db67828c39e94e15ee2bad12e3411471dc8a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ding-hu-2020-hyperdisk-regular-subset-finite-cover","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Ding and X. Hu, *Masking quantum information on hyperdisks*, Phys. Rev. A 102, 042404 (2020), arXiv:1909.11256v3, state in the conclusion at line 651 of `masking.tex`:\n\n> Nevertheless, it is an open question for a nondegenerate masking machine whether the number of hyperdisks in the set of maskable states is always finite. A related open question is whether a regular subset of a hyperdisk can be covered with finite number of hyperdisks.\n\nThe settled question is the second sentence. In the module, a hyperdisk is a set of unit vectors in `EuclideanSpace ℂ (Fin n)` represented by an orthonormal family `φ`, positive real coefficient moduli `r`, equality of the span of `φ` with the span of the set, and equality with the full unit-vector level set having those coefficient moduli. A regular subset `c` of `s` satisfies `IsHyperdisk s` and `(Submodule.span ℂ c : Set _) ∩ s = c`. The claim asks, for every natural `n` and every such `c` and `s`, for a natural `k` and a map `h : Fin k → Set (EuclideanSpace ℂ (Fin n))` such that every `h i` is a hyperdisk contained in `c` and `c = ⋃ i, h i`.","title":"A regular subset of a four-dimensional hyperdisk has no finite hyperdisk cover","url":"https://the-omega-institute.github.io/trureturing-pages/research/ding-hu-2020-hyperdisk-regular-subset-finite-cover/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a37e1f6cdfbc7e0d4d8b56151f7149ae8a2352376991f01c4e28c77792779d7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dinov-2026-equipartition-duality-minimum-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dinov, arXiv:2607.07851v1, Conjecture 3.22 (Equipartition duality):\n\n> Fix $n\\geq2$, an entropy value $s$, and an action marginal $\\rho_{\\mathbf J}$ on $(0,\\infty)^n$. Among all states on $\\mathbb T^n\\times(0,\\infty)^n$ with entropy $\\geq s$ and action marginal $\\rho_{\\mathbf J}$, the phase-equipartitioned product state (unique when it exists) simultaneously (i) maximizes the entropy, (ii) minimizes every within-DOF uncertainty $u_j$, and (iii) is the unique state at which the per-DOF conjectured bound of Problem 3.17(b) is saturated for all $j$; moreover it is the unique fixed point, with the given marginal, of the multi-DOF kime-deformed semigroup $\\partial_t\\tilde\\rho=\\sum_j(-\\omega_j\\partial_{\\theta_j}+\\varepsilon\\partial^2_{\\theta_j})\\tilde\\rho$.\n\nThis dossier anchors the necessary minimum clause (ii). The other clauses\nare quoted to identify the original simultaneous assertion and receive no\nseparate settlement.","title":"The within-DOF minimum clause of equipartition duality","url":"https://the-omega-institute.github.io/trureturing-pages/research/dinov-2026-equipartition-duality-minimum-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/474d193457c404efe435c6b75aa1125c336e75ac3c60a2b4aed0c4b4a92af64d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dirr-helmke-kleinsteuber-schulte-herbrueggen-2008-block-shift","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Dirr, U. Helmke, M. Kleinsteuber and T. Schulte-Herbrüggen,\n*Relative C-numerical ranges for applications in quantum control and quantum information*,\nLinear and Multilinear Algebra 56 (2008), 27–51, arXiv:math-ph/0702005v1,\nConjecture 3.6, pp. 26–27:\n\n> Every element of $E(t_{\\mathrm{loc}})$ is similar to a block-shift matrix via an element of\n> $\\Pi^{\\mathrm{ex}}_{\\mathrm{loc}} := \\Pi_{\\mathrm{loc}}\\cdot\\Pi_{\\mathrm{out}}$.\n\nThe definitions and source quotations are in\n[the literature note](../Library/QuantumBounds/dirr2008relative.md).\nIssue [#14776](https://github.com/the-omega-institute/trureturing/issues/14776)\nfixes DHKS-1 and the literal Lean conventions. The universal claim ranges over every\n$n\\ge2$ and every complex matrix in the union of the nonzero root spaces of the local\ndiagonal torus. The tensor basis has its first factor as its most significant bit.","title":"Extended local permutations do not suffice for block-shift form","url":"https://the-omega-institute.github.io/trureturing-pages/research/dirr-helmke-kleinsteuber-schulte-herbrueggen-2008-block-shift/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e927b2afd39561d76e842bcc52f47cf440f28e2c182e64bf2ffdec1df8d4e9e0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dive-koukoulekidis-mousafeiris-mintert-2020-three-coherence-ramsey-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Dive, N. Koukoulekidis, S. Mousafeiris and F. Mintert, *Characterisation of multi-level quantum coherence without ideal measurements* (Phys. Rev. Research 2, 013220 (2020); arXiv:1901.08599), certify k-coherence from the interference pattern p(t) = ⟨χ|e^{−iHt} ρ e^{iHt}|χ⟩ of a Ramsey-type experiment.\n- The certifier is the moment ratio R3 = M3/M1², with M_q = (1/2π)∫₀^{2π} p(t)^q dt.\n- R3 is convex in ρ.\n- Its maximum over states that are not (k + 1)-coherent is a threshold: a measured R3 above it proves at least (k + 1)-coherence, whatever the Hamiltonian and the measurement vector χ.\n- Table I gives the thresholds 1 (k = 1) and 5/4 (k = 2). For k = 3 it gives only the analytic upper bound 179/96 ≈ 1.86 and the numerical value 1.77, with the remark \"In the cases where the exact threshold values are not known, we use numerical methods to approximate them\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/dive2020multilevelcoherence.md).\n\nIssue [#18378](https://github.com/the-omega-institute/trureturing/issues/18378) fixes the reading. The k = 3 threshold is the maximum of R3 over:\n- all finite dimensions;\n- integer spectra;\n- unit measurement vectors;\n- finite mixtures of unit vectors with at most three nonzero amplitudes, with M1 > 0.\n\nIt is proved to equal N(α)/D(α) ≈ 1.7731757563.","title":"The exact three-coherence threshold of the Ramsey moment ratio R3","url":"https://the-omega-institute.github.io/trureturing-pages/research/dive-koukoulekidis-mousafeiris-mintert-2020-three-coherence-ramsey-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/230269e1e663c69fa59a2a8ebc4e49a3ff7f52f19026c5084139bfee2cfaf979.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:divianszky-et-al-2023-lk-bound-nontightness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Diviánszky, I. Márton, E. Bene and T. Vértesi, *Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant*, arXiv:2211.17185v2 (Sci. Rep. 13, 13200, 2023), Methods, after Lemma III.4:\n\n> It is an open question whether Lemma~\\ref{LkkL} is tight or not.\n\n> Hence, for $k=2$ the upper bound in Lemma~\\ref{LkkL} is tight. We conjecture that the bound is not tight for greater values of $k$.\n\nHere, for a real $n\\times m$ matrix $M$:\n- $L(M)=\\max_{v\\in\\{\\pm1\\}^n}\\|vM\\|_1$, the classical bound of the correlation expression $M$;\n- $L_k(M)=\\max_W\\sum_i\\|(W^{\\mathsf T}M)_i\\|_1$ over the $n\\times k$ matrices $W$ whose rows are standard basis vectors, which groups the inputs into $k$ classes;\n- Lemma III.4 states $L_k(M)\\le kL(M)$.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/divianszkymartonbenevertesi2023certification.md).\n\nIssue [#20049](https://github.com/the-omega-institute/trureturing/issues/20049) fixes the reading: for every $k\\ge3$ there is $c_k a(n) is the largest multiple of n with fewer than twice as many divisors as n.\n\n> Conjecture: a(n) is always a divisor of n^2.\n\nThe entry's worked example: \"a(6) = 18 because 6 has 4 divisors and term must have fewer than 8\ndivisors. Only 6, 12, and 18 are multiples of 6 with fewer than 8 divisors.\"","title":"A Multiple with Fewer than Twice as Many Divisors Divides the Square","url":"https://the-omega-institute.github.io/trureturing-pages/research/divisor-doubling-multiple-divides-square/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/63f36a2b6a642224d4b793501377e8b59a75a6c8ad437a948d7eb2621f9d73d0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dmitrovic-2024-mixed-norm-point-evaluation-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Dmitrović, *Norm of composition operator on mixed norm spaces*,\nOperators and Matrices 18(3) (2024), 697–710,\n[doi:10.7153/oam-2024-18-42](https://doi.org/10.7153/oam-2024-18-42),\nend of section 2, page 705, states:\n\n> CONJECTURE. From the previous discussion, we conjecture that for all 0 < p, q ⩽ ∞, 0 < α < ∞ and f ∈ H^{p,q,α}: |f(z)| ⩽ ‖f‖_{p,q,α} / (1 − |z|²)^{α + 1/p}.\n\nThe definitions on page 697 are:\n\n$$\n\\|f\\|_{p,q,\\alpha}\n=\\left(2\\alpha q\\int_0^1 r(1-r^2)^{\\alpha q-1}M_p^q(r,f)\\,dr\\right)^{1/q}\n\\qquad(0 Hence, we expect this walk to always yield the maximum entanglement.\n> This is indeed the observed behavior for all graph sizes and time steps we have tested.\n\nThe preceding sentence, line 652, identifies the initial coin by $\\theta=\\phi=\\pi/2$. The walk, measure and initial states are specified in Sections II A–C and the localized initial-state equation at TeX lines 423–425, respectively; see [the literature note](../Library/Entropy/donkerslootsondenheimersperling2026walks.md).\n\nFor $P\\ge1$, the modes are $(b,x)\\in\\{0,1\\}\\times\\mathbb Z/P\\mathbb Z$. The evolution is $T=S(H\\otimes I)$, where $H=\\frac1{\\sqrt2}\\begin{pmatrix}1&1\\\\1&-1\\end{pmatrix}$ and $S$ shifts coin component $0$ left and component $1$ right, periodically. A normalized coin $c=(\\alpha,\\beta)$ starts at position zero. Write $\\lambda_{P,n,c}=T^n(c\\otimes|0\\rangle)$ and $c_*=(1,i)/\\sqrt2$.\n\nFor a finite amplitude family $\\lambda$, the encoded single-photon vector is $|\\psi_\\lambda\\rangle=\\sum_k\\lambda_k|1_k\\rangle$, with one tensor factor per mode. The module defines $g_{\\max}(\\lambda)=\\sup_{\\|z_k\\|=1}|\\langle\\bigotimes_k z_k|\\psi_\\lambda\\rangle|^2$, with $z_k\\in\\mathbb C^2$, and $E_g(\\lambda)=1-g_{\\max}(\\lambda)$.\n\nIssue [#22527](https://github.com/the-omega-institute/trureturing/issues/22527) fixes the reading: for every natural $P\\ge1$, every $n\\in\\mathbb N$ and every $c\\in\\mathbb C^2$ with $|\\alpha|^2+|\\beta|^2=1$, $E_g(\\lambda_{P,n,c})\\le E_g(\\lambda_{P,n,c_*})$. This is proved, including $P=1$ and $n=0$.","title":"The symmetric coin maximizes full-mode entanglement of the Hadamard walk on every finite circle","url":"https://the-omega-institute.github.io/trureturing-pages/research/donkersloot-sondenheimer-sperling-2026-symmetric-coin-entanglement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a7ab47c054924f3d8294e5658fbb0ef6f8e13c62dfa6083e9cf0a39221d8edc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:doolittle-chitambar-2021-signaling-dimension-certificate-size","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Doolittle and E. Chitambar, *Certifying the Classical Simulation Cost of a Quantum Channel*, arXiv:2102.12543v1 (Physical Review Research 3, 043073, 2021), define a classical channel by its conditional probabilities $P(y|x)$ and define d-simulation by\n\\[\nP(y|x)=\\sum_\\lambda q(\\lambda)\\sum_{m\\in[d]}R_\\lambda(y|m)T_\\lambda(m|x).\n\\]\nTheir signaling polytope $\\mathcal C_d^{n\\to n'}$ is the set of channels with such a simulation. Immediately after Theorem `Thm:main2`, the source states (main.tex, line 409):\n\n> We conjecture that in general, no more than $\\binom{n'}{d}$ inputs are needed to certify that a channel $\\mbf{P}\\in\\mc{P}^{n\\to n'}$ has a signaling dimension larger than $d$.\n\nThe settled finite formulation uses natural numbers $m,k,d$, a channel `p : \\mathrm{Fin}\\,m \\to \\mathrm{Fin}\\,k \\to \\mathbb R`, the imported predicate `IsRowStochastic p`, and the finite shared-randomness predicate `IsSimulable d p`. For $1\\le d$ and $d Another useful general relation\n>\n> $K^{(d)}(n) = 0 \\cup K^{(0)}(n-d)$\n>\n> is conjecturally valid for all $n \\geq d + 1$ and $d \\geq 1$.\n\nHere $K^{(d)}(n)$ lists the distinct upper degrees in ascending order, without multiplicities (TeX lines 415–425); the $0$ on the right denotes adjoining the value zero. The growth rule and upper degrees are also described in [the literature note](../Library/StatisticalMechanics/dorogovtsevkrapivsky2025deterministic.md).\n\nIssue [#20715](https://github.com/the-omega-institute/trureturing/issues/20715) fixes the precise reading. Let $V$ be a type with decidable equality, let $v_0\\in V$, and let $v:\\mathbb N\\to\\operatorname{Finset}(V)\\to V$ be injective as a function of the pair $(s,\\sigma)$, with $v(s,\\sigma)\\ne v_0$ for every $s,\\sigma$. Define the finite families of nonempty faces by\n\n$C_0=\\{\\{v_0\\}\\}$,\n\n$C_{s+1}=C_s\\cup\\bigcup_{\\sigma\\in C_s}\\{\\{v(s+1,\\sigma)\\}\\cup\\beta:\\beta\\subseteq\\sigma\\}$.\n\nEvery existing nonempty face receives its own fresh vertex, all subsets $\\beta$ including the empty subset are used, and earlier faces persist. The labeling hypotheses ensure that a new vertex differs from every earlier vertex and from the other vertices added in the same step.\n\nFor a finite family $C$ and a finite vertex set $\\tau$, define\n\n$\\operatorname{upperDegree}(C,\\tau)=\\#\\{\\rho\\in C:\\tau\\subseteq\\rho,\\ |\\rho|=|\\tau|+1\\}$,\n\n$\\operatorname{degreeSet}(C,d)=\\{k\\in\\mathbb N:\\exists\\tau\\in C,\\ |\\tau|=d+1,\\ \\operatorname{upperDegree}(C,\\tau)=k\\}$.\n\nWrite $K^d(n)=\\operatorname{degreeSet}(C_n,d)$. The claim is that, for every such $V,v_0,v$ and all natural numbers $d,n$ with $d\\geq1$ and $n\\geq d+1$,\n\n$K^d(n)=\\{0\\}\\cup K^0(n-d)$.\n\nSubtraction is natural subtraction; the hypotheses imply $n-d\\geq1$. Equality of these sets expresses equality of the source's lists of distinct values. This claim is proved.","title":"The distinct upper degrees of deterministic simplicial complexes shift by dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/dorogovtsev-krapivsky-2025-deterministic-simplicial-degree-shift/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8d5b789d17f0688139da44f9ef04a18e188f0fc2f961054f8b00c07ae438d07.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ducci-modular-cycle-factorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fellman and Klyve, *INTEGERS* **23** (2023), Article #A86, Section 7, state:\n\n> Conjecture 1. Every cocomposite cycle is a product of a constant cycle and two\n> generating cycles.\n\nand add: \"This conjecture has been verified for all moduli up to 161. One\npossible direction of proving this conjecture is to show that all cocomposite\ncycles are a product of a cocomposite constant cycle and a coprime cycle, but we\nhave been unsuccessful on this front.\"\n\nThe game replaces the difference map of the classical Ducci game by\n`[a b c d] ↦ [ab bc cd da]` on `(Z_n)^4`. A 4-tuple is in a cycle when iterating\nreturns it; by their Lemma 2 the entries of such a tuple are then either all\nunits or all non-units, and the cycle is called coprime or cocomposite\naccordingly. A cycle is constant when its tuples have four equal entries. A\ngenerator 4-tuple is `[1 x 1 x^{-1}]` with `x` a unit lying on a cycle, and a\ncycle containing one is a generator cycle. Products are taken entry by entry.","title":"Ducci Modular Cycle Factorization","url":"https://the-omega-institute.github.io/trureturing-pages/research/ducci-modular-cycle-factorization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/720171d8c4b2d5747c653872be6d616cd31e490b5523e5d3266df180d155fc44.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ducoffe-dumitru-2026-bipartite-minimal-two-edge-connected","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ducoffe–Dumitru, *Towards a more structured search for Erdős–Gyárfás\ncounter-examples*, arXiv:2609.28594v1, §1:\n\n> for any class of graphs F, a counter-example within F refers to any graph\n> G ∈ F such that G has minimum degree at least three but no cycle of length\n> some power of two. It is called a minimal counter-example within F if it\n> minimizes (|V(G)|, |E(G)|) with respect to the lexicographic order.\n\nThe bipartite clause in §6 reads:\n\n> we also conjecture … that any minimal bipartite counter-example is 2-edge\n> connected.\n\nFor a nonempty finite simple graph G, let HasPowTwoCycle(G) mean that G has\nan IsCycle walk of length 2^k for a natural k ≥ 2. A bipartite\ncounterexample is a graph admitting a proper two-colouring, with degree at\nleast three at every vertex and no such cycle. Minimality compares\n(|V(G)|, |E(G)|) lexicographically against every finite nonempty bipartite\ncounterexample, independently of its vertex type. The target conclusion\nis that G is connected and deleting any edge of G leaves a connected\ngraph.\n\nThis is a structural question about potential counterexamples to Erdős\nProblem #64, the Erdős–Gyárfás conjecture. It asserts neither that a\ncounterexample exists nor that every graph of minimum degree three has a\npower-of-two cycle.","title":"Minimal bipartite Erdős–Gyárfás counterexamples are 2-edge-connected","url":"https://the-omega-institute.github.io/trureturing-pages/research/ducoffe-dumitru-2026-bipartite-minimal-two-edge-connected/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40db65941d0b064719184171860f02536c38f6d150b6cfd55c81f9cf7d106354.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ducuara-susa-skrzypczyk-2024-f3-steering-ellipsoid-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. F. Ducuara, C. E. Susa and P. Skrzypczyk, *Locally inaccessible hidden quantum correlations*, Phys. Rev. A **110**, 022435 (2024), arXiv:2406.03945v1, state the QSE conjecture in `QC_LI.tex`, lines 282–284:\n\n> For any two-qubit state $\\rho \\in D(\\mathds{C}^2 \\otimes \\mathds{C}^2)$, its maximal violation of the $\\rm CHSH$-inequality \\eqref{eq:HC} and the $\\rm F_3$-inequality \\eqref{eq:CAC} are upper bounded by functions that depend on the magnitude of the QSE's centre as ${\\rm B}(\\rho) \\leq f_{\\rm CHSH}(c_B)$ and ${\\rm F_3}(\\rho)\\leq f_{\\rm F_3}(c_B)$, respectively. With $c_B=|\\bm{c}_B|$.\n\nLine 288 asks:\n\n> Exploring the form of function $f_{\\rm F_3}(c_B)$ is an interesting task which we leave for future research.\n\nThe same line describes the numerical thresholds:\n\n> In particular, we highlight that there seem to exist values $c_{\\rm \\, CHSH}=0.5$ and $c_{\\rm \\, F_3} = 0.66$, such that states with QSE centres' magnitudes greater than these values, can no longer lead to a violation of the $\\rm CHSH$-inequality and the $\\rm F_3$-inequality, respectively.\n\nFor a positive semidefinite trace-one complex two-qubit matrix $\\rho$, write $\\boldsymbol a$, $\\boldsymbol b$ for its Bloch vectors and $T_{ij}=\\operatorname{Re}\\operatorname{tr}[\\rho(\\sigma_i\\otimes\\sigma_j)]$. The module defines Bob's centre by\n\n$\\boldsymbol c_B=(\\boldsymbol b-T^{\\mathsf T}\\boldsymbol a)/(1-|\\boldsymbol a|^2)$ when $|\\boldsymbol a|\\ne1$, and $\\boldsymbol c_B=\\boldsymbol b$ when $|\\boldsymbol a|=1$; put $c_B=|\\boldsymbol c_B|$.\n\nIts $\\mathrm F_3(\\rho)$ is the real supremum of\n\n$\\displaystyle \\frac1{\\sqrt3}\\left|\\sum_{i=1}^3\\operatorname{tr}[\\rho\\,((\\boldsymbol\\alpha^i\\cdot\\boldsymbol\\sigma)\\otimes(\\boldsymbol\\beta^i\\cdot\\boldsymbol\\sigma))]\\right|$\n\nover real unit Alice directions $\\boldsymbol\\alpha^i$ and real orthonormal Bob frames $\\boldsymbol\\beta^i$. The absolute value here is the complex norm. The source's Eq. (CAC), lines 209–214, gives its optimized value as $\\sqrt{s_1^2+s_2^2+s_3^2}$, where the $s_i$ are the singular values of $T$; the relationship to the frame formulation is discussed below. The source definitions and locator are in [the literature note](../Library/Entropy/ducuarasusaskrzypczyk2024hidden.md).\n\nIssue [#26164](https://github.com/the-omega-institute/trureturing/issues/26164) fixes the settlement as the conjunction of\n\n$\\displaystyle \\mathrm F_3(\\rho)\\le\\max\\{\\sqrt{3(1-c_B)},1\\}$\n\nfor every positive semidefinite trace-one two-qubit matrix, and the existence, for every real $0\\le t<2/3$, of such a matrix with $c_B=t$ and $\\mathrm F_3(\\rho)>1$. This conjunction is proved.","title":"The $\\mathrm F_3$ steering inequality has the sharp ellipsoid-centre threshold two thirds","url":"https://the-omega-institute.github.io/trureturing-pages/research/ducuara-susa-skrzypczyk-2024-f3-steering-ellipsoid-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca922095f89933566fb9852ee915ab8f9612b994c9fdca760f00f29afae82b3f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dukes-2023-permaspin-eigenvalue-order","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mark Dukes, *An Ising model having permutation spin motivated by a permutation complexity measure*, Physica A 626 (2023) 129090, DOI 10.1016/j.physa.2023.129090, arXiv:2211.08908v3, Section 4, `permaspin.tex`, lines 510–513:\n\n> Let $\\lambda^{*}$ be the largest root of $t(\\lambda)=0$.\n> It remains to see which of the eigenvalues $\\lambda^{*}$, $\\lambda_4$, and $\\lambda_5$ is largest.\n> Numerical investigations indicate that $\\lambda^{*}$ appears to be the largest and this is supported by the evidence presented by\n> in Figures~\\ref{figblue1} and \\ref{figblue2}.\n\nThe source sets $c=a^4$ and $d=b^2$, where $a=e^{H'/2}$ and $b=e^{J'}$ (lines 458 and 490). These parameters are positive for real $H',J'$. Its two explicit expressions are $\\lambda_4=c(1-d)^2$ and $\\lambda_5=c(1-d^2)$ (lines 500–503), and its cubic is (lines 504–509)\n\n$t_{c,d}(x)=x^3-(cd^2+c^2+2cd+c+1)x^2-c(cd^3+3c^2d+cd^2+c^2+cd+c+3d+1)(d-1)x+(d^2+4d+1)c^3(d+1)(d-1)^3$.\n\nThe definitions and source context are in [the literature note](../Library/StatisticalMechanics/dukes2023permaspin.md). The module's `dukesCubic c d` is this polynomial, defined for all real $c,d$, and `largestRoot c d` is $\\lambda^*(c,d)=\\sup S_{c,d}$, where $S_{c,d}=\\{x\\in\\mathbb R:t_{c,d}(x)=0\\}$.\n\nIssue [#22514](https://github.com/the-omega-institute/trureturing/issues/22514) fixes the reading: for every real $c>0$ and $d>0$, $S_{c,d}$ is nonempty and bounded above, and $\\max(\\max(\\lambda^*(c,d),c(1-d)^2),c(1-d^2))=\\lambda^*(c,d)$. This is proved.","title":"The largest root of Dukes's permaspin cubic dominates the two explicit eigenvalues","url":"https://the-omega-institute.github.io/trureturing-pages/research/dukes-2023-permaspin-eigenvalue-order/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/67c50bd81a4f16ca16ce628d05d48ec18b5756fc72babfb3cc94033cc79a217d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dumitru-franco-merino-2026-alternative-mean-linearization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Dumitru, J. Franco, and A. Merino, *On The Linearization of Alternative Means*, arXiv:2606.25150v1, §1, ask in the supplied verbatim text:\n\n> Question : Can we characterize all functions f : (0, ∞) → (0, ∞) such that Aσ̂_f B is linearizable if and only if AB = BA and |Spec(A⁻¹♯B)| ≤ 2 ?\n\nRemark 4.4 restates this question. Definition 2.3 specifies $A\\widehat\\sigma_fB=f(Y)Af(Y)$ with $Y=A^{-1}\\sharp B$, for continuous positive $f$ with $f(1)=1$. Remark 2.2 calls a mean linearizable when it equals $aA+bB$ for real $a,b$.","title":"Characterizing rigid alternative means","url":"https://the-omega-institute.github.io/trureturing-pages/research/dumitru-franco-merino-2026-alternative-mean-linearization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9743d7ed7aed09eb034d031569144ab8fb47d6e97d0a8a084cc54cbc1f59e8ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:duschenes-garciamartin-holmes-cerezo-2026-chaar-concatenation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Duschenes, D. García-Martín, Z. Holmes and M. Cerezo, *Moments of Quantum Channel Ensembles*, PRX Quantum **7**, 033050 (2026), DOI [10.1103/grdp-jph9](https://doi.org/10.1103/grdp-jph9), [arXiv:2511.12700v2](https://arxiv.org/abs/2511.12700v2), Appendix B, Conjecture 1, asks whether the cHaar moment norm is non-increasing with concatenations:\n\n$$\\|\\widehat T_{\\mathrm{cHaar}}^{(t)k'}\\|_{HS}^2\\leq\\|\\widehat T_{\\mathrm{cHaar}}^{(t)k}\\|_{HS}^2\\quad(k'\\geq k).$$\n\nThe verbal source clause is: “Further, it is conjectured that the cHaar norm is monotonically non-increasing with environment dimension and concatenations,”. Only the concatenation clause is refuted. [Preregistration #15553](https://github.com/the-omega-institute/trureturing/issues/15553) specifies the conventions and witness. [The Library note](../Library/QuantumChannels/duschenesgarciamartinholmescerezo2026moments.md) contains the source definitions, norm convention and locators.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Haar unitary on system and environment, with fixed pure ancilla; partial trace over environment. | `cHaarChannel d e U X`. | Reuses `PartialTraceMutualInformation.partialTraceRight`; its sum is exactly the preregistered sum. `Matrix.single 0 0 1` is the pure coordinate-zero ancilla. |\n| The order-$t$ ensemble moment of the channel tensor power. | `momentOp d e t`. | All tensor factors use the same Haar sample, for every $t$. |\n| Hilbert–Schmidt norm, squared. | `hsNormSq`, used in `claim`. | The definition is the double sum of `Complex.normSq` specified by #15553. Private `hsNormSqCorrespondence` equates this sum with the frozen `RHLinalg.frobSq` used by `normReindex`, `normNormal` and `result`. |\n| All positive dimensions, moment orders and concatenation counts, with $k'\\geq k$. | `claim`. | Positive $d,t,k$ and nonzero $e$; positive counts suffice for the witness. |\n| Refutation at $d=e=t=2$, $k=4$, $k'=5$. | `result : ¬ claim`. | Literal universal claim negation; exact instance values are `instanceN4`, `instanceN5`. |","title":"Concatenation can increase the cHaar moment norm","url":"https://the-omega-institute.github.io/trureturing-pages/research/duschenes-garciamartin-holmes-cerezo-2026-chaar-concatenation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01edcaf02534324224dd2e709df3f911cdfba8b94675e8a4083be4de8293f4b0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dutta-tushar-2026-wigner-distance-equatorial-multiplicativity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Soumyojyoti Dutta and Tushar, *A Phase-Space Geometric Measure of Magic in\nQubit Systems*, arXiv:2603.20792v3, printed p. 8, Conjecture 5.6:\n\n> For $\\langle Z\\rangle_\\rho=\\langle Z\\rangle_\\sigma=0$: $C(\\rho\\otimes\\sigma)=C(\\rho)+C(\\sigma)+C(\\rho)C(\\sigma)$.\n\nThe quantities use the fixed Wootters product frame, the convex hull of\nWigner functions of actual Pauli stabilizer states, and the L1 distance.\nIssue #11541 preregisters the fully quantified density-matrix statement,\nTier 1 classification and literature reading before the Lean probe.","title":"Equatorial multiplicativity of the qubit Wigner distance","url":"https://the-omega-institute.github.io/trureturing-pages/research/dutta-tushar-2026-wigner-distance-equatorial-multiplicativity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b427cf683d1674e07ea0ff496860991c7106fc7eee0dcadbb8e5296a2d8c020.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dutta-tushar-2026-wigner-distance-factored-deficit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Dutta and Tushar, *A Phase-Space Geometric Measure of Magic in Qubit Systems*, arXiv:2603.20792v4,\nSection 5.4, Conjecture 5.5 (Factored deficit): \"For equatorial magic states $\\rho$ and qubit states\n$\\sigma$ with $s(\\sigma)>0$: $\\mathrm{deficit}(\\rho,\\sigma)=C(\\rho)\\cdot f(\\sigma)$ for a universal function\n$f\\geq0$ of the Pauli invariants $(|r_x|,|r_y|,|r_z|)$ of $\\sigma$, vanishing iff $s(\\sigma)\\leq0$.\" Here\n$C$ is the $\\ell_1$ distance of the Wootters Wigner function to the stabilizer Wigner polytope,\n$s(\\sigma)=\\operatorname{sgn}(r_xr_yr_z)$ and $\\mathrm{deficit}(\\rho,\\sigma)=(1+C(\\rho))(1+C(\\sigma))-1-C(\\rho\\otimes\\sigma)$.\nThe verbatim statements are in [the literature note](../Library/QuantumStates/dutta2026wignerdistance.md).\n\nIssue [#14651](https://github.com/the-omega-institute/trureturing/issues/14651) reads the factorization\nclause: there is a function $f$ of $\\sigma$ with $\\mathrm{deficit}(\\rho,\\sigma)=C(\\rho)f(\\sigma)$ for every\ndensity matrix $\\rho$ with $\\langle Z\\rangle_\\rho=0$ and $C(\\rho)>0$ and every density matrix $\\sigma$\nwith $r_xr_yr_z>0$. Any $f$ satisfying the conjecture satisfies this clause.","title":"The positive-branch Wigner-distance deficit does not factor through the equatorial input","url":"https://the-omega-institute.github.io/trureturing-pages/research/dutta-tushar-2026-wigner-distance-factored-deficit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9534eb98ef33f80e196339d4922000e8f8778b222c5ac09add615ab1b48468c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dutta-tushar-2026-wigner-distance-self-tensor-superadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Soumyojyoti Dutta and Tushar, *A Phase-Space Geometric Measure of Magic in\nQubit Systems*, arXiv:2603.20792v3, printed p. 8, Conjecture 5.7:\n\n> For any qubit state $\\rho$ with $s(\\rho)\\leq0$: $C(\\rho\\otimes\\rho)\\geq2C(\\rho)$.\n\nThe quantities use the fixed Wootters product frame, the convex hull of\nWigner functions of actual Pauli stabilizer states, and the L1 distance.\nIssue #11541 preregisters the fully quantified density-matrix statement,\nTier 1 classification and literature reading before the Lean probe.","title":"Self-tensor superadditivity on the nonpositive Bloch-product branch","url":"https://the-omega-institute.github.io/trureturing-pages/research/dutta-tushar-2026-wigner-distance-self-tensor-superadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/821e688384a3dd8dd7c3a208916bc271855093f2399365a4459ceef8d17cfed2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dyson-mckay-cycle-clique-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul W. Dyson and Brendan D. McKay, *Ramsey numbers for regular induced subgraphs*, arXiv:2604.08215v3, Section 4:\nafter Theorem 4.2 the authors write that they believe the graphs G_p of Theorem 4.2 are optimal for p ≥ 13 within the\nclass of disjoint unions of lexicographic products of cycles and cliques. Precisely: for every prime p ≥ 13, every\ndisjoint union of products C_r[K_s] (r ≥ 3, s ≥ 1) with no induced regular subgraph on exactly p vertices has at most\n9(p−1)²/8 vertices when p ≡ 1, 5 (mod 12), (p−1)(9p−7)/8 when p ≡ 7 (mod 12) and (p−1)(9p−11)/8 when p ≡ 11 (mod 12),\nand G_p attains the bound.","title":"Optimal Cycle–Clique Unions Without Regular Induced Subgraphs of Prime Order","url":"https://the-omega-institute.github.io/trureturing-pages/research/dyson-mckay-cycle-clique-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/301ab563e56650605070ff8a3fccc1f6e605c41209f53066b79c25d24a2fa68c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:dzhenzher-sakbaev-2025-random-semigroup-slln","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. V. Dzhenzher and V. Zh. Sakbaev, *The Strong Law of Large Numbers for random semigroups on uniformly smooth Banach spaces*, arXiv:2507.07658v3 (Advances in Operator Theory 11 (2026), article 4), state Conjecture 2.6 in `mains/slln_smooth.tex`, lines 224–229:\n\n> This leads us to the following conjecture. Let $\\mathcal{X}$ be a Banach space. Then under the assumptions of Theorem~\\ref{t:slln-sot}, $e^{A_1t/n} \\ldots e^{A_nt/n}$ converges a.s. in SOT of $\\norm{\\cdot}_\\mathcal{X}$ to $e^{\\Expect At}$ uniformly for $t>0$ in any segment.\n\nThe formal declaration `randomProduct` multiplies, in increasing index order, the factors $\\operatorname{exp}(((t/n : \\mathbb R):\\mathbb K)\\mathbin{\\cdot}a_i(\\omega))$ for $i\\in\\operatorname{Fin}(n)$. The declaration `claim` quantifies over a scalar type $\\mathbb K$ with `RCLike`, a normed additive commutative group $e$ with a `NormedSpace` and `CompleteSpace` structure, a measurable type $\\alpha$, a probability measure $\\mu$, and a sequence $a : \\mathbb N\\to\\alpha\\to(e\\toL[\\mathbb K]e)$. It assumes `AEStronglyMeasurable` for every $a_i$, pairwise `IndepFun`, `IdentDistrib (a_i) (a_0)` for every $i$, and a real $m$ such that $\\|a_i(\\omega)\\|\\le m$ almost everywhere for every $i$. For every $x:e$ and $t\\ge0$, almost every $\\omega$ satisfies: for every $\\varepsilon>0$, eventually in $n$ and for every $s\\in[0,t]$,\n\n$\\left\\|\\left(\\operatorname{randomProduct}(a,n,s,\\omega)-\\operatorname{exp}\\left(((s:\\mathbb R):\\mathbb K)\\mathbin{\\cdot}\\int z,\\,a_0(z)\\,d\\mu\\right)\\right)x\\right\\|<\\varepsilon$.\n\nThe operator measurable space used by `claim` is the Borel space on $e\\toL[\\mathbb K]e$, and its algebra structure is `NormedAlgebra.restrictScalars ℝ \\mathbb K`.","title":"The Dzhenzher–Sakbaev random semigroup SLLN holds on every Banach space","url":"https://the-omega-institute.github.io/trureturing-pages/research/dzhenzher-sakbaev-2025-random-semigroup-slln/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e050ff9b3409636dceb2fc594d00c6089f19d9c7589567672e4d4042995658d8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:easlick-2026-minimal-advantage-cycle-conjecture-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Terry Easlick, *Computer-assisted results for n-sided non-transitive dice taking face values in [n]*, arXiv:2610.40321v1, Conjecture 1, states:\n\n> n=6 is the largest n≥3 such that the set of non-standard n-sided dice admits a non-transitive cycle of maximum length n with uniform and minimal winning advantage.\n\nThe exact target is the conjunction $S(6)$ and $\\forall n\\in\\mathbb N,\\ n\\ge7\\Rightarrow\\neg S(n)$.\nFor $n\\ge3$, a die is a nondecreasing list of exactly $n$ labels in $[1,n]$, with repetitions permitted. The standard tuple is included, as in the source's Example 1 and its full graph enumeration. Each of the $n^2$ independent face-index pairs has equal weight. A strict win excludes ties from the numerator and retains them in the denominator.\nLet $m_n$ be the least attainable strict-win probability greater than one half. An edge $d\\to e$ means that the actual unconditional probability that $d$ beats $e$ equals $m_n$. A cycle is a nonempty list of pairwise distinct dice, with every successive edge and its closing edge. The proposition $S(n)$ says that a cycle of length $n$ exists and that no cycle of any length $k>n$ exists. The theorem quantifies over all such $k$ and all $n\\ge7$; it introduces no finite cutoff.","title":"Easlick's complete minimal-advantage dice cycle conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/easlick-2026-minimal-advantage-cycle-conjecture-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11fa2c16637900e1c86b60d46fc91e34d6835d7123e95f934ebc1f0555f6000b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egecioglu-gaiser-yin-rotation-wilf-classes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ömer Eğecioğlu, Collier Gaiser and Mei Yin, *Pattern avoidance in permutations and their rotations*,\narXiv:2607.20750v1, Section 6, Open Question 6.1: for k ≥ 4, are the Wilf classes of S_n^{(k)}(q), the permutations\nof [n] whose first k rotations all avoid q, over the patterns q ∈ S_4 exactly the eight complement–reverse orbits?\nWriting a(n,k;q) for the number of such permutations, the statement is: for every k ≥ 4 and q, s ∈ S_4,\na(n,k;q) = a(n,k;s) for every n ≥ k if and only if s lies in the orbit of q under complement and reverse.","title":"Wilf Classes of Rotation Avoidance for Patterns of Length Four","url":"https://the-omega-institute.github.io/trureturing-pages/research/egecioglu-gaiser-yin-rotation-wilf-classes/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50018b2013f2b2b44702104aeec7c6bfd49b0b3983d2aac6020ab2113cc280c2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-10","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.10. For all n ≥ 0, |Fn(2143, 1423, 3124)| = |Sn(321, 2143, 3124)| = |Sn(231, 4132, 2134)|.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding 2143, 1423 and 3124","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-10/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf434fd253c3347753c4fa384ee61d5fc79e6a7844e31e1247fcf0ba5fb2708e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-11","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.11. For all n ≥ 0, |Fn(1243, 2134)| = |Sn(123, 3241)|.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding 1243 and 2134","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-11/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cfca69738d8d48da1fbd3f8d680ef28e915c082f72a1769e074fca1a34961232.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-12","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.12. For all n ≥ 0, |Fn(1243, 3124)| = |Sn(231, 4123)|.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding 1243 and 3124","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-12/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab546adf7da6c293312e719d95f7c6014256ef7e0b3a8a2811148e910201cb35.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-13","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.13. For all n ≥ 1, |Fn(2413, 2431)| = |Fn(2431, 3241)| = Σ_{k=1}^{n} C(n−1, k−1) C_{n−k}.\n\nHere C_m is the m-th Catalan number. A permutation p is Fishburn when there are no indices i and j > i + 1\nwith p_i = p_j + 1 < p_{i+1}; F_n(B) is the set of Fishburn permutations of length n avoiding the classical\npatterns in B.","title":"Fishburn Permutations Avoiding 2413 and 2431, or 2431 and 3241","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-13/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84888edb34b61442f1dcc7ed7f58f0bc357f7faf2c1d76b9c4c70605754a2599.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.4. For all n ≥ 1, |Fn(1324, 2143)| = |Fn(1423, 2143)| = |Fn(1423, 3124)| = (n − 1)2^{n−2} + 1.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding Pairs Counted by (n − 1)2^(n−2) + 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be0d020f0645f27f5485065772616fb2ab94ba5a913f3c2de197fb41ab6ce62d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.5. For all n ≥ 1, |Fn(1324, 1423)| = |Fn(1324, 3124)| = F_{2n−2}.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding Pairs Counted by Odd-Indexed Fibonacci Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5811331ed758a1d30133e27daf06c655ba291a03ddb6b827c5843e55bd4ccb8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-7","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.7. For all n ≥ 1, |Fn(1324, 2143, 1423)| = |Fn(1324, 2143, 3124)| = |Fn(1324, 1423, 3124)|\n> = 2^n − n.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nis the set of Fishburn permutations of length n avoiding the classical patterns in B. The source notes that\n2^n − n also counts the Grassmannian permutations of length n.","title":"Fishburn Permutations Avoiding Three Patterns from {1324, 2143, 1423, 3124}","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-7/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/703bbc3ea804a4903d28164c223098f07cfee512c85038abd81361568ebbb5c4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:egge-fishburn-conjecture-10-9","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eric S. Egge, *Pattern-Avoiding Fishburn Permutations and Ascent Sequences*, arXiv:2208.01484v1, Section 10:\n\n> Conjecture 10.9. For all n ≥ 1, |Fn(2143, 3124)| = |Sn(231, 4123)|.\n\nA permutation p is Fishburn when there are no indices i and j > i + 1 with p_i = p_j + 1 < p_{i+1}; F_n(B)\nand S_n(B) are the Fishburn permutations and all permutations of length n avoiding the classical patterns\nin B.","title":"Fishburn Permutations Avoiding 2143 and 3124","url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-9/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2595e286ec96f33a438ca3e2196df86a2b6ec32ad64a8c79d1203d5b79107d71.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:eidesen-2025-nice-error-basis-non-normal-stabilizer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eidesen models quantum errors by a projective error model `(G, π)` on a\nfinite-dimensional Hilbert space `V`: a projectively faithful irreducible\nprojective representation of a finite group. For a subspace `W` with orthogonal\nprojection `P_W`, the logical operators and the stabilizers are\n\n> L_{(G,π)}(W) := { x ∈ G : P_W π(x) = π(x) P_W },\n> S_{(G,π)}(W) := { x ∈ G : P_W π(x) P_W ∈ 𝕋 P_W }.\n\nAfter recording that no code has been found that is a Clifford code and a\nweak stabilizer code but not a stabilizer code, §11 of arXiv:2506.01843v3\nasks Question 11.3:\n\n> Does there exist a Hilbert space V, a subspace W ⊂ V, and a projective error\n> model (G, π) ∈ PEM_V with |G| = (dim V)^2, such that the equation\n> |G| = |L_{(G,π)}(W)| · |S_{(G,π)}(W)| holds, but S_{(G,π)}(W) is not normal\n> in G?\n\nIssue #10156 fixes the readings: `V = EuclideanSpace ℂ (Fin d)`; `π` takes\nvalues in `d × d` complex matrices acting on `V`; a projective representation\nhas unitary values with `π(x)π(y) = c π(xy)` for some `|c| = 1`; projective\nfaithfulness is the injectivity of `q ∘ π`; irreducibility means that the only\ninvariant subspaces are `0` and `V`; `P_W` is the orthogonal projection; the\norders are `Set.ncard`; and \"not normal\" means that some conjugate `g s g⁻¹`\nof an element of `S_{(G,π)}(W)` lies outside it. The formal `claim` is the\nnegative answer, and the formal `result` refutes it.","title":"A nice-error-basis code whose stabilizer group is not normal","url":"https://the-omega-institute.github.io/trureturing-pages/research/eidesen-2025-nice-error-basis-non-normal-stabilizer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a899302960ba725d4edf1f13ac301dcd4d8855914306798fa44faeb42788fdd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elder-lafreniere-conjecture-4-12-signed-cardinality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Elder, Lafrenière, McNicholas, Striker and Welch, *Toggling, rowmotion, and\nhomomesy on interval-closed sets*, arXiv:2307.08520v2, Conjecture 4.12,\nstates:\n\n> If m = 2 or m = 3, then the signed cardinality statistic is 0-mesic\n> under rowmotion on interval-closed sets of [m]×[n] whenever m + n − 1\n> is even.\n\nThe signed point statistic is 1 on even rank and −1 on odd rank, and the\nstatistic of an interval-closed set is the sum over its members. The\nfollow-up by Lafrenière, Lewis, McNicholas, Striker and Welch,\narXiv:2505.04000v1, retains the m = 3 even-n case as an open conjectural\nstatement in Remark 3.31.","title":"Signed cardinality is not 0-mesic on [3] × [12]","url":"https://the-omega-institute.github.io/trureturing-pages/research/elder-lafreniere-conjecture-4-12-signed-cardinality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/216e3df77940bf25fc309e5ee09bcb6f759167529f4eba1792c55d192222dd60.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elizalde-luo-nonnesting-1132-2213","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sergi Elizalde and Amya Luo, *Pattern avoidance in nonnesting permutations*, arXiv:2412.00336v6,\nSection 4, Table 4, row `{1132, 2213}`: the conjectured ordinary generating function is\n`((1-x)^2 - √((1-x)^4 - 4x(1-x)^2))/(2x)` (OEIS A006319), with the remark\n\n> All the conjectures have been checked for n up to 8.","title":"Nonnesting Permutations Avoiding 1132 and 2213","url":"https://the-omega-institute.github.io/trureturing-pages/research/elizalde-luo-nonnesting-1132-2213/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb003fa8fb40677a27d49e5e151fc6a75211512c692a41d4018a0e1284a13dc1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elizalde-luo-nonnesting-1231-1312-2231-3221","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sergi Elizalde and Amya Luo, *Pattern avoidance in nonnesting permutations*, arXiv:2412.00336v6,\nSection 4, Table 4, row `{1231, 1312, 2231, 3221}`: the conjectured ordinary generating function of\n`c_n({1231, 1312, 2231, 3221})` is `(1 - 3x + 2x^2)/((1 - 3x)(1 - x - x^2))`, with the remark\n\n> All the conjectures have been checked for n up to 8.\n\nHere a nonnesting permutation of size `n` is a permutation of `{1, 1, 2, 2, …, n, n}` avoiding `1221` and\n`2112`, and pattern containment keeps equal letters equal.","title":"Nonnesting Permutations Avoiding 1231, 1312, 2231 and 3221","url":"https://the-omega-institute.github.io/trureturing-pages/research/elizalde-luo-nonnesting-1231-1312-2231-3221/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1416fb90ed4bc3ce714c2d1e6c7d8f27b3658204e90f335d9c69b53ed7f9fd3d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elizalde-luo-nonnesting-1233-1322","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sergi Elizalde and Amya Luo, *Pattern avoidance in nonnesting permutations*, arXiv:2412.00336v6,\nSection 4, Table 4, row `{1233, 1322}`: the conjectured ordinary generating function is\n`((1-x)^2 - √((1-x)^4 - 4x(1-x)^2))/(2x)` (OEIS A006319), with the remark\n\n> All the conjectures have been checked for n up to 8.","title":"Nonnesting Permutations Avoiding 1233 and 1322","url":"https://the-omega-institute.github.io/trureturing-pages/research/elizalde-luo-nonnesting-1233-1322/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02d7f1d7b60f1c892a0fb4220a24f6645cbddcb637bde7251264594ce2b9c013.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elizalde-luo-nonnesting-1322","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sergi Elizalde and Amya Luo, *Pattern avoidance in nonnesting permutations*, arXiv:2412.00336v6,\nSection 4, Table 4, row `{1322}`: the conjectured number of nonnesting permutations of\n`{1, 1, 2, 2, …, n, n}` avoiding `1322` is `(1/n) Σ_{k=0}^{n-1} C(3n, k) C(2n-k-2, n-1)` (OEIS A007297),\nwith the remark\n\n> All the conjectures have been checked for n up to 8.","title":"Nonnesting Permutations Avoiding 1322","url":"https://the-omega-institute.github.io/trureturing-pages/research/elizalde-luo-nonnesting-1322/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5fb24589b1f17c8c671a109ee454bde84739b213c7875b3e04355873624e0fc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elliott-2025-rebit-minimum-distance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. J. Elliott, *Strict advantage of complex quantum theory in a communication task*, arXiv:2505.16523v2, Phys. Rev. A **111**, 062401 (2025), studies a four-input task with target distribution P(y|x) = (1 − δ_xy)/3 on X = Y = {0, 1, 2, 3}. A complex qubit realizes it exactly, while no real qubit (rebit) strategy does. For the robustness of this separation the source measures the error by d = (1/|X|) Σ_x (1/2) Σ_y |P(y|x) − P̃(y|x)|. It shows that d = 1/12 is achievable with a rebit (Appendix D), and writes: \"We therefore only weakly conjecture that d = 1/12 the the minimum distance for a rebit strategy.\" The verbatim statements are in [the literature note](../Library/Quantum/elliott2025complex.md).\n\nIssue [#16523](https://github.com/the-omega-institute/trureturing/issues/16523) fixes the reading. A rebit strategy is:\n- four real symmetric positive-semidefinite 2 × 2 states of trace 1;\n- a four-outcome POVM of real symmetric positive-semidefinite effects summing to the identity.\n\nMixed states and POVMs cover private randomness and post-processing on either side. The claim is that 1/12 is the least distance attained by such strategies. It is proved.","title":"The minimum average trace distance of a rebit strategy for Elliott's exclusion task is exactly 1/12","url":"https://the-omega-institute.github.io/trureturing-pages/research/elliott-2025-rebit-minimum-distance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ee20ba903107f7cda3340a2f9c53024128e51c97cb64f1f3335f6594ed986ce0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ellis-thanatipanonda-2022-cut-family-a-conjecture-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul Ellis and Thotsaporn Aek Thanatipanonda, *The Arithmetic-Periodicity\nof cut for C = {1, 2c}*, arXiv:2203.02457v1, Section 7, page 18;\nDiscrete Applied Mathematics 322 (2022), 391–403,\nDOI 10.1016/j.dam.2022.08.027:\n\n> Let X to be a non-empty set of even numbers, each of which is at least 4. Let Y to be a non-empty set of odd numbers, each of which is at least 5. Let x = 2c and y be the smallest elements of X and Y, respectively.\n\n> Family A: C = {1, 3} ∪ X, or C = {1, 3} ∪ X ∪ Y.\n\n> The nim-sequence for all games of cut in this family are all precisely (0, 1)^c (+2).\n\nThis is Conjecture 1. Preregistration:\n[#19674](https://github.com/the-omega-institute/trureturing/issues/19674).\nThe settling theorem is\n`D5.S0.Certificates.Games.EllisThanatipanondaCutFamilyA.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A move cuts a pile into $d+1$ positive parts, $d\\in C$ | `G`: `Fin (d + 1)` option, positivity and exact sum | All legal options occur |\n| Nim-sum of the part values; mex of option values | `G`: `List.ofFn`, right fold of natural xor, existing `mex` | The guard only ensures well-founded recursion and is inactive for Family A options |\n| Nonempty even $X$, all elements at least $4$ | First two hypotheses of `claim` | Arbitrary natural sets, including infinite sets |\n| No extra odd cuts, or nonempty odd $Y$ with elements at least $5$; its least element $y$ | Disjunction in `claim` | $Y=\\varnothing$ gives the first alternative; nonempty natural $Y$ has a least member, which the conclusion does not use |\n| Smallest even cut $2c$ | $2c=\\operatorname{sInf}X$ | Nonempty natural sets have a least member; $c\\ge2$ follows |\n| Sequence $(0,1)^c(+2)$, beginning at $n=1$ | Conclusion of `claim`, proved by `result` | $G_C(n)=2\\lfloor(n-1)/(2c)\\rfloor+((n-1)\\bmod2)$ for every $n\\ge1$ |\n\nThe natural subtraction, integer quotient and remainder are the literal\nbinding conventions. The recursion also defines values outside Family A;\n`guard_value` and `G_equation` identify its legal-option recursion on the\npositive cut-sets used in the proof. No assertion about an invalid zero\ncut is needed.","title":"ellis thanatipanonda 2022 cut family a conjecture 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/ellis-thanatipanonda-2022-cut-family-a-conjecture-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a46950b126883d0bdf4f018c11e622aef6f14d13a342bb605c1d4798cc7a6ce4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:elzein-mortada-saturated-packing-conjecture-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"El Zein and Mortada, arXiv:2603.25113v1, Conjecture 3, printed page 27,\nstate: \"Every 2-saturated subcubic graph is (1, 1, 2)-packing\ncolorable.\" A subcubic graph has maximum degree at most three. It is\n2-saturated when each degree-three vertex has at most two degree-three\nneighbors. A `(1,1,2)`-packing coloring partitions the vertices into two\nindependent classes and one class whose distinct vertices have distance\ngreater than two. The printed conjecture contains no local-girth hypothesis.","title":"El Zein-Mortada Saturated Packing Refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/elzein-mortada-saturated-packing-conjecture-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b4c2b0f671ebbf0357afa5c666af37060befc62db5f5d7567b85aa6cc590c80.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:emeriau-2020-torpedo-perfect-classical","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P.-E. Emeriau, M. Howard and S. Mansfield (arXiv:2007.15643, quant-ph; PRX\nQuantum 3, 020307 (2022)) introduce the dimension-`d` Torpedo Game: Alice\nreceives `x, z ∈ Z_d` and sends one dit; Bob is asked\n`q ∈ {∞, 0, …, d − 1}` and must answer `a ≠ x` for `q = ∞` and `a ≠ qx − z`\notherwise. They find the classical values `3/4` and `11/12` for `d = 2, 3`,\nperfect classical strategies for `5 ≤ d ≤ 23`, and state\n\n> Conjecture: θ^C_{d≥5} = 1.\n\nIssue #11350 fixes the reading: classical strategies use finite shared\nrandomness with stochastic encodings and decodings, and `θ^C_d = 1` means that\n`1` is the greatest classical winning probability.","title":"A perfect classical strategy for the Torpedo Game","url":"https://the-omega-institute.github.io/trureturing-pages/research/emeriau-2020-torpedo-perfect-classical/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1618958ca08a350c6c56abab15c58c52fd9005bd71ac8d0ed8489f6eb3c0b045.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:enciso-finkel-gonzalez-lopez-rodriguez-2007-freezing-unique-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Enciso, F. Finkel, A. González-López and M. A. Rodríguez,\narXiv:0704.3046v1, §3, printed p. 13, after Eqs. (31)–(32), state:\n\n> The first one is the requirement that ξ be the unique minimum of the potential U in the domain C. Although our numerical calculations suggest that this is indeed the case, we have not been able to provide a rigorous proof of this fact.\n\nThe chamber is $C=\\{x\\in\\mathbb R^N:x_1<\\cdots It is also of interest to determine whether the position of the last spin tends to infinity as N → ∞, since according to our interpretation of the chain’s geometry the number 2ξN/π is the radius of the circle on which the spins lie.\n\nThe sites are increasing real solutions of Eq. (6),\n\n$$\n\\xi_i=\\frac1{\\xi_i-\\xi_{i-1}}+\\frac1{\\xi_i-\\xi_{i+1}},\n\\qquad \\xi_0=\\xi_N,\\quad \\xi_{N+1}=\\xi_1.\n$$\n\nIssue #11858 preregisters the fully quantified assertion: for every real\n$B$ there is a natural $N_0$ such that every $N\\geq N_0$, $N\\geq3$, and\nevery strictly increasing cyclic site solution satisfy $B<\\xi_N$.\nThe source motivates the assertion by Eq. (19), then cautions that the\ninverse-error-function argument in Eq. (18) is accurate only up to terms\nof order $1/N$.","title":"Divergence of the last site of the nearest-neighbor QES chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/enciso-finkel-gonzalez-lopez-rodriguez-2007-last-site-divergence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/145538c9cb78327b60d47c614495cd9d9bae65444d91c8482548f011820d7b7f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-197-order-gadget-all-residues","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"William Kasel, *Structural rigidity in the Erdős–Graham two-set permutation problem*,\narXiv:2609.02939v1, Remark 30:\n\n> The full order gadget is nevertheless conjectured infeasible at every M ≥ 16 (machine-verified for\n> 16 ≤ M ≤ 200 and M = 512, Proposition 18); at every swept scale with M ≢ 0 (mod 8) the C3 core was\n> satisfiable, and other attack subsets were observed computationally to take over (Proposition 19);\n> a proof beyond the swept scales would need per-residue analogues of Theorems 27–28 built from the\n> same flood toolkit.\n\nThe order gadget OG(M) of Definition 15 asks for a linear order of `(M, 2M]` in which no in-block\nthree-term arithmetic progression is monotone and, for `x ∈ {15, 16}` and `1 ≤ j ≤ x/2`, the guard\n`2M + 2j − x` precedes the bottom `M + j`.","title":"The Order Gadget of the Erdős–Graham Two-Set Problem Is Infeasible at Every Scale","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-197-order-gadget-all-residues/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39b31b9dc3a23287b4ae492a3c00bd8ce71bbad54eecf4da6b6f8460b3aeb490.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-1979-exactly-one-divisor-unimodality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Erdős, *Some unconventional problems in number theory*, Astérisque 61 (1979),\nprinted page 78:\n\n> Perhaps ε₁(n,m) is unimodular for m > n+1, but I know nothing about this.\n\nHere `ε₁(n,m)` is the natural density of integers `N` having precisely one\ndivisor `d` with `n < d < m`. Tenenbaum, *Some of Erdős' Unconventional Problems\nin Number Theory, Thirty-four Years Later* (2013), printed page 18, states:\n\n> To my knowledge, the question of unimodality of ε₁(y,z) as a function of z is still open.\n\nTenenbaum writes the upper endpoint as `d <= z`; the formal refutation follows\nthe strict upper endpoint in Erdős's original sentence.","title":"Refutation of the one-divisor interval-density unimodality suggestion","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-1979-exactly-one-divisor-unimodality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4df1e889cdc61eb90475f9e9d230b23f2424e0557ce8206922a751bb865ee874.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-1985-consecutive-product-squarefree-factor-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Erdős, Rocky Mountain Journal of Mathematics 15(2) (1985), page 361\n(verbatim from the rendered scan at\nhttps://users.renyi.hu/~p_erdos/1985-27.pdf):\n\n> Put (21) ∏_{i=1}^{k} (x + i) = u_k(x) v_k(x), (u_k(x), v_k(x)) = 1 where v_k(x) is squarefree and all prime factors of u_k(x) occur with an exponent greater than 1. The representation in (21) is clearly unique. Clearly for k > k_0(x), u_k(x) > v_k(x). Perhaps one can estimate the smallest k_0(x) so that u_k(x) > v_k(x) for all k > k_0(x) quite well. I have not done this. For small values of k usually v_k(x) > u_k(x). I thought that for every x there is a k for which v_k(x) > u_k(x). x = 7 seemed a likely counterexample but if k = 7, u_7(7) = 2^7 3^3 < v_7(7) = 5·7·11·13. On the other hand a simple computation shows that n = 23 is a counterexample, i.e., for every k, v_k(23) < u_k(23). The reason for this is the existence of 24, 25, 27 and 32. I would not be surprised if 23 is the only counterexample. Perhaps in fact there is a k_0 so that for every k > k_0 and all n > n_0(k) (22) v_k(n) > u_k(n). (22) is perhaps too optimistic.\n\nThe literal refuted statement is\n\n```text\n∀ x ≥ 1, x ≠ 23 → ∃ k ≥ 1, u(x,k) < v(x,k).\n```\n\nFor `P(x,k) = ∏_{i=1}^k (x+i)`, `v(x,k)` is the product of the\nprimes whose exponent in `P(x,k)` is exactly one. It is not the squarefree\npart: a prime occurring to exponent at least two is absent from `v`.\nThe factor `u(x,k)` is the product of every prime power whose exponent in\n`P(x,k)` is at least two, with its full exponent. Thus `u(x,k)·v(x,k)=P(x,k)`.\n\nThe result refutes only the sentence proposing 23 as the unique\ncounterexample. It makes no claim about conjecture (22), an estimate of\n`k_0(x)`, the assertion that 23 and 47 are the only counterexamples, or the\nstatement that 23 itself is a counterexample.","title":"Starting value 47 refutes Erdős's proposed unique counterexample","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-1985-consecutive-product-squarefree-factor-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc2ae1f7d1c84d8719e853f30dd17a627ccb65d065f1d7e8ec55e807f34cb274.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-1989-mahler-binary-digit-square-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Erdős's 1989 reminiscence, page 2 (verbatim):\n\n> He told me about his last paper. Let e_i = 0 or 1. Then Σ_{i=1}^{n} e_i k^i = x^2 has infinitely many solutions for k = 2, 3 and 4. He conjectured that it has only a finite number of solutions for k > 4. In fact, the only nontrivial solution he found was 7^3 + 7^2 + 7 + 1 = 20^2, and perhaps there are no others (1 + (k^2 − 1) = k^2 counts as trivial).\n\nThe literal refuted statement is the proposition `claim`:\n\n```text\n∀ (k x : ℕ) (S : Finset ℕ),\n 5 ≤ k → 1 < x → BinaryBaseSquare k x S →\n (k = 7 ∧ x = 20) ∨ k = x ^ 2 - 1\n```\n\nHere `BinaryBaseSquare k x S` means\n`Nat.Coprime k x ∧ ∑ i ∈ S, k ^ i = x ^ 2`. The exponent set includes zero\nbecause the printed example contains `k^0 = 1`. \"Trivial\" means the two-digit\nfamily `x^2 = k + 1`, equivalently `k = x^2 − 1`, matching Erdős's\nparenthetical after renaming the base variable. The coprimality condition\n`gcd(k, x) = 1` comes from Mahler's setting and excludes digit-shift scalings\n`x ↦ k·x`. The counterexamples `(12, 521)` and `(8, 11677)` are in no trivial\nfamily under either reading.\n\nThis refutation makes no claim about Mahler's finiteness conjecture for any\nfixed `k > 4` and no claim about infinitude for `k = 2, 3, 4`.","title":"Refutation of Erdős's report of Mahler's binary-digit square uniqueness","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-1989-mahler-binary-digit-square-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0cf77b313a3ccab42e3f4b8f9e9d41244563ddf39f1d1314c0e5992f11e0af8d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-7-odd-covering-systems","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The [problem page](https://www.erdosproblems.com/7) asks:\n\n> Is there a distinct covering system all of whose moduli are odd?\n\nThe negative target is the following unrestricted assertion. For every finite\nset `D ⊂ ℕ` with `d > 1` and `d` odd for every `d ∈ D`, and every assignment\n`a : D → ℤ`, there exists `z ∈ ℤ` such that\n\n\\[\n \\forall d\\in D,\\qquad d\\nmid z-a(d).\n\\]\n\nDivisibility is in the integers. Set membership enforces distinct moduli;\nthere is no bound on their sizes, exponents, number, or total prime support.\nA refutation requires a finite family satisfying exactly these conditions\nwhose classes cover every integer. The page remained open when read on\n28 September 2026. The results below do not settle this unrestricted assertion.\n\nThe complete results and proofs are organized below. Each link opens the corresponding mathematical section; all original assumptions, bounds and open obligations are retained.\n\n### Results and proofs\n\n- [Passing every complete prime marginal requires at least six support primes; with exactly six, five monotone survivor-profile exclusions leave only seventeen prime sets, with arbitrary heights and residues. These are necessary conditions, not feasible constructions or a solution of the unrestricted problem](../Library/Arith/lettlsun2008cosets.md#existing-survivor-laws-restrict-simultaneous-complete-marginal-feasibility)\n- [Actual tight rows can be individually coverable yet admit no common original phases: a complete odd-wheel CRT graph passes every clique budget and has a sharp positive defect on its tight region; other rows remain below one, so this is not a new global noncoverage range](../Library/Arith/lettlsun2008cosets.md#common-original-phases-can-obstruct-all-tight-rows)\n- [A blocked joint-loss tree forces private leaves or a doubled multiplicity budget; removing the height factor requires one distribution across blocked fibres. Two odd noncovers defeat the local geometric shortcut, while the complete-source projection count rejects both](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/375-deep-prime-prefix-projections-and-tree-contraction.md#9-liability-multiplicity-and-the-missing-distribution-of-blocked-fibres)\n- [The true joint loss after deleting low-height originals gives a necessary-and-sufficient prime-absorption test; extremal covers require a common blocking law avoiding every retained event, and an actual even control separates this from both the old filter and an unsafe individual-private shortcut](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/375-deep-prime-prefix-projections-and-tree-contraction.md#8-joint-liability-gives-the-exact-absorption-criterion)\n- [Two different complete, exactly valued prime-coordinate averages recover every original numerical label and phase when moduli are distinct; finite-field height peeling preserves the actual source, while a period24 even control separates support-prime average lower bounds from whole coverage](../Library/Arith/lettlsun2008cosets.md#complete-prime-coordinate-averages-identify-distinct-original-labels)\n- [Absorbing a larger prime into new higher digits of a smaller prime gives a live-cofactor blocker and the necessary bound q-r<=H*tau(M)-1; an actual irredundant even cover contracts14 to8, and a filtered-only control separates retained from discarded events](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/375-deep-prime-prefix-projections-and-tree-contraction.md#7-live-cofactors-obstruct-absorbing-a-larger-prime-into-new-higher-digits)\n- [Original descendants raise the private-hull closure threshold from d to the largest original multiple of d; a same-phase group gives an exact repair budget for the complete private region, and a ten-class odd noncover passes every old PH5 test but admits the stronger reduction](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/385-private-congruence-hulls-and-crossed-modulus-closure.md#10-descendant-assisted-private-hull-closure-and-phase-group-repair-budgets)\n- [A reciprocal private-hull pair admits an exact source-preserving swap: after repairing both complete private regions, only the points with exactly those two old owners remain at risk; any surviving descendant then gives a strict cardinality descent, while comparable pairs only simplify one hull condition and crossed pairs remain open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/385-private-congruence-hulls-and-crossed-modulus-closure.md#176-reciprocal-private-hull-swaps-leave-one-exact-joint-liability)\n- [If a reciprocal private-hull pair has at least two receiving descendants, its occupied lcm transfers the exact remaining liability to the lcm's complete private region; every repair needs at least as many classes as deleted descendants, and every divisor of the lcm hull must remain in the retained label palette](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/385-private-congruence-hulls-and-crossed-modulus-closure.md#177-an-occupied-lcm-transfers-a-reciprocal-swap-to-complete-private-liability)\n- [An irredundant odd noncover with every normalized prime class satisfies all scalar directional inequalities at every actual private point and every divisor cut, but still has a hole; numerical moduli repeat, and thinning to distinct labels destroys the prime13 demand](../Library/Arith/lettlsun2008cosets.md#prime-normalization-and-all-private-scalar-cuts-do-not-force-line-coverage)\n- [The463-class obstruction also excludes every mixed-prime common-center relocation batch:1603 actual parent-phase options are contradicted by private and joint-owner point constraints; one five-parent control preserves all858 supplied private points but loses a jointly covered integer, and arbitrary pure-prime tails retain the exclusion](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#138-mixed-child-primes-also-forbid-every-common-center-relocation-batch)\n- [At one original private point, every affine quotient cut reduces exactly to a prime-depth suffix; arbitrarily high divisor-closed, irredundant odd families pass all those cuts and every shell-mean test at that point and prime while leaving holes on the same prime line](../Library/Arith/lettlsun2008cosets.md#arbitrary-quotient-cuts-collapse-to-the-same-depth-suffixes)\n- [Mixed parent-divisor and crossed carriers have an exact allocation threshold: a26-class divisor-closed odd family contracts to24 using only three proper cofactor divisors where the all-crossed route requires seven](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/385-private-congruence-hulls-and-crossed-modulus-closure.md#9-mixing-three-carrier-types-lowers-the-exact-divisor-requirement)\n- [The463-class obstruction resists every size of same-prime centered parent relocation, even with all children retained:858 verified private points exclude1658 options without enumerating subset sizes; arbitrary pure-prime tails preserve this at arbitrarily small positive hole density](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#137-every-size-of-same-prime-centered-parent-relocation-is-blocked)\n- [Two actual composite-parent patterns permit joint two-class reductions, including one at every ternary height; divisor-closed irredundant families contract26 to24 and18 to16 with no lost covered points although either separate contraction fails, while occurrence in every hypothetical cover remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/385-private-congruence-hulls-and-crossed-modulus-closure.md#6-a-mixed-root-cross-permits-a-joint-composite-parent-contraction)\n- [Every original literal4555 height-two maximum74 source admits one common law with Gamma1225 at most9-7/11026; shared-row reduction and one fixed mixture close the final stratum of this fixed-head source class, while arbitrary heights and original outside-cofactor lifting remain open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#one-common-law-below-nine-for-the-complete-original-maximum74-class)\n- [Every current maximal mass18 entry can be lowered globally or yields an already supplied cut; outside those branches, one shared unit of capacity reduction can be assigned to all current bad entries, with RC74 supplying the additional common-query regularization](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#current-maximal-mass18-entries-admit-global-capacity-reductions)\n- [Two safe half-unit transports decrease one common balanced74 potential in at most four steps; exact complete-source controls verify both moves, while terminal actual-support traps and universal accessibility remain unresolved](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#safe-half-unit-transports-decrease-one-common-balanced74-potential)\n- [A jointly minimal set of dangerous mass18 blocks forces one-inactive77, two-inactive80 or three-inactive83 cuts; only the five exact83 shapes lie outside the listed suppliers, but lowering a sum is not yet a common-law proof](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#joint-selected-block-minimization-yields-a-controlled-inactive-root-cut)\n- [Every actual cut through80 with at least two inactive roots admits one law bounded by2865/319; exact three-inactive83 cuts leave five explicit complete-fibre profiles requiring further suppliers](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#two-inactive-cuts-through80-and-the-exact-three-inactive83-boundary)\n- [A463-class irredundant divisor-closed noncover defeats both low-excess suppliers under every outside law; CRT extensions preserve the obstruction with arbitrarily many support primes, arbitrary tail heights, and arbitrarily small positive uncovered density, leaving exact whole-cover structure essential](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#13-irredundancy-and-divisor-closure-do-not-supply-a-low-excess-law)\n- [Deleting189 individually contained classes from the original452-class phase obstruction preserves its entire covered set and leaves zero phase excess at one outside point; this applies to that input, while the different463-class irredundant obstruction admits no such reduction](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#124-containment-reduction-removes-this-particular-obstruction)\n- [A selected balanced74 mass18 block either lowers globally to at most17.5 or yields an already supplied one-inactive77/three-inactive82 actual cut; a complete282-point source verifies lowering18 to0, with simultaneous common-law handling supplied by RC74](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#a-selected-mass18-block-can-be-lowered-or-supplies-an-existing-actual-source-law)\n- [T3-good whole block support permits one mass18 transport with fine bound2175/7; together with mass18.5 repair this gives the conditional balanced74 law9313/1036, with the remaining T3-bad blocks handled by the later RC74 whole-source mixture](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#t3-good-actual-support-repairs-the-mass18-half-integral-obstruction)\n- [Balanced value74 admits support-uniform mass18.5 transport with fine bound1835/6; a unique mass18 flow refutes universal fixed-external repair, leaving a precise joint-flow condition for the conditional1331/148 law](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#balanced-value74-admits-mass185-transport-but-fixed-external-mass18-repair-can-fail)\n- [Every original height-two4555 source of maximum flow75 or76 admits one law with Gamma1225 at most3572/397; three-inactive cuts through82 are supplied, while its particular source-capacity obstruction at maximum74 is bypassed by the later RC74 theorem](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#every-original-maximum-flow75-or76-source-has-a-common-law-below-nine)\n- [Four classified original children and an arbitrary fifth admit one actual law bounded by2779/323; all2,985,984 coarse queries and complete-source controls support the supplier for both remaining partial and11113 public3 profiles](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#an-unrestricted-original-fifth-child-supplies-the-two-remaining-public3-profiles)\n- [Fourteen actual private points and one joint public law supply eight nonpartial public-cost3/4 cut profiles with bound3572/397; two original-source controls exclude the automatic three-exterior-label bridge for11113](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#fourteen-private-points-supply-eight-nonpartial-public3-and-public4-profiles)\n- [Two public labels and a complete clean full root admit one actual law; the shared-column bound4252/477 supplies all five one-inactive public-cost2 profiles through77, with public costs3–4 supplied by the later fourteen-point consumers; the unrestricted lift remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#two-public-labels-on-a-clean-full-root-supply-one-common-law)\n- [One-inactive cuts through77 with public prefix cost0 or1 admit a common law below nine; three tight full11222 roots have an explicit law bounded by80/9 via the shared49/7/35 queries. The subsequent two-public-label supplier handles cost2; the later fourteen-point consumers handle costs3–4; the unrestricted lift remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#one-inactive-root-with-public-prefix-cost-zero-or-one)\n- [Every actual cut below80 with at least two inactive roots supplies one law at most2865/319; complete fibres and a four-owner public-singleton argument close the entire two-inactive C75/C76 obstruction branch](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#any-actual-cut-below80-with-at-least-two-inactive-roots-supplies-one-law)\n- [A452-class literal noncover has merged phase excess above both reserve thresholds at every outside survivor, refuting an unconditional low-excess supplier under every supported law; whole-cover structure is still needed](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#12-a-literal-noncover-forbids-every-unconditional-low-excess-law)\n- [Retained numerical weights replace depth multiplicity by outside-support incidence under one common law; explicit conditional criteria allow arbitrary retained heights and positive phase collisions, while shared-cylinder examples separate merged envelopes from divergent row costs](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#11-retained-weights-pay-arbitrary-depth-multiplicity-under-one-outside-law)\n- [Every complete full12222 source admits one original-source law with Gamma1225 at most2865/319; graph reductions and a complete128-layout classification finish the specified C75/C76 three-inactive-root lists, with the whole75/76 classes supplied by the later RC76 theorem; RC74 supplies maximum74, while arbitrary-height lifting remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#every-complete-full12222-source-admits-a-common-law-below-nine)\n- [Sharp mass19 transport and maximum-flow terminal synchronization give conditional74–76 laws; complete actual sources refute universal terminal-cap feasibility despite good common laws, motivating one joint coarse-cap flow criterion](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#mass19-transport-terminal-synchronization-and-conditional-balance-consumers)\n- [Every complete monochromatic anchor supplies a law below nine; two singleton and two double fibres allow an arbitrary fifth fibre, and gap12/gap222 permit arbitrary remaining fibres, supplying three further C75 shapes](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#every-complete-monochromatic-anchor-supplies-a-common-law-below-nine)\n- [Every complete full11222 source admits one common law with Gamma1225 at most503/56; distinct-label punctures and all64 root-query layouts supply the final C76 three-inactive-root shape, with the remaining one-inactive branches addressed by later reductions](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#every-complete-full11222-support-has-one-common-law-below-nine)\n- [A full root with one public and five private candidates admits one common law with Gamma1225 at most503/56; exact shared-query certificates supply the entire full/public1/private11111 family](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#a-public-singleton-and-five-private-candidates-at-a-full-root)\n- [Four complete double fibres at the gap admit one actual common law with Gamma1225 at most719/80; normalized original trees retain the complete shared-label exclusion, supplying gap2222 without a minimum-cut or standalone premise](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#four-complete-double-fibres-at-the-gap-admit-one-common-law)\n- [Small complete anchors supply seven further three-inactive-root cut shapes; retaining global fine-label exclusion gives common laws for three to eight sparse labels, with the whole75/76 classes supplied by the later RC76 theorem; RC74 supplies maximum74, while the arbitrary outside-cofactor lift remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#small-complete-anchors-supply-additional-common-laws-below-nine)\n- [One fixed anchor per retained numerical modulus supplies an actual outside law with zero phase excess when its row budget is below one; a pure-survivor bound and bad-depth incidence at most48 give an explicit sufficient class, without solving the arbitrary-family lift](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#10-anchor-phases-give-a-same-law-arithmetic-supplier)\n- [Every literal4555 height-two actual source of maximum flow77 has one common law with Gamma1225 at most2078/231: asymmetric capacities and exact-cut uncrossing close this finite stratum; the later RC76 theorem supplies75/76; RC74 supplies maximum74, while the arbitrary outside-cofactor lift remains unresolved](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#asymmetric-root-capacities-supply-the-complete-maximum-flow77-class)\n- [Whole coverage forces weighted excess of distinct retained phases at every outside survivor; one-law phase merging retains the exact subtraction. Section12 rules out an unconditional strict reverse for arbitrary families, leaving a whole-cover-specific bridge open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/450-499/450-weighted-original-depths-and-the-uniform-lift-boundary.md#9-whole-coverage-needs-weighted-excess-of-distinct-projected-phases)\n- [Every exact77 cut with one inactive root admits one actual common law below nine: a bounded residual-cycle step handles the gap root, and integer budgets reduce the full-root case to IF77](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#one-inactive-full-root-has-no-remaining-partial-child-exact77-profile)\n- [Avoiding each five-block set is sufficient for one common phase law; failure forces a minimal two-to-five-block hitting family, without making that family a negative law certificate](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#avoiding-every-five-block-set-is-a-sufficient-joint-bridge)\n- [Two remaining capacity78 profiles admit actual common laws: gap2222/full22222 has bound249/28, and full12222/full12222 has1385/154 including its fourth-column exception](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#the-private-gap2222full22222-profile-admits-a-common-law)\n- [An exact capacity77 cut with two inactive roots admits one actual common law with Gamma1225 at most1385/154; capacity78 reduces to nine private cost shapes, while the unrestricted cofactor lift remains open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/449-equality-sources-have-a-private-law-below-nine.md#two-inactive-roots-at-exact-capacity77-admit-one-common-law)\n- [Occupancy-aware restrictions and weighted root caps give one law below nine for four-root height-(2,2) product-tree blockers with at most one full five-child fibre](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/445-occupied-branch-restrictions-and-weighted-root-caps.md)\n- [Uniform subtree restrictions couple both prefix trees under one actual law; three robust first-five roots give Gamma6125 at most2024/225 without a fixed good-fibre selector](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/444-uniform-subtree-restrictions-couple-two-prefix-trees.md)\n- [One finite flow couples row and tree-prefix caps with optimal joint factor q/m; isolated ternary five-prefix sources admit one law below nine through five-height22 and every finite seven-height](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/443-one-supported-law-couples-rows-and-tree-prefixes.md)\n- [Minimum coarse sources admit matching lifts and two height strips below nine; joint mod25 fibres give a separate bound eight, while a 540-point blocker refutes automatic mod175 fibre matching](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/442-minimum-coarse-sources-lift-through-actual-fibres.md)\n- [Integer joint moments improve actual-family height lifting with full cofactor tests; a four-atom joint witness limits what the four retained budgets alone can certify](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/441-integer-joint-moments-improve-two-axis-height-lifting.md)\n- [A literal mod-35 mixture separates equal coarse-marginal and scalar-Gamma summaries; joint layout price profiles retain exact finite-fibre contraction and expose the additional conditions for dynamic boundary sufficiency](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/440-joint-test-profiles-as-composable-boundaries.md)\n- [Fixed coarse marginals have an exact coprime free-coordinate completion cost; actual residual lifting retains a joint minimax obligation, and the original H1 scalar formula fails to reach nine at finite heights](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/439-actual-residual-lifting-and-exact-free-coordinate-cost.md)\n- [Separated capacity-six flows and shared weak/grid blocks give common laws at most five; a distinct star construction remains below 46/9, with exact limits on support extraction and weak-child deletion](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/438-separated-weak-supports-and-common-laws.md)\n- [Four roots with three nonsingleton children each and cross-root disjoint original exact pairs admit a common law below five; an exact capacity-flow formula removes the pair-extraction requirement](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/437-four-root-nonsingleton-sources-admit-a-common-law-below-five.md)\n- [For one fixed original family, identical clique-DP keys count correctly but cannot answer a newly authorized archive read; actual branch records and CRT cells give distinct task-specific repairs using existing task-completion interfaces](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/436-self-observation-refines-a-count-sufficient-clique-state.md)\n- [Minimum height-two sources are exactly three five-matchings and admit the sharp universal budget 68/15; every admissible source has an actual nine-point joint flow, while the weighted common-law condition remains additional](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/435-minimum-height-two-sources-and-a-nine-point-flow.md)\n- [Every four-root, three-child, two-neighbor source with disjoint root pair sets admits one explicit original-label law of cost at most 149/30; larger-neighborhood controls delimit extraction and correctly aligned height lifting retains its actual-residual premise](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/434-four-root-pair-sources-admit-a-common-law-below-five.md)\n- [Chordal overlap certificates have an exact active-component remainder; on all nonunit divisors of 945 their forced-coprime optimum is 175 survivors, improving the optimal forest bound 141 while leaving a gap to the existing sharp minimum 191](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/433-chordal-overlap-certificates-and-their-exact-finite-limits.md)\n- [Actual four-root matching, rectangle and shared-column selections give common laws at five-height two; keeping the inherited seven phase shared yields the stronger bound five for two weak-root geometries, while an arbitrary-height source compression preserves every five-column projection](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/432-four-root-source-selections-and-the-shared-column-boundary.md)\n- [At five-height two and seven-height one, three actual first-five fibres each meeting every child-by-column 3-by-5 rectangle admit one probability with complete independent-layout cost at most 46/9; this includes every three-root source satisfying the full product-tree test](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/431-three-robust-five-roots-admit-a-height-two-common-law.md)\n- [The existing height-lifting formulas cannot carry the supplied first-five-layer constants to the all-height target nine; a precise same-law coarse/fine tradeoff is also refuted for the existing seven prescribed cap components, leaving free source-law selection open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/430-same-law-height-lifting-needs-a-new-seed-tradeoff.md)\n- [A sharp height-two cap-flow bound of 46/9 closes the remaining finite-height case: at five-height one, every source on four allowed rows with the full five-ary projection tree and all six pairwise ternary trees admits one probability meeting the complete independent-layout target at every seven-height](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/429-phase-conflict-cap-flow-closes-the-height-two-bound.md)\n- [At five-height one, freely weighting any seven fixed full-five and pair-ternary cap laws gives a universal bound 407237/70076 below six and the finite target at every seven-height K at least three; a literal height-two fixed-component game has exact value 46/9](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/427-shared-row-cap-mixtures-break-the-six-barrier.md)\n- [At five- and seven-heights three, legal fixed cap components can defeat every mixture of their weights even though the source has a good law; two actual laws with identical complete coordinate marginals have exact joint moments 328/27 and 5776/729](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/428-fixed-cap-components-do-not-transport-across-five-adic-depth.md)\n- [Two exact eight-prime cores omitting 7 admit arbitrary seven-vertex attachments under the inherited Chapter 31 and ordinary-source premises; extendible core Haar density exceeds 1/2200000](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/69-eight-prime-cores-omitting-seven.md)\n- [An eight-prime core {3,5,7,13,17,19,23,29} omitting 11 admits arbitrary seven-vertex attachments under the inherited Chapter 31 and ordinary-source premises; extendible core Haar density exceeds 1/2200000](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/68-eight-prime-core-omitting-eleven.md)\n- [Complete shared-root query tables, an exact binary flow/cut certificate and rational first9-prefix charges strengthen the unchanged actual-law credit to16.8681002893431; a feasible relaxed assignment bounds the remaining room in this refinement.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/67-shared-root-prefix-bounds-for-the-actual-survivor-law.md)\n- [An exact shared-phase triangle correction sharpens the unchanged actual-law credit; uniform layout averaging caps the complete seven-prime triangle replacement, and finite positive measures refute a universal common-CRT-center restriction.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/66-shared-phase-triangles-and-common-center-obstruction.md)\n- [Complete survivor-cylinder queries and exact small-modulus controls certify a fixed-law credit, while independent support counts place the entire E154 independent-marginal-max tau1 method strictly below the credit required at B16384; joint layout improvements remain outside this ceiling.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/65-independent-survivor-marginal-ceiling-at-the-fixed-cutoff.md)\n- [An attained full-history adaptive order among core7 coordinates lowers the exact seven-phase154 bad mass; its complete positive policy is reconstructed on actual leaves, and the three retained stop-loss thresholds are unique real-threshold minima of their fixed directed upper envelopes.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/64-adaptive-core-policy-and-continuous-stoploss-optima.md)\n- [A common squared-load stop-loss threshold strengthens same-law conditioning for every complete divisor layout; all1362 exact square-threshold bounds are independently reproduced from the complete verified product tables.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/63-squared-load-stoploss-continuation.md)\n- [Enlarging the stop certifies the attained seven-phase head atB32768, and its full outside-D7 smooth extension atB65536, with every directed stage independently reproduced.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/62-expanded-stopping-cutoffs-for-the-seven-phase-head.md)\n- [Coordinatewise retention transports the exact auxiliary budget lower bound; combined with the fixed positive cylinder polynomial it excludes a full relative1/50000 box of all finite head-depth caps.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/61-coordinate-retention-and-profile-neighborhood-obstruction.md)\n- [A fixed positive cylinder cover yields a38-term multiaffine survivor bound at new caps on all20 head coordinates, with every coefficient independently reproduced.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/60-positive-cylinder-covers-across-head-profiles.md)\n- [Globally injective terminal phases reduce core-first optimization; a seven-phase family defeats the fixed balanced-profile budget for every adaptive order while retaining an uncovered integer and154 private class witnesses.](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/59-terminal-phase-elimination-and-uniform-balanced-profile-obstruction.md)\n- [An adaptive five-prime initial block recovers fixed-phase noncoverage where all 120 fixed block orders fail the same budget; a cylinder dual excludes scheduling-only recovery of the fixed D7 allowance](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/58-adaptive-initial-block-recovery-and-all-schedule-budget-obstruction.md)\n- [A balanced depth profile enlarges the fixed 154-class noncoverage result to the degree-seven exponent frontier with unrestricted original tails](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/57-balanced-depth-profile-and-degree-seven-frontier.md)\n- [Joint exponent frontier D9 extends the fixed 154-class noncoverage law; a cylinder-cover dual sharply bounds a literal 105 family](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/56-exponent-frontiers-and-cylinder-cover-certificates.md)\n- [Adaptive read-once head orders retain conditional comparison and strictly improve a family of period 315; the joint atom-cap obstruction persists](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/55-adaptive-read-once-head-orders-with-unrestricted-tail-comparison.md)\n- [Depth-profile laws prove noncoverage for the literal 154-class head, arbitrary sixth-power smooth head additions and unrestricted original tails](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/54-depth-profile-head-laws-with-unrestricted-original-tails.md)\n- [Exact optimization of full-prefix capped head laws, arbitrary fixed head order, and complete counterexamples to the universal 9/20 threshold](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/04c-full-history-capped-laws-and-exact-global-optimization.md#full-history-capped-laws-and-exact-global-optimization)\n- [Shared small-prime budgets prove noncoverage through six or seven predecessor supports, with full Haar density and two exact limits of the complete support-Shearer envelope](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/53-shared-small-prime-budgets-for-six-or-seven-predecessor-supports.md)\n- [Two exact predecessor supports in arbitrary order, or four in numerical order, give noncoverage with full Haar density; a large-prime inventory may grow as p to the one-quarter](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/52-two-or-four-exact-predecessor-supports.md)\n- [At most two cycles per original incidence component gives noncoverage at arbitrary ranks and heights, full Haar density and unrestricted large-prime continuation](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/51-incidence-bicyclic-components-with-arbitrary-original-heights.md)\n- [At most five distinct mixed supports per prime gives noncoverage at arbitrary ranks and heights, with full Haar density from actual conditional avoidance ratios](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/50-prime-support-incidence-at-most-five.md)\n- [Arbitrary-rank prime-support incidence pseudoforests are noncovering, with height-independent full Haar density and unrestricted large-prime continuation](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/49-incidence-pseudoforests-with-arbitrary-original-heights.md)\n- [Prime-support incidence forests admit simultaneous original-label extensions, a full Haar density bound and an unchanged-source attachment budget](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/48-incidence-forests-with-arbitrary-original-heights.md)\n- [An anchored order with at most four earlier neighbours is noncovering: a record-minimum potential pays for arbitrary prime order and gives full Haar density greater than (107/240000)(5/12)^M](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/44-anchored-four-predecessor-orders-via-record-minimum-potential.md)\n- [In arbitrarily many naturally ordered prime coordinates, identical complete AP event laws leave a sharp nonzero interval of half-threshold first-hit costs](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/43-sharp-first-hit-ambiguity-with-identical-complete-event-laws.md)\n- [The complete original-AP event law can agree while first-hit costs differ; a two-level cap-aware LP has exact optimum 97/5775](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/42-two-level-fixed-colors-and-first-hit-moment-obstruction.md)\n- [Natural increasing prime order with at most four smaller neighbours gives full Haar density greater than (3/800)(11/24)^M, including books with three private primes per page](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/41-natural-four-predecessor-orders-are-noncovering.md)\n- [The fixed-order independent-run scalar ledger has global minimum 1.002553… over all real thresholds; actual cofactor colors require a stronger joint comparison](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/40-fixed-order-scalar-threshold-barrier-and-cofactor-colors.md)\n- [An order anchored at 3,5 with 7 first and at most three earlier neighbours per later prime gives explicit full Haar density, allowing arbitrary overlap and large-prime continuation](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/39-anchored-three-predecessor-orders-are-noncovering.md)\n- [An eight-prime core containing 3,5,7,11 admits arbitrary seven-vertex attachments; extendible core Haar density exceeds 1/2200000](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/38-eight-prime-cores-containing-three-five-seven-eleven.md)\n- [All eight-prime cores containing 3,5 admit arbitrary seven-vertex attachments under the inherited Chapter 23/31 and ordinary-source premises; extendible core Haar density exceeds 1/2200000](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/70-all-eight-prime-cores-containing-three-five.md)\n- [Audit of a claimed global nonexistence proof: the kernel-localization step is unproved, the extension allocation omits prime-power reuse, and the renormalization inequality has the wrong sign](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/71-audit-of-global-nonexistence-claim.md)\n- [SRCT exact-two-9 nonexistence would imply the distinct odd-covering conjecture, but its repeated-layer gain bound fails; exact transport and every fixed-size nonnegative fractional union-capacity refinement lose positivity under some actual fresh-prime extensions](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/75-srct-exact-two-quantifier-audit.md)\n- [An abstract 36-point source satisfies all selected prime-chain tree conditions and saturated-prefix caps, but every supported law fails the proposed full-history cap bridge and a finite-depth stop-loss comparison](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/388-saturated-chain-blockers-do-not-supply-conditional-caps.md)\n- [The same 36-point source has exact complete-layout second moment 21/4 and all-height supremum 256543/32400 under uniform lifts; the direct second-moment comparison survives although the stronger conditional and convex bridges fail](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/389-saturated-chain-source-has-small-complete-layout-second-moment.md)\n- [Every 5-by-7 root set meeting all three-by-five rectangles and using at least five columns admits one supported law with full-layout second moment at most 4; a support classification and Hall argument also bound independent uniform tail lifts](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/390-two-prime-root-blockers-admit-a-common-second-moment-law.md)\n- [A diagonal depth-two source meets both full-tree conditions but has minimax second moment 33/5 above the proposed 529/81: the all-root finite-height extension fails, while the excluded-root arithmetic subclass remains unresolved](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/391-diagonal-prefix-sources-obstruct-the-finite-height-moment-comparison.md)\n- [A normalized seven-point source has root minimax 631/166, but its identical optimal product fails at height two while a different product succeeds; the exact column-cap tradeoff also shows that root cost at most 4 requires a column of mass at least 1/4](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/392-root-optimal-laws-do-not-tensorize-the-full-layout-bound.md)\n- [One compatible seed-and-tail probability on repeated type-B sources satisfies the full arbitrary-layout comparison at every pair of positive heights; its uniform bound is 67803771/7658816 below 9](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/393-two-digit-seed-controls-repeated-type-b-sources-at-all-heights.md)\n- [Standalone tree blocking need not pass to joint prefix fibres; an explicit 116-point source still admits a common full-layout law bounded by 335/54](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/394-standalone-tree-blocking-need-not-pass-to-joint-prefix-fibres.md)\n- [Every 5-by-7 source meeting all three-by-five rectangles admits a common root law with second moment at most 4, without any standalone column-count premise](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/395-root-rectangle-blockers-need-no-standalone-projection-condition.md)\n- [A row containing a complete five-ary 7-tree, with ternary trees in each pair of the other three rows, supplies a common full-layout law at every depth K at least 2; its uniform bound is 60709/10800 below 6](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/396-a-heavy-row-controls-all-seven-adic-depths.md)\n- [If every union of three of the four rows contains a five-ary 7-tree, a stationary mixture gives one common full-layout law at every positive depth, with uniform bound 51/10; no pairwise ternary premise is needed](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/397-stationary-row-mixtures-control-all-seven-adic-depths.md)\n- [For each of the six exact root types A–F, the full product and standalone tree conditions supply a common law satisfying the finite comparison at every seven-adic depth, without repeated tail geometry; arbitrary larger root supports remain outside this theorem](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/398-six-exact-root-types-control-arbitrary-seven-adic-tails.md)\n- [A five-ary projection tree and at most two second seven-digits in each joint root cell give a common law with uniform bound 231/50; an explicit all-height family satisfies the full tree tests while containing no same-height component from the heavy-row, stationary-mixture or exact-root classes](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/350-399/399-sparse-second-digits-control-sources-outside-fixed-subclasses.md)\n- [An exact tree-rank dual removes the source variable from the common-law problem; a literal-layout counterexample rules out averaging deterministic bounds, and a constructive tail-coupling criterion keeps full joint realizability](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/400-literal-layout-mixtures-and-exact-tree-rank-duality.md)\n- [A recursive family attaining the minimum source size has one law for all original divisor phases at every height, with uniform comparison margin 43/1764; translated leaf colourings extend the construction beyond fixed-row private branches](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/401-a-recursive-minimum-source-has-one-law-at-every-height.md)\n- [For every source with the six pairwise ternary trees and full five-ary projection, one actual law has independent-layout second moment below 168/25; this universal estimate still exceeds the required limiting constant six](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/402-one-common-law-with-a-universal-row-compatible-bound.md)\n- [Sources of sharp minimum size at height at least two have a five-column root with either one two-surplus child or two one-surplus children; exact partition rules constrain the latter tails without supplying a new second-moment law](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/403-root-structure-of-sharp-minimum-sources.md)\n- [A single full-and-pair-tree mixture has universal second moment below 309/50; actual witness choices rule out reaching six by a fixed mixing coefficient alone](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/404-a-sharper-common-law-and-its-witness-choice-boundary.md)\n- [Four private terminal row children give one law with exact moment at most 19/5 at height at least two; refinement preserves a sharp two-surplus source family without proving inheritance for arbitrary sources](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/405-terminal-private-rows-control-sharp-two-surplus-sources.md)\n- [Two admissible continuing root fibres and three distinct-row private full trees admit one law below the finite target; four admissible root fibres give another sufficient condition](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/406-actual-root-fibres-admit-one-law-below-the-target.md)\n- [Actual sources obstruct a stationary row-weight recipe for every mixing coefficient; eight clean-tree signatures cannot constrain fixed row polytopes under uniformly faster-than-ternary prefix decay](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/407-limits-of-stationary-row-mixtures-and-finite-signature-summaries.md)\n- [A balanced labelled five-tree supplies one law below the finite target at every height at least two; admissible sources need not admit that selection](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/408-a-balanced-five-tree-selection-controls-all-heights.md)\n- [Two finite tail budgets preserve one actual common law through arbitrary ternary prefix depth; a nonuniform law resolves the balanced-selection boundary family with margin greater than 3/25](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/409-ternary-prefixes-preserve-terminal-common-laws.md)\n- [Three initial ternary levels permit independently varying stopping depths and one actual law with margin greater than 7/225; a height-eight law lies outside every synchronized 409 certificate](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/410-an-initial-ternary-budget-allows-asynchronous-stopping.md)\n- [Exact future survivor counts have canonical joint histogram at gcd(L,lcm(future)); actual odd-distinct histories refute dynamic sufficiency of separate next-action histograms](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/411-exact-continuation-state-is-a-joint-residue-histogram.md)\n- [A larger finite-height five-tree criterion; a sharp minimum source blocks every uniform ternary root law but admits one five-tree law with margin at least 16/225](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/412-a-sharper-five-tree-law-crosses-the-ternary-root-obstruction.md)\n- [One actual mixed centre and four independently shaped monochromatic ternary tails admit a common law; fixed root weights 65/197 and 33/197 allow central row cap 18/65](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/413-an-actual-mixed-centre-and-four-ternary-tails-admit-one-law.md)\n- [Same-projection row transport gives one actual law for an explicit concentrated sharp source at every height K at least 10, with margin 9/100; a literal K=2 layout refutes the same fixed recipe](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/414-same-projection-transport-controls-concentrated-sharp-sources.md)\n- [Even witness-dependent mixture coefficients fail the finite target on one fixed family of actual components, at every height; two original layouts give an exact segment obstruction](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/415-adaptive-coefficients-cannot-meet-the-finite-target.md)\n- [Exact arbitrary-height separation retains all independent original phases; different actual laws succeed on the concentrated sharp sources at heights three and four](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/416-exact-independent-layout-tree-separation-and-actual-laws.md)\n- [Exact prefix disagreement improves the same concentrated-source law to every height K at least seven, with margin 1/15 and no change of source or phase quantifiers](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/417-prefix-local-disagreement-extends-the-same-law-to-height-seven.md)\n- [Every concentrated sharp source in the explicit family admits one actual common law, for all K at least two with uniform margin 107/2700; exact heights five and six close the finite gap](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/arithmetic/400-449/418-concentrated-sharp-sources-admit-a-common-law-at-every-height.md)\n- [A finite-prefix transport lemma makes the missing-3 and missing-5 source-coordinate maps explicit; exact finite checks pass, while the new source and attachment budgets remain open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/72-uniform-prefix-transport-and-missing-anchor-interface.md)\n- [Exact source-row and attachment-budget audit for selected eight-prime cores missing 3 and/or 5; transport, root orientation and gluing remain open](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/73-missing-anchor-eight-core-source-audit.md)\n- [A book of N four-prime pages has full Haar density greater than (67/4000)(3/16)^N, supplying a computable cutoff for arbitrary large-prime constraints joining head pages](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/37-full-density-and-large-prime-continuation-for-spine-books.md)\n- [Arbitrarily many four-prime pages sharing the spine {3,5} are noncovering, with a uniform positive measure of common boundary configurations admitting all page extensions](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/36-common-spine-books-of-four-prime-pages.md)\n- [One actual block on the first eight odd primes admits arbitrarily many seven-vertex attachments; extendible core configurations have Haar density greater than 1/1200000](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/35-eight-prime-core-with-seven-vertex-attachments.md)\n- [Fixed prime graphs with blocks of at most seven vertices have a uniform head-density bound and a computable cutoff for unrestricted large-prime tails, including tail classes joining the whole head](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/34-uniform-head-density-from-thick-block-domains.md)\n- [At most seven prime divisors up to 100000, or eight up to 100000000 using the attributed eight-prime density theorem, admit arbitrarily many larger primes without any graph restriction](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/33-seven-small-primes-with-an-unrestricted-large-prime-tail.md)\n- [Six specified root-block heads admit arbitrarily many primes above 10000, with arbitrary original heights and supports; all other graph blocks may have up to seven vertices](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/32-conditional-root-measures-and-unrestricted-prime-tails.md)\n- [Every prime-interaction graph with blocks of at most seven vertices is noncovering, with arbitrary total prime count, original heights, and residues](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/31-seven-vertex-block-noncoverage-with-actual-prime-measures.md)\n- [Every prime-interaction graph with blocks of at most six vertices is noncovering: transported six-prime prefix measures pay all actual attachments, with arbitrary total prime count and heights](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/30-six-prime-prefix-measures-close-all-six-vertex-blocks.md)\n- [Actual conditioned heads admit explicitly bounded sparse prime tails of arbitrary finite size, retaining every mixed support](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/27-sparse-tail-extensions-of-conditioned-heads.md)\n- [Literal modulus 5 or 15 absence, or agreement of their residues modulo 5, pays every remaining six-vertex block under the actual pure-class invariant](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/26-literal-pure-classes-and-six-vertex-residue-branches.md)\n- [Six-vertex block fees reduce the remaining root-3 cases to fifty explicit prime cores, with arbitrary original heights and residues retained](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/25-six-vertex-block-fees-and-a-finite-prime-core-frontier.md)\n- [The unique root reserve and shared descendant budget reduce the fifty six-vertex prime cores to twelve, retaining arbitrary original heights and residues](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/28-unique-root-reserve-and-shared-descendant-budget.md)\n- [Outside-root fee savings and actual prime ownership reduce the twelve six-vertex core types to six, preserving original heights and residues](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/29-outside-root-savings-and-prime-ownership.md)\n- [Literal prime-power residue conflicts bound queries under the actual conditional survivor law, with dependent digits retained](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/24-laminar-prefix-conflicts-under-actual-conditioning.md)\n- [Actual conditional kernels close every five-vertex block and an unbounded large-prime block regime under arbitrary recursive attachments](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/23-conditional-kernels-and-recursive-block-noncoverage.md)\n- [Two shared parent layers give fees for exceptional 3,5,11,13,17 and 3,7,11,13,17 blocks with established block descendants](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/22-two-layer-profiles-for-two-further-five-prime-blocks.md)\n- [A shared first-root profile gives a fee for one exceptional 3,5,7,11,13 block, with arbitrary established block descendants](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/21-coupled-first-root-profiles-and-an-exceptional-five-prime-block.md)\n- [Original AP three-color intersections can exceed the proposed residual threshold: a complete 510-label control](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/20-original-ap-three-color-intersections-can-exceed-the-residual-threshold.md)\n- [Five-prime parent envelopes and the limits of scalar prefix budgets](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/19-five-prime-parent-envelopes-and-the-cofactor-allocation-barrier.md)\n- [Five-prime cores with singly attached trees or specified cactus graphs: all-height noncoverage and exact weighted extension bounds](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/18-five-prime-cores-with-tree-and-cactus-attachments.md)\n- [Two-prime separator kernels: exact gluing, a common-law sufficient criterion, and original-AP probes that recover the joint extension counts](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/17-two-prime-separator-interfaces-and-exact-count-probes.md)\n- [Four-vertex blocks and arbitrary cycle blocks: noncoverage from a common descendant budget, retaining all four-prime labels and heights](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/06c-four-vertex-blocks-and-cycle-breaking-vertices.md#four-vertex-blocks-with-a-common-descendant-budget)\n- [Every original prime graph with at most one cycle per component is noncovering, at arbitrary heights and residues](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/06-block-saturation-and-the-actual-crossing-budget.md#arbitrary-odd-pseudoforests-are-noncovering)\n- [Every original cactus prime graph is noncovering: arbitrarily many cycles joined at articulation primes, with all heights and residues retained](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/06b-arbitrary-odd-cactus-graphs-are-noncovering.md#arbitrary-odd-cactus-graphs-are-noncovering)\n- [Current bounds and comparisons](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/problem-details/01-current-bounds-and-comparisons.md)\n- [Actual AP(4,5): layout costs and complete finite-core tails](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/001-064/35-ap45-layout-costs-and-complete-core-tails.md)\n- [Original-layout distance forces a Jensen loss](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/001-064/34-original-layout-distance-forces-a-jensen-loss.md)\n- [Retained and removed events: the complete convex-cost bound](https://github.com/the-omega-institute/trureturing-experiments/blob/main/docs/reports/erdos7-odd-covering/profile-notes/001-064/33-retained-and-removed-original-events-control-convex-costs.md)","title":"Erdős #7: odd covering systems","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-7-odd-covering-systems/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ffa58f1d09bff4523e94ee4769ac43d15243458c3b1134a891459e3ba7eed4c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erdos-deep-triple-classification-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dukes and Gaede, *INTEGERS* 23 (2023), Article #A25, work in `Z_n` with\n`|x|_n := min(x, −x)` and `dist(x, y) = |x − y|_n`. For a family\n`F = {A_1, …, A_s}`, `ΔF` is the multiset of distances `dist(x, y)` over pairs\n`{x, y} ⊆ A_i`, and `F` is Erdős-deep when \"the multiplicities of distances that\noccur in `ΔF` are precisely `1, 2, …, k − 1` for some integer `k`\". They write\n`AP_n(g, k) := {0, g, 2g, …, (k−1)g}`, and Theorem 2 fixes the conventions for\nfamilies of APs: lengths at least three and `gcd(n, g_1, …, g_s) = 1`. Section 5\nstates:\n\n> Conjecture 1. An Erdős-deep family of three APs of lengths `k1 ≥ k2 ≥ k3` in\n> `Z_n` exists if and only if `(k1, k2, k3) ∈ {(4, 4, 3), (6, 3, 3)}`, each for\n> infinitely many n, or `(k1, k2, k3) ∈ {(6, 5, 3), (6, 6, 4), (6, 6, 6),\n> (7, 7, 3), (9, 4, 3), (8, 7, 4), (8, 8, 5), (10, 6, 4), (12, 4, 4), (13, 5, 3),\n> (13, 7, 4), (16, 5, 4), (21, 6, 4)}`, each for a finite number of values of n.\n\nConjecture 4.5 of Gaede's University of Victoria thesis states the same with one\nfurther hypothesis, `3 ≤ k3 ≤ k2 ≤ k1 ≤ ⌊n / (2·gcd(n, g1))⌋ + 1`.","title":"Erdos Deep Triple Classification Refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-deep-triple-classification-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c072769c624ceec3e267e438490f967ad3b658ecc99b89c4c7eb23ee2f7fd72.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:erew-goldstein-2025-sic-clifford-stabilizer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Erew and M. Goldstein, *Extremizing Measures of Magic on Pure States by Clifford-stabilizer\nStates*, arXiv:2512.19657v2, prove that states uniquely stabilized by a subgroup of the finite\neigenphase-extended Clifford group extremize several magic measures, and state:\n\n> **Conjecture 1** (Stabilizer nature of SIC fiducials). Every SIC-POVM fiducial state is a\n> Clifford-stabilizer state.\n\nThe verbatim definitions are in [the literature note](../Library/Quantum/erewgoldstein2025magic.md).\nIssue [#14173](https://github.com/the-omega-institute/trureturing/issues/14173) reads it for every\nprime $d$ and every normalized single-qudit fiducial.","title":"A qutrit SIC fiducial that is not a Clifford-stabilizer state","url":"https://the-omega-institute.github.io/trureturing-pages/research/erew-goldstein-2025-sic-clifford-stabilizer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d32f9f29817e1580e986d502997741a0432b2e268c76388851b696d4c9165ce1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ernst-lia-obrien-sheekey-zumbragel-2026-alternating-sign-margins","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ernst, S. Lia, C. O'Brien, J. Sheekey and J. Zumbrägel,\n*Generalising Latin square orthogonality and Frobenius-König with alternating\nsign matrices*, arXiv:2606.25884v1, Section 8, printed p. 27:\n\n> Problem 8.3. For which (0, ±1)-vectors R and S of order n does there exist X ∈ W_n with row-sums R and column-sums S?\n\nThe same page defines the class:\n\n> Section 4 introduces the set W_n consisting of all (0, ±1)-matrices in which the non-zero entries of each row and column alternate in sign, and the sum of each row/column is in {0, ±1}.\n\nFor every natural number n and signed integer vectors R,S on Fin n, the exact\nanswer is\n\n$$\n\\left(\\exists X\\in W_n:\\ \\forall i,\\ \\sum_j X_{ij}=R_i,\\quad\n\\forall j,\\ \\sum_i X_{ij}=S_j\\right)\n\\iff \\sum_i R_i=\\sum_j S_j.\n$$\n\nAlternation is with respect to the natural row and column order. First and last\nsigns are unrestricted. The zero order and zero lines are included.","title":"Equal totals characterize margins of alternating signed square matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/ernst-lia-obrien-sheekey-zumbragel-2026-alternating-sign-margins/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a13cff43174d2697dca98afdd78796794f8bec9756d411d9456f951273c66344.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:escobar-garcia-2025-seven-site-early-state-exclusion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. G. Escobar, V. Garcia and A. Minenkova, *Early State Exclusion in 7-Qubit\nSpin Chains*, arXiv:2507.18767v1, Section 4. A Jacobi matrix is a real\nsymmetric tridiagonal matrix with positive off-diagonal entries (Eq. (1.1):\n\"for $a_i \\in \\R$ and $b_i > 0$\"). The source defines (quoted with its\ndisplayed equations set inline and without `\\emph` and `\\label` markup):\n\n> A Jacobi matrix $J$ realizes \\textbf{perfect state transfer} (PST) between the end-vertices of the weighted path in Figure \\ref{fig:nearestneighbor} at time $T > 0$ if $e^{-iJT}\\e_0 = e^{i\\phi}\\e_N$ for some phase $\\phi \\in \\R$.\n\n> We consider a special case where $J$ has symmetric spectrum in section 3, meaning that for $\\{\\lambda_k\\}_{k = 0}^N$, $\\lambda_j = -\\lambda_{N - j}$ for all $1 \\leq j \\leq N - 1$.\n\n> Let $J$ be a Jacobi matrix realizing PST at earliest time $T > 0$. If there exists some $\\tau \\in (0, T)$ such that $\\langle e^{-iJ\\tau}\\e_0, \\e_0 \\rangle = 0$, then we say that $J$ exhibits \\textbf{early state exclusion} (ESE) at time $\\tau$.\n\nand conjectures:\n\n> Let $J$ be a $7 \\times 7$ Jacobi matrix with symmetric spectrum realizing PST. Then $J$ does not have ESE if and only if its positive eigenvalues are integer multiples of the smallest positive eigenvalue.\n\nIssue #14487 fixes the reading. \"Symmetric spectrum\" is read as closure of the\nspectrum under negation, the definition of arXiv:2606.04353 (\"if\n$-\\lambda \\in \\sigma(J)$ whenever $\\lambda \\in \\sigma(J)$\") and the sense in\nwhich the source uses it for $\\{0, \\pm x, \\pm y, \\pm z\\}$. The matrix is `pathHamiltonian r q` with\n`r : Fin 6 → ℝ` positive and `q : Fin 7 → ℝ` arbitrary; perfect state transfer\nat time `t` is `HasPST J t 0 (Fin.last 6)`, the modulus-one form of the\nsource's condition; the eigenvalues are Mathlib's\n`(pathHamiltonian_isHermitian r q).eigenvalues`, and the spectrum is symmetric\nwhen the negative of every eigenvalue is an eigenvalue; `T` is an earliest time\nwhen `0 < T`, there is transfer at `T` and at no time in `(0, T)`; early state\nexclusion is a `τ ∈ (0, T)` with `hamiltonianPropagator J τ 0 0 = 0`; the\ndivisibility condition asks for a positive eigenvalue that is the least\npositive one and of which every positive eigenvalue is an integer multiple.\nThe statement is universally quantified over `r`, `q` and `T`.","title":"Early state exclusion in seven-site chains with symmetric spectrum","url":"https://the-omega-institute.github.io/trureturing-pages/research/escobar-garcia-2025-seven-site-early-state-exclusion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0447779a21ef1e669e764145746b3847f69c7c88e1218b1d50bc6f279ad9146b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:espinosa-garcia-2026-hiv-wheel-extinction-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Espinosa-García, Figueroa, Fresán-Figueroa, Maldonado and Sánchez-Solís\n(arXiv:2608.00340v1) study Mukwembi's graph model of HIV infection. States\ngive each vertex the value `0` (healthy), `1` (infected) or `2` (dead). With\n`d_{t,I}(v)` the number of infected neighbours of `v` and a positive\nreplacement parameter `R`, a healthy vertex becomes infected when\n`d_{t,I}(v) ≥ 1`, an infected vertex dies, and a dead vertex is replaced by an\ninfected one when `d_{t,I}(v) ≥ R` and by a healthy one otherwise. The\nextinction set `𝓔(G)` consists of the `R` for which every admissible initial\nstate (values in `{0, 1}`) reaches the all-healthy state. Conjecture 1 states:\n\n> The extinction sets of the wheel graphs satisfy $\\mathcal{E}(W_n) = \\{3\\}\\cup\\{R\\in\\mathbb{Z^+}:R\\geq n-1\\}$ for every even $n\\geq12$, and $\\mathcal{E}(W_n) = \\{4\\}\\cup\\{R\\in\\mathbb{Z^+}:R\\geq n-1\\}$ for every odd $n\\geq17$.\n\nIssue #10263 fixes the reading: `W_n = K₁ ∨ C_{n−1}` on `Fin n` with hub `0`\nand the cycle `1, 2, …, n − 1`, and `𝓔(W_n)` as defined, quantified over all\n`2^n` admissible initial states.","title":"The extinction set of the wheel W_18 contains 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/espinosa-garcia-2026-hiv-wheel-extinction-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15bde9565ce3d53bb81beafccbb11aee0471b9b022f7af763b7b90d7dcbc9d6b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:evans-francis-jones-2025-katz-cutoff-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Evans, Francis and Jones, *Katz similarity index comparisons*, arXiv:2509.02874v1, Conjecture 4:\n\n> The cut-off values α_n converge to 1/√5 from above.\n\nThe source defines the Katz matrix by $K_{P_n}(\\alpha)=(I-\\alpha A_n)^{-1}-I$ and the cut-off values as roots of\n$p_{n,1}(\\alpha)=\\min_i K_{i,i+1}-\\max_i K_{i,i+2}$, with each extremum restricted to existing vertex pairs.\nTheorem 19 locates a root between $1/\\sqrt5$ and $1/2$ for every $n\\ge6$.\nThe preregistration is [#20122](https://github.com/the-omega-institute/trureturing/issues/20122).\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Scope |\n| --- | --- | --- |\n| The path $P_n$ and its adjacency matrix | `SimpleGraph.pathGraph n`, in `Reduction.katz` | Vertices are `Fin n`, indexed from zero. |\n| $(I-\\alpha A_n)^{-1}-I$ and its entries | `Reduction.katz`, `Reduction.katzEntry` | The matrix inverse is Mathlib's inverse. Out-of-range natural indices are extended by zero and excluded from both extrema. |\n| The minimum for separation one and maximum for separation two | `Reduction.p1` | The ranges are `Fin (n-1)` and `Fin (n-2)`. |\n| Theorem 19's existence | First conjunct of `Result.claim` | Every $n\\ge6$ has a root in $(1/\\sqrt5,1/2)$. |\n| Convergence from above | Second conjunct of `Result.claim`, proved by `Result.result` | Every sequence choosing roots in $(0,1/2)$ for $n\\ge6$ lies above $1/\\sqrt5$ there and tends to it. The first six values are unrestricted. |\n\nAll abbreviated declaration names in this dossier have prefix `D5.S1.Recurrence.KatzCutoff.`.","title":"The Katz cut-off limit for path graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/evans-francis-jones-2025-katz-cutoff-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51445009707985eb5d39b8a9e67aea51575e6d60fc4cf4af3c5f06306e1a6a5a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:even-hypergraph-inverse-positivity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 8.1 of arXiv:2507.20847v1, PDF p. 23, states: “Let 𝒢 be a simple\nhypergraph. Then we have I(𝒢,−x)⁻¹ ≥ 0 if and only if all edges of 𝒢 must\nhave even number of elements.” Quotations preserve the prose verbatim;\nmathematical glyphs and whitespace are normalized.\n\nSection 3.1, p. 7, begins: “Let n be a positive integer.” Definition 2(1),\np. 7, says: “A hypergraph 𝒢 is called simple if for any e,f ∈ ℰ, we have\n|e| ≥ 2 and e ⊆ f implies e = f.” Definition 2(5), p. 7, says: “A subset\nI ⊆ 𝒱 is called independent if no edge of 𝒢 is entirely contained in I,\ni.e., e ⊈ I for all e ∈ ℰ.”\n\nDefinition 3, p. 7, includes every independent subset, including the empty\nset, in the ordinary independence polynomial. The coefficient ring is rational\n(p. 1) and positivity is coefficientwise (p. 2). Relabel `[n]` as `Fin n`.\nHere `indicator S` is one on `S` and zero elsewhere, and `signedIndependence E`\nis the sum, over all independent `S ⊆ Fin n`, of the monomial with exponent\n`indicator S` and coefficient `(-1 : ℚ)^S.card`. The precise claim is\n\n```lean\n∀ n : ℕ, 0 < n → ∀ E : Finset (Finset (Fin n)),\n ((∀ e ∈ E, 2 ≤ e.card) ∧\n (∀ e ∈ E, ∀ f ∈ E, e ⊆ f → e = f)) →\n ((∀ m : Fin n →₀ ℕ,\n 0 ≤ MvPowerSeries.coeff m (signedIndependence E)⁻¹) ↔\n ∀ e ∈ E, Even e.card)\n```\n\nThe inverse is in `MvPowerSeries (Fin n) ℚ` and all multiindices are quantified.\nThere is no marked-vertex parameter. The scope is the full Section 8.1\nequivalence, also introduced by Conjecture 1 on p. 5.","title":"Even-hypergraph inverse-positivity refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/even-hypergraph-inverse-positivity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6066569ab805cd54eb75e69ab436575ae6dc671216ea4c5e07b4b49046abd766.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fabbri-marcelli-giuliani-2026-half-flux-hofstadter-mass-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Fabbri, G. Marcelli and A. Giuliani, *The conductivity matrix at the topological phase transition*, arXiv:2609.03832v1 (2026), Section 3.1, leave an analytic gap assertion supported by numerical evidence. In the source member `FMG_v1.tex`, lines 689–690, they write (mathematical macros expanded and the assumption cross-reference rendered):\n\n> In the `MathematicaNotebook1.nb` sheet we provide numerical evidence of the validity of Assumption 2.1, and specifically of the fact that for all $m\\in[-2,2]\\setminus\\{0\\}$, $\\min_{\\vec{k}\\in\\mathbb{B}}(\\lambda_4(\\vec{k};m)-\\mu)>0$ and $\\max_{\\vec{k}\\in\\mathbb{B}}(\\lambda_3(\\vec{k};m)-\\mu)<0$. We expect that, with some additional technical effort, this can be fully proved analytically, but we prefer not to belabor the details of such a proof here.\n\nTheir Bloch Hamiltonian is Eq. `eqn:Hofst_Hamiltonian`, lines 663–671:\n\n$$\nH(\\vec k;m)=-\\begin{pmatrix}\nd&1-i e^{2ik_1}&1-i e^{2ik_2}&0\\\\\n1+i e^{-2ik_1}&0&0&-i+e^{2ik_2}\\\\\n1+i e^{-2ik_2}&0&0&1+i e^{2ik_1}\\\\\n0&i+e^{-2ik_2}&1-i e^{-2ik_1}&-d\n\\end{pmatrix},\\qquad d=2+m.\n$$\n\nThe substitution $d=2+m$ and the Fermi energy are specified at lines 686–687. Write $\\lambda_1\\le\\lambda_2\\le\\lambda_3\\le\\lambda_4$ for its real eigenvalues, counted with multiplicity, $\\vec k_F=(\\pi/4,\\pi/4)$, and $\\mu:=\\lambda_3(\\vec k_F;0)$.\n\nIssue [#22177](https://github.com/the-omega-institute/trureturing/issues/22177) fixes the reading: $\\lambda_4(\\vec k_F;0)=\\mu$, and for every real $m\\in[-2,2]\\setminus\\{0\\}$ there is $\\delta>0$ such that\n\n$$\n\\mu+\\delta\\le\\lambda_4(\\vec k;m),\\qquad\n\\lambda_3(\\vec k;m)\\le\\mu-\\delta\n\\quad\\text{for every }\\vec k\\in\\mathbb R^2.\n$$\n\nThe same $\\delta$ works for all momenta at a fixed mass; it may depend on the mass. This is proved.","title":"A uniform mass gap for the π/2-flux Hofstadter model","url":"https://the-omega-institute.github.io/trureturing-pages/research/fabbri-marcelli-giuliani-2026-half-flux-hofstadter-mass-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da61ada4c7629aa8e69fbe183f063157358da5f339e2cab66eb78077d5ef7e35.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:faifman-polterovich-2026-centred-gaussian-dilation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Faifman and I. Polterovich, *The Faber–Krahn position of convex bodies and Gaussian measure inequalities*, arXiv:2607.21539v2, §5.2, p. 38, state verbatim:\n\n> The B-theorem requires central symmetry. To prove Theorem 1.1, we circumvent this by proving a weaker concavity property of the Gaussian measure in a special case. However, we do not know a convex counterexample to the B-conjecture with its linear barycenter at the origin. Question 5.3. Let K ⊂ R^n be a convex body whose Lebesgue barycenter is at the origin. Is t ↦ γ_n(e^t K) log-concave? More generally, is t ↦ γ_n(e^{tS} K) log-concave for every symmetric matrix S?","title":"Centred convex bodies and Gaussian dilation","url":"https://the-omega-institute.github.io/trureturing-pages/research/faifman-polterovich-2026-centred-gaussian-dilation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4fd6032bda4333628b54995e590d46de4cf77f557410853a03891993af927b5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fan-2026-circle-capacity-conjecture-5-9","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 5.9 of Qiuling Fan, *Riesz capacity ratios with negative\nexponents*, arXiv:2609.11186v1, printed page 19, states:\n\n> Suppose r ≥ 2. Let A, B, C be three points on the unit circle. Fix B and C\n> such that they are symmetric with respect to the x-axis, having polar angles\n> φ and −φ, respectively, where φ ∈ (π/2, 2π/3]. Move A along the\n> circle with polar angle ψ ∈ [0, 2π − 3φ], so that BC remains the\n> longest side of the triangle △ABC. Then the capacity of the three-point set\n> {A, B, C} is maximized when ψ = 2π − 3φ, i.e., when AC = BC.\n\nThe formal target keeps that original domain exactly. For every real `r >= 2`,\n`pi/2 < phi <= 2*pi/3`, and `0 <= psi <= 2*pi-3*phi`, it compares the set of\nactual complex unit-circle points at angles `psi`, `phi`, and `-phi` with the\nisosceles endpoint. Capacity is Fan's negative-exponent Riesz capacity: the\n`1/r` power of the supremum of the positive `r`-power distance energy over all\nprobability measures on the actual point subtype. The target does not restrict\n`r` to integers, impose positive singleton masses, use a finite grid, or claim\nstrictness or uniqueness.","title":"Fan Conjecture 5.9: moving three-point circle capacity","url":"https://the-omega-institute.github.io/trureturing-pages/research/fan-2026-circle-capacity-conjecture-5-9/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8b5a8e216f44818d23ab835092517417614a258c19021fb8bb0f40b24fa8492.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fan-wu-fei-2024-channel-coherence-domination","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Fan, Z. Wu and S.-M. Fei, *Quantifying coherence of quantum channels based on the generalized α-z-relative Rényi entropy*, Quantum Information Processing 23, 100 (2024), DOI 10.1007/s11128-024-04309-w; arXiv:2506.10796v1, Section 3, page 8:\n\n> From the examples in the next section and numerical results, it is conjectured that C_{α,1}(φ) ≥ C̃_{α,1}(φ) holds for all quantum channels φ, but we have not yet found a proof.\n\n[Preregistration #20020](https://github.com/the-omega-institute/trureturing/issues/20020) fixes the source conventions and the settlement as Refuted. The [Library note](../Library/Divergence/fanwufei2024channelcoherence.md) gives the source locators. The parameter range is $\\alpha\\in(0,1)\\cup(1,2]$ from Eq. (8).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Eqs. (1)–(2), $z=1$ | `renyiDiv` | The real part of the trace of the three-factor power sandwich, outer power $1/\\alpha$, subtraction of one and division by $\\alpha-1$ are literal. |\n| Eq. (7), incoherent states on $A\\times B$ | `coherence` | Infimum over positive semidefinite trace-one diagonal matrices; normalized Choi entries retain the factor $1/|A|$. |\n| Eq. (9), computational-basis dephasing | Diagonal expressions inside `coherence` and `commutativityCoherence` | The public formulas inline the source expression; proofs use `ProjectedUnistochasticDynamics.projectiveReadout`. |\n| Eq. (10), supremum over every input state | `commutativityCoherence` | The full positive semidefinite trace-one domain is retained. |\n| Every CPTP channel between finite systems | `claim` | Finite index types with decidable equality, `MatrixMap`, complete positivity and trace preservation on every matrix. The counterexample uses two nonempty two-dimensional systems. |\n| Every allowed $\\alpha$ and $\\widetilde C_{\\alpha,1}\\le C_{\\alpha,1}$ | `claim` | Both intervals and the non-strict comparison are retained. |\n| Refutation | `result` | The theorem proves the negation of that universal proposition, with no hypotheses. |\n\nFinite enumerations and decidable equality implement the matrix coordinates. Matrix powers use the pinned continuous functional calculus. Density and Choi expressions are inlined literally, with private consumed correspondences to existing owners. No additional channel hypothesis restricts the conjecture.","title":"Fan–Wu–Fei channel coherence domination is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/fan-wu-fei-2024-channel-coherence-domination/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b5712db86d99355f960ec7885b57cc5928355c2d37c0db2c82014b2190af3c6e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fang-fu-kitaev-li-su-sun-2026-class69-involutions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Q. Fang, S. Fu, S. Kitaev, H. Li, X. Su and Z. Sun, *On mesh patterns of short length: Equidistribution and enumeration*, arXiv:2606.14367v1, Concluding remarks, Conjecture 1:\n\n> The patterns in the set {\\pattern{scale=0.5}{2}{1/1,2/2}{1/2,1/1,2/1,0/0}, \\pattern{scale=0.5}{2}{1/1,2/2}{2/2,0/1,1/1,1/0}, \\pattern{scale=0.5}{2}{1/1,2/2}{0/2,1/1,2/1,1/0}, \\pattern{scale=0.5}{2}{1/1,2/2}{1/2,0/1,1/1,2/0}} are equidistributed on involutions. (The first two patterns, as well as the last two patterns, are trivially equidistributed via the composition of reverse and complement.)\n\nThe macro shades the unit box with lower-left corner `(x,y)` for each `x/y` in its fourth argument; the dots are at `(1,1)` and `(2,2)`. Let `R0`, `R1`, `R2`, `R3` denote those shaded-cell sets in that order. Equidistribution requires equal numbers of involutions having exactly `k` occurrences, for every length `n`, count `k`, and pair of pattern indices.","title":"Class 69 Mesh Patterns Are Not Equidistributed on Involutions","url":"https://the-omega-institute.github.io/trureturing-pages/research/fang-fu-kitaev-li-su-sun-2026-class69-involutions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e71bebf0d124a22d5b0bd71b822a5179862463ad7dd9dbf41b3f5df57c962457.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fannes-demelo-roga-zyczkowski-2012-root-fidelity-holevo-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Fannes, F. de Melo, W. Roga and K. Życzkowski, *Matrices of fidelities for ensembles of quantum states and the Holevo quantity*, arXiv:1104.2271v1, page 10, Conjecture 1:\n\n> For any ensemble E₃ = {(pᵢ, ϱᵢ)} with K = 3 three states of arbitrary dimension d the corresponding Holevo quantity χ(E₃) is bounded from above by the entropy of the matrix C√F(E₃) defined in (16): χ(E₃) ≤ S(C√F(E₃)).\n\nHere $S(X)=-\\operatorname{Re}\\operatorname{tr}(X\\log X)$, $\\chi=S(\\sum_i p_i\\rho_i)-\\sum_i p_iS(\\rho_i)$, and $C_{ij}=\\sqrt{p_ip_j}\\operatorname{tr}|\\sqrt{\\rho_i}\\sqrt{\\rho_j}|$. The logarithm is natural. The matrices are positive semidefinite of trace one; weights are nonnegative and sum to one.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Three states of arbitrary dimension | `claim` quantifies `d : ℕ`, three matrices indexed by `Fin 3` on `Fin d` | No invertibility assumption; dimension zero has no trace-one state |\n| Density matrices | Each matrix is positive semidefinite and has trace one | Singular matrices included |\n| Ensemble probabilities | Each weight is nonnegative, and their sum is one | Zero weights included |\n| Fidelity and matrix (16) | The literal trace-norm/square-root expression in `rootFidelityMatrix` | The trace norm is the existing `FiniteTraceDistance.traceNorm` |\n| Holevo quantity and entropy | `holevo`; the literal trace-log expression in `claim` | Ordinary complex matrices, with the L2 operator norm for functional calculus |\n| Upper bound | `FannesRootFidelityHolevo.result` | The full expanded claim |","title":"The three-state root-fidelity Holevo bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/fannes-demelo-roga-zyczkowski-2012-root-fidelity-holevo-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1cc61f58121df3b5575e94d1119fcec83d9911d08472df824eb55471b07b53b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:farhan-2026-complete-subdivision-edge-multiset","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 24, section 7.4, printed page 27 of the versioned source asks whether the edge\nmultiset dimension of K_n^1 is infinity for every integer n at least three. The definition\non printed page 21 inserts one midpoint into every complete-graph edge. Section 6.1 on\nprinted page 20 uses minimum actual shortest endpoint distances and arbitrary landmark\nsubsets of the full vertex set, including originals and midpoints. Profiles are multisets\nwith multiplicity; infinity means that no resolving subset exists.","title":"Once-subdivided complete graphs have infinite edge multiset dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/farhan-2026-complete-subdivision-edge-multiset/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2c7e0f6f6c4bb4d78749cb90f63d8ff7dc9b261af4ce7c9020c8cce629beeca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:farhan-kuziak-yero-2026-direct-square-weak-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mohammad Farhan, Dorota Kuziak and Ismael G. Yero, *The weak k-metric dimension\nof the direct product of complete graphs*, arXiv:2605.22307v1,\nSection 3, paragraph `S3.p17.1`:\n\n> The result above clearly leads to a question concerning the exact value of\n> wdim4(Kn × Kn) when n ≥ 4.\n\nSection 6, first bullet, asks for the remaining exact formula at k = 4 and\nwhether the upper bound in Theorem 3.3 is the exact value.\nThis is one question over all integer orders at least four.\n\n### Exact value and semantics\n\nFor every integer $n\\ge4$,\n\n$$\n\\operatorname{wdim}_4(K_n\\times K_n)\n=2n+\\left\\lceil\\frac{2n}{11}\\right\\rceil+\\mathbf 1_{n=5}.\n$$\n\nThe vertex set is $[n]\\times[n]$. Two vertices are adjacent exactly when\nboth coordinates differ. This is the direct product in the source.\nFor $n\\ge3$ the actual shortest-walk distance is zero at equal vertices,\none when both coordinates differ, and two otherwise. In the last case,\na third row and a third column give a common neighbor. The graph is connected.\n\nA finite landmark set $S$ is weak four-resolving if\n\n$$\n\\sum_{w\\in S}|d(x,w)-d(y,w)|\\ge4\n\\quad\\text{for every distinct }x,y.\n$$\n\nThe introductory source definition omits the word “distinct.” Proposition 2.2\nexplicitly restricts to distinct pairs and supplies the coordinate identities\nused by the paper. Self-pairs have separation zero and cannot be intended\nwhen the paper discusses positive k. The formal predicate retains this\nexplicit distinct-pair intent.\n\nThe dimension is the least cardinality of such a set. The formal construction\nproves that its feasible family is nonempty for every order in the question;\nthe universal lower bound and an equality-size set prove attainment. Natural\nnumber division represents the ceiling as $(2n+10)/11$.\n\nAt order nine the answer is twenty. The source upper bound is twenty-one\nthere, so Theorem 3.3 is not always sharp. The answer covers every order;\nthis strict inequality alone is not the whole result.","title":"Weak-four metric dimension of complete graph direct squares","url":"https://the-omega-institute.github.io/trureturing-pages/research/farhan-kuziak-yero-2026-direct-square-weak-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3923565e578b8dd81ba6989e9de4028bf3943098533e9ef1d9e55cec12c8ab0e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:faust-friedman-omalley-ramos-sharma-2026-spectral-invariants-conjecture-3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Faust, L. Friedman, G. O'Malley, R. Ramos and A. Sharma,\n*Algebraic Properties of the Ideal of Spectral Invariants for the Discrete\nLaplacian*, arXiv:2602.08304v1, Conjecture 3, p. 11:\n\n> When q is odd, then V(I′) has only regular nonzero points, and when it is an even number greater than 6 there is a single singular point, up to symmetry, of multiplicity 2^{q/2−2}. Moreover, the multiplicity at 𝟎 is 2^{⌊q/2⌋}.\n\nThis delivery refutes the odd clause under the Jacobian regularity criterion\nspecified in [preregistration #19188](https://github.com/the-omega-institute/trureturing/issues/19188).\nThe [Library note](../Library/GraphInvariants/faust2026spectralinvariants.md)\nrecords the source definitions. Period means a period, without a least-period\nrequirement. The even and origin-multiplicity clauses are not settled here.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $L_V=L_V(1)$, diagonal $V$, cyclic off-diagonal entries one | `floquetMatrix` | Literal cyclic-adjacency formula; the claim uses sizes at least five |\n| $D_V(\\lambda)=\\det(L_V-\\lambda I)$ | `floquetCharPoly` | This sign convention is retained |\n| $U=(u_1,\\ldots,u_m,0,-u_m,\\ldots,-u_1)$ | `specializedPotential` | Zero-based indices; no additional constraint on the coordinates |\n| $\\zeta_U=D_U-D_{\\mathbf0}$ | `zeta` | Full polynomial difference |\n| $p'_k=(-1)^{n+k}[\\lambda^{n-2k}]\\zeta_U$ | `pPrime` | Literal coefficient and sign; only $1\\leq k\\leq m$ generate the ideal |\n| A point of $V(I')$ | `InVariety` | Every generator vanishes |\n| Regular nonzero points | `jacobianAt`, `IsRegularPoint`, `u ≠ 0` in `claim` | Regularity is precisely nonvanishing of the generator Jacobian determinant |\n| Every odd $q=2m+1>3$ | `claim`, with $2\\leq m$ | No least-period hypothesis |\n| Refutation | `result` | Negates this full universal claim |\n\nAll names in this table belong to\n`D5.S3.Combinatorics.SpectralGraph.FaustSpectralInvariantsOddRefutation`.\nSection 3, p. 7 refers to multiplicity for the defining ideal rather than its\nradical. No separate local-algebra equivalence between multiplicity one and\nJacobian invertibility is formalized in this delivery.","title":"A singular nonzero point of the odd spectral-invariant variety","url":"https://the-omega-institute.github.io/trureturing-pages/research/faust-friedman-omalley-ramos-sharma-2026-spectral-invariants-conjecture-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/87918b62dd98e442ddf496cb7a96fcd5449369be7b3e5cc5d872a5aa20a40d32.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fazekas-abelian-square-equality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 4 of Fazekas, Mammoliti, Mercaş, and Simpson,\n*Binary Words Containing Few Abelian Squares*, arXiv:2604.23188v1,\nstates that every binary word of length n contains at least floor(n/4)\ndistinct abelian-square factors and that equality forces all of those factors\nto be trivial even powers of one letter.\n\nThe formal claim quantifies over every finite binary word, counts distinct\nfactors whose equal-length halves have the same Parikh vector, and expresses\nboth the lower bound and the equality conclusion.","title":"Refutation of the equality conclusion in Fazekas et al. Conjecture 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/fazekas-abelian-square-equality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/acc592e83dbf7e719598d2c112341dee9c1259219974bcebfc348ad208130462.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:feldman-wilce-countably-compact-unbounded-cliques","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David V. Feldman and Alexander Wilce, *A Note on Compactness and Clique Size*,\narXiv:2608.13320v1, Question 1, printed page 5:\n\n> Is there a countably compact T1 (or Hausdorff) space with a closed\n> orthogonality relation having finite but unbounded cliques?\n\nAn orthogonality relation is symmetric, irreflexive and closed in the entire\nproduct, including the diagonal. A clique is a subset with every distinct\npair related. The requested relation must have every clique finite and no\nnatural number bounding all clique sizes.","title":"Feldman–Wilce Question 1: finite, unbounded cliques","url":"https://the-omega-institute.github.io/trureturing-pages/research/feldman-wilce-countably-compact-unbounded-cliques/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/068b9f4ad5095e55abacc019af3970f47ce2c665db86969308737c5611a41744.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:feldman-zigzag-conjecture-nine-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David V. Feldman, *The Missing Zigzag: Cycles of Semitone Trichords and a\nConservation Law in Equal Temperament*, arXiv:2609.26114v1, Section 5,\nequation (2), Conjecture 9.3. For each integer `t>=2`, set `n=3t`. For every\n`k=2,...,n-2`, choose one labelled directed form in `ZMod n`:\n\n| Label | From | To |\n| --- | --- | --- |\n| I | `1` | `k-1` |\n| II | `k` | `1-k` |\n| III | `-1` | `k` |\n| IV | `k-1` | `-k` |\n| V | `-k` | `1` |\n| VI | `1-k` | `-1` |\n\nAt `k=2`, exactly II, III, IV, and V are permitted. Labels remain different\nchoices when their endpoint pairs coincide. Balance means integer\noutdegree minus indegree zero at each nonzero residue; no closedness,\nconnectivity, Eulerian-circuit, or cycle condition is added. Let `a(t)` be\nthe number of these actual balanced labelled choices. The full conjecture is\n\n`a(2)=4`; for every `r>=2`, `a(2r)=4a(2r-1)`; and for every `r>=1`,\n`(2r)a(2r+1)=6(2r-1)a(2r)`.","title":"Feldman's Conjecture 9.3: balanced zigzag choices","url":"https://the-omega-institute.github.io/trureturing-pages/research/feldman-zigzag-conjecture-nine-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a484aea53a0b367c2604c1cf8c979c789ecf90b6b4c07ee9d1c3ecf24a274da4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ferenczi-zamboni-2023-arnoux-rauzy-longest-clustering","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Ferenczi and L. Q. Zamboni, *Clustering and Arnoux–Rauzy words*,\narXiv:2305.18986v1, Advances in Applied Mathematics 153 (2024) 102621,\nDOI 10.1016/j.aam.2023.102621, state Conjecture 1:\n\n> In a given AR language $\\Lambda$ satisfying $(Rabc)$, the word in Proposition\n> [14] is the longest clustering word, or, if this fails, the longest clustering\n> word conjugate to a standard AR word.\n\nThe formal statement uses the source's three-letter Arnoux–Rauzy blocks,\nlanguages, standard words, conjugacy, Burrows–Wheeler transform and clustering\ndefinitions. It records the second alternative of the conjecture as `claim`.\n\n### Statement echo\n\nThe alphabet is $\\mathrm{Fin}\\,3$, with $a=0$, $b=1$, and $c=2$; words\nare lists. The three AR rules are the pointwise update that fixes the selected\nblock and appends it to each other block. `arBlocks` uses `Nat.rec` and\n`Function.update` for that recurrence. `InfOften` is the universal occurrence\ncondition expressed using `Filter.Frequently` at `Filter.atTop`, equivalent to\n$\\forall x\\,N,\\ \\exists k\\ge N,\\ d(k)=x$. The boundary $\\mu$ is\n`Nat.findGreatest` of the positions $j$ with $d(j-1)\\ne d(\\mu_y-1)$,\nbounded by $\\mu_y$; the value is $0$ when no such positive position exists.\nStage blocks are enumerated by `List.ofFn` and sorted by length. For BWT,\nthe non-strict permutation-rank order is the negation of reverse `List.lex`;\n`List.insertionSort` sorts the rotations and `List.getLast?` extracts their\nlast letters. The quantified conditions in `claim` are exactly the weaker\nstandard-conjugacy alternative of Conjecture 1.","title":"Ferenczi–Zamboni Conjecture 1 on longest clustering words","url":"https://the-omega-institute.github.io/trureturing-pages/research/ferenczi-zamboni-2023-arnoux-rauzy-longest-clustering/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65772bcaca28478f550ce46c254b4c91077641b97ec3e073b9f5cc66d32eeb4f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ferenczi-zamboni-2025-palindromic-complexity-no-connection","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sébastien Ferenczi and Luca Q. Zamboni, *Clustering, order conditions, and languages of interval exchanges*, arXiv:2507.17370v2, Section 4, Conjecture 1, page 14:\n\n> A language on $k$ letters satisfying a symmetric order condition and with no connection has palindromic complexity $1$ for $n$ even, $k$ for $n$ odd.\n\nThe source language is factorial and two-sided extendable. A symmetric pair has arrival order opposite to departure order. Connections are the consecutive-letter, missing-cross-edge configurations of Definition 7. Recurrence and closure under reversal are not hypotheses of this conjecture. Preregistration: [#15331](https://github.com/the-omega-institute/trureturing/issues/15331).","title":"Symmetric Order and No Connections Do Not Force Full Palindromic Complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/ferenczi-zamboni-2025-palindromic-complexity-no-connection/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e60dbf0b056366069b8a094fd50faa8a99c9155e5291fd9255d6b7674aaec04.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fernandes-2026-intermediate-group-invariant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 6.2 of Fernandes–Qureshi–Reis–Ribas, arXiv:2609.20516v1, asks\nfor a partition P on all finite groups with P₁ We believe that similar impossibility prevails for any other periodic trajectory with non-exhaustive firing in this graph.\n\nThe graph is the right transition graph in Figure 1, for the Section 4 system with $N=3$, $\\epsilon>1/2$ and coupling matrix\n\n$$\nW=\\begin{pmatrix}1-\\epsilon&0&\\epsilon\\\\ \\epsilon&1-\\epsilon&0\\\\ 0&\\epsilon&1-\\epsilon\\end{pmatrix}.\n$$\n\nThe source assumes $0<\\eta<1$, $\\epsilon<1-\\eta$, initially $(Wx)_j>\\eta$ for every cell, and restricts to trajectories with distinct coordinates and no simultaneous firings (Section 4, lines 239–255). Exhaustiveness means that every three consecutive firing labels form a permutation of the three cells (line 68). The quoted belief extends the source's exclusion of a particular six-firing itinerary to every non-exhaustive periodic trajectory in this graph.\n\nIssue [#25922](https://github.com/the-omega-institute/trureturing/issues/25922) fixes the formal reading. Write $\\theta$ for the source's $\\eta$. For every real $\\theta,\\epsilon$ with $0<\\theta<1$ and $1/2<\\epsilon<1-\\theta$, and every state sequence $s:\\mathbb N\\to\\mathbb R^3$, label sequence $\\mathrm{lab}:\\mathbb N\\to\\mathrm{Fin}\\,3$ and waiting-time sequence $\\tau:\\mathbb N\\to\\mathbb R$, `claim` says that initial repressors strictly above $\\theta$, `IsRun ε θ s lab τ`, and `IsPeriodic s` imply `IsExhaustive lab`. Lean labels $0,1,2$ correspond to the source's $1,2,3$. This assertion is refuted.","title":"A non-exhaustive periodic orbit in Fernandez's three-cell degrade-and-fire chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/fernandez-2018-degrade-fire-non-exhaustive-orbit-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a2275c64a1393ccb2dc190bf02c1ec2a61c2422746b85847e190384cc72c9efc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fibonacci-pythagorean-perimeter-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A134492 is the sequence `a(n) = Fibonacci(6n)`. Its COMMENTS section carries,\nat revision #62 of Sep 22 2025:\n\n> Conjecture: For n >= 2, the terms of this sequence are exactly those Fibonacci\n> numbers which are the sum of the three numbers of a Pythagorean triple (checked\n> up to F(80)). - _Felix Huber_, Nov 03 2023\n\nWith quantifiers written out, and `P(N)` for `∃ a b c > 0, a² + b² = c² ∧ a + b + c = N`:\n\n ∀ N, (∃ k, F(k) = N) → (P(N) ↔ ∃ n ≥ 2, F(6n) = N).\n\nThe comment says \"a Pythagorean triple\" with no qualifier, so the triples are all\ntriples of positive integers, primitive or not.","title":"Fibonacci Numbers That Are Pythagorean Perimeters","url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-pythagorean-perimeter-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec52c8e73692be592caf566ba4e208add3fdd9d95964743d804f0568738b4241.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fibonacci-second-bit-run-length","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A272170 is the sequence of second most significant binary digits of the Fibonacci\nnumbers exceeding one. Its COMMENTS section carries, at revision #34 of Dec 08 2025:\n\n> It is conjectured that there are no more than two consecutive \"0's\" or \"1's\" (tested up to\n> n=10^5). The sequence looks quasiperiodic and its Fourier spectrum seems to have a fractal\n> structure. - _Andres Cicuttin_, Apr 21 2016\n\nWriting `s(m) = m / 2 ^ (size m - 2) % 2` for that digit, the assertion is\n\n ∀ n ≥ 3, ¬ ( s(F n) = s(F (n+1)) ∧ s(F (n+1)) = s(F (n+2)) ).","title":"Runs in the Second Binary Digit of the Fibonacci Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-second-bit-run-length/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1bdf19d72320e5f7048e79d6c22253cfbda0716a7c6042db6bb9ab45ee62d0c7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiebig-mbirika-spilker-ballot-extended-even","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Question 5.3 of arXiv:2408.14632v2, as printed:\n\n> Subsection 3.4 begins with a convention restricting the `p` and `m` values to either of the\n> following classes: (1) `p` is odd, or (2) `p` is even and `m` is odd. ... However, data\n> generated with Mathematica provides ample support that when `q = -1`, the corollary holds even\n> in the setting when both `p` and `m` are even. This remains an open problem.\n\nThe corollary being extended reads: \"Let `m > 2` be given and set `V_n := V_n(p,q)`. If `e_V(m)`\nexists, then we have `pi_U(m) = pi_V(m)` if either condition holds: (1) `p` is odd, or (2) `p` is\neven and `m` is odd.\"","title":"Equality of Lucas periods when p and m are both even, at q = -1","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiebig-mbirika-spilker-ballot-extended-even/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/df670e188fdecdedd83628e56527cc4d978a88b9fa7161e4670bc661d322d62e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiebig-mbirika-spilker-even-descent-dichotomy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 5.2 of arXiv:2408.14632v2, as printed:\n\n> Set `U_n := U_n(p, q)`, and let `m >= 1` and `n` in `Z`. Assume that both `p` and `m` are even.\n> If `U_{2n} = 0 (mod m)` and `U_{2n+1} = q^n (mod m)`, then exactly one of the following two\n> conclusions occur: (i) `U_n = 0 (mod m)`, or (ii) `U_n = m/2 (mod m)` where `n = c * e_U(m)/2`,\n> where `c` is some odd integer and `e_U(m)` is the entry point of `m` in `(U_n)`.\n\nThe paper adds a second sentence: \"Can we give conditions for when Conclusion (i) versus\nConclusion (ii) occurs?\"","title":"The even-parameter descent dichotomy for Lucas sequences","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiebig-mbirika-spilker-even-descent-dichotomy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e97b3c29b7acb165364fda1b826c757c5b2037410c96e58210a121b18e9eaed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiebig-mbirika-spilker-even-period-exception","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Question 5.4 of arXiv:2408.14632v2, as printed:\n\n> ... when `p = 0 (mod 4)`, then it appears that the corollary holds for all even values `m > 2`\n> except for the single value of `m = 4`. And in that case, we have `e_V(4) = 1` but\n> `pi_U(4) = 4 != 2 = pi_V(4)` ... However, when `p = 2 (mod 4)` and `m > 2` is even, the\n> existence of `e_V(m)` appears to always guarantee that `pi_U(m) = pi_V(m)`. Can we prove that\n> the corollary holds for all even values `p` and `m` with `m > 2`, except in the singular case\n> when `m = 4` and `p = 0 (mod 4)`?\n\nThe excerpt is taken from a subsection whose standing hypothesis is `q = 1`, so the exceptional\ncase has **three** conditions, not two: `q = 1`, `m = 4`, and `4` dividing `p`. That matters\nbecause with `q = -1` the same modulus and the same divisibility give **equal** periods, which is\nwhat the sibling entry `fiebig-mbirika-spilker-ballot-extended-even` records. The theorem below\ncarries all three conditions in its exceptional clause.","title":"The single exceptional modulus for equality of even Lucas periods","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiebig-mbirika-spilker-even-period-exception/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f1d62c53b8febd629732cb2042bec310e4d54e05aa9e8e62eb8c27db91740c2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiebig-mbirika-spilker-vertical-slice","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Write `V_n = V_n(p, q)` for the companion Lucas sequence and `pi_V(m)` for its least period\nmodulo `m`. Drawing one fundamental period around a circle, Fiebig, Mbirika and Spilker call it\nthe *vertical slice result* when a line through the circle meets two terms that are additive\ninverses of one another. Their Remark 4.12 records that it holds in many cases where the entry\npoint `e_V(m)` does not exist at all, and their Theorem 4.11 proves it for `q = 1` and order\nstatistic `omega_V(m) = 4`, where the axis passes through the two zeros. Question 5.5 of\narXiv:2408.14632v2 asks:\n\n> Can we find necessary and sufficient conditions for the \"vertical slice result\" to hold for\n> (V_n(p,q)) modulo m with q = 1 and omega_V != 4?\n\nThe sentence quoted from Remark 4.12 is looser than the property the paper actually tracks.\nModulo 28 with `p = 7` and `q = 1` the period is `2, 7, 19, 14, 23, 7, 26, 7, 23, 14, 19, 7`, and\nthe chord joining positions 0 and 6 does join two additive inverses, `2 + 26 = 28`; the paper\nnonetheless lists `m = 28` as a case where the result fails. So the property is that **every**\npair reflected in one axis cancels, not that **some** chord joins a cancelling pair.","title":"Necessary and sufficient conditions for the vertical slice result","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiebig-mbirika-spilker-vertical-slice/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e773cd0efbeea8e824ac9386bac22b6844196cdb43745eaa9450a8c13112760.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:filimonova-puzynina-2026-rank-one-iteration-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Filimonova and Puzynina ask in Section 4 of *On abelian periodicity of purely\nmorphic words* (arXiv:2605.30306v1) whether the iteration variable in their\nprimitive binary rank-one criterion has an upper bound making the criterion\nalgorithmic.\n\nThe source's fixed word is allowed an arbitrary finite preperiod. The target\nis the complete nonerasing, prolongable, primitive, rank-one binary domain,\nwith exact Parikh equality in the cyclic block witness.","title":"Filimonova–Puzynina's rank-one iteration-bound question","url":"https://the-omega-institute.github.io/trureturing-pages/research/filimonova-puzynina-2026-rank-one-iteration-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba060c9a70b5c616ea6b5bc4d104dff7702c946f950257eeb828318343d4908b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiol-martinezmontoya-riosfukelman-2020-tree-lee-yang","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Fiol, J. Martínez-Montoya and A. Rios Fukelman, *The planar limit of\nN=2 superconformal quiver theories*, JHEP 08 (2020) 161,\narXiv:2006.06379v2, Section 2.2, Conjecture 1:\n\n> For any tree with $e$ edges, any fixed positive integers $n_1,\\dots,n_e$\n> and arbitrary $\\rho(k_i,n_i)>0$ the polynomials\n> $P(\\tau,x,n_1,\\dots,n_e)$ have all their $x$ roots on the unit circle.\n\nEquation (2.30) sums over positive edge splits $1\\le k_i\\le n_i-1$\nand all Boolean spin states. The mass at a vertex is the sum of the integers\nassigned to its incident endpoints. A state contributes\n$\\tau^{e_\\pm}x^{\\sum_{v\\text{ up}}m_v}$, where $e_\\pm$ counts\nopposite-spin edges. The Lee–Yang parameter range is $-1\\le\\tau\\le1$.\nThe source's one-edge case is proved through the Lakatos–Losonczi criterion.\n\nThe source sentence at `quivers_n_2.tex` lines 343–345 makes $\\rho$ a\ndistribution on the whole split configuration. Thus Conjecture 1 permits\nan arbitrary positive joint function; it does not impose edgewise factorization.\nThe physical specialization in (2.31) includes vertex factors $\\mathcal V_i$\nthat depend jointly on the incident splits, as defined at source lines 198–203.\n\n### Statement echo\n\nThe public definitions are `splits`, `mass`, `treePoly` and `claim`.\nThe tree condition is the frozen\n`D5.S3.Observer.AgencyHolonomy.NamedTreePhaseTransport.NamedSpanningTree`\nwith `Set.univ`, selecting every edge: its nonloop, distinct and isTree fields\nstate exactly the finite-type tree convention. The subtype of all edges is\nin bijection with the edge type, so its injectivity and support conditions\nare the same as the source convention.","title":"Positive joint weights and all-vertex physical weights in the tree Lee–Yang conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiol-martinezmontoya-riosfukelman-2020-tree-lee-yang/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92429729ff6306b1c2220c28aeed7331a7684bf546343e4f2008900e799538f8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fiore-nasr-stone-2026-midpoint-negativity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The complete Conjecture 6.8 of `fiore2026squaresideddice` states:\n\n> In the remaining 33 pairs of the polynomials A_abcd and B_(2-a,2-b,2-c,2-d) (including the case where p != 2 for when a=b=c=d=1), one of the two polynomials will have a negative coefficient.\n\nDefinition 6.1 uses distinct natural primes p,q,r and exponents a,b,c,d\nin {0,1,2}. At the midpoint a=b=c=d=1 both complementary tuples are\n(1,1,1,1). The two complete integer polynomials are\n\n$$A(p,q,r)=X\\Phi_p^2\\Phi_q^2\\Phi_{pq}\\Phi_{pr}\\Phi_{qr}\\Phi_{pqr},$$\n\n$$B(p,q,r)=X\\Phi_r^2\\Phi_{pq}\\Phi_{pr}\\Phi_{qr}\\Phi_{pqr}.$$\n\nThe literal closed `claim : Prop` universally quantifies over natural\np,q,r with `Nat.Prime p`, `Nat.Prime q`, `Nat.Prime r`, `p != q`,\n`p != r`, `q != r` and `p != 2`, and asserts that some natural-index\ncoefficient of A or of B is strictly negative. There is no ordering\npremise, prefix bound or scalar proxy.","title":"An odd midpoint refutes Fiore-Nasr-Stone Conjecture 6.8","url":"https://the-omega-institute.github.io/trureturing-pages/research/fiore-nasr-stone-2026-midpoint-negativity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bd01d2a83317678f511c896a025fc346c4ce4ed7afe6160095f221ca3870aa0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fisk10-toeplitz","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Steve Fisk, *Questions about determinants and polynomials*, arXiv:0808.1850v1,\nPDF p. 4, Question 10 and matrix (5), asks:\n\n> Suppose f∈P⁺₂, and write f = ∑ f_i(x)y^i. Form the infinite matrix\n\n> Show that the resulting matrix is totally stable. That is, every minor is a stable polynomial.\n\nThe matrix has $T_f(r,c)=f_{c-r}$ for $r\\le c$ and zero otherwise.\nThe definitions on PDF p. 1 require upper-half-plane nonvanishing and\npositive coefficients on the support, with Hurwitz stability meaning all\nroots have nonpositive real part. The related *totally stable matrix\nconjecture* is Conjecture 21.84, Section 21.10, printed p. 648 of\narXiv:math/0612833v2. Its input class is defined through real substitution,\nhomogeneous positivity and coefficient positivity; no equality of the two\nsource classes is assumed.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Real bivariate polynomial, with outer variable $y$ | `Polynomial (Polynomial ℝ)` | Literal carrier; coefficient variable is $x$. |\n| $U_2$: nonzero at every pair in the open upper half plane | `Upper` | Both complex variables quantified; nested `Polynomial.eval₂`. |\n| $P_2^+$: upper and positive coefficients | `PositiveUpper` | Strict positivity on support; absent coefficients are zero. |\n| Matrix (5) | `toeplitz` | Zero-based upper-triangular coefficient shifts. |\n| Every minor | `minor`, `claim` | All natural sizes and strictly increasing row and column maps; no sign reversal. |\n| All roots in the closed left half plane | `HurwitzStable` | Roots after mapping coefficients to ℂ. |\n| Universal assertion is false | `result` | Negation of the literal `claim`. |\n\nThe size-zero determinant is one. Structural zero minors are admitted by\nthe empty root multiset of the zero polynomial. The exhibited minor is\nnonzero, so either convention about structural zero stability gives the\nsame refutation.","title":"Fisk Question 10: coefficient-Toeplitz minors","url":"https://the-omega-institute.github.io/trureturing-pages/research/fisk10-toeplitz/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17d3dd69e132a91e26aeb7197dc837aa596e449d18ebdca7e7cecd9befb6a454.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:flores-peltola-2018-jones-wenzl-generation-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. M. Flores and E. Peltola, *Generators, projectors, and the Jones-Wenzl algebra* (2018), arXiv:1811.12364v2, Section 1, Conjecture 1.2, TeX source `WJalgebra_gen-sec1-intro.tex`, lines 544–546, with macros expanded and references resolved:\n\n> Item 1 in theorem 1.1 holds whenever $\\max \\varsigma < \\bar{\\mathfrak p}(q)$.\n\nHere $q\\in\\mathbb C^\\times$, $\\nu=-q-q^{-1}$ and $\\varsigma=(s_1,\\ldots,s_d)$ has positive entries, with total size $n_\\varsigma=\\sum_i s_i$. The source also permits the exceptional multiindex $(0)$. The bound $\\bar{\\mathfrak p}(q)$ is infinite at $q=\\pm1$ and otherwise is the order of $q^2$, interpreted as infinite when that order is infinite. The composite Jones–Wenzl projector $P_\\varsigma$ places projector boxes of sizes $s_i$ side by side, and\n$\\mathrm{JW}_\\varsigma(\\nu)=P_\\varsigma\\mathrm{TL}_{n_\\varsigma}(\\nu)P_\\varsigma$.\n\nItem 1 of Theorem 1.1, at lines 502–516, states that, assuming $n_\\varsigma<\\bar{\\mathfrak p}(q)$, the unit $P_\\varsigma$ and the projected boundary generators\n$U_i^{\\mathrm{JW}}=P_\\varsigma U_{s_1+\\cdots+s_i}^{\\mathrm{TL}}P_\\varsigma$, $1\\le i We use C_n^±(k) to mean the set of all paths of length k in the n-dimensional cubic lattice. We divide C_n^±(k) into subfamilies depending on the behavior of the path. We now give definitions and notation for those families. If P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}), then we define V_r := (±e_{j_1}) + (±e_{j_2}) + · · · + (±e_{j_r}), the algebraic combination of the first r components of P for 0 < r ≤ k, i.e., V_r is the sum of the components of any initial subpath of P with r steps. We denote C_n(k) the subset of C_n^±(k) formed by all paths P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}) that satisfy that the nth coordinate of V_k is zero. We use C_n^≥(k) to denote all paths in C_n^±(k) with P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}) and that nth coordinate of V_r is non-negative for all 0 < r ≤ k. We now let C_n^+(k) be C_n^≥(k) ∩ C_n(k). […] For example, Figure 1 depicts the 14 paths in C_2^+(3). Figure 2 depicts the 17 paths in C_3^+(2).\n\nThe paper also specifies that paths start at `p_0 = (0, ..., 0)` and each step\nis a positive or negative coordinate direction.\n\nSection 6 on printed page 23 states:\n\n> **Conjecture 1:** For k ≥ 1, the number of paths in C_3^+(k) that are completely contained in the xz−plane is (see Table 4 first line) Σ_{i=1}^{k+1} \\binom{2i}{i}\\binom{k}{i−1}/(i+1).\n\nThe formal `claim1` retains the\nuniversal quantifier, lower bound, printed path predicate, plane-containment\ncondition, and binomial sum. The theorem `result1` proves its negation.","title":"Flórez-Junes-Ramírez xz-plane path-count refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-xz-plane-path-count-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ceaba53e39c1a9c559e3ca5fda2dd3a877b4814b7b0ad846ccfd42b7741c665.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Flórez, Junes, and Ramírez, Journal of Integer Sequences 21 (2018), Article\n18.1.2, Section 2 (Background) on printed page 3, define the path family as follows:\n\n> We use C_n^±(k) to mean the set of all paths of length k in the n-dimensional cubic lattice. We divide C_n^±(k) into subfamilies depending on the behavior of the path. We now give definitions and notation for those families. If P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}), then we define V_r := (±e_{j_1}) + (±e_{j_2}) + · · · + (±e_{j_r}), the algebraic combination of the first r components of P for 0 < r ≤ k, i.e., V_r is the sum of the components of any initial subpath of P with r steps. We denote C_n(k) the subset of C_n^±(k) formed by all paths P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}) that satisfy that the nth coordinate of V_k is zero. We use C_n^≥(k) to denote all paths in C_n^±(k) with P = (±e_{j_1})(±e_{j_2}) · · · (±e_{j_k}) and that nth coordinate of V_r is non-negative for all 0 < r ≤ k. We now let C_n^+(k) be C_n^≥(k) ∩ C_n(k). […] For example, Figure 1 depicts the 14 paths in C_2^+(3). Figure 2 depicts the 17 paths in C_3^+(2).\n\nThe paper also specifies that paths start at `p_0 = (0, ..., 0)` and each step\nis a positive or negative coordinate direction.\n\nSection 6 on printed page 23 states:\n\n> **Conjecture 2:** For k ≥ 1, the number of paths in C_3^+(k) that are completely contained in the yz−plane is Σ_{i=1}^{k+1} \\binom{2i}{i}\\binom{k}{i−1}/(i+1).\n\nThe formal `claim2` retains the\nuniversal quantifier, lower bound, printed path predicate, plane-containment\ncondition, and binomial sum. The theorem `result2` proves its negation.","title":"Flórez-Junes-Ramírez yz-plane path-count refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ed8e802233797da81d8a772b5dadde024dc3ebf250689b515c651a83efd24566.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:florian-dvorakova-2016-generalized-pseudostandard-periodicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Florian and Ľ. Dvořáková, *On periodicity of generalized pseudostandard words*, Electronic Journal of Combinatorics 23(1), P1.2, Conjecture 29, asks whether a generalized pseudostandard word over any finite alphabet is periodic exactly when its normalized directive sequence satisfies the two tail conditions. The source and literal definitions are in [the Library note](../Library/Rewriting/floriandvorakova2016periodicity.md). The named assertion is proved by `D5.S1.Words.Palindromes.GeneralizedPseudostandardPeriodicity.result`. Preregistration: [#15626](https://github.com/the-omega-institute/trureturing/issues/15626).\n\nFor some common tail offset $k$, the conditions are:\n\n1. $\\vartheta_{k+i}(\\delta_{k+i+1})=\\vartheta_{k+j}(\\delta_{k+j+1})$ for all $i,j\\ge1$.\n2. For all $i\\ge1$, the image under $\\vartheta_{k+i+1}$ of every $\\vartheta_{k+i}$-palindrome is a $\\vartheta_{k+i+2}$-palindrome.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every finite alphabet and every involutory antimorphism | `claim` quantifies over `α : Type`, `Fintype α`, `DecidableEq α`; `InvAntimorphism` has the literal concatenation-reversal and involution laws | No alphabet-cardinality restriction |\n| Successive shortest pseudopalindromic closures, starting with the empty word | `Generates`, with `IsClosure` expressing prefix, fixed word, and shortest length | Directive indices begin at one; the entries at zero are unused |\n| Normalized directive generating the same infinite word | The first conjunct of `claim`; `Generates.normalization_exists` | Every prefix fixed by any involutory antimorphism occurs as a stage |\n| The word is $d$-ary | `Function.Surjective u` | Every alphabet letter occurs |\n| Periodicity | A positive natural period in `Function.Periodic` | Pure periodicity; the sufficiency proof constructs a pure period directly |\n| A common finite directive prefix | One existential natural offset in `TailConditions` | Both directive sequences have the same offset |\n| Condition 1, all tail indices | The first conjunct of `TailConditions`, with images of singleton words and both indices at least one | Literal letter-image equality |\n| Condition 2, every fixed finite word | The second conjunct of `TailConditions` | Literal universal palindrome transport |\n| Both directions | The second conjunct of `claim` is an equivalence; `result` proves `claim` | No source clause is dropped |\n\nPalindromes use Mathlib's `Function.IsFixedPt` and infinite prefixes use the literal `List.ofFn` expression. Finite decidable equality supports the finite-alphabet formalization and does not restrict finite alphabets. The normalization conjunct makes the normalized sequence in the source assertion available for every generated word.","title":"Periodicity of generalized pseudostandard words","url":"https://the-omega-institute.github.io/trureturing-pages/research/florian-dvorakova-2016-generalized-pseudostandard-periodicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd17cce67485b561f39e05b3c5ca06c7b77d4e7bfcb80e5fed734a58ac0bf01e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fokkink-joshi-2025-continued-fraction-tail-beatty","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Fokkink and G. Joshi, *On Cloitre's hiccup sequences*, arXiv:2507.16956v4, Section 4 after Equation (14), state the two-parameter family\n\n$$s_n=nu+v+\\frac{n^2}{s_{n+1}}$$\n\nand its asymptotic line\n\n$$\\alpha=\\frac{u+\\sqrt{u^2+4}}2,\\qquad\n\\beta=\\frac{v\\alpha^2-\\alpha}{\\alpha^2+1}.$$\n\nThey state that positive tails satisfying $s_n=\\alpha n+\\beta+O(1/n)$ round down to the Beatty sequence with slope $\\alpha$ and offset $\\beta$. The settling declaration is `D5.S1.Recurrence.ContinuedFractionTailBeattyRefutation.result`.\n\n### Statement echo\n\nThe Lean `beta`, `IsTail` and `claim` definitions retain the source's radical slope, offset, positive-tail recurrence, uniform $C/n$ error bound, and floor equality. The `claim` body is displayed in the Scribe and the `result` proves its negation.","title":"Continued-fraction tails with u = 2 and v = 1 miss the Beatty sequence","url":"https://the-omega-institute.github.io/trureturing-pages/research/fokkink-joshi-2025-continued-fraction-tail-beatty/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04d9a1f527771212873b11abf5fb233f8f2f92c481c2a00df0d27dac87d3a6ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fokkink-joshi-whole-antirecurrence-automaticity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Robbert Fokkink and Gandhar Joshi, *Anti-Recurrence Sequences*,\narXiv:2506.13337v1, Conjecture 4; journal version, INTEGERS 26, A24,\nDOI 10.5281/zenodo.18305140, Conjecture 2:\n\n> Let a = (a₁,…,aₖ) be positive and integral of dimension k > 1.\n> Let Aₙ be the anti-recurrence sequence generated by the linear form\n> f(x) = a·x. Then Aₙ − κn is τ-automatic for κ = kτ+1 and τ the\n> trace of the linear form.\n\nHere τ = Σ aⱼ. The strictly increasing positive sequences A and B\npartition the positive integers and satisfy\nAₙ = Σⱼ₌₁ᵏ aⱼ Bₖ₍ₙ₋₁₎₊ⱼ for every n ≥ 1.","title":"The whole positive coefficient anti-recurrence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/fokkink-joshi-whole-antirecurrence-automaticity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d03929394e55e89b2a45e84f5c6d459027491efe5595632ab77f96e38489f01a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fokkink-rust-2024-splythoff-grundy-rows","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Robbert Fokkink and Dan Rust, *Queen reflections: a modification of Wythoff\nNim*, arXiv:1904.08339v2, Section 9, printed page 16; International Journal\nof Game Theory 53 (2024), 1099–1117, DOI 10.1007/s00182-022-00824-1:\n\n> For Splythoff Nim, it appears that every row or column contains each Sprague-Grundy value once and no more. We do not have a proof for that.\n\nPreregistration: [#15902](https://github.com/the-omega-institute/trureturing/issues/15902).\nThe statement is proved by `D5/S0/Certificates/Games/SplythoffGrundyRows.result`.\nIts bound theorem `row_exists_bound` supplies an occurrence at\n$y\\le 5x+2g+3$ for every row $x$ and value $g$.\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Two piles, including an empty pile | Natural coordinates in `splythoffMove` and `grundy` | No positivity restriction on positions |\n| Reduce one pile | First two disjuncts of `splythoffMove` | The other coordinate is unchanged |\n| Remove an equal positive number from both piles | Third disjunct of `splythoffMove` | The same $k$ satisfies $1\\le k\\le\\min(x,y)$ |\n| Split only after removing counters from both piles | Fourth disjunct of `splythoffMove` | The smaller pile is positive and the target sum equals $\\max(x,y)-\\min(x,y)$ |\n| Mex of all legal follower values | `followers`, `grundy`, private `followers_iff` and `grundyPair_eq` | Complete legal-follower set and its exact mex equation |\n| Every value in each row once and no more | First conjunct of `claim`; `result` | Universal natural row and value, unique natural column |\n| Every value in each column once and no more | Second conjunct of `claim`; `result` | Universal natural column and value, unique natural row |\n| The preregistered finite bound | `row_exists_bound` | Additional quantitative conclusion, with no extra hypothesis |\n\nThe bounded-follower convention repair leaves the literal move relation\nunchanged, including the unbounded existential for double moves. The\nfinite rectangle with both coordinates at most $\\max(x,y)$, filtered by\nthat relation, is exactly its follower set: `follower_bound` proves the\ncoordinate bound and `followers_iff` proves the equivalence. Every move\nstrictly decreases the total pile size, and `grundyPair_eq` supplies the\nmex equation used by the proof. No legal move is discarded.","title":"fokkink rust 2024 splythoff grundy rows","url":"https://the-omega-institute.github.io/trureturing-pages/research/fokkink-rust-2024-splythoff-grundy-rows/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eaeb852e2b2e208debf1036a44ea8b6cdab811de9f2335d234a6b3c226d6d377.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:foucart-liao-2024-chebyshev-center-regularization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Foucart and C. Liao, *S-Procedure Relaxation: a Case of Exactness\nInvolving Chebyshev Centers*, arXiv:2310.09677v1, in *Explorations in\nthe Mathematics of Data Science* (Springer, 2024),\nDOI 10.1007/978-3-031-66497-7_1, §5 “Some Open Questions”, Question 2:\n\n> Is the Chebyshev center always obtained via regularization?\n\nThe authors' gloss is:\n\n> From [10], we know that the answer is affirmative under the specific conditions mentioned above, since the Chebyshev center equals the regularizer $f_{\\tau_\\sharp}$ with parameter $\\tau_\\sharp\\in(0,1)$ satisfying the implicit equation (9). Without the specific conditions, we also know from [8] that the answer is affirmative if one considered the complex setting rather than the real setting.\n\nThe section takes $Q=\\mathrm{Id}_H$ and, in (25),\n$\\mathcal K=\\{f\\in H:\\|Pf\\|_H\\leq\\varepsilon\\}$ and\n$\\mathcal E=\\{e\\in\\mathbb R^m:\\|e\\|_2\\leq\\varepsilon\\}$.\nThe “specific conditions” are that $P$ is an orthogonal projector\nand $\\Lambda\\Lambda^*=\\mathrm{Id}_m$.\nEquation (8) defines\n\n$$\nf_\\tau=\\operatorname{argmin}_f\n\\bigl((1-\\tau)\\|Pf\\|^2+\\tau\\|y-\\Lambda f\\|^2\\bigr).\n$$\n\nThe Lean `claim` quantifies over finite-dimensional real Euclidean\nspaces, linear $P$ and $\\Lambda$, data $y$, and positive bounds\n$\\varepsilon,\\eta$. For every nonempty bounded feasible set\n$C=\\{f:\\|Pf\\|\\leq\\varepsilon,\\ \\|y-\\Lambda f\\|\\leq\\eta\\}$,\nit asks whether every Chebyshev center minimizes this objective for\nsome $\\tau\\in(0,1)$. The counterexample uses $\\varepsilon=\\eta=1$,\nso it also addresses the equal-bound setting of (25).","title":"Foucart–Liao's Chebyshev-center regularization question","url":"https://the-omega-institute.github.io/trureturing-pages/research/foucart-liao-2024-chebyshev-center-regularization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3dd71dd59ac62a0e38379d5007b1e9ed225706e9fd92e74e3de487f50b78aa5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:franklin-indecomposable-321-1342-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Atli Fannar Franklín, *Pattern avoiding permutations enumerated by inversions*, arXiv:2410.07467v4, Section 1,\nprinted pages 2–3:\n\n> Another case is I_k(321, 1342), which we conjecture to have k(k + 1)/2 + 1 elements.\n\nHere I_k is the set of direct-sum indecomposable permutations, of any length, with exactly k inversions, and\nI_k(321, 1342) is its subset avoiding the classical patterns 321 and 1342.","title":"The Number of Indecomposable 321- and 1342-Avoiders with k Inversions","url":"https://the-omega-institute.github.io/trureturing-pages/research/franklin-indecomposable-321-1342-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4dc243ca065e9dfca2ba3d03ed716f39d6f4e7fa9091759b05022b63865df20b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fresan-figueroa-goldfeder-lozano-2026-mutual-visibility-maximal-connectivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Julián A. Fresán-Figueroa, Ilán A. Goldfeder and Aldo Lozano-Piña,\n*Reconfiguration graphs of mutual-visibility sets*, arXiv:2610.24324v1,\nSection 3.3 immediately after Theorem 3.6, asks:\n\n> A natural question is whether $\\mathcal{R}_{\\mathcal{M}}(G)$ is maximally connected for every connected $G$, that is, whether $\\kappa(\\mathcal{R}_{\\mathcal{M}}(G))=\\delta(\\mathcal{R}_{\\mathcal{M}}(G))$ holds, where $\\delta$ denotes minimum degree?\n\nThis dossier anchors exactly this question, repeated in Section 7. The source's\nstanding convention is finite, simple, undirected, connected graphs. Connected\nhosts are nonempty; orders one and two are included. The preceding theorem's\norder-at-least-three hypothesis does not restrict the question.\n\nFor every distinct selected pair, some shortest path in the original host\nmust avoid the other selected vertices. Different pairs may use different\npaths. The configuration graph contains every feasible subset, including empty\nand all feasible cardinalities. Distinct configurations $S,T$ are adjacent\nwhen $|S\\mathbin{\\triangle}T|=1$, or when $S\\setminus T=\\{x\\}$,\n$T\\setminus S=\\{y\\}$ and $xy$ is an actual host edge. Thus a slide replaces\nan occupied vertex by an unoccupied neighbor.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Every finite nonempty simple undirected host | `V`, `[Fintype V]`, `[DecidableEq V]`, `[Nonempty V]`, `H : SimpleGraph V` in `FullMaximalConnectivityClaim` |\n| Connected, with no order or rank restriction | `H.Connected` is the only graph hypothesis |\n| Some original-host geodesic avoids the other selected vertices | `Visible` uses an existential `H.Walk` of length `H.dist`; shortest walks are paths |\n| All feasible subsets, including empty | `Configuration H` is the subtype of all visible `Finset V` |\n| Addition/removal and actual host-edge slides | The two disjuncts of `MixedMove`, used by `reconfiguration` |\n| Standard vertex connectivity | `Separates` deletes to at most one vertex or to an induced graph that is not preconnected; `vertexConnectivity` minimizes cut size |\n| Equality to minimum degree | `result : FullMaximalConnectivityClaim` |\n\nThe separator convention gives $\\kappa(K_q)=q-1$ for every positive $q$;\n`Connectivity.connectivity_complete` verifies this convention. No geodesic,\nseparator, sparse-slide, girth or diameter premise is added to the conclusion.","title":"Maximal connectivity of full mutual-visibility reconfiguration graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/fresan-figueroa-goldfeder-lozano-2026-mutual-visibility-maximal-connectivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de56174151353897a4e8cabd413cd40e05868664af6ee5e8601f9e8eea8def59.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:frid-2018-fibonacci-prefix-palindromic-length","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. E. Frid, *Representations of palindromes in the Fibonacci word*, Numeration 2018 (Paris), Conjecture 2, states:\n\n> For every k ≥ 1, the prefix of the Fibonacci word of length (100)²ᵏ⁻¹101 cannot be decomposed as a concatenation of at most 2k palindromes.\n\nThe numeral `(100)^(2k-1)101` is interpreted in the Fibonacci numeration system with weights `G(0)=1` and `G(1)=2`. `PL` is the minimum number of nonempty palindrome factors of a finite word.","title":"Frid's Fibonacci-prefix palindromic-length conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/frid-2018-fibonacci-prefix-palindromic-length/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a2237694a329b04388cacdbfb762dd3642d9c06f6b27862eb6ab0a2e8e72df48.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:frid-laborde-peltomaki-2021-period-doubling-c17","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Frid, Laborde and Peltomäki, *On prefix palindromic length of automatic words*,\nTheoretical Computer Science 891, 13–23, arXiv:2009.02934v2,\nSection 5.1, printed page 13, Conjecture 17:\n\n> The sequence $d_{pd}$ of the period-doubling word $\\mathbf{u}_{pd}$ is not\n> $2$-automatic, and so the prefix palindromic length $\\operatorname{PPL}_{pd}(n)$ of\n> $\\mathbf{u}_{pd}$ is not $2$-regular.\n\nThe word is the fixed point beginning with a of a → ab, b → aa.\nWrite P(n) for the minimum number of nonempty palindrome factors of its\nlength-n prefix, P(0)=0, and d_pd(n)=P(n+1)−P(n) for n≥0.\nC17 asks whether the set of sequences n ↦ d_pd(2^e n+r), over all natural\ne and r<2^e, is infinite. No finite-prefix restriction is imposed.","title":"Period-doubling PPL-difference: Conjecture 17","url":"https://the-omega-institute.github.io/trureturing-pages/research/frid-laborde-peltomaki-2021-period-doubling-c17/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff063c48f5598914ab14b53280ae5b2263aa88635b024aa08a28a96f972f7acc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fried-fibonacci-shift-non-fibonacci","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fried, *Journal of Integer Sequences* **28** (2025), Article 25.4.3, Section 6,\nstates:\n\n> Nevertheless, we were not able to show that the two sets\n> `{ nF(n/2 + 3) − (n − 1)F(n/2 + 2) : n ≥ 1 is even }`,\n> `{ F((n+1)/2 + 2) : n ≥ 1 is odd }`\n> are disjoint, or, equivalently, that for every `n ∈ N`, the number\n> `F(n + 2) + 2nF(n + 1)` is not a Fibonacci number. We conjecture that this is\n> so.\n\nHere `F(1) = F(2) = 1`. Both sets are indexed by `n ≥ 1`, so the `N` of the\nequivalent form starts at one; `n = 0` gives `F(2) = 1`, which is a Fibonacci\nnumber, and is outside the range.","title":"Fried's Fibonacci Shift Conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/fried-fibonacci-shift-non-fibonacci/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a85059d30dcf68f4e6552f2ae8b2654dfdc6020a1a72ace891a9aee37741218.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fu-luo-decoherence-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Fu and S. Luo, *Quantifying Decoherence via Increases in Classicality*, Entropy 23 (2021), 1594, Section 3, after Proposition 3:\n\n> We conjecture that D(ρ_S|Π, U) is convex in ρ. It seems that a proof may require some deep mathematics.\n\nThe [source note](../Library/Entropy/fuluo2021decoherence.md) identifies DOI 10.3390/e23121594. Issue [#19026](https://github.com/the-omega-institute/trureturing/issues/19026) fixes the universally quantified reading and the literal conventions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Eq. (17), $J(\\rho,\\Pi)=1+\\sum_i\\langle i|\\sqrt\\rho|i\\rangle^2$ | `J` | Standard pointer basis of `Fin d`; real part of the squared diagonal entry. Positive matrix square roots have real diagonal entries. |\n| Eq. (18), $H_E=\\bigotimes_i H_{E_i}$, each $U_{E_i}$ acts only on its own part | `EnvIdx`, `liftE` | Arbitrary local dimensions `k`; equality on every other coordinate. |\n| Product initial state $\\rho_E=\\bigotimes_i\\rho_{E_i}$ | `rhoE` | Each factor is the reused `DensityState`. |\n| Eq. (21), $\\omega_{ij}=\\operatorname{tr}(U_{E_i}\\rho_EU_{E_j}^\\dagger)$ | `omega` | Exact lifted-operator trace. |\n| Eq. (22), $\\rho'_S=\\rho_S\\circ\\Omega$ | `finalState` | Entrywise multiplication. |\n| Eq. (27), $D=J(\\rho'_S,\\Pi)-J(\\rho_S,\\Pi)$ | `D` | Same environment and detector family for each input. |\n| Convexity in $\\rho$ | `claim` | Every dimension, product environment, local-unitary family, two density states and $0\\le t\\le1$; Jensen inequality. |\n| Refutation | `result` | Negation of that complete claim. |\n\nThere is no extra physical hypothesis. `CStarMatrix.ofMatrix.symm` is the coordinate equivalence from the reused density-state representation to matrices. The conventions in `claim` are the literal preregistered expressions; `DensityState` and `ProbeComponentRigidity.probe` are reused in the proof construction.","title":"Fu–Luo decoherence convexity is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/fu-luo-decoherence-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39616279ee56747ff14541f45edb9057120f48dc9003ee61c7b653a949811cdc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fucci-stanfill-2024-poschl-teller-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Fucci and J. Stanfill, *The exotic structure of the spectral ζ-function\nfor the Schrödinger operator with Pöschl–Teller potential*, arXiv:2411.17860v1,\nAppendix B, Remark B.3, state:\n\n> From calculations performed for particular choices of $p$, $q$, and $\\beta$,\n> we in fact conjecture that all of the $g_{m,j}(p/q,\\beta)$ are nonzero for\n> the choices of $p,\\ q,$ and $\\beta$ considered here.\n\nThe source range is relatively prime integers $0 An interesting open question is whether there exists a basis in which to represent the operators ρ̂₀, ρ̂₁, etc., so that the matrix appearing in the logarithm above is diagonal. If so, this would greatly simply the computation of L(t).\n\nThe [source note](../Library/Entropy/fuchs1996thesislogarithmbasis.md) identifies the version and equations. [Preregistration #18588](https://github.com/the-omega-institute/trureturing/issues/18588) fixes the literal Lean conventions. The universal physical-basis assertion is **Refuted** by `D5.S3.Quantum.Foundation.FuchsLogarithmBasisRefutation.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Kronecker sum, equations (3.237), (3.242) | `ksum` | $I\\otimes X+\\overline X\\otimes I$, with the preregistered column-vectorization order |\n| Matrix inside the logarithm, equation (3.244) | `logArg` | $I+t\\widetilde\\rho_0^{-1/2}\\widetilde\\Delta\\widetilde\\rho_0^{-1/2}$; inverse of `CFC.sqrt` on both sides |\n| Density operators | `d`, `rho0`, `rho1`, positive-definiteness and trace-one hypotheses in `claim` | Faithful finite-dimensional states; a counterexample here refutes a universal claim covering these states |\n| Invertible difference in the source derivation | `(rho1-rho0).det ≠ 0` in `claim` | Preserved, with witness determinant $-1/4608$ |\n| A basis in which to represent both operators | `U ∈ Matrix.unitaryGroup (Fin d) ℂ`, and both conjugations by $U^*\\rho U$ | A physical basis induces $\\overline U\\otimes U$ on matrix space |\n| Source's basis independent of $t$ | `∀ t, 0 < t → t ≤ 1 → ∃ U, …` | Permitting a separate basis at each $t$ weakens the positive assertion; its negation at $t=1$ also refutes a single basis for all $t$ |\n| Diagonal logarithm argument | `Matrix.IsDiag` in `claim` | The entire matrix, in the induced basis |\n| Negative answer | `result : ¬ claim` | Negation of the whole quantified assertion |\n\nThe explicit dimension, faithfulness, trace, determinant and interval binders are the preregistered finite density-state interpretation. The construction and every binder are retained literally. The proof uses the frozen `PathSpectralClassical.relativeCongruence` sandwich and `EntropyProductionCoherenceDeletionIdentity.cfc_diagonal` for square roots of diagonal matrices.","title":"Fuchs's physical-basis logarithm question","url":"https://the-omega-institute.github.io/trureturing-pages/research/fuchs-1996-thesis-logarithm-basis-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73d773801c067465ebdc269a975ebc5b55ab0f17af170d50b2d1a0f01fdc7c0e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fuentes-2025-mmi-forbidden-subgraph","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fuentes, Keeler, Munizzi and Pollack (arXiv:2511.19585v1) conjecture:\n\n> **Conjecture 1 (Forbidden-Subgraph).** Any graph state that violates MMI has\n> a graph representation that is $LC$-equivalent to a graph $H$ containing a\n> $K_4$ subgraph. $H$ is $\\mathcal{G}$ with respect to *some* partition.\n\nIssue #10534 fixes the readings from the paper: the entropy of a set of\nqubits `A` is the rank over `Z_2` of the adjacency block with rows in `A` and\ncolumns outside `A`; an MMI violation is a triple of pairwise disjoint sets\n`I`, `J`, `K` with `S_IJ + S_IK + S_JK < S_I + S_J + S_K + S_IJK`; LC\nequivalence is a finite sequence of local complementations; `K_4` is the\nfour-vertex star `K_{1,3}`, read as an induced subgraph, which is the stronger\nreading; and a generalized star with centre `C` and parts `V^1, …, V^k` has\nnonempty, pairwise disjoint parts that with `C` partition the vertices, no\nedge between different parts, and an edge from every part to `C`.","title":"The forbidden-subgraph conjecture for MMI in graph states holds","url":"https://the-omega-institute.github.io/trureturing-pages/research/fuentes-2025-mmi-forbidden-subgraph/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a57d3db8f3f64ffca8a40a9c22f0cae008340399038508334c2b4bc3bf7fb2c7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fukai-yoshida-katsura-2026-dissipative-ffd-upper-half-plane","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Fukai, H. Yoshida and H. Katsura, *Dissipative free fermions in disguise*, arXiv:2603.22163v1, printed p. 4, after Eq. (17):\n\n> We expect Im ũ_k > 0 to hold for general ECF graphs, as observed numerically in the boundary-driven Fendley model, so that ε̃_k ≡ 1/ũ_k satisfies Im ε̃_k < 0.\n\nThe roots ũ_k are those of the plus polynomial in Eq. (16):\n\n$$\n\\widetilde P_G^+(u)=P_G(u^2)+i\\gamma u P_{G\\setminus K_s}(u^2).\n$$\n\nEquation (7), printed p. 3, defines the actual independent-set sum:\n\n$$\nP_G(x)=\\sum_{S\\in\\mathcal S_G}(-x)^{|S|}\\prod_{j\\in S} b_j^2.\n$$\n\nThe couplings are real, γ > 0 is the dissipation rate, and ECF means even-hole-free and claw-free. The simplicial clique K_s satisfies the source's closed-neighborhood condition, printed p. 3:\n\n> Equivalently, denoting the closed neighborhood of j by Γ[j] ≡ {j} ∪ {ℓ ∈ V(G) | A_jℓ = 1}, K_s is simplicial if and only if Γ[j] \\ K_s is a clique for all j ∈ K_s.\n\nThe standing exclusion after Eq. (26), printed p. 7, is retained:\n\n> Equation (26) shows that N_k = 0 if P_G and P_{G\\K_s} share the root u_k². Throughout this work, we exclude this nongeneric case [80].\n\nThe quantified statement ranges over every finite graph on Fin n, every real coupling vector, every simplicial clique, every positive γ, and every complex root. The shared-root exclusion ranges over all complex x. Neither nonzero individual couplings nor simple roots are assumed.","title":"Fukai–Yoshida–Katsura dissipative free-fermion root expectation","url":"https://the-omega-institute.github.io/trureturing-pages/research/fukai-yoshida-katsura-2026-dissipative-ffd-upper-half-plane/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72eb5f8f31ca8c98e83c092da06e24367fb11bfa6c124c7c908fc0231f11d73f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fuks-2019-ternary-density-classification","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Henryk Fukś and Roman Procyk, *Explorations of Ternary Cellular Automata and\nTernary Density Classification Problems*, arXiv:2002.08924v1 (Conjecture 1 of\nActa Phys. Pol. B Proc. Suppl. 12(1) (2019) 75–89), write:\n\n> Let $F$ be the ternary nearest-neighbour rule with Wolfram number 6478767664173, and $G$ be the rule with Wolfram number 7580606234490. For any finite ternary string $\\mathbf{x}$ of length $L$, containing at least one zero, let $\\displaystyle \\rho=\\frac{1}{2L}\\sum_{i=0}^{L-1} x_i$. Then $G^LF^L(\\mathbf{x})=0^L$ if $\\rho(\\mathbf{x}) \\in [0, 2/3)$, $G^LF^L(\\mathbf{x})=1^L$ if $\\rho(\\mathbf{x}) \\in (2/3, 3/4)$, $G^LF^L(\\mathbf{x})=2^L$ if $\\rho(\\mathbf{x}) \\in (3/4, 1)$.\n\nConfigurations are periodic, $(F(\\mathbf{x}))_i=f(x_{i-1},x_i,x_{i+1})$ with\nindices in $\\mathbb{Z}/L$, and the rule with Wolfram number $N$ has\n$f(a,b,c)=\\lfloor N/3^{9a+3b+c}\\rfloor \\bmod 3$. The precise question,\npreregistered in issue #11845, is whether the three implications hold for\nevery $L$ and every configuration containing a zero.","title":"A misclassified configuration for a ternary two-rule density classifier","url":"https://the-omega-institute.github.io/trureturing-pages/research/fuks-2019-ternary-density-classification/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0acd3ef186e493e19967c818a3b2a75b54746d8665a41664a45f53d5e29304e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fuks-fates-2026-diploid-10-245-density","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Fukś and N. Fatès, *Analysis of the asymptotic density of endogamous diploid cellular automata*, arXiv:2607.01917v1 (EPJ ST 2026), study probabilistic cellular automata on ℤ in which every cell independently applies elementary rule f with probability 1 − λ and rule g with probability λ: w(1|x₁x₂x₃) = (1−λ) f(x₁,x₂,x₃) + λ g(x₁,x₂,x₃) (Eq. (2)). Starting from the symmetric Bernoulli measure, d_t = Pr(s_i(t) = 1) is the density and d∞ = lim d_t (§2). For rule 10-245, third-order local structure approximation predicts d∞ = 1/4 + λ/2 (§3.2), while mean-field and second-order approximation give a nonlinear curve. §5 asks: \"for rule 10-245, the expression for the asymptotic density, d = 1/4 + λ/2, appears to be exact. Could this be formally demonstrated?\" The verbatim statements are in [the literature note](../Library/StatisticalMechanics/fuksfates2026diploid.md).\n\nIssue [#16027](https://github.com/the-omega-institute/trureturing/issues/16027) fixes the reading: for every λ ∈ [0, 1], with the lattice process encoded by its cylinder probabilities (independent simultaneous updates of each interval from its n + 2 parents, fair Bernoulli start), d_t → 1/4 + λ/2. The settlement proves the stronger d_t = 1/4 + λ/2 for every t ≥ 1.","title":"The diploid cellular automaton 10-245 has exact density 1/4 + λ/2","url":"https://the-omega-institute.github.io/trureturing-pages/research/fuks-fates-2026-diploid-10-245-density/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ee4e4ba53787e28eac8f655103dd27f4404d8fd2dbb5e0ca297a544cf657e0e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fullwood-ma-wu-2026-spatiotemporal-born-dichotomy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Fullwood, Z. Ma and Z. Wu, *The spatiotemporal Born rule is quasiprobabilistic*, arXiv:2507.16919v2; published as *Quasiprobabilistic nature of the spatiotemporal Born rule*, Phys. Rev. A 113, 032204 (2026). Section VI, final paragraph (`PRA_rev.tex`, line 577), asks the following. The source macro `\\E` denotes $\\mathcal E$.\n\n> It is then natural to ask whether $\\rho$ being maximally mixed or $\\E$ being a discard-and-prepare channel is also a necessary condition for the existence of the spatiotemporal Born rule for the LvN distribution. While it is straightforward to show that this is indeed the case when $A$ consists of a single qubit, for systems of arbitrary dimension we leave it as an open question.\n\nIssue [#21996](https://github.com/the-omega-institute/trureturing/issues/21996) fixes the statement for all positive finite input and output dimensions. For every $d,e\\ge1$, every density matrix $\\rho\\in M_d(\\mathbb C)$ and every CPTP complex linear map $\\mathcal E:M_d(\\mathbb C)\\to M_e(\\mathbb C)$,\n\n$$\n\\begin{aligned}\n&\\exists W\\in M_{de}(\\mathbb C)\\;\\forall P,Q\\text{ orthogonal projections},\\quad\n\\operatorname{tr}[W(P\\otimes Q)]\n =\\operatorname{tr}[\\mathcal E(P\\rho P)Q] \\\\\n&\\qquad\\Longleftrightarrow\\quad\n\\rho=I_d/d\\quad\\text{or}\\quad\n\\exists\\sigma\\in M_e(\\mathbb C),\\quad\n\\sigma\\succeq0,\\quad\\operatorname{tr}\\sigma=1,\\quad\n\\forall X\\in M_d(\\mathbb C),\\quad\\mathcal E(X)=\\operatorname{tr}(X)\\sigma.\n\\end{aligned}\n$$\n\nHere $P$ acts on the input and $Q$ on the output; an orthogonal projection is a self-adjoint idempotent. A density matrix is positive semidefinite with trace one. Complete positivity means positivity of every finite amplification, and trace preservation is required on every matrix. The same $W$ must work for all projections, without a positivity, Hermiticity or normalization assumption on $W$. This equivalence is proved.\n\nThe source defines the LvN distribution by $\\mathbf P(i,j)=\\operatorname{tr}[\\mathcal E(P_i\\rho P_i)Q_j]$ in Section II (`PRA_rev.tex`, lines 247–250), and its universal Born representation in Section VI, lines 534–537. The [literature note](../Library/Observer/fullwoodmawu2026spatiotemporalborn.md) supplies the corresponding source locators.","title":"The spatiotemporal Born rule for the LvN distribution has exactly two cases","url":"https://the-omega-institute.github.io/trureturing-pages/research/fullwood-ma-wu-2026-spatiotemporal-born-dichotomy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84ffe03cbb8995b1ea4eb48a1bed8c016dafaf4aee79cb672257a71df0485407.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:fullwood-parzygnat-2025-states-over-time-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Fullwood and A. J. Parzygnat, *On Dynamical Measures of Quantum Information*, Entropy **27**(4), 331\n(2025), Remark 1:\n\n> At present, we do not know of any examples of quantum states over time that violate subadditivity,\n> which leads us to conjecture that the entropy functional S restricted to quantum states over time\n> does in fact satisfy subadditivity.\n\nA quantum state over time is $\\varrho=\\tfrac12\\{\\rho\\otimes I,\\mathscr J[\\mathcal E]\\}$ for a density\nmatrix $\\rho$ and a CPTP map $\\mathcal E$, with the Jamiołkowski matrix\n$\\mathscr J[\\mathcal E]=\\sum_{i,j}|i\\rangle\\langle j|\\otimes\\mathcal E(|j\\rangle\\langle i|)$, and\n$S(X)=-\\mathrm{tr}(X\\log|X|)$. The verbatim texts, including the restatement in J. Fullwood and B. Yang,\narXiv:2608.28946 §6, are in [the literature note](../Library/QuantumChannels/fullwood2025dynamical.md).\nIssue [#14994](https://github.com/the-omega-institute/trureturing/issues/14994) preregisters the reading\nover finite Kraus channels and the refutation.","title":"Subadditivity of the entropy of quantum states over time","url":"https://the-omega-institute.github.io/trureturing-pages/research/fullwood-parzygnat-2025-states-over-time-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0afe8b3ccede1c3b2e4a5a7141d1d05b22136de027df57027be1c6b919dbe415.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:furuichi-eghbali-amlashi-2022-hyperbolic-ratio-not-positive-definite","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Furuichi and M. Eghbali Amlashi, *On bounds of logarithmic mean and\nmean inequality chain*, arXiv:2203.01134v2, Remark 3.2(i), proposes the\npositive definite function conjecture. The supplied source identifies\nthis with Remark 2(i) in Mathematical Inequalities & Applications 27(2)\n(2024), 379–399.\n\nThe ratio is\n\n$$\n\\frac{K_r(e^{2t},1)}{L(e^{2t},1)}\n=\\frac{t(\\cosh t)^{1-r}}{\\sinh t}=: \\phi_r(t),\\qquad \\phi_r(0)=1,\n$$\n\nwhere $K_r=G^r A^{1-r}$ and $L$ is the logarithmic mean. The source states:\n\n> Thus these numerical computations support the conjecture that the function ϕr (t) for r < 1, is not positive definite.\n\nThe exact target is: for every real $r<1$, the complex cast of $\\phi_r$\nis not positive definite. Positive definiteness means that for every\nnatural $m$ and every $x:\\operatorname{Fin}(m)\\to\\mathbb R$, the matrix\nwith entries $f(x_j-x_i)$ is positive semidefinite. The existing\n`FiniteInversePoissonCriterion.PositiveDefinite` supplies this definition.","title":"Furuichi–Eghbali Amlashi: the hyperbolic ratio below one","url":"https://the-omega-institute.github.io/trureturing-pages/research/furuichi-eghbali-amlashi-2022-hyperbolic-ratio-not-positive-definite/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0dc703ca51d9ba04ad27b9bf86b1be07ffb7a791e69a319fce7611e237f50ec5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gabric-shallit-2023-block-palindrome-expected-width","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Gabrić and J. Shallit, *Smallest and Largest Block Palindrome Factorizations*, arXiv:2302.13147v2, §3, p. 5, Conjecture 5; WORDS 2023, DOI 10.1007/978-3-031-33180-0_14:\n\n> Let $k\\geq2$. Then $E_k=1+\\sum_{i=1}^{\\infty}a_i k^{-i}$ where the sequence $(a_n/2)_{n\\geq1}$ is A274199 in the On-Line Encyclopedia of Integer Sequences (OEIS).\n\nA274199 has offset zero. The paper's displayed coefficients $2,4,6,10,16,\\ldots$ fix $a_i=2\\,\\mathrm{A274199}(i)$ for $i\\geq1$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A word is over a $k$-letter alphabet | Lists over `Fin k`, enumerated by `DepthBudgetIidGreedyOptimality.words` | Uniform cardinality is $k^n$. |\n| $w=w_m\\cdots w_1w_0w_1\\cdots w_m$, with nonempty outer blocks and possibly empty centre | `GreedyRenewal.IsBPFactorization` | `bs` lists the outer blocks from the centre outwards; the two flattenings have the source order. |\n| Width $2m$ or $2m+1$ according to the centre | `GreedyRenewal.bpWidth` | The empty word has width zero, extending the source without affecting a limit at infinity. |\n| Largest width over all valid factorizations | `GreedyRenewal.largestBPWidth` | The supremum is over valid widths; its boundedness is proved. |\n| $E_{n,k}=k^{-n}\\sum_{i=1}^n i\\,\\mathrm{LBP}_k(n,i)$ | `LargestFactorizationExpectedWidth.expectedWidth` | The definition is the uniform function-space average. The private `expectedWidth_weighted_count` identity gives the source's weighted width-count expression. |\n| A274199 is the limiting reverse row of A274190 | `CompositionInversion.a274190_stable` and `a274190_diagonal_limit` | For $N\\geq2d$, the diagonal equals $g(2d,d)$. The triangle includes $g(n,0)=1$ and the recurrence for every positive column. |\n| Every integer $k\\geq2$ and convergence to $1+2\\sum_{i\\geq1}\\mathrm{A274199}(i)k^{-i}$ | `LargestFactorizationExpectedWidth.claim` and `result` | `claim` states existence of every reverse-row limit, summability of the positively indexed series, and convergence of the uniform average to one plus that series. `result` applies `coefficient_limit_results` to all three clauses. |\n\nThe auxiliary parameters in border and peeling statements range over arbitrary list alphabets. They strengthen the supporting statements and add no hypothesis to the settling result.","title":"Expected largest block-palindrome width","url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/912a9189763db00e7f8656e25270d4e510ae24c25b359995a935e272517ff3b5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gachechiladze-wyderka-guhne-2018-three-product-observables","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Gachechiladze, N. Wyderka and O. Gühne, *The structure of ultrafine entanglement witnesses*, J. Phys. A: Math. Theor. 51, 365307 (2018), arXiv:1805.06404v2, Summary, TeX source `uew-iop-revision.tex`, lines 750–752:\n\n> For instance, it would be interesting to identify three product observables, for which each pair is useless for entanglement detection, but the whole triple is useful.\n\nThe same source defines separable states as the convex hull of pure product states (lines 113–114) and describes a useless pair by the fact that every possible pair of expectation values can originate from a separable state (lines 504–507).\n\nIssue [#25940](https://github.com/the-omega-institute/trureturing/issues/25940) fixes the existential reading: find three Hermitian product observables $M_i=A_i\\otimes B_i$ such that, for each distinct pair $i,j$ and every density matrix $\\rho$, one separable density matrix $\\sigma$ reproduces both expectations, whereas some density matrix has no separable density matrix reproducing all three expectations. The pairwise matching state may depend on the pair and on $\\rho$. The module fixes both local dimensions to three. This existence statement is proved.","title":"Three product observables with useless pairs and a useful triple","url":"https://the-omega-institute.github.io/trureturing-pages/research/gachechiladze-wyderka-guhne-2018-three-product-observables/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/effa86f35d00e43704dbcf13fd3e9b3448a7f5e60fa7208c77a9d4f1b0814ebd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gadde-jain-krishna-kulkarni-sharma-2023-multi-entropy-positivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Gadde, S. Jain, V. Krishna, H. Kulkarni and T. Sharma, *Monotonicity conjecture for multi-party entanglement. Part I*, JHEP 02 (2024) 025, arXiv:2308.16247v1, Section 7:\n\n> “Although we found that E_n is positive in the examples we studied, it would be nice to prove this in general and to see if there is any relation between the positivity of E_n and the coarse-graining monotonicity.”\n\nFootnote 2 makes positivity the assumption needed for the logarithm in Eq. (1.11). The [literature note](../Library/QuantumChannels/gaddejainkrishnakulkarnisharma2024multientropy.md) gives the full quotation and the replica convention in Eqs. (1.10)–(1.11). Issue [#17540](https://github.com/the-omega-institute/trureturing/issues/17540) fixes the finite-dimensional statement with all quantifiers before formalization.\n\nThe question is refuted. For the normalized totally antisymmetric three-qutrit state\n$\\psi_{ijk}=\\varepsilon_{ijk}/\\sqrt{6}$,\n\n$$E_4^{(3)}(\\psi)=\\frac{-1680}{6^{16}}=\\frac{-35}{58773123072}<0.$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $n^{q-1}$ replicas on $(\\mathbb Z/n)^{q-1}$ | `E`, replica index `Fin (q - 1) → ZMod n` | Literal contraction in Eq. (1.10). |\n| Independent translations for the first $q-1$ parties; last party fixed | `StabilizerMultiEntropyCollapse.shift` | The source labels parties from one; Lean labels from zero. |\n| Arbitrary finite local bases and complex pure-state amplitudes | `E`, dependent family `L`, its `Fintype` instances and amplitude `ψ` | No dimension restriction in `claim`. |\n| Normalization | `IsUnitState` | Sum of `Complex.normSq` is one, exactly the convention in #17540. The proof uses the existing `unitVector` predicate through the private consumed correspondence `unit_norm_iff`. |\n| Every $n,q\\ge2$ has positive real invariant | `claim` | Positive real part and zero imaginary part; `NeZero n` is implied by $n\\ge2$ and supports `ZMod n`. |\n| Negative answer at $n=4,q=3$ | `result : ¬ claim` | One normalized instance disproves the universal assertion. |","title":"Positivity of the multi-entropy replica invariant","url":"https://the-omega-institute.github.io/trureturing-pages/research/gadde-jain-krishna-kulkarni-sharma-2023-multi-entropy-positivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2466587e0545468b10a81355d1bdfb6dac0ff72e27c60849d7e9c17f91a038ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gaiser-restricted-schur-three-colours","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Collier Gaiser, *Restricted generalized Schur numbers*, arXiv:2608.08789v1, Section 6: S_r(k; l) is the least n such\nthat every r-colouring of {1, …, n} has a monochromatic solution of x_1 + ⋯ + x_k = x_{k+1} with exactly l + 1\ndistinct integers. Proposition 6.1 proves S_3(k; 2) ≥ k^3 + 3k^2 + k − 1, and \"Open Question 6.2. Is it true that\nS_3(k; 2) = k^3 + 3k^2 + k − 1 for all large enough k?\"","title":"The Three-Colour Restricted Schur Number S_3(k; 2) Exceeds k^3 + 3k^2 + k − 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/gaiser-restricted-schur-three-colours/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0edc81f74d019643ecb296e0bbf07e3254a99aafb3f10f4106e5ebaeaaa23249.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:galindo-rowell-2026-pauli-rigidity-mixed-sign-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Galindo and E. C. Rowell, *Unitary Yang–Baxter Operators: Towards a Classification*, arXiv:2608.16865v1, Conjecture 10.6 (§10.2.7), state:\n\n> Let $d\\geq2$ with $4\\nmid d$, let $\\alpha\\in(\\mathbb Z/d\\mathbb Z)^\\times$, and put $P_\\alpha=X\\otimes Z^\\alpha$, $\\tau_\\alpha(s,t)=w^{\\alpha st}$. If $a=(1,a_1,\\ldots,a_{d-1})\\in\\mathbb T_d^{\\mathrm{coef}}$, $R_\\alpha(a)$ satisfies the Yang–Baxter equation, and $R_\\alpha(a)$ is projectively unitary, then there is one sign $\\epsilon\\in\\{+1,-1\\}$ with $a_{s+t}=a_sa_t\\tau_\\alpha(s,t)^\\epsilon$ for every $s,t\\in\\mathbb Z/d\\mathbb Z$.\n\nThe Lean `claim` is this quantified statement with `ZMod d`, the displayed braid equation, and Mathlib projective unitarity. The theorem `result` proves its negation.","title":"A mixed-sign Gaussian refutes Galindo–Rowell Conjecture 10.6","url":"https://the-omega-institute.github.io/trureturing-pages/research/galindo-rowell-2026-pauli-rigidity-mixed-sign-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c9666e8020c197dc916b421ae62c7a41b967dbca3393710e8946ab10a7c372a5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gallardo-rahavandrainy-2023-binary-collatz-3-11","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gallardo and Rahavandrainy, *On Collatz Conjecture for binary polynomials*,\narXiv:2308.01181v2, Conjecture 3.11, p. 15, state:\n“If $A=(M^2+M+1)^n$ with $n\\ge2$, then $\\ell_A$ equals $n+1$.”\nHere $M=x^2+x+1$ in $\\mathbb F_2[x]$. The source calls a polynomial odd when\nit is coprime to $x(x+1)$, removes all powers of these two linear factors to\nobtain its odd part, and iterates the odd part of $1+MA$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Binary polynomials and $M=x^2+x+1$ | `Polynomial (ZMod 2)` and `M` | Literal coefficient ring and polynomial. |\n| Remove $x^{val_x(S)}(x+1)^{val_{x+1}(S)}$ | `oddPart` | `rootMultiplicity` at 0 and 1; `X-C 1=X+1` in characteristic two. |\n| The odd subsequence $A_1,A_3,\\ldots$ | `oddSeq A k` | Source index $2k+1$; its initial term is the odd part of $A$. |\n| Length counts the initial odd term and terminal 1 | `collatzLength` | First hitting index of 1, plus one. |\n| Every natural $n\\ge2$ | `claim` | The quantified natural exponent has hypothesis $2\\le n$. |\n| $\\ell_{(M^2+M+1)^n}=n+1$ | `result : claim` | Proved, with no extra hypothesis. |\n\nThe total definition of `oddPart` also accepts zero. The proof only transports\nnonzero polynomials with the required oddness; this total extension does not\nweaken or change the stated conjecture.","title":"Gallardo–Rahavandrainy Conjecture 3.11","url":"https://the-omega-institute.github.io/trureturing-pages/research/gallardo-rahavandrainy-2023-binary-collatz-3-11/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29453752cc9fda267902c576c9a21cf21222a5031ffda79935afd4d46bbb4c52.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:galvao-brod-2020-overlap-interval-propagation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. F. Galvão and D. J. Brod, *Quantum and classical bounds for two-state overlaps*, Phys. Rev. A 101, 062110 (2020), arXiv:1902.11039v3, Section II.B, `overlap_pra_v3.tex`, line 165:\n\n> We leave it as an open question whether this procedure is optimal, but emphasize that even the initial iteration that produces a complete graph already provides bounds for every pair of vertices, which may be useful even if not tight.\n\nThe procedure is specified at lines 151–156: use the three-state bounds through each common neighbour of an unmeasured pair, intersect the inferred intervals, add the edge, and continue. Line 165 also allows revisiting inferred edges. The source's definitions and bounds are at lines 97–131; see [the literature note](../Library/Observer/galvaobrod2020overlaps.md).\n\nIssue [#23699](https://github.com/the-omega-institute/trureturing/issues/23699) fixes the reading to graphs with exactly one unknown pair. Write\n$U(x,y)=(\\sqrt{xy}+\\sqrt{(1-x)(1-y)})^2$, and let $L(x,y)=(\\sqrt{xy}-\\sqrt{(1-x)(1-y)})^2$ when $x+y>1$, and $L(x,y)=0$ otherwise. These are the module's `upper` and `lower`.\n\nFor $m$ labelled states, distinct indices $i_0,j_0$, and an array $r$, a realization is a family of unit vectors $\\psi_i\\in\\mathbb C^n$ satisfying $|\\langle\\psi_i,\\psi_j\\rangle|^2=r_{ij}$ for every ordered pair of distinct indices except the unordered pair $\\{i_0,j_0\\}$. Diagonal entries impose no conditions. The set $F$ of feasible unknown overlaps ranges over every finite dimension $n$. The procedure's endpoints are\n$\\ell=\\sup\\{L(r_{ki_0},r_{kj_0}):k\\ne i_0,j_0\\}$ and $u=\\inf\\{U(r_{ki_0},r_{kj_0}):k\\ne i_0,j_0\\}$, encoded by `procLower` and `procUpper`.\n\nThe proposition `claim` asserts that, for every $m\\ge3$, every distinct $i_0,j_0$, and every $r$ with $F\\ne\\varnothing$, both $\\ell=\\inf F$ and $u=\\sup F$. This universal tightness statement is refuted.","title":"Three-state overlap interval propagation is not always optimal","url":"https://the-omega-institute.github.io/trureturing-pages/research/galvao-brod-2020-overlap-interval-propagation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f78d13a012ce0f4dcd316fc30305342e3f1188116499abbfe7a290de9003d794.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gan-kim-2024-wasserstein-weak-log-majorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Luyining Gan and Sejong Kim, *Weak log-majorization between the geometric and Wasserstein means*, arXiv:2301.07934v4, §5, Conjecture 5.7. See [gankim2024wasserstein](../Library/Eigenstructure/gankim2024wasserstein.md).\n\n> Conjecture 5.7. Let A, B ∈ Pm and t ∈ (0, 1). Then (A^p ⋄t B^p)^{1/p} ց≺w log exp((1 − t) log A + t log B) as p ց 0.\n\n> A ⋄t B := (1 − t)² A + t² B + t(1 − t)[A(A^{−1} #B) + (A^{−1} #B)A]\n\nHere $A\\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$ and $\\mathbb P_m$ is the cone of complex Hermitian positive definite matrices. Weak log-majorization compares every initial product of decreasingly sorted eigenvalues, including the product of all $m$ eigenvalues.\n\nFor $F_t(p)=(A^p\\diamond_tB^p)^{1/p}$, the decreasing arrow is the conjunction of the comparisons $F_t(p)\\prec_{w\\log}F_t(q)$ for all real $0 Let $\\rho$ be a density matrix. Then\n> $\\inf_{\\sigma \\in \\mathrm{PPT}} d_T(\\rho, \\sigma) = N(\\rho)$.\n\nHere $d_T(\\rho,\\sigma)=\\frac12\\|\\rho^{T_B}-\\sigma^{T_B}\\|_1$ is the partial\ntranspose distance, $N(\\rho)=\\frac12(\\|\\rho^{T_B}\\|_1-1)$ is the negativity,\n$T_B$ is the partial transpose on the second system, and PPT is the set of\nstates with $\\sigma^{T_B}\\ge0$. The triangle inequality gives\n$N(\\rho)\\le\\inf_\\sigma d_T(\\rho,\\sigma)$, so the conjecture asserts the\nreverse inequality.","title":"Negativity and the partial transpose distance to the PPT states","url":"https://the-omega-institute.github.io/trureturing-pages/research/ganardi-2022-negativity-ppt-distance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fca83986ecfb39d019d0d4a06965bfaadf49d68d965f1a67d471d4f79120cca2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garcia-2026-erdos64-agl29-orientation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Daniel Garcia, arXiv:2609.04686v1, §§3 and 5, leaves the order-812 affine-group orientation instance undecided after several CPU-hours. The settlement refutes the solvability of this instance, using the universal obstruction preregistered in issue #14828.\n\nLet $V=\\operatorname{AGL}(1,29)=(\\mathbb Z/29\\mathbb Z)^\\times\\times\\mathbb Z/29\\mathbb Z$, with $(x,y)(z,w)=(xz,xw+y)$. For each $a\\in\\{3,8,10,11\\}$, put $t=(-1,0)$, $g=(a,1)$ and $r=g^{-1}$. Let $B_a$ be the simple undirected Cayley graph joining $x$ to $xt,xg,xr$.\n\nThe claim is: some orientation of Garcia's p = 29 instance avoids 64-cycles. Precisely, there exist one of these four multipliers, a function $\\sigma:V\\to\\{t,g,r\\}$, and bijections $\\tau_x:\\{t,g,r\\}\\to\\{u,v,w\\}$ with $\\tau_x(\\sigma(x))=u$, such that the literal $H_{15}$ replacement has no simple cycle of length 64. The closed Lean theorem `result : ¬ claim` refutes this existence assertion. Every independent assignment of the remaining incident edges to $v,w$ is included.","title":"garcia 2026 erdos64 agl29 orientation","url":"https://the-omega-institute.github.io/trureturing-pages/research/garcia-2026-erdos64-agl29-orientation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea876eece3e8f4078f81f058796a010929bf20fce052e547bf1100e27272cd60.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garcia-2026-erdos64-agl31-orientation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Daniel Garcia, arXiv:2609.04686v1, §5, leaves the order-930 affine-group orientation instance open. The settlement refutes the solvability assertion preregistered in issue #14838.\n\nLet $V=\\operatorname{AGL}(1,31)=(\\mathbb Z/31\\mathbb Z)^\\times\\times\\mathbb Z/31\\mathbb Z$, with $(x,y)(z,w)=(xz,xw+y)$. For each $a\\in\\{11,17,22,24\\}$, put $t=(-1,0)$, $g=(a,1)$ and $r=g^{-1}$. Let $B_a$ be the simple undirected Cayley graph joining $x$ to $xt,xg,xr$.\n\nThe claim is: some orientation of Garcia's p = 31 instance avoids 64-cycles. Precisely, there exist one of these four multipliers, a function $\\sigma:V\\to\\{t,g,r\\}$, and bijections $\\tau_x:\\{t,g,r\\}\\to\\{u,v,w\\}$ with $\\tau_x(\\sigma(x))=u$, such that the literal $H_{15}$ replacement has no simple cycle of length 64. The closed Lean theorem `result : ¬ claim` refutes this existence assertion. Every independent assignment of the remaining incident edges to $v,w$ is included.","title":"garcia 2026 erdos64 agl31 orientation","url":"https://the-omega-institute.github.io/trureturing-pages/research/garcia-2026-erdos64-agl31-orientation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61268aa709e32ce227307c65d8a712320480be420113fd5e1e411d75cacdd32d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garcia-fernandez-2026-open-circuit-integrability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"García Fernández, Paletta and Retore, arXiv:2607.02093v1,\n§3.1.2, “From the circuit to $\\vec{n}$”, state:\n\n> For the open-boundary case, similarly to the periodic setting \\cite{Paletta:2025sap}, we conjecture that any circuit in which each gate $U_{i,i+1}$ (constructed from an $\\check{R}$-matrix) appears exactly once per period to every nearest-neighbor pair of spins, and where each boundary gate is constructed from a $K$-matrix, is integrable.\n\nSection 5 restates the conjecture and says that an analytic proof remains\nopen. The bulk gate is $\\check R(\\kappa,-\\kappa)$, where\n$\\check R(u,v)=P R(u,v)$; the boundary gates are $K^R_1(\\kappa)$ and\n$\\widetilde K^L_L(\\kappa)=\\operatorname{tr}_a(K^L_a(\\kappa)\n\\check R_{La}(\\kappa,-\\kappa))$.\n\nThe formulation in [issue #13232](https://github.com/the-omega-institute/trureturing/issues/13232#issuecomment-5987481348)\nuses the explicit double-row product and states mutual commutation of its\ntransfer matrices as the hypothesis of the commutation conclusion.\nFor every length $L\\ge2$ and permutation of the $L+1$ gate labels, there\nare signed inhomogeneities $\\theta_i\\in\\{\\kappa,-\\kappa\\}$ and $c\\ne0$\nsuch that $M_\\pi=c\\,t(\\kappa;\\theta)$. Whenever the transfer matrices\n$t(u;\\theta)$ commute pairwise, $M_\\pi$ commutes with every $t(u;\\theta)$.\nThe hypotheses of the identity are $R(u,u)=g(u)P$ and\n$g(\\kappa)\\ne0$, $g(-\\kappa)\\ne0$.","title":"Every gate ordering in an open-boundary integrable quantum circuit","url":"https://the-omega-institute.github.io/trureturing-pages/research/garcia-fernandez-2026-open-circuit-integrability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06314ed9fe9226dc1cc194948ec0d697869fb34b0128b7b6d02300ddb27eb4d8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garcia-fernandez-2026-open-circuit-min-depth-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"García Fernández, Paletta and Retore (arXiv:2607.02093v1) build open-boundary\nintegrable quantum circuits for every set of sites carrying the inhomogeneity\n`-κ` and group them by the number `κ₋` of such sites. They conjecture:\n\n> **Conjecture 1** For odd $N$ and $0<\\kappa_-\\le \\frac{N-1}{2}$, the minimum\n> depth is given by $d=\\frac{1}{2}(N+3)-\\kappa_-.$\n>\n> **Conjecture 2** For even $N$ and $0<\\kappa_-\\le \\frac{N}{2}$, the minimum\n> depth is given by $d=\\frac{1}{2}(N+4)-\\kappa_-.$\n\nIssue #10379 fixes the readings: the circuits are those of the paper's\nTheorems 1 and 2, with `U_{j,j+1}` on the sites `j, j + 1`, `K₁^R` on site 1\nand `K̃^L_N` on site `N`, and the rightmost factor of the operator product\nacting first; the depth is the least number of layers into which the gates can\nbe placed so that any two gates sharing a site keep their time order; the\nminimum is taken over all sets of `κ₋` sites among `1, …, N`.","title":"The minimum depth of open integrable circuits","url":"https://the-omega-institute.github.io/trureturing-pages/research/garcia-fernandez-2026-open-circuit-min-depth-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af402d0f65fd74b84bb0a799f9207718be4624a6405eace8dc5e6ed6227c54b0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garcia-marco-2025-uniform-toeplitz-factorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ignacio García-Marco, Irene Márquez-Corbella and Daniel Seco, *On the minimum number of Toeplitz factors of a matrix*, [arXiv:2506.16432v1](https://arxiv.org/abs/2506.16432v1), Section 5.2, Conjecture 5.1, original archive `Toeplitz-GMS.tex`, lines 1170–1175 (`super-conjecture`), states:\n\n> There exists a constant C > 0 such that every matrix can be expressed as a product of an invertible matrix and C Toeplitz matrices. In particular, Toep_n ≤ Toep_n^Inv + C.\n\nThe source's introduction, equation (1), defines ordinary complex Toeplitz matrices by equality of adjacent southeast diagonal entries. Section 4 explicitly writes the left-multiplier constructions `M = P T_1 T_2` and `M = P_0 rref(M)`. The ambient matrices are complex and square, including singular matrices. The version was submitted 2025-06-19; the inspected arXiv history lists v1 only and no journal reference. The [literature note](../Library/Eigenstructure/garciamarco2025toeplitzfactors.md) gives the original definitions and neighboring questions.\n\nThe target F is the complete first assertion:\n\n$$\n\\exists C\\in\\mathbb N,\\quad 0 Let $n,d\\ge 2$. For all tuples of hermitian operators $X_1,\\dots,X_n$ on a Hilbert space,\n> $\\ker(H_{2d}(X_1,\\dots,X_n)-\\mu_{n,d}(X_1^{2d}+\\cdots+X_n^{2d}))=\\ker X_1\\cap\\cdots\\cap\\ker X_n$.\n\nThe settled assertion is exactly the bounded complex Hilbert-space statement\npreregistered in [#13735](https://github.com/the-omega-institute/trureturing/issues/13735).\nHere $H_k(X)$ is the ordered word sum, with each word $w$ weighted by\n$\\prod_i m_i(w)!/k!$. This is the reciprocal cardinality of the word's\nabelianization fibre. The constant $\\mu_{n,d}$ retains the source's $n=1$,\nodd-degree and even-degree branches.","title":"Garcia–Volčič Conjecture 4.5","url":"https://the-omega-institute.github.io/trureturing-pages/research/garcia-volcic-2025-nc-hunter-kernel-rigidity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/996803f8746ec8d0602e47ee1c0c5b2949cdc888d17a725fc11f768362cdcfac.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garg-2019-fibonacci-prefix-antipowers","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Swapnil Garg, *Antipowers in Uniform Morphic Words and the Fibonacci Word*,\narXiv:1907.10816v4, DMTCS 23(3), article 14, Conjecture 18, printed page 8:\n\n> Let $F_n$ be an even Fibonacci number. Then, there is an $(F_n-1)$-antipower with block length $\\frac{F_n}{2}+F_{n-1}$ that is a prefix of $\\mathbf f$.\n\nThe Fibonacci word is the fixed point starting with $0$ of $0\\mapsto01$,\n$1\\mapsto0$; $F_1=F_2=1$. A $k$-antipower consists of $k$ consecutive,\npairwise distinct blocks of equal length. [The source note](../Library/Words/garg2021antipowers.md)\nquotes the definitions from §3 and the conjecture. [Issue #14479](https://github.com/the-omega-institute/trureturing/issues/14479)\nfixes the universal statement GARG-1: every $n\\ge1$ with $F_n$ even,\n$K=F_n-1$, $L=F_n/2+F_{n-1}$, and the first $K$ blocks starting at $jL$.","title":"Garg's Fibonacci prefix antipower conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/garg-2019-fibonacci-prefix-antipowers/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2651d1938c6dc859b77da9bb54471a0ffc0da7487b218d672c1378230d96f4a6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:garrappa-gerhold-popolizio-simon-2025-kummer-modulus","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Garrappa, S. Gerhold, M. Popolizio and T. Simon,\n*On some inequalities for the two-parameter Mittag-Leffler function in the complex plane*,\nJ. Math. Anal. Appl. 551(1) (2025), 129588,\nDOI 10.1016/j.jmaa.2025.129588, arXiv:2410.11852v4, Remark 6:\n\n> In view of the preceding result and Remark 2, one may ask if for all α, β > 0 one has |M(α, α−β, z)| ≥ M(α, α−β, ℜz), z ∈ ℂ. (2.4) … In particular, (2.4) holds globally for α > 1 ≥ β > 0 … The question whether (2.4) holds globally for all α, β > 0 seems to be still open.\n\nRemark 2(a) identifies M as \"a Kummer confluent hypergeometric function\": M(a, b, z) = Σ_{n≥0} (a)_n z^n / ((b)_n n!), b ∉ {0, −1, −2, …}.\n\nThe verified locators are Remark 6 and Remark 2(a) of v4; a page number\nis not asserted from the supplied extract.\n\n### Formal reading\n\nNames without a prefix belong to `D5.S3.Analytic.KummerModulusLowerBound.Result`.\n\n| Source object | Lean object | Correspondence |\n| --- | --- | --- |\n| $(a)_n/((b)_n n!)$ | `D5.S3.Analytic.KummerRatio.Defs.kummerCoeff` | The frozen real coefficient, cast into ℂ for complex arguments. |\n| $M(a,b,z)$ | `kummerC` | Infinite sum of those coefficients times $z^n$. |\n| $M(a,b,x)$ for real $x$ | `D5.S3.Analytic.KummerRatio.Defs.kummer` | The original real series. |\n| The universal question | `claim` | All positive real $\\alpha,\\beta$, nonpole second parameter, and all complex $z$. |\n| Negative answer | `result` | Exactly `¬ claim`. |","title":"A counterexample to the universal Kummer modulus lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/garrappa-gerhold-popolizio-simon-2025-kummer-modulus/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38d9af818c4b82e63022e47e704da526fe19cb822b0e67987ab7f15e51bddfe8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gaunt-2019-bessel-tail-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. E. Gaunt, *Inequalities for some integrals involving modified Bessel\nfunctions*, Proc. Amer. Math. Soc. 147 (2019), 2937–2951,\nDOI 10.1090/proc/14433, arXiv:1806.00524v2, Conjecture 2.4,\nmanuscript page 7. The source states:\n\n> “Conjecture 2.4. Let n < 1. For ν ≥ 1 − n + √(2(1 − n)), define α_{ν,n} := ν + n − √((ν + n)² − 2ν − 1). Then, for all x > 0,\n> ∫_x^∞ K_{ν+n}(t)/t^ν dt > K_{ν+n−α_{ν,n}}(x)/x^ν, ν ≥ 1 − n + √(2(1 − n)). (2.17)\n> We have equality in (2.17) when n = 1, and this equality is valid for all ν ∈ ℝ. Let α′_{ν,n} := ν + n + √((ν + n)² − 2ν − 1), so that ν + n − α′_{ν,n} = −(ν + n − α_{ν,n}) and thus K_{ν+n−α′_{ν,n}}(x) = K_{ν+n−α_{ν,n}}(x). Then, the index ν + n − α_{ν,n} (and equivalently ν + n − α′_{ν,n}) in inequality (2.17) is best possible, in the sense that, for any β ∉ {α_{ν,n}, α′_{ν,n}}, either K_{ν+n−β}(x)/x^ν < K_{ν+n−α_{ν,n}}(x)/x^ν for all x > 0, or there exists a y > 0 such that ∫_y^∞ K_{ν+n}(t)/t^ν dt < K_{ν+n−β}(y)/y^ν.”\n\nThe quotation is the source statement. The Lean correspondence and its\nliteral definitions are given below.\n\n### Formal reading\n\nLean names have prefix `D5.S3.Analytic.GauntBesselTail.Result`.\n\n| Object | Lean declaration | Correspondence |\n| --- | --- | --- |\n| Bessel K | `D5.S3.Analytic.GammaGammaNBU.Result.besselK` | The existing hyperbolic integral, DLMF 10.32.9, on positive arguments. |\n| Two shifts | `alpha`, `alphaPrime` | Literal source square-root formulas. |\n| Weighted upper tail | `tail` | Real-power weight on t>x>0. |\n| Entire conjecture | `claim` | The conjunction of the strict bound, its either/or optimality, and the unrestricted real-order equality. |\n| Settlement | `result` | Exactly `claim`. |","title":"The sharp weighted Bessel K tail bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/gaunt-2019-bessel-tail-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c027523ee14be8276162ea302f459b2627d6f6212483d0c2996b335da5a7eaf3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gaussian-atemporality-entanglement","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Song et al., “Gaussian Atemporality: When Gaussian Quantum Correlations Imply Common Cause”, arXiv:2508.11804v2, Section III.D (the fourth subsection of Section III), p. 6, state:\n\n> From Fig. 3, Gaussian-atemporality appears to imply entanglement. The converse, on the other hand is not generally true.\n\nPreregistration [#19029](https://github.com/the-omega-institute/trureturing/issues/19029) supplies the source quotations, Tier 1 classification, binding covariance conventions and literature screen. The assertion concerns every physical two-mode Gaussian covariance matrix in standard form, in quadrature order $(q_A,p_A,q_B,p_B)$ and units $\\hbar=2$.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Eq. (2), $V_{AB}=\\begin{pmatrix}V_A&C\\\\C^T&V_B\\end{pmatrix}$ | `covariance` | Arbitrary real two-by-two blocks, indexed by `Fin 2`; the combined index is `Fin 2 ⊕ Fin 2`. |\n| Definition 1, “with locally diagonal covariance matrices” | `IsStandardForm` | Both local blocks satisfy `Matrix.IsDiag`. |\n| Spatial uncertainty inequality | `IsSpatiallyPhysical` | $V_{AB}+i(\\Omega\\oplus\\Omega)\\succeq0$, including Hermiticity. |\n| Eqs. (7)–(8) | `forwardX` | $T=C^TV_A^{-1}$, $N=V_B-C^TV_A^{-1}C$, $X_f=N+i\\Omega-iT\\Omega T^T$. |\n| Reverse temporal direction | `forwardX VB VA Cᵀ` | Exchange the local blocks and transpose $C$. |\n| Section III.B, “Gaussian-atemporal in both temporal direction” | `IsGaussianAtemporal` | Both directional matrices fail to be positive semidefinite. |\n| Section III.D, the two-mode Gaussian PPT criterion | `partialTransposeA`, `IsEntangled` | $\\Lambda=\\operatorname{diag}(1,-1,1,1)$ and failure of $\\Lambda V_{AB}\\Lambda+i(\\Omega\\oplus\\Omega)\\succeq0$. |\n| Section III.D, the quoted implication | `claim`, `result` | All three hypotheses imply the covariance NPT predicate. |\n\nThe symplectic form is the existing `D5.S3.Quantum.Thermal.PositiveOneModeWilliamson.physicalJ2`, namely $\\Omega=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}$. Real-to-complex conversion is `Matrix.map Complex.ofReal`; no additional conversion or reverse-direction alias is introduced. The inverse is Mathlib's total nonsingular matrix inverse. Physicality supplies the invertibility used by the proof. No assumption beyond the displayed source hypotheses is added to `claim`.","title":"Gaussian atemporality implies entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/research/gaussian-atemporality-entanglement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a1b24f994981eb8ff79056260662e5add1f720456a4273100f90a752b68b096.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gawron-ulas-2022-factorial-sign-conjecture-4-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gawron and Ulas, “Signs behaviour of sums of weighted numbers of partitions”,\narXiv:2202.05221v1, Conjecture 4.4, p. 11, asks for two eventual sign inequalities\nfor $U_m=\\{i!:1\\le i\\le m\\}$, $m\\ge2$. With $c_A(i,n)$ counting unordered\npartitions of $n$ into exactly $i$ positive parts from $A$, put\n$S_{A,k}(n)=\\sum_{i=0}^n(-1)^i i^k c_A(i,n)$.\n\nThe first assertion is $(-1)^{\\lfloor(n+1)/2\\rfloor}S_{U_m,k}(n)\\ge0$\nfor every sufficiently large $n$ when $0\\le k\\le m-2$.\nThe second assertion is $(-1)^nS_{U_m,k}(n)\\ge0$ for every sufficiently large\n$n$ when $k\\ge m-1$. Each threshold may depend on $m$ and $k$.\nThe settlement is **Refuted**, specifically for the v1 statement.\nThe preregistration is [#19687](https://github.com/the-omega-institute/trureturing/issues/19687).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Unordered positive-part partitions with exactly $i$ parts in $A$ | `c` filters `Nat.Partition n` by part membership and `Multiset.card` | Literal count; allowing zero in $A$ adds no partition |\n| Sum from $0$ through $n$ with weight $(-1)^i i^k$ | `S` uses `Finset.range (n+1)` and integer casts | Both endpoints; $0^0=1$ |\n| $U_m=\\{i!:1\\le i\\le m\\}$ | `U` uses the existential factorial membership condition | Literal set |\n| Every $m\\ge2$, every $k\\le m-2$, eventual floor-exponent inequality | First conjunct of `claim` | Natural division is the floor; no least-period assertion |\n| Every $m\\ge2$, every $k\\ge m-1$, eventual alternating inequality | Second conjunct of `claim` | Independent existential threshold for each pair |\n| Negation of the whole conjecture | `result : ¬ claim` | Contradicts the second conjunct at $m=3,k=2$ |","title":"gawron ulas 2022 factorial sign conjecture 4 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/gawron-ulas-2022-factorial-sign-conjecture-4-4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1928a1b5210e720f7237586b079e0bf4ab1e929c6813d9f5bc2a4db95e1f9f4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gayrard-2025-mixed-memory-converse-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Véronique Gayrard, *Mixed memories in Hopfield networks*, arXiv:2504.04879v2,\nSection 1.2.1, Conjecture 1.4, printed p. 6:\n\n> Given any $n\\in\\mathbb N$ odd, $\\xi^{(N)}(m)$ is an $n$-mixed memory of type $F$ if and only if $m\\in\\mathcal M_{n,F}$.\n\nThe source uses a doubly infinite jointly independent family of fair signs,\na smooth $F$ with $F'(x)>0$ for $x>0$, fixed odd $n$, a nondecreasing pattern\ncount $M(N)$, and deterministic coefficients in $[-1,1]$. Definition 1.1\nrequires exactly $n$ nonzero coefficients, binary configurations\n$\\xi_i(m)=\\operatorname{sign}(\\sum_{\\mu=1}^{M}\\xi_i^\\mu F'(m_\\mu)\n\\mathbf1_{m_\\mu\\ne0})$, almost-sure overlap limits $m_\\mu$ on that support,\nand zero overlap limits on available coordinates outside it. The sign at zero\nis zero (Section 2.2, printed (2.25)).\n\nThe set $\\mathcal M_{n,F}$ is defined by allowable ordered compositions:\nall nonfinal blocks are even and at least two, and the final block is positive\nand odd. With\n$\\alpha^{(a)}=2^{1-a}\\binom{a-1}{\\lfloor(a-1)/2\\rfloor}$ and\n$\\gamma^{(k)}=\\prod_{l=1}^k\\alpha^{(n_l)}$, only compositions satisfying\n$2F'(\\gamma^{(k)})>\\sum_{r>k}n_r F'(\\gamma^{(r)})$ for every nonfinal\nblock are retained. Their block vectors repeat $\\gamma^{(k)}$ on $n_k$\ncoordinates and are padded by zeros. All permutations and all coordinate\nsigns raised to exponent one for odd $F'$ and two otherwise are included.\n\nFor every positive block size $a$, the formalization reuses `D5.S3.Quantum.Entanglement.PrecessionSpinOneSeparableBound.c`, equal to the source's $\\alpha^{(a)}=2^{-a+1}\\binom{a-1}{\\lfloor(a-1)/2\\rfloor}$.","title":"An asymmetric mixed memory outside Gayrard's hierarchy","url":"https://the-omega-institute.github.io/trureturing-pages/research/gayrard-2025-mixed-memory-converse-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e8df1c9105fca3614580677a0f1c2d3c102b27c7abb02354ec117f4afb5d8f63.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gazeau-2026-problem-4-3-hilbert-realization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jean-Pierre Gazeau, *Bi-Oriented Interlaced Matrix Multiplication and an\nOperator Factorisation of the Quantum Harmonic Oscillator*, arXiv:2609.28511v1,\nProblem 4.3, asks:\n\n> Is D essentially self-adjoint, skew-adjoint, or normal with respect to a natural\n> Hilbert-space structure on L2(R) direct sum L2(R)?\n\nThe ordinary action on the full complex Schwartz core is\n$D_0(f,h)=(xf+i h',-i f'+xh)$. The fixed vector space is the product of the\nusual complex Lebesgue L2 spaces, including their almost-everywhere equivalence\nclasses. A candidate positive Hermitian metric must make that entire vector\nspace a complete Hilbert space.","title":"Gazeau Problem 4.3: no normal Hilbert realization","url":"https://the-omega-institute.github.io/trureturing-pages/research/gazeau-2026-problem-4-3-hilbert-realization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ecbcee3f08a98e685710efd5ffa48622545b243d6f68ed4a59f8a1108afc7f1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:geiger-kedem-2022-spin-one-entropy-phase-independence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Geiger and Z. M. Kedem, *Spin Entropy*, Entropy 24, 1292 (2022), arXiv:2111.11605v3, define the spin-entropy of a pure spin state as the Shannon entropy of the S_z probabilities plus the differential entropy of the polarization-angle density (Eq. (4)).\n- **Spin 1.** For the state (15), with amplitudes sin θ_α cos θ_β e^{iφ_x}, cos θ_α and sin θ_α sin θ_β e^{iφ_z}, Proposition 2 gives the spin-entropy (17). Its density is ρ_1(φ) = (1/2π)|sin θ_α cos θ_β e^{i(φ_x+2φ)} + cos θ_α e^{iφ} + sin θ_α sin θ_β e^{i(φ_z−2φ)}|².\n- **The conjecture.** \"Conjecture 1. The spin-entropy depends only on the variables that define the spin-entropy component along the z axis. In particular, the spin-entropy (17) of a state given by (15) can be simplified to\" its zero-phase form (18).\n\nThe verbatim statements are in [the literature note](../Library/Entropy/geigerkedem2022spinentropy.md).\n\nIssue [#17162](https://github.com/the-omega-institute/trureturing/issues/17162) fixes the reading: for all θ_α, θ_β ∈ [0, π/2] and φ_x, φ_z ∈ [0, 2π), Eq. (17) equals its value at φ_x = φ_z = 0. The conjecture is refuted.","title":"The Geiger–Kedem spin-1 spin-entropy depends on the relative phases","url":"https://the-omega-institute.github.io/trureturing-pages/research/geiger-kedem-2022-spin-one-entropy-phase-independence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2c02b7aa6269f56825d95177a71990025cd0c7e0c439cc45ee16d3d2dc39d57c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:geisser-2021-squares-prefix","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source identified by `geisser2021squares`, Section 3.3, Conjecture 1,\nprinted pages 25–26, states:\n\n> If ⟨a₁′,...,aₗ′⟩ is a solution with l ≥ 2, then ⟨a₁′,...,aₗ₋₁′⟩ is also a solution.\n\nFor a finite list of natural numbers, form each block as 4 followed by\nexactly that entry's number of nines, concatenate the blocks, and append 5.\nLet `encode` be the resulting ordinary decimal natural number, and let\n`GoodSquare n` mean that every actual decimal digit of `n²` belongs to\n{0,2,5}. The closed claim is exactly:\n\n```text\n∀ runs : List ℕ, 2 ≤ runs.length →\n GoodSquare (encode runs) → GoodSquare (encode runs.dropLast)\n```\n\nZero runs are included. Neither list length nor entries have an upper\nbound. Deletion removes the entire final 4/nines block and preserves the\nfinal 5. All three allowed digits need not occur.","title":"Prefix closure for squares with three digits","url":"https://the-omega-institute.github.io/trureturing-pages/research/geisser-2021-squares-prefix/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4aa031d417bc09838e584ca899ac4c3da9d7193bd48492452093c7c10b35d934.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gervacio-2026-vertex-identity-seidel-switches","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. V. Gervacio, *On identity Seidel switches*, arXiv:2601.04530v1,\nSection 6, Problem 6.1, page 13:\n\n> Characterize graphs $G$ for which every vertex is a vertex-ISS. Lemma 4.4 gives a necessary condition in terms of the minimum and maximum degree; can this be strengthened to a complete characterization?\n\nThe source's graphs are simple graphs on a nonempty finite vertex set.\nSwitching across $S$ complements exactly the pairs with one endpoint in\n$S$. An identity Seidel switch (ISS) means $S(G)\\cong G$; a vertex-ISS uses\n$S=\\{v\\}$. The complete answer is $|V(G)|=1$.","title":"Every vertex an identity Seidel switch","url":"https://the-omega-institute.github.io/trureturing-pages/research/gervacio-2026-vertex-identity-seidel-switches/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32aa152bd2105594480bdb7b55284fb3470a1ac14d3bd66c85a2a905451bb368.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gesmundo-lysikov-steffan-2022-bridge-graph-max-flow","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Gesmundo, V. Lysikov and V. Steffan, *Quantum max-flow in the bridge graph*,\narXiv:2212.09794v2, Conjecture 1.4 (also Conjecture 3.14), state:\n“Let $(a,b),(a',b')\\in U_w\\cup V_w\\cup W_w$. Then”\n$\\operatorname{QMaxFlow}=\\operatorname{QMinCut}=\\min\\{ab',a'b\\}$.\nConjecture 3.15 is the width-three case. The\n[literature note](../Library/QuantumBounds/gesmundo2025bridge.md) gives the\nverbatim source clauses and locators. [#14652](https://github.com/the-omega-institute/trureturing/issues/14652)\nrecords the complete target and literature scope.","title":"Quantum max-flow equals min-cut in the symmetric bridge regions","url":"https://the-omega-institute.github.io/trureturing-pages/research/gesmundo-lysikov-steffan-2022-bridge-graph-max-flow/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8b2b18c2c4ebb3d25bf59f4edf09b76eed0b2d0c99da21f974bf1247f067324.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ggps2025","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Garrappa, Gerhold, Popolizio and Simon, *On some inequalities for the\ntwo-parameter Mittag-Leffler function in the complex plane*, arXiv:2410.11852v4,\np. 18, Conjecture 25 (journal DOI 10.1016/j.jmaa.2025.129588):\n\n> For every α ∈ (1, 2), one has\n> The inequality (1.4) holds globally ⇐⇒ β ∈ [α − 1, α] ⇐⇒ 1/Eα,β(x) is CM on (0, ∞).\n\nHere $\\beta\\ge0$, and (1.4) is\n$|E_{\\alpha,\\beta}(z)|\\ge E_{\\alpha,\\beta}(\\operatorname{Re}z)$ for every\n$z\\in\\mathbb C$. Equation (1.1), p. 1, defines\n$E_{\\alpha,\\beta}(z)=\\sum_{k=0}^{\\infty}z^k/\\Gamma(\\alpha k+\\beta)$ for\n$\\alpha>0$ and every real $\\beta$. The reciprocal Gamma value at a pole is\nzero. Complete monotonicity means a smooth positive-valued function on\n$(0,\\infty)$ whose signed derivatives of all natural orders are nonnegative.\n\n### Statement echo\n\nUnqualified Lean names below belong to\n`D5.S3.Analytic.Asymptotics.MittagLefflerModulusRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (1.1), all positive $\\alpha$, real $\\beta$, complex $z$ | `IsSourceSeries` | `HasSum` includes convergence; Gamma and ordinary complex division are Mathlib's. |\n| $1/E_{\\alpha,\\beta}(x)$ | `reciprocal` | The real argument is embedded in $\\mathbb C$ and the real part reads the real series value. |\n| Global inequality | First equivalence in `claim` | Every complex argument is quantified; its real part is embedded without rescaling. |\n| $1<\\alpha<2$, $\\beta\\ge0$, $\\alpha-1\\le\\beta\\le\\alpha$ | Parameter binders and interval in `claim` | Both endpoints and all domain restrictions are literal. |\n| Positive smooth reciprocal with signed derivatives | Second equivalence in `claim` | Existing `D5.S3.Analytic.ExpRatioCM.Defs.CompletelyMonotoneOn` is used, with a separate positivity conjunct. |\n| Existence of the source family | Leading existential in `claim` | Explicit convergence/existence requirement; it does not weaken the universal conjecture. |\n| Negative answer | `result` | Exactly the negation of the complete claim. |","title":"The reverse Mittag-Leffler modulus characterization is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/ggps2025/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e633f8882f2b8691692f46f1ffde2e0dfd60d38d3e9bf6d5ad77e9b595116c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ghanbari-2022-sombor-energy-integer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Ghanbari, *On the Sombor characteristic polynomial and Sombor energy of a graph*,\nComputational and Applied Mathematics 41 (2022), article 242, Conjecture 3.8:\n\n> There is no graph with integer-valued Sombor energy.\n\nThe paper studies finite simple graphs. For a graph $G$, the Sombor matrix has entry\n$\\sqrt{d_i^2+d_j^2}$ on an adjacent pair and zero otherwise, and the Sombor energy is the\nsum of the absolute values of its eigenvalues.","title":"Ghanbari's integer Sombor energy conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/ghanbari-2022-sombor-energy-integer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/180e6b6b7b424facbb5aa402884cc994d630e271ddb039dc71375136ecaa3cef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ghosal-das-banerjee-2021-critical-concurrence-universal-teleportation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Ghosal, D. Das and S. Banerjee, *Characterizing qubit channels in the context of quantum teleportation*, Phys. Rev. A 103, 052422 (2021), arXiv:2102.02054v2, send one half of |Ψa⟩ = √a|00⟩ + √(1 − a)|11⟩ (1/2 < a < 1, concurrence C = 2√(a(1 − a))) through a qubit channel Λ and ask when the final state (id ⊗ Λ)(|Ψa⟩⟨Ψa|) is useful for universal quantum teleportation (UQT).\n- **The criterion.** A two-qubit state is useful for UQT if and only if, in its canonical form under product unitaries, |t₁₁| = |t₂₂| = |t₃₃| > 1/3 (§II.B–§II.C).\n- **Unital and non-unital channels.** Unital channels need C > 1/2. For the two-Kraus non-unital family of Eq. (50), the final state is useful for UQT exactly when C > √((2/3)(4 + √3)(1 − (4 + √3)/6)) ≈ 0.41.\n- **The question.** Conjecture 1 states that the final state is useful for UQT for some non-unital channels if and only if C_Critical < C < 1, \"where 0 < C_Critical ≤ 0.41. We could not determine the precise value of C_Critical; further investigations are needed.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/ghosal2021qubitchannelsteleportation.md).\n\nIssue [#16624](https://github.com/the-omega-institute/trureturing/issues/16624) fixes the reading: \"useful for UQT\" means that some product of unitaries brings the state to correlation matrix −tI₃ with t > 1/3, and channels are finite Kraus families. The question settled is the precise value of C_Critical: C_Critical = C_* = √(5 − 2√3)/3 = 0.4131045583…, the exact value of the source's radical. The source's printed bound \"C_Critical ≤ 0.41\" is the rounding of that radical; taken literally as the rational 41/100 it would be false, so the settlement answers the precise-value question and does not assert the printed inequality. Conjecture 1's further clause that only a strict subset of the non-unital channels succeeds is not part of the formal statement.","title":"The exact critical concurrence for universal teleportation through qubit channels","url":"https://the-omega-institute.github.io/trureturing-pages/research/ghosal-das-banerjee-2021-critical-concurrence-universal-teleportation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ed57a7416b0657da15f6086034a2df93495b8a9de377d832f5f1c7540d624c85.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ghosh-birbonshi-ojha-2026-weighted-shift-root-uniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 2, Open question 1 of arXiv:2608.17486v1 asks:\n\n> For $s,t>0$ and $0 While our initial investigation with $N\\leq 10$ indicates that $W_{(N,K^\\prime,K)}$ remains unchanged as long $\\Delta = |K-K^{\\prime}|$ is constant for $N\\neq 4N^\\prime$ with $N^\\prime=1,2,\\cdots$, no such trend is observed in the case of $N\\neq 4N^\\prime$. However, concrete conclusion requires extensive numerical analysis with large $N$, which is challenging.\n\nHere:\n- **Hamiltonians:** $H_{(N,K)}=\\sum_iH_{(i,K)}$ is the stabilizer Hamiltonian of the circulant $K$-regular graph on $N$ sites, $H_{(i,K)}=[\\bigotimes_{j\\le K/2}Z_{i-j}]X_i[\\bigotimes_{j\\le K/2}Z_{i+j}]$.\n- **Battery and charger:** the battery $H_B=H_{(N,K')}$ starts in a ground state $\\rho(0)$, and the charger $H_C=H_{(N,K)}$ drives $\\rho(t)=e^{-iH_Ct}\\rho(0)e^{iH_Ct}$.\n- **Work:** $W_{(N,K',K)}(t)=\\mathrm{Tr}[(\\rho(t)-\\rho(0))H_B]$.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/ghoshkonarfrancicapalsende2025stabilizer.md).\n\nIssue [#20091](https://github.com/the-omega-institute/trureturing/issues/20091) fixes the reading: for every $N>2$ with $4\\nmid N$ and all even interior degrees with $K'\\ne K$, $L'\\ne L$ and $|K-K'|=|L-L'|$, every pair of ground states gives $W_{(N,K',K)}(t)=W_{(N,L',L)}(t)$ for all $t\\ge0$. Excluding the endpoint degrees $0$ and $K_{\\max}$ makes the claim weaker than the literal one. It is refuted.","title":"The work stored in a stabilizer battery is not a function of the degree difference","url":"https://the-omega-institute.github.io/trureturing-pages/research/ghosh-konar-francica-pal-sen-de-2025-stabilizer-battery-degree-difference-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c0ecf7ca463b26458787d0d7f341378df2ecb0c56916f3e5cac8e445cdb052e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ghosh-ramachandran-2026-capacity-one-decay-rate","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Ghosh and K. Ramachandran, *A note on the Erdős minimal area problem*,\narXiv:2604.03036v3, beginning of Section 4:\n\n> This shows that while $\\vartheta(K)=0$ may hold for all compacts $K$ with $\\mathrm{cap}(K)=1$, the decay rate of $A_n(K)$ is not universal. It will be interesting to determine whether $A_n(K)$ can only decay like $\\frac1n$ or $\\frac1{\\log n}$, or whether other rates are possible.\n\nFor compact $K\\subseteq\\mathbb C$, let $P_n(K)$ consist of the monic\ndegree-$n$ polynomials whose zeros all lie in $K$. Write\n$\\Lambda_p=\\{z:|p(z)|<1\\}$ and\n$A_n(K)=\\inf_{p\\in P_n(K)}m(\\Lambda_p)$, where $m$ is planar Lebesgue\nmeasure. The paper defines capacity as logarithmic capacity,\nequivalently the transfinite diameter.\n\nThe Lean `claim` says that for every nonempty compact $K$ of transfinite\ndiameter one, the real-valued minimal-area sequence is either\n$\\Theta(1/n)$ or $\\Theta(1/\\log n)$ as $n\\to\\infty$.\nThus `result : ¬ claim` refutes the proposed two-rate restriction.","title":"Ghosh–Ramachandran's capacity-one area decay question","url":"https://the-omega-institute.github.io/trureturing-pages/research/ghosh-ramachandran-2026-capacity-one-decay-rate/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbb1bbcc285070c485461dbd7f9eee792ef31c5ab22479638998406035eda72a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ghosh-ramachandran-2026-fekete-lemniscate-area","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Ghosh and K. Ramachandran, *A note on the Erdős minimal area problem*,\narXiv:2604.03036v3, immediately after Proposition 3.6:\n\n> While the above proposition proves that Fekete polynomials are not area minimizers in general, it would be of interest to know if they are minimzers when the set $K$ is connected. We leave this as an open question.\n\nIn Section 1, $P_n(K)$ consists of monic degree-$n$ polynomials whose\nzeros all lie in the compact set $K\\subseteq\\mathbb C$.\nWrite $\\Lambda_p=\\{z:|p(z)|<1\\}$ and\n$A_n(K)=\\inf_{p\\in P_n(K)}m(\\Lambda_p)$, with $m$ planar Lebesgue measure.\nA Fekete tuple $a=(a_1,\\ldots,a_n)$ maximizes\n$\\prod_{i (Cone of initial data for variable coefficients.) Proposition 3.4 gives a matrix condition on v₀ for log-concavity of a general P-recursive sequence. Characterize the geometry of the cone C∞ = {v₀ : Lⁱ(aₙ) ≥ 0 ∀ n, i} for d ≥ 3. Is it convex? Simply connected?\n\nThe convexity question is GIA-1 in preregistration\n[#18698](https://github.com/the-omega-institute/trureturing/issues/18698), a tier-1\npublished question. The [Library note](../Library/Certificates/giacomelli2026logconcavity.md)\ncontains the source's recurrence and operator definitions.\n\n### Statement echo\n\nThe Lean namespace is `D5.S0.Certificates.GiacomelliLogConcaveConeNonconvex`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (3): $a_{n+1}=\\sum_{k=0}^{d-1}(p_kn+q_k)a_{n-k}$ for $n\\ge d-1$ | `seq` | Literal recurrence, with argument $m=n+1$, increasing-order initial coordinates, and the natural subtraction performed before the real cast. |\n| Definition 2.1: $L(a)_k=a_k^2-a_{k+1}a_{k-1}$ | `logConcaveOp` | Literal expression; natural subtraction is truncated. |\n| $C_\\infty=\\{v_0:L^i(a_n)\\ge0\\ \\forall n,i\\}$ | `coneInf` | Every $i,n\\in\\mathbb N$ with $i\\le n$; these are the interior indices where the complete dependence window exists. |\n| Infinitely log-concave initial data | Iteration zero in `coneInf` | Requires original nonnegative terms. Both witnesses also satisfy the convention starting at $i=1$; the midpoint fails at $i=1$. |\n| “for d ≥ 3. Is it convex?” | `claim` | Every natural $d\\ge3$ and every pair of real coefficient vectors, using Mathlib's `Convex`. |\n| Negative answer to convexity | `result` | Negates the complete affirmative assertion; $d=3$ supplies the counterexample. |\n| Characterization and simple connectedness | Separate open items in Triage | Neither is included in the settled assertion. |\n\nThere is no additional coefficient hypothesis. The binding definitions are\nliteral source expressions with the index convention above.","title":"Giacomelli's cone of infinitely log-concave initial data","url":"https://the-omega-institute.github.io/trureturing-pages/research/giacomelli-2026-log-concave-cone-convexity-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1805c94bb6b370c75dd3bbc659a0e2add42e1defadc2178408f076e8bc0f6bfa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gibilisco-isola-2004-log-geometry-scalar-curvature","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gibilisco and Isola, *On the monotonicity of scalar curvature in classical and quantum information geometry*, arXiv:math-ph/0407007v2, Conjecture 4.1, printed page 11:\n\n> Suppose $n>2$. For the function $\\mathrm{Scal}_p(\\cdot):\\mathcal P^1_n\\to\\mathbb R$ one has the following properties: if $p\\in(1,2)$ then $\\mathrm{Scal}_p$ is strictly Schur-decreasing; if $p\\in(2,+\\infty]$ then $\\mathrm{Scal}_p$ is strictly Schur-increasing.\n\nThis settlement proves the $p=\\infty$ endpoint only. The positive simplex is $\\mathcal P^1_n=\\{x\\in\\mathbb R^n: x_i>0,\\ \\sum_i x_i=1\\}$. The source says $x$ is more mixed than $y$ when the decreasingly sorted initial sums of $x$ are at most those of $y$, with equal total sums. Strict Schur increase means weak curvature increase for all such pairs and strict increase unless $x$ is a permutation of $y$.\n\nDefinitions 4.3–4.4 define the $\\alpha$-geometry as the Euclidean pull-back under $A_p(x)=p x^{1/p}$ for finite nonzero $p$ and $A_\\infty(x)=\\log x$, and $\\mathrm{Scal}_p$ as its scalar curvature. The literal logarithmic pull-back, Christoffel symbols, Ricci contraction and decreasing-sort majorization are the conventions of [#15145](https://github.com/the-omega-institute/trureturing/issues/15145).","title":"Logarithmic scalar curvature is strictly Schur-increasing","url":"https://the-omega-institute.github.io/trureturing-pages/research/gibilisco-isola-2004-log-geometry-scalar-curvature/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/629a2a6012560dd4fdba7895ed013f25ea85307cf6ea5b8ec76010a69c93dd92.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gibilisco-isola-2004-quantum-cube-root-scalar-curvature","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gibilisco and Isola, *On the monotonicity of scalar curvature in classical and quantum information geometry*, arXiv:math-ph/0407007v2, Conjecture 4.2, printed page 11:\n\n> Suppose $n\\geq2$. For the function $\\mathrm{Scal}_p(\\cdot):\\mathcal D^1_n\\to\\mathbb R$ one has the following properties: if $p\\in(1,2)$ then $\\mathrm{Scal}_p$ is strictly Schur-decreasing; if $p\\in(2,+\\infty]$ then $\\mathrm{Scal}_p$ is strictly Schur-increasing.\n\nThis settlement proves the quantum $p=3$ slice, equivalently $\\alpha=1/3$, in every dimension $n\\ge2$. The faithful density states are positive definite complex Hermitian matrices of trace one. The metric is the Euclidean pull-back of $A\\mapsto3A^{1/3}$, with real trace inner product. Scalar curvature is defined by coordinate Christoffel symbols and Ricci contraction, rather than by the spectral formula proved below. The source's order calls $x$ more mixed than $y$ when the decreasingly sorted prefix sums of $x$ are at most those of $y$, with equal total sums.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every dimension $n\\ge2$ | `claim` quantifies natural `n` with hypothesis `2 ≤ n`. | Exact. |\n| Faithful complex density states | `densityStates` requires `PosDef` and trace one for both matrices; positive definiteness supplies Hermiticity. | Exact. |\n| Cube-root pull-back geometry | `HermitianCubeDerivative.alphaEmbed3`, `HermitianDensityCoordinates.rho`, and `metric` give the literal derivative pull-back. `scal3` contracts coordinate curvature at `chartPoint`. | Exact; the fixed trace-one chart is proved equivalent to the Hermitian trace-one space. |\n| Eigenvalue majorization | `eig` selects the Hermitian eigenvalue vector; `PositiveSimplexGeometryAndMixing.moreMixed` is the existing decreasing-sort owner. | Exact and independent of permutation order. |\n| Schur increase | `result` concludes `scal3 B ≤ scal3 A` when `eig A` is more mixed than `eig B`. | Exact source convention. |\n| Strictness outside permutations | The second conjunct concludes `scal3 B < scal3 A` unless the eigenvalue vectors differ by a permutation. | Exact. |\n| Other values of $p$ and the decreasing clause | No corresponding conclusion in `claim`. | Open; outside this slice. |","title":"Quantum cube-root scalar curvature is strictly Schur-increasing","url":"https://the-omega-institute.github.io/trureturing-pages/research/gibilisco-isola-2004-quantum-cube-root-scalar-curvature/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/969ef8af582a28ca5d12dbcbff2b2f3349c3835df226d37a5343f50cef6d750c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gil-liang-odetola-weiner-antidiagonal-traffic","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 7.4 of arXiv:2609.01562v1 states:\n\n> For every n ≥ 496 and every obstruction B on the antidiagonal x+y=n,\n> the maximum of f_B is attained at (1,1) and (n-1,n-1).\n\nHere f_B counts monotone north-east paths from (0,0) to (n,n) avoiding B\nand passing through a grid point other than those two endpoints. The\nobstruction ranges over all (a,n-a), 0≤a≤n. Existing preregistration:\nhttps://github.com/the-omega-institute/trureturing/issues/8249.\n\nThe formal arithmetic proposition is exactly\n\n```text\nforall n a : Nat, 496 <= n -> 1 <= a -> 2*a < n ->\n n*(n-2*a+1)/(n-a)*choose(n,a)*choose(n-2,a-1)\n /choose(2*n-2,n-1) < 1\n```\n\nThe divisions in this expression are rational. The domain includes a=1\nand the odd near-central pair (497,248); it is not natural `a $\\sigma_{ABC} = 3 - (I_1 + I_2 + I_3) I_4$ and numerical results suggest\n> that it is an EM. ... For the rest of the paper I will assume that\n> $\\sigma_{ABC}$ is an EM.\n\nFor $\\psi=\\sum t_{ijk}|ijk\\rangle$, the polynomial invariants are\n$P_{\\sigma,\\tau}=\\sum t_{i_1j_1k_1}\\cdots t_{i_nj_nk_n}\\,\n\\bar t_{i_1j_{\\sigma(1)}k_{\\tau(1)}}\\cdots\\bar t_{i_nj_{\\sigma(n)}k_{\\tau(n)}}$\n(Eq. (8)), with $I_1=P_{e,(12)}$, $I_2=P_{(12),e}$, $I_3=P_{(12),(12)}$ (the\nthree one-qubit purities) and the Kempe invariant $I_4=P_{(123),(132)}$. An\nentanglement monotone (EM) in the paper's sense satisfies\n$E(\\psi)\\ge\\sum_kp_kE(A_k\\psi/\\sqrt{p_k})$ for every complete instrument\n$\\{A_k\\}$ on one party, $p_k=\\|A_k\\psi\\|^2$.","title":"Gingrich's proposed three-qubit monotone built from the Kempe invariant","url":"https://the-omega-institute.github.io/trureturing-pages/research/gingrich-2002-sigma-abc-monotone/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0a63e65126ffa75255132ff8dd67ab82d5153263015c82a816623a5bb3e3610.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girard-2017-binegativity-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. W. Girard and G. Gour, *The binegativity of two qubits*, arXiv:1701.02724v3 (2017), Eq. (8):\n\n> Based on numerical evidence, we conjecture that the binegativity is bounded below by $\\frac{\\nu}{2} \\left(1+\\frac{c_\\nu}{\\sqrt{(1-c_\\nu)^2+c_\\nu^2}}\\right) \\leq N_2(\\sigma)$ where we define $c_\\nu=\\sqrt{2\\nu(\\nu+1)}-\\nu$.\n\nHere $\\nu = N(\\sigma) = 2\\operatorname{Tr}[(\\sigma^\\Gamma)_-]$ and $N_2(\\sigma) = \\operatorname{Tr}[(\\sigma^\\Gamma)_-] + 2\\operatorname{Tr}[(((\\sigma^\\Gamma)_-)^\\Gamma)_-]$, with $X_-$ the negative component of a self-adjoint $X$. The verbatim statements are in [the literature note](../Library/QuantumStates/girard2017binegativity.md).\n\nIssue [#18490](https://github.com/the-omega-institute/trureturing/issues/18490) fixes the reading: for every two-qubit density matrix σ, with the partial transpose on the second qubit, the inequality holds. This is proved.","title":"The two-qubit binegativity lower bound at fixed negativity","url":"https://the-omega-institute.github.io/trureturing-pages/research/girard-2017-binegativity-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d07d0b86dd9e5ee2945dc9e1888f00a6f8348273c9aabe01310b338d0278b280.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girard-2017-binegativity-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. W. Girard and G. Gour, *The binegativity of two qubits*,\narXiv:1701.02724v3, introduction, p. 1:\n\n> That is, we conjecture that for any two-qubit state $\\sigma$ it holds that\n> $N_2(\\mathcal{E}(\\sigma))\\leq N_2(\\sigma)$ for any LOCC (or PPT) channel\n> $\\mathcal{E}$ that outputs states of two qubits.\n\nHere $N_2(\\sigma)=\\operatorname{Tr}[(\\sigma^\\Gamma)_-]+\n2\\operatorname{Tr}[(((\\sigma^\\Gamma)_-)^\\Gamma)_-]$, with partial\ntransposition on Bob and the positive and negative components of\nself-adjoint matrices. S. Sazim and N. Awasthi, *Binegativity of two qubits\nunder noise*, arXiv:1711.03717v2, introduction, p. 1, restate:\n\n> On the basis of numerical evidence, it is conjectured that the\n> binegativity behaves monotonically under both LOCC and PPT channels [15].\n\nTheir abstract also states: \"Our study supports the conjecture that the\nbinegativity is a monotone.\"","title":"Binegativity monotonicity for two qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/girard-2017-binegativity-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9f5110656d84216a9a15960baad36282ffa81704dbbf177d712d681f03d0a56.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girard-2017-binegativity-upper-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. W. Girard, G. Gour, *The binegativity of two qubits*, arXiv:1701.02724v3\n(2017), Eq. (9):\n\n> In general, based on numerical evidence from randomly generated states,\n> we conjecture that the the binegativity of two-qubit states is bounded by\n> $\\nu\\frac{(c+\\nu)(\\nu+1)}{(c+\\nu)^2+2c(1-c)}\\leq N_2(\\sigma)\\leq\n> \\frac{\\nu}{2}\\frac{(c+\\nu)^2}{c^2+\\nu^2}$ for all states $\\sigma$ with\n> fixed negativity $N(\\sigma)=\\nu$ and concurrence $C(\\sigma)=c$.\n\nHere $N(\\sigma)=2\\operatorname{Tr}[(\\sigma^\\Gamma)_-]$ and\n$N_2(\\sigma)=\\operatorname{Tr}[(\\sigma^\\Gamma)_-]+2\\operatorname{Tr}[\n(((\\sigma^\\Gamma)_-)^\\Gamma)_-]$, with $X_\\pm$ the positive and negative\ncomponents of a self-adjoint $X$ and $C$ the two-qubit concurrence. The\ntarget is the upper inequality; it is a preprint conjecture (no journal\nversion is listed).","title":"The upper binegativity bound for two qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/girard-2017-binegativity-upper-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74fdcd0f1968c6cdf8e7e4381eb987346bd07fbb7b1a43f7a3358bda96d8e402.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girard-plagne-2026-box-davenport-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Girard and Plagne, *The Davenport constant of low-dimensional boxes*,\narXiv:2610.16562v1, Section 2, printed page 4, ask:\n\n> It appears tempting to conjecture that for every integer d ≥ 1, the ratio D^(d+1)(⟦−m, m⟧^d)/m^d has a limit when m tends to infinity. This is clear when d = 1, and it follows from Theorems 1 and 5 when d ∈ {2, 3}. However, this problem remains open when d ≥ 4.\n\nHere $D^{(d+1)}$ is the maximal length of a minimal zero-sum sequence\nin $\\llbracket-m,m\\rrbracket^d$ with exactly $d+1$ distinct support vectors.\nThe result is **Proved** under the literal conventions of preregistration\n[#17348](https://github.com/the-omega-institute/trureturing/issues/17348).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Integer box | The coordinatewise absolute-value condition in `CofactorKernel.lengths` | Every coordinate is bounded by the natural radius |\n| Minimal zero-sum sequence | `CofactorKernel.IsMinimalZeroSum` | Nonzero sequence, zero total, no nonempty proper zero-sum subsequence |\n| Support of size $d+1$ | Support cardinality in `CofactorKernel.lengths` | Exactly $d+1$, not at most |\n| Maximal length | `GirardPlagneBoxLimit.D` | Natural supremum of the literal sum-of-multiplicities length set |\n| Every positive dimension | `GirardPlagneBoxLimit.claim` and `result` | Every natural $d\\ge1$ |\n| A limit as the radius tends to infinity | `GirardPlagneBoxLimit.girard_plagne_box_limit` | Natural radii, real ratios, limit $\\Delta_d$ |\n| Existence of the maximum | The nonemptiness and boundedness conjuncts of `claim` | Every $m\\ge1$; makes the supremum the intended maximum |\n\nThe box and total are literal expressions; proofs use `Finsupp.degree`\nfor length. Their correspondence to the preregistered formulas is definitional.\nThe auxiliary constructions `augment`, `cofactor`, `fromWeights`,\n`signMatrix` and `delta` are repository constructions. In particular,\n$\\Delta_d$ is the finite maximum of $|\\det T(V)|$ over Boolean sign matrices,\nwhere $T$ adds a top row of ones.","title":"The fixed-support Davenport limit of integer boxes","url":"https://the-omega-institute.github.io/trureturing-pages/research/girard-plagne-2026-box-davenport-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/34f83ffe6d918fdef8e0d1f06bd25ffa7cbb02116414a92bf4ae8686ee27cdbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girard-plavala-sikora-2021-jordan-compatibility-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Girard, M. Plávala and J. Sikora, *Jordan products of quantum channels and their compatibility*, Nat. Commun. 12, 2129 (2021), arXiv:2009.03279v1, study when two channels can be implemented jointly.\n- **Jordan compatibility.** For a Hermitian A on X ⊗ X_1 ⊗ X_2 with Tr_{X_1}(A) = Tr_{X_2}(A) = J(1_{L(X)}), the generalized Jordan product Φ_1 ⊙_A Φ_2 has Choi matrix (1 ⊗ Φ_1 ⊗ Φ_2)(A) (Definition 25). The channels are Jordan compatible if this product is completely positive for some such A (Definition 27). Jordan compatibility implies compatibility (Proposition 26), and for invertible channels the converse holds (Theorem 30).\n- **The conjecture.** After Theorem 30: \"the question of whether compatibility of channels is equivalent to Jordan compatibility remains open. We conjecture that such a pair of channels does not exists.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/girardplavalasikora2021jordan.md).\n\nIssue [#17171](https://github.com/the-omega-institute/trureturing/issues/17171) fixes the reading: for every d, every compatible pair of channels on M_d is Jordan compatible. The conjecture is refuted at d = 4.","title":"Compatible quantum channels need not be Jordan compatible","url":"https://the-omega-institute.github.io/trureturing-pages/research/girard-plavala-sikora-2021-jordan-compatibility-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a285d88abf8be25ca58fe2092bc55e20baf4ceb7de11f53738af7419896c7188.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:girling-cirstoiu-jennings-2022-unital-block-overlap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Girling, C. Cîrstoiu and D. Jennings, *Estimation of correlations and\nnon-separability in quantum channels via unitarity benchmarking*,\narXiv:2104.04352v3, *Physical Review Research* 4, 023041 (2022),\nAppendix C.5 after Lemma C.3, state:\n\n> However, we conjecture a stronger result that for any two quantum channels E₁, E₂ that ⟨T₁, T₂⟩ ≥ −1.\n\nThey also state:\n\n> Since it is not essential to our work we leave it as an open problem.\n\nSection II.C chooses a Hilbert-Schmidt orthonormal operator basis with\n$X_0=I/\\sqrt d$. Its positive-index Liouville entries are\n$T_{ij}=\\operatorname{tr}(X_i^\\dagger\\Phi(X_j))$.\nThe overlap is $\\operatorname{tr}(T_\\Phi^\\dagger T_\\Psi)$.\nThe [Library note](../Library/QuantumChannels/girlingcirstoiujennings2022unitarity.md)\ngives the literal source and locators.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Any two quantum channels | `claim` quantifies over `d`, `Phi`, `Psi`; both satisfy the frozen `CoPRelativeQuantumnessRefutation.IsCPTP` | Complete positivity and trace preservation on all complex matrices |\n| Orthonormal operator basis | `IsLiouvilleBasis` | All $d^2$ inner products, and normalized identity at zero |\n| Positive-index unital block | `unitalBlock`, restricted in `overlap` | Both index values are nonzero |\n| Hilbert-Schmidt overlap | `overlap` | Sum of conjugate products; `claim` compares its real part |\n| Lower bound $-1$ | `claim` | Universal implication, with all channel and basis hypotheses |\n| Refutation | `result` | Negation of `claim`, using one Hermitian qutrit basis |\n\nThe natural dimension and proof `h0` encode a nonempty finite-dimensional\nsystem. Defining entries at every index does not add zero-index terms to\nthe overlap. The real-part comparison agrees with the paper on Hermitian\noperator bases. Quantifying over complex orthonormal bases adds no\nrestriction on the explicit Hermitian witness.","title":"The unital-block overlap lower bound fails for qutrit channels","url":"https://the-omega-institute.github.io/trureturing-pages/research/girling-cirstoiu-jennings-2022-unital-block-overlap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72f9b87da857403fb8a6489ff6216baeb28f5a1a8bf3e57f900ea2166a6b30f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:giuffrida-squartini-grunwald-garlaschelli-2023-ensemble-equivalence-converse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Giuffrida, T. Squartini, P. Grünwald and D. Garlaschelli, *Description length of canonical and microcanonical models*, arXiv:2307.05645v4, Physical Review Research 7, 043057, Section IV.C, following (71):\n\n> i.e., whenever ensemble equivalence holds, the corresponding microcanonical and canonical models are equivalent on the model level. The reverse implication, verified here in the specific case of one-sided local constraints, will be investigated in future work.\n\nIssue [#18580](https://github.com/the-omega-institute/trureturing/issues/18580) preregisters its refutation. The source and equations are recorded in the [Library note](../Library/StatisticalMechanics/giuffrida2025descriptionlength.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Binary $n\\times m$ matrices, (12) | $\\mathrm{Fin}(n)\\to\\mathrm{Fin}(n)\\to\\mathrm{Bool}$ | Specialize $m=n$; a counterexample in this subclass refutes the general implication. |\n| Scalar statistic and fiber cardinality, (15) | `omega` | Count every binary matrix with the observed statistic. |\n| Microcanonical uniform law, (16) | `pMic` | Reciprocal fiber size on the fiber, zero off it. |\n| Canonical exponential law, (20), and matching, (21) | Exponential expression in `IsMatched` | Sum over every binary matrix; the parameter is real and finite. |\n| Matched measure divergence, (22), and equivalence scale, (35) | `measureDiv` and the conclusion of `claim` | Matching holds eventually; divergence is divided by $n$. |\n| Maximized likelihoods, (5), (17) | `lCan`, `lMic` | The real supremum includes limiting endpoint likelihoods. |\n| Normalized maximum likelihood, (8) | `nmlMic`, `nmlCan` | The literal likelihood-over-sum expressions. |\n| Model divergence and its vanishing rate, (69)–(70) | `modelDiv` and the premise of `claim` | The normalization scale is $n$, not $n^2$. |\n| Reverse implication after (71) | `result : ¬ claim` | Universal statistics and observation sequences, restricted to eventual finite matching. |\n\nThe exponential formula is literal in the public definitions. Private proofs use the existing `gibbs` and `meanEnergy` constructions; private consumed bridges relate the literal NML quotients to the existing `normalized` construction. Restricting the source assertion to square matrices, one scalar statistic and eventually matched observations weakens it, so refuting this restriction refutes the general converse.","title":"Model-level ensemble equivalence does not imply measure-level equivalence","url":"https://the-omega-institute.github.io/trureturing-pages/research/giuffrida-squartini-grunwald-garlaschelli-2023-ensemble-equivalence-converse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b6782ba603eac1c1e83dc9c74f22b5cb4c9345c4ca4f5febddd02753d845574.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:glab-2025-achievement-residue-converse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Szymon Głąb and Franciszek Prus-Wiśniowski, *Achievement sets – current results and open problems*, arXiv:2512.17285v1, Section 2, Problem 9, PDF p.6. The complete source question, transcribed in preregistration [#23055](https://github.com/the-omega-institute/trureturing/issues/23055), is:\n\n> Let (k₁,…,kₘ;1/n) be a multigeometric series such that kᵢ∈ℕ, Σ={σ₀<⋯<σₙ₋₁} has n elements, none of k₁,…,kₘ is divisible by n, and E(k₁,…,kₘ;1/n) has positive Lebesgue measure. Is it true that the set of remainders tᵢ0$, indexed coefficients $k:\\operatorname{Fin}(m)\\to\\mathbb N$ and $n\\in\\mathbb N$, define\n\n$$\n\\Sigma(k)=\\left\\{\\sum_{i\\in\\operatorname{Fin}(m)}\\varepsilon_i k_i:\\varepsilon:\\operatorname{Fin}(m)\\to\\{0,1\\}\\right\\},\\qquad\n a_t=\\frac{k_{t\\bmod m}}{n^{\\lfloor t/m\\rfloor+1}},\n$$\n\n$$\nE(k,n)=\\left\\{x\\in\\mathbb R:\\exists A\\subseteq\\mathbb N,\\ \\operatorname{HasSum}\\bigl((a_t)_{t\\in A},x\\bigr)\\right\\}.\n$$\n\nHere the coefficient subscript includes the proof that $t\\bmod m0\\ \\forall k:\\operatorname{Fin}(m)\\to\\mathbb N\\ \\forall n\\in\\mathbb N,\\quad\n\\Bigl[(\\forall i,\\ k_i>0)\\land \\operatorname{Antitone}(k)\\land n>1\\land\n |\\Sigma(k)|=n\\land(\\forall i,\\ n\\nmid k_i)\\land\\lambda(E(k,n))>0\\Bigr]\n\\Longrightarrow \\{s\\bmod n:s\\in\\Sigma(k)\\}=\\{0,\\ldots,n-1\\}.\n$$\n\n`Antitone` means weakly decreasing, so repeated entries are allowed. The $n>1$ condition expresses the source ratio $1/n\\in(0,1)$. There is no primitive, gcd-one, normalization or distinct-coefficient hypothesis.\n\nThe closed theorem `result : ¬ claim` proves exactly\n\n$$\n\\neg\\!\\left(\\forall m>0\\ \\forall k:\\operatorname{Fin}(m)\\to\\mathbb N\\ \\forall n\\in\\mathbb N,\\quad\n\\Bigl[(\\forall i,\\ k_i>0)\\land \\operatorname{Antitone}(k)\\land n>1\\land\n |\\Sigma(k)|=n\\land(\\forall i,\\ n\\nmid k_i)\\land\\lambda(E(k,n))>0\\Bigr]\n\\Longrightarrow \\{s\\bmod n:s\\in\\Sigma(k)\\}=\\{0,\\ldots,n-1\\}\\right).\n$$","title":"The original unnormalized achievement-set residue converse","url":"https://the-omega-institute.github.io/trureturing-pages/research/glab-2025-achievement-residue-converse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7c64268f36d69f85a4010aff4b9a1153a08ba83143d410c4a1e4a911e7b25073.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:glasby-paseman-powered-ratio-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The version-1 source, D5/L/Analytic/byun2026unimodality, page 2, defines\nthe sequence (1.4) by\n\n\\[\nR_{m,l,a}(r)=\n\\frac{\\sum_{i=0}^{r}(\\binom mi a^i)^l}\n {\\sum_{i=0}^{r}(\\binom ri a^i)^l},\\qquad 0\\le r\\le m.\n\\]\n\nConjecture 1.1(d) asks, for every fixed positive real a and positive\ninteger l, whether\n\n\\[\n\\lim_{m\\to\\infty}\\frac{\\max_{r\\in\\{0,\\ldots,m\\}}R_{m,l,a}(r)}{A_m}=1,\n\\qquad\nA_m=\\frac{\\sqrt l}{\\sqrt{2\\pi m}}\n\\frac{\\sqrt{1+2a}(1+a)a^{(l-2)/2}}{(1+a)^l-1}\n\\left(\\frac{1+2a}{1+a}\\right)^{(m+1/2)l}.\n\\]\n\nThe displayed constant transcribes clause (d); source footnote 1 specifies\nthe ratio-to-one meaning. Both weighted products are raised to l before\nsummation. The denominator is a complete row-r powered sum, not\n(1+a)^(rl). All integer r with 0<=r<=m are included, and ties are allowed.\nThe exponent (l-2)/2 uses real subtraction and equals -1/2 at l=1.\nThe quantifiers fix a and l before taking m to infinity; uniformity in\nvarying a or l is not asserted.","title":"Generalized Glasby–Paseman Conjecture 1.1(d)","url":"https://the-omega-institute.github.io/trureturing-pages/research/glasby-paseman-powered-ratio-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c547b6e5640dad844e1bfe0786eebe4a9979d086fb4f605bf64337bae8e0edda.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gledhill-hanany-2023-coulomb-branch-poisson-bracket-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kirsty Gledhill and Amihay Hanany, *Poisson brackets for some Coulomb branches*, JHEP 03 (2023) 154, arXiv:2210.02966v2, conjecture bracket tables for the second E6 family and the truck family. The v2 TeX source `main.tex` has md5 `e6664eac5bf9635a3684d84de4e80a97`.\n\nFor the $E_{6,n}^{II}$ family, `main.tex`, lines 1845–1856, display labelled `final e62 pbs`:\n\n> We conjecture that the non-zero Poisson brackets of (\\ref{gens of e6 II fam}) are given by:\n\nThe spinor entries of that display, lines 1851–1853, are:\n\n> $\\{C_1,s^{\\alpha}\\}=+ \\, s^{\\alpha}$\n>\n> $\\{C_1,s_\\alpha\\}=- \\, s_\\alpha$\n>\n> $\\{s^{\\alpha},s_\\beta\\} = {C_1}^{n-1} \\, {\\delta^{\\alpha}}_{\\beta}$\n\nThe sentence following the display is:\n\n> where $C_1$ is normalised such that there are no additional numerical coefficients in (\\ref{final e62 pbs}). All other Poisson brackets between (\\ref{gens of e6 II fam}) that are not listed in (\\ref{final e62 pbs}) vanish.\n\nFor the truck family, `main.tex`, lines 2185–2197, display labelled `final truck pbs`:\n\n> We conjecture that the non-zero Poisson brackets of (\\ref{gens of truck fam}) are given by:\n\nThe two weight brackets are the same, and the mixed entry, lines 2193–2194, is:\n\n> $\\{ s^{\\alpha}, s_{\\beta} \\} = {C_1}^{n+\\sigma-1} \\, {\\delta^{\\alpha}}_{\\beta}$\n\nThe following sentence specifies:\n\n> All other Poisson brackets between (\\ref{gens of truck fam}) that are not listed in (\\ref{final truck pbs}) vanish.\n\nThe second E6 table concerns $n\\ge3$ (line 1807); the truck quiver has $n\\ge2$ and hypermultiplet multiplicity $\\sigma$ (line 2149). The spinor indices have $N=2^{2n}$ values (lines 1843 and 2185). Appendix A, line 2237, states:\n\n> A Poisson bracket $\\{ \\cdot , \\cdot \\}$ is simply a bilinear map which obeys anticommutativity, Leibniz and the Jacobi identity.\n\nThe source context is collected in [the literature note](../Library/QuantumContext/gledhillhanany2023poissoncoulomb.md).\n\nIssue [#21464](https://github.com/the-omega-institute/trureturing/issues/21464) fixes the proposition `claim` as $\\mathrm{ConjE6}\\lor\\mathrm{ConjTruck}$. Here `ConjE6` requires, for every natural $n\\ge3$, a reduced commutative complex algebra $R$, a Poisson bracket, a nonzero element $h$, and two spinor families indexed by $\\mathrm{Fin}(2^{2n})$ satisfying the displayed spinor rules with exponent $e=n-1$. `ConjTruck` requires the same for every natural $n\\ge2$ and every natural $\\sigma\\ge1$, with $e=n+\\sigma-1$. The algebra and bracket may depend on these parameters. This disjunction of universal family assertions is refuted.","title":"Jacobi and reducedness obstruct the second E6 and truck Poisson bracket conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/gledhill-hanany-2023-coulomb-branch-poisson-bracket-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca14f16793ee19e3aa7d3fc928b86b6825a95ffc60d7c321d104b84ad320e158.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:glen-simpson-smyth-2017-lyndon-factors-open-problem-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Glen, Simpson and Smyth, *Counting Lyndon factors*, Section 6, p. 9,\nOpen Problem 1 (arXiv:1701.00928v1; DOI 10.37236/6915), ask:\n\n> Is it true that the minimum number of distinct Lyndon factors over all\n> Lyndon words of the same length is attained by at least one Christoffel\n> word of that length?\n\nThe resolution is **Refuted**, under the literal finite-word conventions\nof [#18581](https://github.com/the-omega-institute/trureturing/issues/18581).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Lyndon words over an ordered alphabet | Frozen `LyndonOrder.IsLyndon`, applied to `List.ofFn w` | Nonempty and strictly least among nontrivial rotations; the imported suffix characterization is used in the proof. |\n| A factor is a contiguous subword | `factor w i l` | Drops the first $i$ letters and takes at most $l$; valid nonempty occurrences satisfy $1\\le l$ and $i+l\\le n$. |\n| Number of distinct Lyndon factors | `distinctLyndon w` | Cardinality after taking the image of valid occurrences; equal factor lists count once, including letters and the whole word. |\n| At every common length | `claim`, quantified over $n\\ge2$ | All lengths remain in the claim; $n=506$ refutes it. |\n| At least one Christoffel word | A numerator $0 For mixtures of Dicke states, are the following two sets equivalent?\n\n$$\n\\mathsf{PPTBExt}_{n}^{(\\mathcal K_r)}\n=\\mathsf{PPTBExt}_{n}^{(r-1)},\\qquad r\\ge2.\n$$\n\nThe target is exactly this versioned question. For every integer $n\\ge1$,\n$r\\ge2$ and every real upper-triangular family $Y_{ij}\\ge0$, put\n\n$$\nD_{ii}=e_i\\otimes e_i,\\qquad\nD_{ij}=\\frac{e_i\\otimes e_j+e_j\\otimes e_i}{\\sqrt2}\\quad(i We conjecture that this algorithm is optimal (i.e., produces an (L, b)-coloring if K_n is (L, b)-colorable).\n\nThe algorithm is ColorSym. An $a$-list assignment associates to every vertex\na set of $a$ integers. An $(L,b)$-coloring assigns a $b$-element subset of\neach vertex's list, with disjoint subsets at adjacent vertices. For $K_n$\nevery pair of distinct vertices is adjacent.\n\n### Statement echo\n\nThe binding semantics are recorded in\n[preregistration #16679](https://github.com/the-omega-institute/trureturing/issues/16679).\nThe public definitions and the claim use those expressions. The paper's\nvertices $1,\\ldots,n$ correspond to Lean's `Fin n` values $0,\\ldots,n-1$.\n\n| Source clause | Lean counterpart | Correspondence |\n| --- | --- | --- |\n| An $a$-list assignment assigns $a$ integers to each vertex. | The `claim` parameters $n,a,b$, the finite integer lists $L$, and the list-cardinality hypothesis. | All natural parameters are quantified, including degenerate cases. |\n| Proper intersections are the intersection of selected lists minus the union of the other lists. | `supp` and `Ip`. | A color is selected exactly when its support is the specified set; the empty support contributes no color from the union. |\n| A coloring assigns $b$ colors from each list, disjoint on every edge. | `IsColoring` and `Colorable`. | The exact cardinality, list inclusion and pairwise disjointness are all retained. |\n| “for each set S with \\|S\\| = i” in steps of increasing $i$. | `AdmissibleOrder` and `runSupports`. | Every support of size at least two occurs once; sizes are nondecreasing; equal-size order is arbitrary. |\n| “Remove a color from Ip(S) and assign it to vj.” | `AdmissiblePick` and `assign`. | Every choice from a nonempty integer pool is allowed; insertion and load increment are explicit. |\n| “let j ∈ S be the index for which w(j) is minimum (if more than one j with minimum w, then take the smallest index)” | `best`. | The minimum-load vertices are sorted by index and the first is selected. |\n| “If wi ≥ b for each i or i = n + 1, then Stop.” | `stopped`, `runPool` and the terminal case of `runSupports`. | Global stopping is rechecked at each color assignment; exhausting the finite support order implements the terminal step. |\n| “Goto Step i until Ip(S) is empty.” | `runPool`. | The pool loses one chosen color per assignment; initial fuel is its cardinality. |\n| “Color every proper intersections of sets of size 1: vi gets \\|Ip({i})\\| colors. Thus wi = \\|Ip({i})\\|.” | `initial`. | Singleton atoms determine both the initial outputs and loads. |\n| “produces an (L, b)-coloring if K_n is (L, b)-colorable” | `colorSym`, `claim` and `result`. | The claim asks for some admissible order and pick whose output contains a valid $b$-coloring; `result` negates it. |\n\nThe existential choice of order and pick and the allowance to trim an output\nto $b$ colors give the algorithm the weakest favorable reading. Every\nadmissible execution fails on the witness, so this allowance does not rescue\nthe conjecture. The fuel parameter, option for an empty support, product\nstate and shifted indices implement the algorithm; they impose no additional\nhypothesis on the public claim. There is no separation bound in the conjecture.","title":"ColorSym optimality is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/godin-grisot-togni-2022-colorsym-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/705c147a463fbd05a385df1958f92bc708ff68cd7eea6605b5e3d30374440c85.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:godreche-luck-2020-moving-average-record-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Godrèche and J.-M. Luck, *Records for the moving average of a time series*, arXiv:1907.07598v2, J. Stat. Mech. (2020) 023201, DOI 10.1088/1742-5468/ab5d08, state after equation (6.33) that the data suggest $3/(2n)$ is an absolute upper bound for the record probability $Q_n$.\n\n### Statement echo\n\nThe source's two-term moving average is $Y_n=X_{n-1}+X_n$, with records counted from the first complete sum $Y_2$. The formal `recordProb` event is the strict comparison $Y_n>\\max(Y_2,\\ldots,Y_{n-1})$ in the preregistered zero-based coordinates. The theorem `movingAverage_record_bound` proves the bound for every atomless probability measure on $\\mathbb R$ and every $n\\ge3$; `result` specializes it to the density-form `claim`. Its `Continuous f` hypothesis is the density restriction fixed by preregistration [#15864](https://github.com/the-omega-institute/trureturing/issues/15864); discontinuous densities in the paper are covered by the public atomless theorem.","title":"Godrèche–Luck moving-average record bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/godreche-luck-2020-moving-average-record-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4143f7f4fedf1f9deab43496f60fcd000a33bd5073aeef01cc3b8bc725807476.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:godsil-2011-average-mixing-discriminant-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Godsil, *Average mixing of continuous quantum walks*, arXiv:1103.2578v3,\nsection 11, Question 1 (PDF page 20), asks:\n\n> Is it true that if $D$ is the discriminant of the minimal polynomial of $X$, then $D\\widehat M_X$ is an integer matrix?\n\nFor a finite simple graph $X$ with adjacency matrix $A$, the minimal\npolynomial is taken over $\\mathbb Q$. The source's Lemma 1.1 identifies\n$\\widehat M_X=\\sum_\\theta E_\\theta\\circ E_\\theta$, with one orthogonal\nspectral projector for each distinct eigenvalue. Lemma 3.1 establishes\nonly the factor $D^2$. Issue #12339 preregisters the exact all-graphs\nassertion, with no connectedness or simple-spectrum restriction.","title":"The discriminant clears average mixing denominators","url":"https://the-omega-institute.github.io/trureturing-pages/research/godsil-2011-average-mixing-discriminant-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/048770e2516a221176ea3153901c66d60cd95637feca7693ad4f0062e3603dc1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:godsil-2017-conference-graph-sedentariness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chris Godsil, *Sedentary quantum walks*, arXiv:1710.11192v1 (2017; Linear Algebra and its Applications 614 (2021), 356–375), Section “Questions?”, TeX source `conemix.tex`, lines 1037–1038:\n\n> We do not know whether, for conference graphs, the diagonal entries of $M(t)$ can be\n> bounded away from zero.\n\nIn the Introduction, `conemix.tex`, lines 190–207, Godsil defines $U(t)=\\exp(itA)$ for the adjacency matrix $A$ and $M(t)=U(t)\\circ\\overline{U(t)}$, where $\\circ$ is the Schur product. Thus $M(t)_{u,u}=|U(t)_{u,u}|^2$.\n\nIssue [#26302](https://github.com/the-omega-institute/trureturing/issues/26302) fixes the reading as a sharp classification, distinguishing a positive bound for each fixed graph from a bound uniform across orders. Write $n=4m+1$, with $m\\in\\mathbb N$ and $m\\ge1$. The module takes a finite vertex type $V$ with decidable equality and a simple graph $G$ with decidable adjacency satisfying `G.IsSRGWith (4 * m + 1) (2 * m) (m - 1) m`. This means that $|V|=n$, every vertex has degree $2m$, adjacent vertices have $m-1$ common neighbours, and distinct nonadjacent vertices have $m$ common neighbours. For $u\\in V$, put $p_u(t)=|\\exp(itA)_{u,u}|^2$.\n\nThe classification gives a sharp positive bound $1/n^2$ exactly at square orders; at nonsquare orders, return probabilities become arbitrarily small at positive times. Its precise quantified statement is given under “Proved by `result`”.","title":"Sharp return-probability bounds for conference graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/godsil-2017-conference-graph-sedentariness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ef042e4b43f61f8381d1acba89d1bf2fe95c6291d0bd47884f39cd136f1cf12.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:godsil-2023-average-mixing-trace-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chris Godsil, Krystal Guo and Mariia Sobchuk, *Diagonal entries of the\naverage mixing matrix*, arXiv:1910.02039v1, Section 9 (Conjecture 9.1 of\nAustralas. J. Combin. 86(3) (2023) 373–386), write:\n\n> Based on the computations summarized in Table 2 and on Corollary 6.3, we also make the following conjecture. The complete graph on $n$ vertices attains the maximum trace with respect to the $\\widehat{M}_{A}$ for all $n$.\n\nFor a graph $X$ with adjacency matrix $A=\\sum_\\theta\\theta E_\\theta$ over\nits distinct eigenvalues, $E_\\theta$ the orthogonal projection onto the\n$\\theta$-eigenspace, the average mixing matrix of the continuous quantum\nwalk is $\\widehat M_A(X)=\\sum_\\theta E_\\theta\\circ E_\\theta$ (Schur\nproduct), and $\\operatorname{tr}\\widehat M_A(K_n)=(n^2-2n+2)/n$.\n\nThe precise question, preregistered in issue #11821, is whether\n$\\operatorname{tr}\\widehat M_A(X)\\le\\operatorname{tr}\\widehat M_A(K_n)$\nfor every $n$ and every connected graph $X$ on $n$ vertices. Over all\ngraphs the statement is false: the empty graph has $\\widehat M_A=I$ and\ntrace $n$. The paper's Laplacian results on the maximum trace are stated\nfor connected graphs, and A. Mohan, C. Tamon, Y. Xu and H. Zhan\n(arXiv:2608.20739, 2026) restate the conjecture over connected graphs.","title":"The complete graph maximizes the trace of the average mixing matrix","url":"https://the-omega-institute.github.io/trureturing-pages/research/godsil-2023-average-mixing-trace-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/153244b7ef72bc697b4a8b7c2d18914b112902db2a1053ce07a8584d057430b0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:goel-2026-pisano-rank-ratio-oq4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Aradhya Goel, *Sophie Germain Primes and the Totient of Fibonacci Numbers*,\narXiv:2604.17847v3, Section 10, OQ4, p. 10, asks:\n\n> **OQ4. Values of $\\pi(q)/z(2q+1)$.** Is $\\{\\pi(q)/z(2q+1):q>5\\text{ with }z(2q+1)\\mid\\pi(q)\\}=\\{\\text{odd integers}\\}$?\n\nThe versioned TeX source reads verbatim:\n\n```tex\n\\item[OQ4.] \\textbf{Values of $\\pi(q)/z(2q+1)$.} Is $\\{\\pi(q)/z(2q+1) : q > 5 \\text{ with } z(2q+1) \\mid \\pi(q)\\} = \\{\\text{odd integers}\\}$?\n```\n\nThe Introduction defines $z(p)$ at primes as the least positive index $k$ with\n$p\\mid F_k$, and $\\pi(n)$ as the period of the Fibonacci residues modulo $n$.\nThe standing domain in Sections 5–7 is Sophie Germain primes: $q$ and $2q+1$\nare both prime. The assertion here uses $q>5$ and $z(2q+1)\\mid\\pi(q)$.\nThe right-hand side is the positive odd integers; reading it as all odd integers\nalso fails because the ratios are positive.","title":"Goel's OQ4: Fibonacci period-to-rank ratios omit five","url":"https://the-omega-institute.github.io/trureturing-pages/research/goel-2026-pisano-rank-ratio-oq4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d33e996b5ae98b36c68fc74a4a20d2266eb6da0029a820ab5fb72e0c20b5d5da.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:goh-2026-entropic-glibichuk-konyagin","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. K. Goh, *On an entropic analogue of additive energy*, Essential Number\nTheory 5 (2026), no. 2, 243–269, DOI 10.2140/ent.2026.5.243,\narXiv:2406.18798v4, §6, Conjecture 15, equation (89):\n\n> **Conjecture 15 (Entropic Glibichuk–Konyagin lemma).** There is an integer $k\\geq 3$ such that the following holds. Let $X$ be a random variable taking values in $\\mathbf F_p$ for a prime $p$. Then for independent copies $X_1,\\ldots,X_k,X'_1,\\ldots,X'_k$ of $X$, we have\n> $$H\\{X_1X'_1+\\cdots+X_kX'_k\\} \\geq \\min(2H\\{X\\},\\log p)-1. \\qquad (89)$$\n\n> In the original additive-combinatorial version of this statement, $k$ can be taken to be $3$, but any fixed integer is enough for our purposes.\n\nThe paper uses binary logarithms and discrete random variables.\nNo additional hypothesis on the law of X is stated.\n\n### Formal reading\n\nAll local Lean names have prefix\n`D5.S3.Entropy.GohGlibichukKonyagin.Result`.\nHere $p$ is prime, $\\mathbf F_p$ is `ZMod p`, $\\delta_0$ is the point mass at zero,\nand $U_p$ assigns mass $1/p$ to each field element. Outside the source quotation,\n$H$ denotes Shannon entropy in nats and $H_{\\mathrm{bits}}=H/\\log 2$;\n`shannonEntropy` is reused from `D5.S3.Entropy.MaxEntropy`.\n\nA law $\\mu:\\mathbf F_p\\to\\mathbb R$ satisfies\n$\\mu(a)\\ge0$ and $\\sum_a\\mu(a)=1$.\n`sumLaw k μ` is the pushforward of the product weights\n$\\prod_{i\\in\\operatorname{Fin}k}\\mu(x_i)\\mu(x'_i)$,\nindexed by $(x,x'):(\\operatorname{Fin}k\\to\\mathbf F_p)^2$,\nunder $(x,x')\\mapsto\\sum_i x_i x'_i$.\nHere $\\operatorname{Fin}k=\\{0,\\ldots,k-1\\}$.\nFor the convolution computation, `iter_product` uses tuples of pairs\n$v:\\operatorname{Fin}k\\to\\mathbf F_p\\times\\mathbf F_p$.\nThe standard equivalence `Equiv.arrowProdEquivProdArrow` sends\n$v$ to its two projection functions; its inverse sends $(x,x')$ to\n$i\\mapsto(x_i,x'_i)$. In `sum_spike`, `Equiv.sum_comp` proves\nthat this change of indexing preserves the entire pushforward sum.\nThus $Z=\\sum_iX_iX'_i$ has exactly this law for the $2k$ independent copies.\n`claim` retains the existential $k\\ge3$, all primes $p$, all such laws,\nand the constant $1$ in bits. `result` states exactly `¬ claim`.","title":"Refutation of the unconditional entropic Glibichuk–Konyagin conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/goh-2026-entropic-glibichuk-konyagin/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/36d6877523b1408950562a7a73881fae443aed7575956c7ffa05ff2f78a08e29.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:golden-ratio-base4-dfao-minimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The paper constructs a DFAO which, on the Zeckendorf representation of `q = b^i`,\noutputs the `i`th base-`b` digit of the golden ratio. The full Walnut automaton\nis minimal on all valid inputs, but only powers of `b` matter for digit\nextraction.\n\nQuoted from arXiv:2405.02727v1:\n\n> “Could it be that there are even smaller automata that answer correctly on\n> inputs of the form \\(b^i\\) (but might give a different answer for other\n> inputs)?”\n\n> “We do not know the answer to this question, in general.”\n\nThe concrete target is the unresolved base-4 phi instance: prove that the\npaper's 22-state Walnut DFAO is minimal among DFAOs which are correct on the\nZeckendorf encodings of all `4^i`, ignore leading zeroes, and obey the\nZeckendorf/Ostrowski validity rules. Equivalently, prove that no such DFAO with\nat most 21 states exists.\n\nThe paper also says:\n\n> “It is conceivable that the automata produced by our method are indeed minimal\n> and unique in general, and we leave this as an open question.”\n\nMinimality of the fixed base-4 instance is narrower and mechanically\nfalsifiable; uniqueness should remain a separate target.\n\nThe paper states the difficulty:\n\n> “The question is likely difficult; in terms of computational complexity, it is\n> a special case of a problem known to be NP-hard, namely, the problem of\n> inferring a minimal DFAO from incomplete data.”\n\n> “For this reason, \\(\\varphi\\) in base 4 ... encountered prohibitively long\n> solving times before the required number of states (22 states ...) could be\n> reached, preventing the minimality of the Walnut solutions from being\n> determined.”\n\n> “For \\(\\varphi\\) in base 4, it took over 25 hours for the 78'th digit set to\n> be declared UNSAT at 13 states...”\n\nThe paper explains that both the digit set needed for a candidate and the\nrepresentation length of each digit position can be arbitrarily large.","title":"Minimality of the base-4 golden-ratio DFAO","url":"https://the-omega-institute.github.io/trureturing-pages/research/golden-ratio-base4-dfao-minimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/705a68777a905021a06dacb9df117fc5e9db6d5c1b0bc12f44212503c1fec887.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gonzalez-dleon-wachs-conjecture-4-13-2-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 4.13(2) of arXiv:2608.08692v1 states that if `G` is a graph on\n`n` vertices with `k` connected components and `H` is a spanning subgraph\nwith the same `k` components, then\n`(-1)^(n-k) (mu_G - mu_H)` is real-rooted. Here `mu_G` is the Section 2\nweighted bond-poset Mobius polynomial. The statement permits disconnected\ngraphs. This dossier concerns only part (2) as written.","title":"Refutation of Gonzalez D'Leon-Wachs Conjecture 4.13(2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/gonzalez-dleon-wachs-conjecture-4-13-2-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e3d0f18f56280c31b5f93b38748e9a4f9149f927b49e32f1484ca6ccdeb134a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:goodenough-landahl-2026-connected-progenitor-adaptive-fusion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Goodenough, Landahl, Lee, Russo and Thompson, *Optimal Fusion Strategies for\nQuantum Computation*, arXiv:2609.02559v2, study logical fusions of\n$[[n,1,d]]$ codes: a failed physical fusion measures both qubits in its\nfailure axis, and failures on a set $W$ cause a logical failure iff the\nsubstrings of the failure-axis string on $W$ contain a non-trivial logical\noperator (Proposition 3.1). A strategy is perfect when one successful fusion\nsuffices. Section 6 reads\n\n> So, a natural conjecture and direction of future work is that all\n> $[[n, 1, d]]$ codes with connected progenitor graphs have perfect adaptive\n> strategies.\n\nIssue #13256 fixes the reading. Pauli strings are taken up to phase as\n$(x,z)$ bits. The stabilizers of the progenitor graph state $G$ with encoding\nvertex $e$ are the products $\\Gamma_U$ of canonical generators. A non-trivial\nlogical operator is the restriction of some $\\Gamma_U$ to $V\\setminus\\{e\\}$\nwith $\\Gamma_U$ not the identity at $e$. An adaptive strategy attempts the\nphysical vertices in a fixed order and chooses each failure axis (X, Y or Z)\nfrom the earlier outcomes.","title":"Connected progenitor graphs give perfect adaptive fusion strategies","url":"https://the-omega-institute.github.io/trureturing-pages/research/goodenough-landahl-2026-connected-progenitor-adaptive-fusion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c14d2e26c772856fde2f5aa29b9d966819987f6346edd7c5ea17c816c53a622.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gottlieb-krnc-mursic-pnim-heavy-interval-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gottlieb, Krnc, and Mursic, *Nim on Integer Partitions and Hyperrectangles*,\narXiv:2506.04991v1, printed page 14, Conjecture 2 states that whenever the\nrectangle `[a+1]^(b+1)` is heavy, every partition in the Young interval from\n`[a+1,a,...,a-b+1]` to that rectangle is heavy.\n\nThe formal claim is the fully quantified natural-number version with\n`b <= a`, the PNim row and column moves, the recursive Grundy value, and the\ndefinition of heaviness from Proposition 2.","title":"Refutation of the PNim heavy-interval conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/gottlieb-krnc-mursic-pnim-heavy-interval-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7c2b61f7ce9bad7f8d4762914d20d2a02054e9e94d344adc18650c7b8f3ca8d3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:graffeo-gubbiotti-2024-nambu-kahan-cremona-decomposition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Graffeo and G. Gubbiotti study the Kahan–Hirota–Kimura (KHK) discretisation of the quadratic three-dimensional Nambu system\n$ \\dot{x}=\\nabla H_1(x)\\mathbin{\\times}\\nabla H_2(x),\\qquad H_i(x)=x^{\\mathsf T}A_i x. $\nIn Remark 7.7 of *Growth and Integrability of Some Birational Maps in Dimension Three*, they write:\n\n> “We conjecture that, up to conjugation by a projectivity, the map $\\Phi_{h}^{(N)}$ is of type (A)-(i), i.e. it exists an analogue of Lemma 2.6 for the map $\\Phi_{h}^{(N)}$.”\n\nThis is quoted from `GG_cremona_v1_clean.tex`, lines 3209–3211, Remark `ex:celledoni`; the same remark says that a complete proof is left for further research.\n\nThe Lean module defines, for $X:\\mathrm{Fin}\\,4\\to\\mathbb C$, the polarised matrix `nambuLinearPart`, the homogeneous KHK matrix\n$ K_{A,B,h}(X)=X(3)I_3-2h\\,\\mathrm{nambuLinearPart}(A,B)(X\\mathbin{\\circ}\\mathrm{Fin.castSucc}), $\nthe homogeneous lift\n$ \\mathrm{kahan}(A,B,h,X) =\\mathrm{Fin.snoc}\\bigl(\\operatorname{adj}(K_{A,B,h}(X))\\,\\mathrm{mulVec}\\, (X\\mathbin{\\circ}\\mathrm{Fin.castSucc}),\\det K_{A,B,h}(X)\\bigr), $\nand\n$ \\mathrm{cremona}(X)= \\bigl(X(1)X(2)X(3),\\,X(0)X(2)X(3),\\,X(0)X(1)X(3),\\,X(0)X(1)X(2)\\bigr). $\nIts `claim` states that for every real $3\\times3$ matrices $A,B$ with `A.PosDef` and `B.PosDef`, with\n$ \\forall t:\\mathbb R,\\quad 2\\le \\operatorname{rank}(B-tA), $\nand every $h:\\mathbb R$ with $0 Suppose that p₁, …, pₙ are distinct and relatively prime to q > (7/4)ⁿ,\n> with ∑ pₖ ≤ q, and for each k ∈ [n],\n> 2/sin(π/q) ≤ ∑ᵢ₌₁ⁿ 1/|sin(π pₖ p̄ᵢ/q)|.\n> Then q = 2ⁿ − 1 and {p₁, …, pₙ} ≡ {1, 2, …, 2ⁿ⁻¹} (mod q).\n\nThe quantifiers range over every positive integer n and q and every n-tuple\nof distinct positive integer representatives coprime to q. The last set\ncontains all n powers 2⁰ through 2ⁿ⁻¹. The barred quantity is a modular\nmultiplicative inverse, and every complete row includes its diagonal.\nThe conclusion compares ordinary residue sets, without independent signs.","title":"Graham–O'Bryant Conjecture 5.2: ordinary residue-set refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/graham-obryant-inverse-sine-conjecture52-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a502096026850e8fe589b3ed07d66d2e6bb216d8c66d1a9c725bd55871a53d5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:grandjean-liang-bancal-brunner-gisin-2012-mermin-cglmp-facet","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Grandjean, Y.-C. Liang, J.-D. Bancal, N. Brunner and N. Gisin,\n“Bell inequalities for three systems and arbitrarily many measurement outcomes”,\n*Physical Review A* 85, 052113 (2012), arXiv:1204.3829v2, Section II,\nstates:\n\n> We conjecture that inequality (1) is indeed facet-defining for all $K\\ge2$.\n\nFor two settings and $K$ outcomes at each of three parties, its functional is\n\n$$\nI_K=\\langle[A_2-B_1+C_1]_K\\rangle+\n\\langle[A_1+B_2-C_1]_K\\rangle+\n\\langle[-A_1+B_1+C_2]_K\\rangle+\n\\langle[-A_2-B_2-C_2-1]_K\\rangle.\n$$\n\nHere $[X]_K$ is the least nonnegative residue and\n$\\langle[X]_K\\rangle=\\sum_{j=0}^{K-1}jP(X=j\\bmod K)$.\nLet $\\mathcal L_K$ be the convex hull of all deterministic full behaviours\n$p_s(a,b,c\\mid x,y,z)=[A_x=a][B_y=b][C_z=c]$.\nThe settled statement is validity $I_K(p)\\ge K-1$ on $\\mathcal L_K$ and\n\n$$\n\\dim\\operatorname{aff}\\{p\\in\\mathcal L_K:I_K(p)=K-1\\}+1\n=\\dim\\operatorname{aff}\\mathcal L_K\n\\qquad(K\\ge2).\n$$","title":"The Grandjean et al. Mermin-CGLMP facet conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/grandjean-liang-bancal-brunner-gisin-2012-mermin-cglmp-facet/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/030f5be9c4c8e1907f341edc23ef46c45ffe5591e7496bbcfdacdc2e38c810aa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:grandjean-liang-bancal-brunner-gisin-2012-sliwa-seven-local-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Grandjean, Y.-C. Liang, J.-D. Bancal, N. Brunner and N. Gisin,\n“Bell inequalities for three systems and arbitrarily many measurement\noutcomes”, *Physical Review A* 85, 052113 (2012), arXiv:1204.3829v2,\nAppendix B, pp. 5–6, gives the arbitrary-output Sliwa seventh inequality\n(B1). It states that its local bound is numerically $6(K-1)$, attained by\nthe all-zero deterministic strategy, and conjectures that both the local\nbound and the facet-defining property hold for every $K$. The settled\nstatement is the local-bound clause:\n\n$$\n\\forall K\\ge 2,\\quad\n\\min\\{J_K(a,b,c,A,B,C):a,b,c,A,B,C\\in\\mathbb Z/K\\mathbb Z\\}=6(K-1).\n$$\n\nwhere $J_K$ is the twelve-term deterministic expression, with each\nbracket interpreted as the least nonnegative residue.","title":"The local-bound clause of the Grandjean et al. conjecture on (B1)","url":"https://the-omega-institute.github.io/trureturing-pages/research/grandjean-liang-bancal-brunner-gisin-2012-sliwa-seven-local-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/edf9c4d9ef5fe922f992ce9466e8e23e61020fea90755b9d1877b786ddc195e1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:granvik-2024-a292273-real-part","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mats Granvik, OEIS A292273, revision 24, Jul 06 2024:\n\n> For odd n: a(n) = 0, and for even n: a(n) = -mu(n), where mu is Moebius function (A008683).\n\n> Conjecture: a(n) = Re(A008683(n)*(i^n)). - _Mats Granvik_, Jul 06 2024\n\nThe offset is one. The assertion concerns every positive integer, with no\nupper bound. Here mu is the standard integer Mobius function and i is the\nordinary complex imaginary unit.","title":"Granvik's A292273 real-part identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/granvik-2024-a292273-real-part/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/10b1550883f026d9f70f9c1de06a3066b6e6d030d82531587b61b591d3abc339.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:graydon-skanes-norman-wallman-2022-qutrit-stochastic-unitality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Graydon, J. Skanes-Norman and J. J. Wallman, *Designing Stochastic\nChannels*, arXiv:2201.07156v1, Section II, ask:\n\n> We leave open the question of whether or not all stochastic channels are unital for d = 3.\n\nEq. (2) defines the normalized Choi state\n$J(\\Phi)=d^{-1}(\\Phi\\otimes\\mathrm{id})(\\operatorname{col}(I_d)\n\\operatorname{col}(I_d)^\\dagger)$. Eq. (11) requires that $\\Phi$ be\ncompletely positive and trace preserving and that\n$J(\\Phi)\\operatorname{col}(I_d)=\\lambda\\operatorname{col}(I_d)$ for\nsome real $0<\\lambda\\leq1$. Unitality means $\\Phi(I_d)=I_d$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Complex-linear maps on qutrit matrices | `claim` quantifies over `MatrixMap (Fin 3) (Fin 3) ℂ` | Exact dimension and carrier |\n| CPTP | Frozen `CoPRelativeQuantumnessRefutation.IsCPTP` within `IsStochastic` | Complete positivity and trace preservation on every matrix |\n| Some real $0<\\lambda\\leq1$ | The existential real parameter and both inequalities in `IsStochastic` | No endpoint or sign clause dropped |\n| Normalized Choi state and identity vector | Their preregistered bodies are inlined in `IsStochastic` | Output-first Choi coordinates; vector entries are one exactly on equal indices |\n| Every stochastic channel is unital | The implication in `claim` concludes $\\Phi(I_3)=I_3$ | Literal universal question |\n| Refutation | `result` has type `¬ claim` | A counterexample suffices |\n\nThe definition also admits dimension zero as a total Lean expression; the\nquestion and its refutation use dimension three. Proofs use the existing\n`ProductUnitaryChoiSpectrum.omega` and\n`PassiveMemoryNonclassicalRefutation.choi`, with correspondence by unfolding.","title":"A stochastic qutrit channel that is not unital","url":"https://the-omega-institute.github.io/trureturing-pages/research/graydon-skanes-norman-wallman-2022-qutrit-stochastic-unitality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/49acf371c9bcf263016544debd1d3b34efb2fc30bd8575d236521f4a74fe1efe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:greaves-iverson-jasper-mixon-2021-gerzon-equality-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. R. W. Greaves, J. W. Iverson, J. Jasper and D. G. Mixon, *Frames over finite fields: Equiangular lines in orthogonal geometry*, arXiv:2012.13642v2, Example 5.8, page 21, asks:\n\n> The authors do not know how to construct an ETF with any of the parameters given in Table 4. In particular, every instance of Gerzon equality that we currently know satisfies the identity a² = (d + 2)b, but this equation does not occur anywhere in Table 4 (where b = 1). Proposition 2.10 implies that a² = (d + 2)b whenever Gerzon’s bound is saturated, d ≢ₚ 0, and d ≢ₚ 1. The authors do not know if these conditions can be removed.\n\nThe answer is affirmative for every $d\\ge2$. Preregistration: [#16342](https://github.com/the-omega-institute/trureturing/issues/16342). The [Library note](../Library/QuadraticForms/greavesiversonjaspermixon2021orthogonalframes.md) records Definition 2.8, Propositions 2.9–2.10 and the source locator.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite field of odd characteristic | `F : Type`, `Field F`, `Fintype F`, `ringChar F ≠ 2` in `claim` | Exact source scope |\n| Dimension $d\\ge2$ | `d : ℕ`, `2 ≤ d` | Explicit boundary of the affirmative answer |\n| Quadratic space | `B : LinearMap.BilinForm F (Fin d → F)`, `B.IsSymm`, `B.Nondegenerate` | Arbitrary symmetric nondegenerate form in coordinates |\n| $N=\\binom{d+1}{2}$ vectors | `φ : Fin ((d + 1).choose 2) → Fin d → F` | Exact Gerzon cardinality |\n| $(a,b)$-equiangular | `IsEquiangularSystem B φ a b` | Every diagonal pairing equals $a$; every distinct squared pairing equals $b$ |\n| $a^2\\ne b$ | `a ^ 2 ≠ b` | Retained hypothesis of Proposition 2.9 |\n| Remove both dimension congruences | No assumption that $d$ or $d-1$ is nonzero in $F$ | Neither restriction retained |\n| Parameter identity | `result : claim` | $a^2=((d:F)+2)b$ |\n\nThe two public definitions are the literal conventions of #16342. `Fin n` indexes from zero instead of one without changing the pairwise conditions. A choice of basis identifies any $d$-dimensional space with these coordinates; that transport is the coordinate interpretation, not an additional public theorem of this module.","title":"Gerzon equality fixes the equiangular parameters in every dimension at least two","url":"https://the-omega-institute.github.io/trureturing-pages/research/greaves-iverson-jasper-mixon-2021-gerzon-equality-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/49c8bdf42b6e9f25101ccac31a8586a595044ff79081773845f7f426c95dd466.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:greedy-fibonacci-average-closed-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fried, *Journal of Integer Sequences* **28** (2025), Article 25.4.3, Section 6,\nwrites:\n\n> As mentioned earlier, we observed the statement of the previous theorem while\n> working on a conjecture stated in A248982, which is defined to be the sequence\n> of distinct least positive numbers such that the average of the first `n` terms\n> is a Fibonacci number. Let `(a_n)_{n≥1}` be this sequence. Refining the\n> conjecture stated in A248982 regarding a closed-form formula for `(a_n)`, it\n> seems that, for `n ≥ 10`, we have\n>\n> a_n = nF(n/2 + 3) − (n − 1)F(n/2 + 2), if n is even;\n> a_n = F((n+1)/2 + 2), otherwise.\n>\n> A proof of this likely proceeds along similar lines as the previous theorem.\n\nHere `F(1) = F(2) = 1`. The sequence entry A248982 carries two further standing\nconjectures, both recorded by Colin Barker on 2014-10-19 and both still labelled\n`Conjecture`: that `a(n) = 2a(n−2) + a(n−4) − 2a(n−6) − a(n−8)` for `n > 17`, and\nthat `a(2n+1) = F(n+3)` for `n > 4`. An `Empirical g.f.` equivalent to the first\nis recorded alongside them.","title":"Closed Form for the Greedy Fibonacci-Average Sequence","url":"https://the-omega-institute.github.io/trureturing-pages/research/greedy-fibonacci-average-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/843e88d9a9a4bad0337df8fec4af432ad597058d8a2714fc53ab0b66a62ca01c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:greedy-three-sumfree-two-parameter","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier deliberately anchors Conjecture 17 of Bosma et al., *Journal of\nInteger Sequences* 28 (2025), Article 25.3.8, printed page 18, which is\nthe earlier Conjecture 6 of arXiv:2503.04122v1 with its third-seed bound\ncorrected from `z > g+d` to `z >= g+d`. The caller-supplied reading, 2026-09-07,\nquotes this single proposition:\n\n> Let d >= 2. For every g >= d + 1 the greedy 3-sumfree sequence S_{1,g,g+d}\n> is characterized as follows: z in S_{1,g,g+d} <=> z in {1, g, 2g+d-1, 2g+d}\n> or z >= g+d and z mod 5g+2d in {g+d-2, g+d-1, ..., 2g+d-2}.\n\nHere `d`, `g`, and sequence entries are integers, with positive seeds\n`1 < g < g+d`. Every subsequent entry is the least larger positive integer\nthat is not a sum of three distinct earlier entries, as defined on printed\npage 16. Conjecture 16 and the paper's other open items are deliberately out\nof scope.","title":"Two-parameter greedy three-sumfree membership","url":"https://the-omega-institute.github.io/trureturing-pages/research/greedy-three-sumfree-two-parameter/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9639c0c0fab310cf784aafe00015912248fd04c232812da3bb1334ed8d116d9d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:groot-nibbelink-2020-four-dimensional-singlet-branching","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Groot Nibbelink, O. Loukas, A. Mütter, E. Parr and P. K. S. Vaudrevange\n(arXiv:1710.09237, hep-th; Fortschr. Phys. 68 (2020) 2000044) reduce the\nquestion of heterotic orbifolds that are supersymmetric in every twisted sector\nbut not globally to a statement about finite groups:\n\n```latex\nThere does not exist any finite group $\\mathbf{H}$ that has a\nfour-dimensional representation $D_\\rep{4}$ with the following three\nproperties: (i) $D_\\rep{4}$ has a trivial determinant … (ii) $D_\\rep{4}$ does\nnot contain the trivial singlet representation of $\\mathbf{H}$ … (iii) but\nthe branchings of $D_\\rep{4}$ to all $\\Z{N} \\subset \\mathbf{H}$ subgroups\nalways contain the trivial $\\Z{N}$-singlet representation.\n```\n\nIssue #11489 fixes the reading: `H` is any finite group, `D` any complex\nrepresentation on `ℂ⁴`, (ii) means that no nonzero vector is fixed by all of\n`D(H)`, and (iii) means that every cyclic subgroup fixes a nonzero vector. The\nformal `claim` is the conjecture; the result proves it.","title":"Four-dimensional determinant-one representations and cyclic singlets","url":"https://the-omega-institute.github.io/trureturing-pages/research/groot-nibbelink-2020-four-dimensional-singlet-branching/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31fd68ff56dee0a865bca9a932486cb3c6d203876fc123153e8cbe1be131e306.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:grosjean-huillet-2016-carleman-jordan-chain-inverse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Grosjean and T. Huillet, *Some combinatorial aspects of discrete non-linear population dynamics*, Chaos, Solitons & Fractals 93, 71–79 (2016), arXiv:1610.05919v1, §2.2, state the following conjecture (`snlmatrixRR.tex`, lines 449–452; the displayed formula is rendered inline):\n\n> Then, we conjecture that $W^{-1}(k,k^{\\prime })=\\sigma (k-1,k^{\\prime })$, $k^{\\prime }\\geq k\\geq 1$.\n\nThe paper defines $\\phi(x)=\\sum_{r\\geq1}\\alpha_rx^r$ and $P(k,k')=[x^{k'}]\\phi(x)^k$ in Eq. (2) (lines 126–141). In the critical case $\\alpha_1=1$, its normalized columns satisfy $P\\mathbf w_1=\\mathbf w_1$, $P\\mathbf w_k=\\mathbf w_{k-1}+\\mathbf w_k$ for $k>1$, $\\mathbf w_1=e_1$, $\\mathbf w_k(1)=0$ for $k>1$, and $\\mathbf w_k(r)=0$ for $r>k$ (lines 403–428). Its recursion divides by $P(r,r+1)=r\\alpha_2$ (lines 420–436), so this construction requires $\\alpha_2\\ne0$.\n\nFor $m\\geq1$, $\\sigma(m,j)$ is the sum of $\\prod_{\\ell=1}^mP(i_\\ell,i_{\\ell+1})$ over $1=i_1<\\cdots As a concrete example, computer experiments indicate that the matrices with ``kite-shaped support'', $A_{ij}=B_{ij}=1$ iff $i+j\\leq n+1$, give a solution for all $n$ not congruent to $1$ modulo $3$.\n\nThe quotation is from the downloaded v2 TeX source `artisanal.tex`, MD5 `29b677f0287fbbb335cd79eb4f5f33d2`. Its surrounding paragraph (lines 2304–2316) takes $A,B$ to be nonsingular symmetric binary matrices with $AB-I$ also nonsingular, and uses $N=\\operatorname{diag}(A,B)$ as the coefficient matrix of a quadratic exponential. See [the literature note](../Library/QuadraticForms/grossgoedicke2025artisanal.md).\n\nFor $n\\in\\mathbb N$, let $K_n\\in M_n(\\mathbb F_2)$ have entry $1$ exactly when $i+j\\le n+1$, with indices $1,\\ldots,n$, and entry $0$ otherwise; at $n=0$ this is the empty matrix. Put $N_n=\\operatorname{diag}(K_n,K_n)$ and $J_n=\\begin{pmatrix}0&I_n\\\\-I_n&0\\end{pmatrix}$.\n\nIssue [#20969](https://github.com/the-omega-institute/trureturing/issues/20969) fixes the following reading, including the converse to the source's observation: for every $n\\ge1$,\n\n- $K_n$ is symmetric and has trivial right kernel;\n- $N_n$ has trivial right kernel;\n- $K_n^2-I_n$ has trivial right kernel if and only if $n\\not\\equiv1\\pmod3$;\n- $N_n+J_n$ has trivial right kernel if and only if $n\\not\\equiv1\\pmod3$.\n\nThis matrix statement is proved.","title":"The binary kite construction works exactly when $n$ is not one modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/gross-goedicke-2025-kite-shaped-quadratic-forms/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a5cbc670652bbaea40a99c01dd9b27b22b65e920d759deba663eaed6db0b8ad7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:grunwald-pales-2024-gini-mean-comparison","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Grünwald and Zs. Páles, *Comparison of Gini means with fixed number of variables*, arXiv:2408.07658v1, §4, state verbatim:\n\n> We conclude this paper by formulating two conjectures that we have not been able to verify. The first conjecture seems to be a natural extension of the formula for Γ2(I), if 0 < inf I < sup I < ∞.\n\nConjecture 1 states verbatim:\n\n> Conjecture 1. Let n ≥ 3 and let I be a subinterval of R+ such that 0 < a := inf I < sup I =: b < ∞. Then Γn(I) = {((r, s), (p, q)) ∈ R² × R² | r + s ≤ p + q, G[n]r,s(a, …, a, b) ≤ G[n]p,q(a, …, a, b), and G[n]r,s(a, b, …, b) ≤ G[n]p,q(a, b, …, b)}.\n\nHere $\\Gamma_n(I)$ is the set of parameter pairs satisfying $G_{r,s}(x)\\le G_{p,q}(x)$ for all $x\\in I^n$. For $r\\ne s$ and positive coordinates,\n\n$$\nG_{r,s}(x)=\\left(\\frac{\\sum_{i=1}^n x_i^r}{\\sum_{i=1}^n x_i^s}\\right)^{1/(r-s)}.\n$$\n\nThe journal bibliographic reference is *Results in Mathematics* 80 (2025), article 54; the quotations and locators refer to the arXiv version.","title":"Endpoint comparison of Gini means on a bounded interval","url":"https://the-omega-institute.github.io/trureturing-pages/research/grunwald-pales-2024-gini-mean-comparison/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac8c07a5798f9b79e741bfdf9fabedf13c1f596a4f570ca3adc3d8cf57575ab9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:grunwald-pales-2024-gini-stationary-test","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Grünwald and Zs. Páles, *Comparison of Gini means with fixed number of variables*, arXiv:2408.07658v1, §4, state:\n\n> Conjecture 2. Assume that 0 = inf I · I⁻¹ and let n ≥ 3. Then ((r, s), (p, q)) ∈ Γ_n(I) if and only if, for all u₁, …, u_n ∈ ℝ₊ such that G^{[n]}_{r,s}(u₁, …, u_n) = 1, ∃ρ ∈ ℝ such that χ′_{p,q}(u_k) + ρχ′_{r,s}(u_k) = 0 (k ∈ {1, …, n}) the inequality 1 ≤ G^{[n]}_{p,q}(u₁, …, u_n) holds.\n\nHere $I$ is a nonempty interval in $(0,\\infty)$ and $\\Gamma_n(I)$ consists of the parameter pairs satisfying $G_{r,s}(x)\\le G_{p,q}(x)$ for every $x\\in I^n$. The stationary test universally quantifies over positive normalized tuples satisfying the common-multiplier equations. It does not assert that every normalized tuple admits such a multiplier.\n\nThe formal reading is:\n\n| Source object or condition | Lean encoding and meaning |\n|---|---|\n| $n\\ge3$, $x\\in I^n$ | `n : ℕ`, `3 ≤ n`, `x : Fin n → ℝ`, and `∀ i, x i ∈ I` |\n| Nonempty positive interval $I$ | `I ⊆ Set.Ioi 0`, `I.OrdConnected`, and `I.Nonempty` |\n| $\\inf(I\\cdot I^{-1})=0$ | `sInf ((fun z : ℝ × ℝ => z.1 / z.2) '' (I ×ˢ I)) = 0`; the set is precisely $\\{a/b:a,b\\in I\\}$, rather than a product of two scalar infima |\n| $G_{p,q}$, $p\\ne q$ | `Defs.giniMean` delegates to `GiniMeanEndpointComparison.giniMean`, namely $(\\sum x_i^p/\\sum x_i^q)^{1/(p-q)}$ |\n| $G_{p,p}$ | $\\exp(\\sum x_i^p\\log x_i/\\sum x_i^p)$, the equal-parameter branch of `Defs.giniMean` |\n| $\\chi_{p,q}$ | `Defs.chi`: $(t^p-t^q)/(p-q)$ for $p\\ne q$, and $t^p\\log t$ for $p=q$ |\n| $\\chi'_{p,q}$ | `Defs.chiDeriv p q t := deriv (Defs.chi p q) t` |\n| $\\Gamma_n(I)$ | `Defs.Gamma n I`: comparison for every tuple from $I$ |\n| Stationary criterion | `Defs.stationaryTest n r s p q`: positivity, $G_{r,s}(u)=1$, and existence of a common real multiplier imply $1\\le G_{p,q}(u)$ |\n| Claimed equivalence | `Defs.claim`: all the displayed interval conditions, all $n\\ge3$, and all real $r,s,p,q$ |","title":"Stationary characterization of Gini mean comparisons","url":"https://the-omega-institute.github.io/trureturing-pages/research/grunwald-pales-2024-gini-stationary-test/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e44548bebf227468a3916b2ac7e3880f79926c4794040a9110cc0f007ed19d52.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gschwendtner-bluhm-winter-2021-pure-program-states","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gschwendtner, Bluhm and Winter, *Programmability of covariant quantum channels*,\nQuantum **5**, 488, DOI 10.22331/q-2021-06-29-488, arXiv:2012.00717v2,\nSection 5, ask:\n\n> We leave it as an open question whether the upper bounds can be improved to match the lower ones also for pure program states.\n\nThe setting is a compact group, an irreducible unitary input representation\n$U$, and an arbitrary unitary output representation $V$. A single exact\nprocessor must implement every $UV$-covariant channel. The mixed optimum is\n$d_c=\\sum_k n_k$, where the commutant of $\\bar U\\otimes V$ is\n$\\bigoplus_k\\mathbb1_{b_k}\\otimes\\mathcal B(\\mathbb C^{n_k})$.\nThe affirmative assertion is that pure programs attain this same dimension.\nIssue [#15518](https://github.com/the-omega-institute/trureturing/issues/15518)\nfixes the source question and its quantifiers.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Compact group $G$ | `claim` quantifies over a group with `Fintype` | A finite group is compact in its discrete topology; this restriction suffices for a counterexample. |\n| Unitary $U,V$ | `MonoidHom` into `Matrix.unitaryGroup` | Every group element acts by its coerced complex matrix. |\n| Irreducible input | Frozen `NiceErrorBasisNonNormalStabilizer.IrredProj` | Its Euclidean-space invariant subspaces are transported by `WithLp.linearEquiv` to coordinate functions. |\n| Completely positive, trace-preserving channel | Supporting `IsChannel` | A finite complete Kraus family and the frozen `MatrixMap.of_kraus` action. |\n| $UV$ covariance | `IsCovariant` | The literal two-sided conjugation equality, for every group element and input matrix. |\n| One fixed processor for every channel | `MixedProgrammable`, `PureProgrammable` | The processor is existentially quantified before the channel. |\n| Mixed density program | Frozen `GHZMeasureBiseparableBound.IsDensity` | The literal positive-semidefinite, trace-one predicate on `Matrix`; `DensityState` uses a different `CStarMatrix` subtype carrier. |\n| Pure density program | Frozen `PureStateHandshake.rankOneDensity` and normalized vector | Its body is `vecMulVec` with the starred vector; normalization is the conjugate dot product equal to one. |\n| Tensor input | Mathlib `Matrix.kronecker` | The product-index matrix has the source tensor expression. |\n| Match the mixed lower bound | `claim` | Every mixed program dimension also admits pure programs. The source's Theorems 25 and 31 identify the mixed optimum with $d_c$. |\n| Negative answer | Settling `result` | A closed negation of `claim`, with $G=\\mathrm{DihedralGroup}\\ 4$, $d_1=d_2=d_P=2$. |\n\nThe arbitrary finite input index types in `IsChannel` include the processor's\nproduct indices. Equality on all matrices expresses exact channel equality.\nThese conventions add no restriction that could exclude the source's counterexample.","title":"Pure programs need more dimension than mixed covariant programs","url":"https://the-omega-institute.github.io/trureturing-pages/research/gschwendtner-bluhm-winter-2021-pure-program-states/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac2f84ca2cfe31519540989125f198686d47b0ca18e24ac4e6f48c5c2d719a9d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gu-2023-cross-correlation-activity-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Gu, *Thermodynamic bounds on the asymmetry of cross-correlations with dynamical activity and entropy production*, Phys. Rev. E 109, L042101 (2024), arXiv:2312.00354, bounds the normalized asymmetry χ_ab = α_ab/√(D_a D_b) of the cross-correlation of two observables in a Markov jump process on N states.\n- Eq. (8) bounds it by the dynamical activity A, the entropy production σ and the flux scale ratio γ: |χ_ab| ≤ γ/tan(π/N) · √(σ/(2A)).\n- The author proves Eq. (8) for unicyclic networks and the weaker Eq. (9) in general, and writes: \"We conjecture that Eq. (8) is valid for arbitrary networks\". The evidence is 10⁶ random samples on three four-state topologies.\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/gu2024crosscorrelationactivity.md).\n\nIssue [#18028](https://github.com/the-omega-institute/trureturing/issues/18028) fixes the reading: Eq. (8) holds for every N ≥ 3, every stationary network with bidirectional support and a positive flux, and all observables with D_a D_b > 0. This conjecture is refuted.","title":"The activity bound on cross-correlation asymmetry fails for general networks","url":"https://the-omega-institute.github.io/trureturing-pages/research/gu-2023-cross-correlation-activity-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f09ade828869d4b53c361a28aaad0ae6eb21be51cbc17ecc46b3ef199d4e18a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gu-2023-pseudo-coarse-grained-tradeoff","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jie Gu, *Speed limit, dissipation bound and dissipation-time trade-off in thermal relaxation processes*, Phys. Rev. E 108, L052103 (2023), DOI 10.1103/PhysRevE.108.L052103, arXiv:2304.08752v2, states the following conjecture in the section “Trade-off between dissipation and time,” Eq. (13). The verbatim sentence and equation are from `relaxation.tex`, lines 328 and 331 (the equation's mathematical content is rendered inline):\n\n> Let us state the third main result: for $N$-state dynamics evolving from $\\boldsymbol{p}^\\text{(i)}$ to $\\boldsymbol{p}^\\text{(f)}$ during $\\tau$ with an total rate $\\mathcal{W}$, the conjecture is that the entropy production $\\Sigma$ is bounded from below by (referred to as trade-off bound)\n>\n> $\\Sigma_\\text{na} \\ge \\Sigma_{\\Omega_i}(\\mathcal W \\tau) \\quad i =1,2,\\ldots,S_N^{(2)}.$\n\nThe displayed inequality concerns non-adiabatic entropy production, despite the sentence's use of $\\Sigma$. Lines 333–337 define each binary comparison by a partition into two nonempty sets, the aggregate endpoints, and the original process's total rate and elapsed time. The source conventions and the two-state steady-state formula are also recorded in [the literature note](../Library/StatisticalMechanics/gu2023dissipationtimetradeoff.md).\n\nIssue [#21842](https://github.com/the-omega-institute/trureturing/issues/21842) fixes the following restricted reading, encoded by `claim`. For every natural number $n$, rate array $a:\\mathrm{Fin}\\,n\\to\\mathrm{Fin}\\,n\\to\\mathbb R$, vector $u$, real curve $P:\\mathbb R\\to\\mathbb R^n$, real time $\\tau$, and finite subset $\\Omega$ of the states, put\n\n$W_{ij}=a_{ij}$ for $i\\ne j$, and $W_{jj}=-\\sum_{i\\ne j}a_{ij}$.\n\nAssume all of the following:\n\n- $a_{ij}>0$ whenever $i\\ne j$;\n- $u_i>0$ for every state, $\\sum_i u_i=1$, and $Wu=0$;\n- $P$ has derivative $WP(t)$ at every real $t$;\n- $\\tau>0$, every coordinate of $P(0)$ is positive, $\\sum_iP_i(0)=1$, and every coordinate of $P(\\tau)$ is positive;\n- $\\Omega$ and its complement are both nonempty.\n\nDefine $D(v\\|w)=\\sum_i v_i\\log(v_i/w_i)$, $d(z\\|b)=D((z,1-z)\\|(b,1-b))$, and\n\n$x=\\sum_{i\\in\\Omega}P_i(0)$, $y=\\sum_{i\\in\\Omega}P_i(\\tau)$, $k=-\\operatorname{Tr}W$, $s=x+(y-x)/(1-e^{-k\\tau})$.\n\nWhenever $0 We conclude from Example~5 that complementarity is a necessary but not a sufficient condition for IC. It is an open problem to find a simple characterization of IC for a pair $\\brac{f,g}\\subseteq\\rscript (X)$.\n\nThe source macros denote $\\{f,g\\}$ and $\\mathcal R(X)$, respectively. Here $X=\\{1,\\ldots,n\\}$ with $n\\ge2$, $\\mathcal R(X)$ consists of all maps $X\\to\\mathbb R$, and a probability weight $\\mu$ has $\\mu(x)\\ge0$ and $\\sum_x\\mu(x)=1$. Its distribution under $f$ is $\\Phi_{f,\\mu}(\\Delta)=\\sum_{x:\\,f(x)\\in\\Delta}\\mu(x)$ for every $\\Delta\\subseteq\\mathbb R$ (TeX lines 435–439). The pair is informationally complete (IC) when, for every two probability weights $\\mu,\\nu$, equality of their distributions under both $f$ and $g$ implies $\\mu=\\nu$ (line 474).\n\nLet $\\Gamma(f,g)$ have vertices in two tagged copies of $\\mathbb R$, with a left value $a$ adjacent to a right value $b$ exactly when some $x\\in X$ satisfies $f(x)=a$ and $g(x)=b$. Issue [#23501](https://github.com/the-omega-institute/trureturing/issues/23501) fixes the reading:\n\n$\\{f,g\\}$ is IC if and only if $x\\mapsto(f(x),g(x))$ is injective and $\\Gamma(f,g)$ is acyclic.\n\nInjectivity is precisely the source's complementarity condition: every nonempty intersection of an $f$-level set and a $g$-level set is a singleton (lines 507–508). The characterization is proved, for every finite domain, without the restriction $n\\ge2$. The definitions and source locators are also recorded in [the literature note](../Library/Observer/gudder2019finiteconvexeffect.md).","title":"Informationally complete pairs are exactly complementary pairs with forest fiber graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/gudder-2019-informationally-complete-pairs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11a91d7c404bc761fa9b7a1bdd406ddd371f650fc3f6a788b006f12585f373a9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gudder-2021-conditionally-random-uniform-attenuation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stan Gudder, *Mutually Unbiased Quantum Observables* (2021), arXiv:2110.03099v1, Section 2, asks:\n\n> We do not know whether the converse of Theorem~\\ref{thm21}(i) or (ii) holds.\n\nThis is the first sentence of line 223 of the downloaded v1 TeX, `MutualUnbiasedQO.tex` (576 lines, MD5 `ea23a0dd97764fa2b7444408b910866f`); the reference resolves to Theorem 2.1. Its first two parts, lines 186–187, read, with typographical commands expanded:\n\n> (i) Condition (1) implies Condition (2).\n\n> (ii) If $A$ and $B$ are sharp, then Condition (1) and Condition (2) are equivalent.\n\nFor finite observables $A=(A_x)_{x\\in\\Omega_A}$ and $B=(B_y)_{y\\in\\Omega_B}$, with $|\\Omega_A|=m\\ge1$ and $|\\Omega_B|=n\\ge1$, the source specifies at line 182:\n\n> Condition~(1) is shorthand for $A_x\\circ B_y=\\tfrac{1}{n}\\,A_x$ and $B_y\\circ A_x=\\tfrac{1}{m}B_y$ for all $x\\in\\Omega _A$, $y\\in\\Omega _B$ and Condition~(2) is shorthand for $(B\\mid A)_y=\\tfrac{1}{n}\\,I$ and $(A\\mid B)_x=\\tfrac{1}{m}\\,I$ for all $x\\in\\Omega _A$, $y\\in\\Omega _B$.\n\nHere $E\\circ F=E^{1/2}FE^{1/2}$, using the positive square root, and $(B\\mid A)_y=\\sum_x A_x\\circ B_y$ (Section 1, lines 97 and 117–120). Each observable consists of positive semidefinite effects summing to $I$. The source calls Condition (1) uniform attenuation and Condition (2) conditional randomness. The definitions and source locators are also in [the literature note](../Library/QuantumContext/gudder2021mutuallyunbiasedobservables.md).\n\nIssue [#21653](https://github.com/the-omega-institute/trureturing/issues/21653) fixes the statement: for every finite-dimensional complex Hilbert space and all finite observables with positive outcome counts, Condition (2) implies Condition (1), without assuming sharpness. This converse of Theorem 2.1(i) is proved. The separately worded, ambiguous question about a converse of part (ii) is outside the registered statement.","title":"Conditional randomness implies uniform attenuation without sharpness","url":"https://the-omega-institute.github.io/trureturing-pages/research/gudder-2021-conditionally-random-uniform-attenuation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8bf1afe311b55319921f526df3f851b8af97947b8c3ca6a79b9f982e3ed620cd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gudder-2023-sob-effect-algebra-luders","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stanley Gudder, *Dual Quantum Instruments and Sub-Observables*, arXiv:2302.12243v1, Section 3, after Theorem 3.1:\n\n> If $\\mathcal I\\in\\operatorname{In}(H)$ and $U_{\\mathcal I}=\\{\\mathcal I_a^*:a\\in\\mathcal E(H)\\}$, then we conjecture that $U$ need not be a Sob effect algebra.\n\nExample 3 asks:\n\n> If $\\mathcal L_x(\\rho)=a_x\\circ\\rho$ is an arbitrary Lüders instrument, we do not know whether $U_{\\mathcal L}$ is a Sob effect algebra. However, in the case where $a_x$ are projections (we then call $\\mathcal L$ sharp) we can show that it is.\n\nThis is a Tier 1, explicitly published conjecture and question, preregistered as [#16759](https://github.com/the-omega-institute/trureturing/issues/16759). The conjecture is **Proved** by a finite counterexample to universal closure; the arbitrary Lüders question is answered negatively. The universal closure assertion is `claim`, and the settling theorem is `result : ¬ claim`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Common outcome space, positive additive outcome values, total at most $I$ | `Fin n`, finite event sums, `IsSubObservable` | Each outcome is PSD and the complement of the total is PSD. Finite all-events space suffices for an existential counterexample. |\n| Effect $0\\le a\\le I$ | PSD body in `ludersImage`; `TypedEscapeNeedsPositivity.IsEffect` in the proofs | Both $a$ and $I-a$ are PSD, equivalently $0 \\le a \\le I$ under `MatrixOrder`. |\n| $U\\subseteq\\operatorname{Sob}(H)$ | First conjunct of `IsSobEffectAlgebra` | Every member is a sub-observable. |\n| (S1) an observable $Z\\in U$ | Existential $Z$, with sub-observability and total $I$ | The observable body is inlined literally; no duplicate `IsObservable` definition is added. |\n| (S2) $Z-A\\in U$ for $A\\in U$ | Complement conjunct with the same $Z$ | No separate choice of $Z$ is allowed. |\n| (S3) $A,B\\in U$ and $A+B\\in\\operatorname{Sob}(H)$ imply $A+B\\in U$ | Final conjunct | Pointwise addition with ambient sub-observability as antecedent. |\n| Lüders action $K_xaK_x$, $K_x=a_x^{1/2}$ | `ludersImage`; PSD $K_x$ and $\\sum_x K_xK_x=I$ in `claim` | A single effect supplies every outcome; the image equality has the preregistered orientation. |\n| Image need not be a Sob effect algebra | `result` | Refutes universal closure at $d=n=2$; it proves the existential conjecture. |\n\nAll binders describe the source's finite matrix representation. There is no additional mathematical hypothesis on the settling theorem and no dropped clause.","title":"An unsharp qubit Lüders image violates ambient Sob sum closure","url":"https://the-omega-institute.github.io/trureturing-pages/research/gudder-2023-sob-effect-algebra-luders/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d3e3efcd11a707af5f5066269a61a6645dc816d391a5f57833d7a6a0e8e5827.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guerrero-2022-vertex-energy-degree-bounds-remark-6","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Guerrero, *Laplacian Energies of Vertices*, arXiv:2201.01252v1,\nRemark 6 (Conjecture), page 13, states:\n\n> According to numerical experiments the analog result is true for the energy\n> of vertices, that is: $d_{\\min}LE_G(v)\\le E_G(v)\\le d_{\\max}LE_G(v)$ for all $v\\in V$.\n\nThe standing assumptions are that $G$ is simple and connected. The definitions\nare $E_G(v)=|A|_{vv}$, $L=I-D^{-1/2}AD^{-1/2}$ and\n$LE_G(v)=|L-I|_{vv}$. The [Library note](../Library/GraphInvariants/guerrero2022laplacianenergies.md)\ncontains the source quotations and locators. The preregistration is\n[#19702](https://github.com/the-omega-institute/trureturing/issues/19702).\nThe v1 conjecture is refuted by the path on six vertices at zero-based index $2$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite simple graph | `V`, `Fintype V`, `SimpleGraph V` in `claim` | Undirected loopless graph; arbitrary finite vertex type |\n| Connected | `G.Connected` | Hypothesis on the entire conjunction |\n| Every vertex | `∀ v : V` | Universal quantifier |\n| $\\lvert A\\rvert$ and $E_G(v)=\\lvert A\\rvert_{vv}$ | `matAbs`, `vertexEnergy`, `SimpleGraph.adjMatrix` | Real symmetric spectral absolute value via `cfc` |\n| $D^{-1/2}=\\operatorname{diag}(1/\\sqrt{d_v})$ | `degInvSqrt` | Natural degrees coerced to real numbers |\n| $L=I-D^{-1/2}AD^{-1/2}$ | `normLap` | Literal real matrix expression |\n| $LE_G(v)=\\varphi_v(\\lvert L-I\\rvert)$ | `normLapVertexEnergy` | $\\varphi_v(M)=M_{vv}$ |\n| Both degree inequalities | Conjunction in `claim` | `minDegree` and `maxDegree` coerced to real numbers |\n| Negative answer | `result : ¬ claim` | One graph and one vertex refute the universal conjunction |\n| Implementation instances | `DecidableEq V`, `DecidableRel G.Adj` | Finite graph operations; no additional mathematical restriction |\n\nReal division fixes reciprocal zero to zero. The counterexample has positive\ndegrees, so this totalization does not affect the refutation.","title":"The upper local degree comparison fails on P6","url":"https://the-omega-institute.github.io/trureturing-pages/research/guerrero-2022-vertex-energy-degree-bounds-remark-6/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d32d4935db9bb28610d4c6229e3a39b74592cf6c85e0f97ad176fc3e5f0ce968.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guha-et-al-2016-multilayer-determinant-root-nonuniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Guha, D. Towsley, P. Nain, Ç. Çapar, A. Swami and P. Basu, *Spanning connectivity in a multilayer network and its relationship to site-bond percolation*, arXiv:1402.7057v4 (Phys. Rev. E 93, 062310):\n\n> Numerically, it is easy to verify that $\\det (\\bA (q))$ has a unique zero in $[0,1]$ for all $M \\ge 1$ as long as $C > 1$. We conjecture that this is always true.\n\nHere A(q) is the source's M × M matrix:\n- A_ij = δ_ij − C (binom(M, j) − binom(M − i, j)) q^j (1 − q)^{M−j} for i, j ∈ {1, …, M};\n- C = Σ_k (k − 1) k p_k / z for a degree distribution p with mean degree z.\n\nThe definitions are quoted in [the literature note](../Library/StatisticalMechanics/guha2016multilayerpercolation.md).\n\nIssue [#19823](https://github.com/the-omega-institute/trureturing/issues/19823) fixes the reading: for every M ≥ 1 and every finitely supported degree distribution with C > 1, det A(q) would have exactly one zero in [0, 1]. The conjecture is refuted.","title":"The multilayer percolation determinant can have three zeros in [0, 1]","url":"https://the-omega-institute.github.io/trureturing-pages/research/guha-et-al-2016-multilayer-determinant-root-nonuniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ee17955935494977efcb5bac2711000aa1e52c8e92d4d0e22ddae4f5b4cee686.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guo-2023-partial-negativity-concavity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Guo, *Partial-Norm of Entanglement: Entanglement Monotones That are not\nMonogamous*, New J. Phys. 25 (2023) 083047, arXiv:2212.06521v6, states:\n\n> We conjecture that $\\hat{h}$ is concave [Suppl].\n\nFor decreasing Schmidt coefficients $\\lambda_1\\ge\\lambda_2\\ge\\cdots$,\nthe paper defines $\\hat N(|\\psi\\rangle)=\\lambda_1\\lambda_2$ and the reduced\nfunction $\\hat h(\\rho)=\\sqrt{\\delta_1\\delta_2}$, where\n$\\delta_j=\\lambda_j^2$. Its definition of concavity is\n\n$$\nh(t\\rho+(1-t)\\sigma)\\ge t h(\\rho)+(1-t)h(\\sigma),\\qquad 0\\le t\\le1.\n$$\n\nIssue [#14495](https://github.com/the-omega-institute/trureturing/issues/14495)\npreregisters the reading over every finite dimension $d\\ge2$ and all\npositive semidefinite complex $d\\times d$ matrices of trace one. The\nlater paper *Complete Genuine Multipartite Entanglement Monotone*,\narXiv:2301.00334v3, explicitly identifies $\\delta_1,\\delta_2$ as the two\nlargest eigenvalues of $\\rho$ and repeats the conjecture in its table\nfootnote. Eigenvalues are counted with multiplicity; their positions are\nchosen separately for each state, rather than tied to fixed eigenvectors.","title":"The reduced function of the partial negativity is not concave","url":"https://the-omega-institute.github.io/trureturing-pages/research/guo-2023-partial-negativity-concavity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa2894b5ea3fc66302dc7def827255b18115b8602176075cfbe62cef14a58d09.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guo-2024-partial-negativity-hhat-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Guo, *Complete Genuine Multipartite Entanglement Monotone*, Results in\nPhysics 57 (2024), 107430, arXiv:2301.00334v3, states the reduced function\n$\\hat h(\\rho)=\\sqrt{\\delta_1\\delta_2}$, where $\\delta_1$ and $\\delta_2$ are\nthe two largest eigenvalues of $\\rho$, and conjectures in equation (24) that\n$\\hat h(\\rho_{AB})\\leq\\hat h(\\rho_A)+\\hat h(\\rho_B)$ for every bipartite\ndensity matrix.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| two largest eigenvalues counted with multiplicity | `hHat` uses positions 0 and 1 of the decreasing `eigenvalues₀` list |\n| every pair of local dimensions | `claim` quantifies `dA dB : ℕ` |\n| every bipartite state | `claim` quantifies `ρ : Matrix (Fin dA × Fin dB) (Fin dA × Fin dB) ℂ` and assumes `IsDensity ρ` |\n| reduced states are the two partial traces | `ρ.traceRight` and `partialTraceFirst ρ` in `claim` |\n| subadditivity inequality (24) | `claim` concludes `hHat ρ ≤ hHat ρ.traceRight + hHat (partialTraceFirst ρ)` |\n| refutation | `result : ¬ claim` |","title":"Guo's partial-negativity subadditivity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/guo-2024-partial-negativity-hhat-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc4211b6b25b88baa5ab2b159f33e13b5fbf1f63646edbbb3206fae2593cc130.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guo-han-automatic-apwenian-one-odd-letter","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ying-Jun Guo and Guo-Niu Han, *On a family of automatic apwenian sequences*, Discrete Mathematics 348 (2025) 114399,\nConjecture 2. A sequence a over ℕ is apwenian if a(0) = 1 and a(n) ≡ a(2n+1) + a(2n+2) (mod 2) for every n. Let Σ be\na finite set of nonnegative integers whose only odd element is 1, and let a be a fixed point of a p-uniform morphism on\nΣ with p ≥ 2. The conjecture states that a is apwenian if and only if a(n) ≡ P(n) (mod 2) for every n, where P is the\nperiod-doubling sequence, the fixed point of 1 ↦ 10, 0 ↦ 11.","title":"Purely Automatic Apwenian Sequences over an Alphabet with One Odd Letter","url":"https://the-omega-institute.github.io/trureturing-pages/research/guo-han-automatic-apwenian-one-odd-letter/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f36e4204d86f7966909c2a1676f582759d5c27cf42ea3c4781f57b6dedf63035.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guo-han-binary-automatic-apwenian-level-generation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ying-Jun Guo and Guo-Niu Han, *On a family of automatic apwenian sequences*, Discrete Mathematics\n348(5), 114399 (2025), Section 3, Conjecture 1, printed page 6 of the author manuscript:\n\n> Let {u(n)}ₙ≥₀ be a 2-automatic apwenian sequence over {0,1}. Then there exist a finite alphabet\n> Σ ⊂ N and a sequence σ over A(Σ) such that {u(n)}ₙ≥₀ = ϕ(σ) (mod 2).\n\nThe full domain consists of every binary sequence computed by a finite DFAO on canonical binary\nexpansions in most-significant-first order, with u(0) = 1 and\nu(n) ≡ u(2n+1) + u(2n+2) modulo two for every nonnegative n. The alphabet Σ is a finite set of\nnatural numbers containing one. Every member of A(Σ) takes each letter of Σ to exactly two letters\nof Σ whose sum is congruent to the original letter modulo two. Set X₀ = [1] and Xⱼ₊₁ = σⱼ(Xⱼ).\nThe generated sequence ϕ(σ) is the concatenation X₀X₁X₂⋯, and the conclusion is equality of its\nparity with u at every nonnegative index.","title":"Binary automatic apwenian sequences have finite level generators","url":"https://the-omega-institute.github.io/trureturing-pages/research/guo-han-binary-automatic-apwenian-level-generation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ba3e2d9b354ed31905a4fceb74a5ed684450a8c08f19b1b2edbc0aaae3ad18b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:guo-jin-2025-max-marginal-entropy-locc","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Guo and Z.-X. Jin, *Measure of entanglement and the monogamy relation: a topical review*, arXiv:2512.21992v1, Section 9.4, p. 88, state:\n\n> Note here that we can not prove here E′^(n) is nonincreasing on average under LOCC, but we conjecture that E′^(n) does not increase on average under LOCC for these cases mentioned above. Hereafter we always assume the conjecture is true.\n\nEquation (189) defines $E'^{(n)}(|\\psi\\rangle)=\\max_i h(\\rho^{A_i})$, and Table 16 includes the von Neumann entropy $h=S$. Section 2 defines $S(\\rho)=-\\operatorname{Tr}(\\rho\\log_2\\rho)$. The settlement refutes the unrestricted entropy assertion using four parties of local dimension nine and one two-outcome local projective measurement.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite pure multipartite state | The amplitude argument of `claim`, indexed by `Fin n → Fin d` | Every finite party count and common local dimension; the witness has four parties and dimension nine. |\n| Reduced state of party $i$ | `marginal` | The literal coordinate partial trace from the binding conventions in #20726. |\n| $\\max_i S(\\rho^{A_i})$ | `maxMarginalEntropy` | The finite real supremum of the literal trace-log expression, divided by $\\log 2$. |\n| One local measurement, an LOCC operation | `applyLocal` and the Kraus-completeness hypothesis in `claim` | One coordinate changes, and the sum of adjoint products is the identity. A violation for this subclass refutes LOCC monotonicity. |\n| Normalized input, probabilities, normalized branches | The finite squared-norm sums, norm-one hypothesis, and inverse-square-root scaling in `claim` | The binding convention is inlined literally; zero-probability branches contribute zero. Proofs reuse `finiteComplexEnergy` for that same sum. |\n| Does not increase on average | The concluding inequality in `claim` | The maximum is inside the outcome average. |\n| Refutation | `result` | Negation of the complete `claim`; no extra hypotheses. |\n\nThe real supremum represents a maximum for the nonempty finite party set in the witness. The claim extends the trace expression to arbitrary matrices, but its counterexample consists of normalized pure states and complete projectors. No positivity or physical-realization gap is used.","title":"Maximum single-party entropy need not decrease on average under LOCC","url":"https://the-omega-institute.github.io/trureturing-pages/research/guo-jin-2025-max-marginal-entropy-locc/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8dd5891f9d4e27228c8717857fe6c08942f8d57a58a0db7e8b4ac1038f2c7761.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gupta-2026-miura-grid-dispersion-inequality-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gupta, *Height functions on the m × n Miura-ori flip graph: degree sequence and diameter*, arXiv:2606.22614v2, Section 4, inequality (4), p. 14, asks for\n\n$\\operatorname{disp}(\\phi)\\le D(m,n)$\n\nfor every integer $1$-Lipschitz function on the rectangular grid, realisable or not. Here\n\n$\\operatorname{disp}(\\phi)=\\min_{K\\in\\mathbb Z}\\sum_v|\\phi(v)-K|$,\n\nand\n\n$D(m,n)=\\min_{K\\in\\mathbb Z}\\sum_{i=1}^m\\sum_{j=1}^n|i+j-K|$.\n\nThe named external problem and literal conventions are preregistered in [#19839](https://github.com/the-omega-institute/trureturing/issues/19839).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Rectangular grid with horizontal and vertical neighbours | `Fin m × Fin n`, adjacency in `SimpleGraph.pathGraph m □ SimpleGraph.pathGraph n` | Source cell $(i,j)$ is represented by $(i-1,j-1)$. |\n| Every integer function, realisable or not | Universal function binder in `claim` | No origami realisability assumption. |\n| Adjacent differences at most one | `IsLipschitz` | Literal edge predicate. |\n| Minimum over integer centres, each vertex counted once | `disp` | Natural-valued infimum of a nonempty integer-cost range, hence its minimum. |\n| One-indexed coordinate sum | `D` | Both zero-based coordinates are incremented by one. |\n| Inequality (4) for all positive rectangles | `result : claim` | All natural dimensions are covered; empty rectangles add the zero-cost case. |\n\nThe public definitions `IsLipschitz`, `disp`, `D` and `claim` display the source expressions. The existing rectangular graph owner `LeeLiuThreeRowIntegerDomination.grid` is used through the private correspondence `lipschitz_owner`; the public predicate retains its literal graph expression.","title":"Gupta's rectangular-grid dispersion inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/gupta-2026-miura-grid-dispersion-inequality-4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b4a19a9624868012605b7d5f4cb021ee67e4a0ac095171a12b755a1b976bbfa1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gupta-2026-miura-residual-degree-conjecture-83","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chakshu Gupta, *One construction for the Miura-ori flip-graph degree sequence*, arXiv:2607.05567v3, Conjecture 8.3 (Degree bound), states: “For d ≥ 5 and every split (a,b) with a+b=d and a≤b, the remainder R_(a,b) has total degree at most d−3.” Lemma 8.2 identifies one bivariate polynomial on all m,n≥2d−1. Section 7, Lemma 7.3 identifies the remainder with the configurations whose minima do not all lie on one common boundary side. The source's splits have positive a and b.\n\nThe original question is preregistered in [issue #22211](https://github.com/the-omega-institute/trureturing/issues/22211), [preregistration](https://github.com/the-omega-institute/trureturing/issues/22211#issuecomment-6412152003).\n\n### Statement echo\n\n| Original clause | Formal counterpart |\n| --- | --- |\n| Every d≥5, positive a+b=d, a≤b | The complete telescope of `MiuraResidualCount.claim` |\n| One rational bivariate polynomial before all dimensions | `∃ p : MvPolynomial (Fin 2) ℚ` precedes `∀ m n`; the witness is `MiuraPolynomialCoreSum.corePolynomial a b` |\n| Total degree at most d−3 | `p.totalDegree ≤ d - 3` |\n| Every m,n≥2d−1 | Both universal dimension binders and their original bounds |\n| Every normalized integer unit-edge height function counted once | `Residual` on the complete function `Fin m × Fin n → ℤ`, normalized at the origin; `residualCount` is the cardinality of that proved finite set |\n| Exactly a strict minima and b strict maxima | Separate exact cardinalities of `minima` and `maxima`, testing every existing physical neighbour |\n| No single common physical side contains all minima | The negation of `∃ sd, AllOnSide ... sd`; corners belong to both incident sides |\n\nNo symmetry quotient, identification of orientations, assumed fiber cardinality or replacement counting object enters the claim. Zero-based `Fin` coordinates are sent to the source's one-based physical coordinates by adding one to each coordinate. All subtractions in the formal bounds are natural-number subtractions.","title":"Gupta's uniform residual degree bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/gupta-2026-miura-residual-degree-conjecture-83/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8de6ab2bd1938dc5ad4fc47e4b4208407970a835c501175a2b7793f12a4b2a60.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gupta-iosevich-2026-edge-complexity-weak-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"V. Gupta, A. Iosevich, J. Iosevich, B. Song and H. Tian, *Edge complexity of graphs*, arXiv:2607.15598v1, Section 3, Remark after Corollary 3.10:\n\n> Remark. The coprimality hypothesis is not merely a technicality in the present argument: it is what permits the product labeling to be viewed as a cyclic labeling. Proposition 3.7 does not prove closure under arbitrary weak products. Consequently, unrestricted closure of the equality class under the weak product has not been established. Whether such closure holds is a natural open question.\n\nThe assertion is that any two finite simple graphs with positive size satisfying $\\operatorname{FR}_{\\min}(G)=E(G)/\\sqrt{2s(G)}$ have a weak product satisfying the same equality. Definition 3.1 gives adjacency precisely when both coordinate pairs are adjacent. The Fourier transform has coefficient $1/N$, the numerator of $\\operatorname{FR}$ is entrywise $\\ell^1$, and the denominator is Frobenius; $E$ is the sum of the absolute adjacency eigenvalues.","title":"Weak-product closure of the edge-complexity equality class","url":"https://the-omega-institute.github.io/trureturing-pages/research/gupta-iosevich-2026-edge-complexity-weak-product/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aca435f71b213fd807749f191cadbdb91447f2a3e41da2e506e77080cb1fbf39.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gurses-pekcan-2025-higher-hirota-four-soliton","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gürses and Pekcan study the Hirota bilinear equations\n\n> D_x(D_x³+α₁D_t+α₂D_y)^{2k+1}{f·f}=0, k=0,1,2,…\n\nFor soliton parameters `p_i = (k_i, ω_i, l_i)` the operator has the polynomial\n`P(p) = k(k³+α₁ω+α₂l)^{2k+1}`, and the dispersion relation is\n`ω_i = −(k_i³+α₂l_i)/α₁`. The paper proves three-soliton solutions for every\n`k` and `(α₁,α₂) ≠ (0,0)`, records the four-soliton condition for `k = 1`, and\nstates, after \"we conjecture that the following lemma is valid for all k\":\n\n> The equation (bilinearProb2) has four-soliton solution only for k=0. For\n> k≥1, it does not satisfy the four-soliton solution condition (4SC)\n> directly.\n\nIssue #10059 fixes the readings: the dispersion relation is\n`k³ + α₁ω + α₂l = 0`, which also covers `α₁ = 0`; the four-soliton condition\nis the eight-term display of the source's introduction with the last factor of\nthe fifth term read as `P(p₁+p₂+p₃−p₄)` (the source prints\n`P(p₁−p₂+p₃−p₄)`, the last factor of the seventh term, and with that factor\nthe condition fails already for `k = 0`); \"has four-soliton solution for\n`k = 0`\" is read as the condition vanishing at all parameters satisfying the\ndispersion relation, and \"does not satisfy (4SC) directly\" as the existence,\nfor each `k ≥ 1`, of such parameters at which it does not vanish.","title":"Gürses and Pekcan's four-soliton conjecture for higher order Hirota forms","url":"https://the-omega-institute.github.io/trureturing-pages/research/gurses-pekcan-2025-higher-hirota-four-soliton/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/083a740cfc3a3caf2974ed66baef69a3fd9b19e6e61da1453406e3191b8a56f5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gurvich-krnc-vyalyi-conjecture-1-total-amoeba-confinement","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vladimir Gurvich, Matjaž Krnc and Mikhail Vyalyi, *Growing Trees and\nAmoebas’ Replications*, arXiv:2401.07484v1/v2 and DOI\n[10.1007/s00025-025-02421-6](https://doi.org/10.1007/s00025-025-02421-6), state\nConjecture 1 in Section 2.1 of arXiv v1 and on page 11 of the published PDF:\nif an amoeba A=(H,m) has maximum degree greater than its number of roots k by\nmore than two, then A is totally confined.\n\nThe formal target keeps the complete ordinary finite-host reading: every finite\nsimple tree H, every multiplicity function m, every extension length l, every\nfinite initial tree T containing H, every multiplicity-preserving copy and\ntransported multiplicity placement, every simultaneous family of distinct arms,\nand every minimum growth ordered by total host edges. Length zero,\nzero multiplicities and k=0 remain included. Total confinement requires a finite\nhost containing T in which every such copy is dead.","title":"gurvich krnc vyalyi conjecture 1 total amoeba confinement","url":"https://the-omega-institute.github.io/trureturing-pages/research/gurvich-krnc-vyalyi-conjecture-1-total-amoeba-confinement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9e798c1b8632cce6185afcc6c59c999bfedfc200b5bfdfb7630de4613a5e70d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gutkin-2013-orthostochastic-dimension-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Gutkin, *On a multi-dimensional generalization of the notions of orthostochastic and unistochastic matrices*, arXiv:1301.2537v1, Conjecture 1:\n\n> Denote by $d_{\\min}(n)$ the smallest value of $d$ such that the set of $d$-orthostochastic matrices coincides with $\\B_n$. Then there exists $n_0$ such that for $n\\ge n_0$ we have $2 \\le d_{\\min}(n) \\le 3.$\n\nA bistochastic real n × n matrix P is d-orthostochastic when P_rc = ‖V_rc‖² for vectors V_rc ∈ ℝ^d satisfying the source's equations (1) and (2): column orthogonality, Σ_r ⟨V_rj, V_rk⟩ = δ_jk, and row orthogonality, Σ_c ⟨V_jc, V_kc⟩ = δ_jk. The definitions are quoted in [the literature note](../Library/QuantumContext/gutkin2013orthostochastic.md).\n\nIssue [#19782](https://github.com/the-omega-institute/trureturing/issues/19782) fixes the reading:\n- d_min(n) is the natural-number infimum of the set of d for which every bistochastic n × n matrix is d-orthostochastic;\n- the claim is that some n₀ has 2 ≤ d_min(n) ≤ 3 for every n ≥ n₀.\n\nThe conjecture is refuted.","title":"The d-orthostochastic dimension is not eventually bounded by 3","url":"https://the-omega-institute.github.io/trureturing-pages/research/gutkin-2013-orthostochastic-dimension-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c0b72dfd4bd426a9ba3a1feff9660440b89942f49a2f0f7dac0d833af696d01f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:gwozdz-jakobczyk-2013-geometric-discord-mutual-correlation-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Gwóźdź and L. Jakóbczyk, *Spontaneous emission can locally create quantum discord out of classical correlations*, arXiv:1306.0130v2, Quantum Inf. Process. 13 (2014) 171–183, Section II:\n\n> C_{M}(\\ro)\\geq \\sqrt{D_{G}^{A}(\\ro)}\\geq N(\\ro).\n\n> The second inequality was shown in Ref. \\cite{Adesso} whereas the first one can be proved analytically for Bell-diagonal states and numerical simulations suggest that it is valid for all mixed two-qubit states \\cite{J}.\n\nD_G^A is the normalized one-sided geometric discord, d/(d − 1) times the least squared Hilbert–Schmidt distance between ρ and its image under a projective measurement of A. C_M is the maximal mutual correlation, the supremum of |ω_ρ(ab) − ω_ρ(a) ω_ρ(b)| over local elements of norm 1. Both definitions are quoted in [the literature note](../Library/QuantumContext/gwozdz2014discordspontaneous.md).\n\nIssue [#19704](https://github.com/the-omega-institute/trureturing/issues/19704) fixes the reading: for every two-qubit density matrix ρ, √(D_G^A(ρ)) ≤ C_M(ρ). The conjecture is proved.","title":"Maximal mutual correlation dominates the square root of geometric discord for all two-qubit states","url":"https://the-omega-institute.github.io/trureturing-pages/research/gwozdz-jakobczyk-2013-geometric-discord-mutual-correlation-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85db18dadfc4c7620bfd2eb505e9ff9f85142a4c54a2853ce09f09958231abcd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:h227051-cossinhmod10","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A227051, revision 10, first FORMULA states:\n\n> a(n) == 3 (mod 10) for n>0 (conjecture).\n\nIts NAME is “E.g.f.: 1/(cos(x) - sin(x)*sinh(x)), omitting the zero-valued coefficients of odd powers of x.” The source offset is `0,2`. Preregistration: [#21582](https://github.com/the-omega-institute/trureturing/issues/21582).\n\n### Statement echo\n\nOver $\\mathbb Q[[x]]$, let $D(x)=\\cos x-\\sin x\\sinh x$, $A(x)=D(x)^{-1}$ and $a_n=(2n)![x^{2n}]A(x)$. The five public definitions `sinhSeries`, `denominator`, `A`, `a` and `claim` retain these literal formal Taylor expressions. The factorial normalization uses the existing `IntegralEGFComposition.eCoeff` owner. The result states that for every natural $n\\ge1$ there is an integer $z$ with $a_n=z$ in $\\mathbb Q$ and $z\\equiv3\\pmod {10}$. The natural-number guard is exactly the source's positive-index condition; the existential integer makes integrality explicit. There is no finite cutoff or convergence hypothesis.","title":"Hanna's mixed trigonometric and hyperbolic congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/h227051-cossinhmod10/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4dab6a7478be12f8524976d35eacccfb5568d95b3a0b4022e19391f9adea8e2c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:h277403-div10","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's sole COMMENTS conjecture in OEIS A277403 states:\n\n> a(n) is divisible by 10 for n>2 (conjecture).\n\nThe NAME field states:\n\n> E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.\n\nThe FORMULA field introduces the primitive:\n\n> Let G(x) = Integral A(x) dx, then e.g.f. A(x) also satisfies:\n\nThe source is revision #21, with offset one and example beginning\n$A(x)=x+2x^2/2!+10x^3/3!+90x^4/4!+\\cdots$.\nThere is no source DOI, arXiv identifier or page number.\nThe preregistration is [#21746](https://github.com/the-omega-institute/trureturing/issues/21746).\n\n### Statement echo\n\nAll declarations below belong to\n`D5.S1.Recurrence.Residue.HannaIntegralCompositionDivTen`.\n\n| Source clause or interpretation | Lean counterpart | Scope |\n| --- | --- | --- |\n| Rational formal EGF | `IsSolution` parameters `A`, `G` | Every pair of rational formal power series |\n| Example has zero constant term | First conjunct of `IsSolution` | $A(0)=0$ |\n| Zero-constant formal integral | Second and third conjuncts of `IsSolution` | $G(0)=0$ and $G'=A$ |\n| Displayed defining equation | Fourth conjunct of `IsSolution` | $A(x-G(x))=x+G(x)$ |\n| Exponential coefficients, offset one | First conclusion of `claim` | $n![x^n]A\\in\\mathbb Z$ for every $n\\ge1$, concluded rather than assumed |\n| Sole COMMENTS conjecture | Second conclusion of `claim`, proved by `result` | $n![x^n]A\\in10\\mathbb Z$ for every $n>2$ |\n\nThe existing `IntegralEGFComposition.eCoeff` supplies the literal factorial-scaled\ncoefficient expression. The existing `QuarticEGFFixedPoint.encode` constructs\nrational series from coefficient sequences. Neither definition is duplicated.\nNo integrality or desired congruence is assumed in `IsSolution`.","title":"h277403 div10","url":"https://the-omega-institute.github.io/trureturing-pages/research/h277403-div10/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ccbde4a72354e7c21e15d2d9cf70c1ed0afcf921882ccbedbd0bf6c6d55862b4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:h369673","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A369673](https://oeis.org/A369673/internal), revision #21,\nstates four conjectures for\n$a(N)=\\prod_{k=0}^{N}(2^k+2^{N-k})$:\n\n- (C.1) $a(N)$ is a square if and only if $4\\nmid N$.\n- (C.2) $5\\nmid a(2n+1)$.\n- (C.3) $v_5(a(4n))=2b(n)$.\n- (C.4) $v_5(a(4n+2))=2b(n+1)$.\n\nHere $b(n)=v_5((10n)!)-v_5((5n)!)$ is the signed integer difference\nin [OEIS A127428](https://oeis.org/A127428/internal). All indices are natural,\nincluding zero. The valuation agrees with [A112765](https://oeis.org/A112765/internal)\non positive arguments. The source, conventions and literature screen are in\n[preregistration #21915](https://github.com/the-omega-institute/trureturing/issues/21915).\n\n### Statement echo\n\n| Source item | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Product from $k=0$ through $k=N$ | `HannaA369673.a` | Literal finite product; natural subtraction has nonnegative arguments on the range. |\n| Signed factorial valuation difference | `HannaA369673.b` | Natural valuations cast into integers before subtraction. |\n| C.1 | First conjunct of `HannaA369673.claim` | Natural square root and both directions of the equivalence. |\n| C.2 | Second conjunct of `HannaA369673.claim` | Every natural index. |\n| C.3 | Third conjunct of `HannaA369673.claim` | Exact integer-valued equality, zero included. |\n| C.4 | Fourth conjunct of `HannaA369673.claim` | Exact integer-valued equality with the source shift. |\n| Entire conjecture package | `HannaA369673.result` | All four conjuncts, with no additional hypothesis. |\n\nNo source clause is dropped. The private products and sums are proof auxiliaries;\nthey impose no hypothesis on the public claim.","title":"Hanna's A369673 square classification and five-adic orders","url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04adc572993b253331ff4b4fbe06594c8afc12860f93f6a8fbd3a1a008be5668.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:h384819-primepower","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A384819, approved revision 14, NAME:\n\n> Nonnegative numbers a(n) < n for n >= 1 such that exp( Sum_{n>=1} (n^2 - a(n))*x^n/n ) is a power series with integral coefficients.\n\nThe sole COMMENTS conjecture is:\n\n> Conjecture: a(p^n) = p - 1 when p is prime for n >= 1.\n\nAll series are formal over the rationals. The offset is 1,3; the unused value\n$a(0)$ is unconstrained. The conclusion quantifies over every integer sequence\nsatisfying the defining condition, every natural prime $p$, and every natural\nexponent $r\\ge1$.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Nonnegative numbers $a(n) It seems reasonable to conjecture that,\n> whenever $|\\alpha | = |\\beta| = \\1rt2$, the channel $\\Phi_{\\alpha,\\beta} $ is a convex combination of a\n> unitary conjugation and a map with Choi-rank $3$ which is extreme in both the UCP and CPT maps.\n> However, one would need a different unitary transformation relating $\\{ A_k \\} $ to $\\{ X_k \\} $.\n\nThe source macro `\\1rt2` means $1/\\sqrt2$ (line 56). For the standard basis of $\\mathbb C^3$, the four operators of Theorem `thm:UCPT` (lines 280–290) are\n\n$A_1=\\alpha|e_1\\rangle\\langle e_1|+|e_2\\rangle\\langle e_3|$, $A_2=\\beta|e_1\\rangle\\langle e_3|+|e_3\\rangle\\langle e_2|$,\n\n$A_3=-|e_1\\rangle\\langle e_2|-\\overline\\beta|e_3\\rangle\\langle e_1|$, $A_4=|e_2\\rangle\\langle e_1|+\\overline\\alpha|e_3\\rangle\\langle e_3|$,\n\nand $\\Phi_{\\alpha,\\beta}(\\rho)=\\tfrac12\\sum_{k=1}^4 A_k^*\\rho A_k$. Here UCP means unital completely positive, CPT means completely positive and trace preserving, and UCPT means both. The definitions and source locators are in [the literature note](../Library/Eigenstructure/haagerupmusatruskai2021extreme.md).\n\nIssue [#22743](https://github.com/the-omega-institute/trureturing/issues/22743) fixes the statement: for every $\\alpha,\\beta\\in\\mathbb C$ with $|\\alpha|=|\\beta|=1/\\sqrt2$, there exist a unitary $U\\in M_3(\\mathbb C)$, a real $\\lambda\\in[0,1]$, and a complex linear map $\\Psi:M_3(\\mathbb C)\\to M_3(\\mathbb C)$ such that $\\Psi$ is completely positive, unital and trace preserving, its Choi matrix has rank $3$, it is separately extreme in the UCP and CPT sets under real convex combinations, and\n\n$\\Phi_{\\alpha,\\beta}(\\rho)=\\lambda U^*\\rho U+(1-\\lambda)\\Psi(\\rho)$\n\nfor every complex matrix $\\rho\\in M_3(\\mathbb C)$. This is proved.","title":"Every equal-modulus HMR qutrit channel has a unitary and doubly extreme decomposition","url":"https://the-omega-institute.github.io/trureturing-pages/research/haagerup-musat-ruskai-2021-qutrit-channel-unitary-extreme-decomposition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d724b2c8fda78468ede9e786f3b4805906dde69a036a97de91bee95c04bb853b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:haapasalo-pellonpaa-2020-extreme-povm-rank-vector-three-three-two-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Haapasalo and J.-P. Pellonpää, *Effective methods for constructing extreme quantum observables*, arXiv:1809.09935v1 (J. Phys. A: Math. Theor. 53, 245301 (2020)), define finite-outcome POVMs as tuples of positive operators summing to the identity and define extremality by the midpoint condition (source TeX `laatikkopaperi_v4.tex`, lines 139 and 148–152). They order rank vectors and omit rank-one outcomes (lines 219–222). Their low-dimensional discussion says:\n\n> In dimensions 1-4, we are able to completely characterize the possible rank vectors of extreme POVMs but in dimensions 5-7 some open questions remain. (line 307)\n\nand, for dimension seven,\n\n> The new problematic case now is $(3_2,2_6)_7$. (line 314)\n\nThe formal question is whether there is an extreme POVM on $\\mathbb C^7$ whose rank-at-least-two multiset is two 3s and six 2s. The module's `claim` states that every `n` and every family $M:\\operatorname{Fin}(n)\\to\\operatorname{Matrix}(\\operatorname{Fin}(7),\\operatorname{Fin}(7),\\mathbb C)$ satisfying `IsPOVM M` and `IsExtreme M` has `bigRanks M` different from $\\{3,3,2,2,2,2,2,2\\}$. The declaration `result` has type $\\lnot\\claim$.","title":"The rank vector $(3_2,2_6)_7$ is realized by an extreme POVM","url":"https://the-omega-institute.github.io/trureturing-pages/research/haapasalo-pellonpaa-2020-extreme-povm-rank-vector-three-three-two-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f6bf3b4eead7c6b559bfa91c5ee34e54b7205bd19f4baf3af76b8c0f30c5ccc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:halliwell-mawby-2019-leggett-garg-six-time-sufficiency","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. J. Halliwell and C. Mawby, *Fine's theorem for Leggett-Garg tests with an arbitrary number of measurement times*, Phys. Rev. A 100, 042103 (2019), arXiv:1906.04865v2, Section 4.B, “Sufficient Conditions: A Conjecture”, state:\n\n> A set of sufficient conditions for the existence of a joint probability matching all possible pairwise probabilities consists of the $n$-gon equalities Eq.(\\ref{neweq}) together with all possible three-time LG inequalities of the form\n>\n> $1 + s_i s_j C_{ij} + s_i s_k C_{ik} + s_j s_k C_{jk} \\ge 0,$\n>\n> where $i Let $\\pi = m_1\\pi_1\\oplus\\dots\\oplus m_d\\pi_d$ be a unitary representation\n> of $G$ such that $\\pi_i$ and $\\pi_j$ are inequivalent irreducible\n> representations for all $i\\ne j$ and $\\mathcal I_k\\le d<\\mathcal I_{k+1}$.\n> Then $pr(\\Phi_\\pi)=\\max\\{\\beta(\\Phi_\\pi),k\\}$.\n\nIssue [#15229](https://github.com/the-omega-institute/trureturing/issues/15229)\npreregistered the source, the finite-group specialization, the route, the\nreadings and the settlement criteria. The frozen Lean declaration\nD5/S3/Quantum/QuantumChannels/TwirlingPhaseRetrievabilityCounterexample.result\nrefutes the universal statement.","title":"Han–Liu's twirling phase-retrievability conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/han-liu-2023-twirling-phase-retrievability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/531f2d23dff92da815534d12eabbf47609a6d5cd2e592e375333776ef2f6fa78.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:han-pedon-metallic-hankel-quasipolynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Guo-Niu Han and Emmanuel Pedon, *Hankel determinants for the (n+2)-shifted q-metallic numbers*, arXiv:2610.36132v1, introduction, Conjecture C. The defining quadratic is equation (2.13); equations (1.4) and (1.5) define the shifted determinant and its empty value. The original preregistration is [issue #23253](https://github.com/the-omega-institute/trureturing/issues/23253). The follow-up is distinct from arXiv:2502.05993v2 and its Conjecture E on bounded-shift periodicity and higher-shift unboundedness.\n\nFor every natural $n\\ge1$, there exists $\\Phi\\in\\mathbb Z[[q]]$ with constant coefficient one satisfying\n\n$$q\\Phi^2+\\bigl((1+q^n)(1-q)-q[n]_q\\bigr)\\Phi=1,\n\\qquad [n]_q=\\sum_{i=0}^{n-1}q^i.$$\n\nFor every series satisfying these conditions and every natural shift $\\ell\\ge n+3$, define\n\n$$\\Delta_j^{(\\ell)}=\\det\\bigl([q^{\\ell+a+b}]\\Phi\\bigr)_{0\\le a,b Glue Markov chains M_i = (S_i, A_i) to a Markov chain M = (S, A), then all the eigenvalues of A lie in the convex hull of the eigenvalues of all A_i on the complex plane.\n\nThe gluing is Definition 8, page 10. The\n[Library note](../Library/Eigenstructure/hanwangyu2025gluing.md) records its four\nentrywise cases. Preregistration: [#19143](https://github.com/the-omega-institute/trureturing/issues/19143).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finitely many Markov chains with selected states | `RootedChain`; families indexed by `Fin k` | States are `Fin (n + 1)`, selected state 0; the stochastic field is `Matrix.colStochastic` |\n| Transition probability from $w$ to $v$ | Matrix entry `A v w` | Column convention of Definition 8 |\n| Common root and otherwise disjoint states | `Option (Σ i, Fin (M i).n)` | `none` is the common root, `some ⟨i,v⟩` corresponds to `v.succ` |\n| Definition 8's four transition cases | `glue` | Root-root weighted sum, private-root weighted entry, root-private unchanged entry, same-chain unchanged entry and cross-chain zero |\n| $0<\\theta_i<1$, $\\sum_i\\theta_i=1$ | Both antecedents of `claim` | No weight hypothesis is dropped |\n| All eigenvalues lie in the input spectral convex hull on the complex plane | `claim` | Complex `spectrum` after entrywise `algebraMap ℝ ℂ`, with `convexHull ℝ` in $\\mathbb C$ |\n| Refutation | `result : ¬ claim` | Two identical 3-state chains give an admissible 5-state counterexample |\n\nThe state and spectrum expressions are expanded directly rather than introduced\nas forwarding definitions. Using labelled finite state sets loses no\ncounterexample: these labelled matrices are themselves instances of the source's\nfinite chains. The finite cardinality encoding and column convention introduce\nno additional hypothesis into the displayed conjecture.","title":"The Han–Wang–Yu gluing spectral hull conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/han-wang-yu-gluing-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c3171b68c306462cbbf50a088ca563c590a38f925eda77975ee7af1b8cdb47b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:han-zhang-zhang-2026-cycle-five-strong-one-resistance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Han, Zhang and Zhang, arXiv:2606.08561v1, page 8, Discussion, ask:\n\n> Several open problems remain. First, do C₅ and C₆ give strongly m-resistant graph states for m = 1 and m = 2, respectively? Here, “strong” means genuine multipartite entanglement rather than mere entanglement.\n\nThis dossier concerns the C₅, m = 1 clause. The five-cycle graph-state\namplitudes are the cyclic controlled-Z phase on five qubits, divided by\n√32. Definition 2 of arXiv:2505.06567v1 requires initial genuine\nmultipartite entanglement (GME), GME after every one-qubit loss, and full\nseparability after every two-qubit loss. The formal claim is\n`IsStrongResistant 1 (cycleGraphState 5)`.","title":"The five-cycle graph state is strongly 1-resistant","url":"https://the-omega-institute.github.io/trureturing-pages/research/han-zhang-zhang-2026-cycle-five-strong-one-resistance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3a28cd79a137181d5b5744d08def23af00730f2bfab9e219c6a0c8ca588b3b2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:han-zhang-zhang-2026-cycle-six-strong-two-resistance-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Han, Zhang and Zhang, arXiv:2606.08561v1, page 8, Discussion, ask:\n\n> Several open problems remain. First, do C₅ and C₆ give strongly m-resistant graph states for m = 1 and m = 2, respectively? Here, “strong” means genuine multipartite entanglement rather than mere entanglement.\n\nThis dossier concerns the C₆, m = 2 clause. On qubits 0,...,5 its graph-state\namplitudes are (-1) raised to the sum of the six cyclic edge products, divided\nby 8. Definition 2 of arXiv:2505.06567v1 requires initial genuine multipartite\nentanglement (GME), GME after every two-qubit loss, and full separability after\nevery three-qubit loss. GME excludes a finite convex mixture of products\nacross nontrivial bipartitions. The formal claim is\n`IsStrongResistant 2 (cycleGraphState 6)`; `result` proves its negation.","title":"The six-cycle graph state is not strongly 2-resistant","url":"https://the-omega-institute.github.io/trureturing-pages/research/han-zhang-zhang-2026-cycle-six-strong-two-resistance-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a04542e0c5359995f593577b4189cd122debd84e18217e1c97330856f9b2bc5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanaoka-konno-2021-riesz-walk-self-similarity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Hanaoka and N. Konno, “Return probability and self-similarity of the Riesz\nwalk,” Quantum Information and Computation 21 (2021), 409–422,\narXiv:2010.04518v3, DOI 10.26421/QIC21.5-6-5, state Conjecture 4.2:\n\n> Let Z_> = {1,2,3,...}. Put T_0(n)={0} and T_k(n)=((k-1)/n,k/n]\n> for 1 <= k <= n. For the Riesz walk with initial state\n> Psi_0=[1,0]^T delta_0, P(X_2t/(2t) in T_k(t)) =\n> P(X_8t/(8t) in T_k(t)) for every t in Z_>.\n\nIssue #15758 preregistered the exact natural-number reading. The delivered\ndeclaration is D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.result : claim,\nand its module state has statement identity\nsha256:4b9935aefe3ec7033ed45b54d5dec573a17d5471a67ae35f412951ac228f6269.\n\n### Statement echo\n\nThe source's t >= 1 is the binder t : Nat with 1 <= t; its 0 <= k <= t\nis k : Nat with k <= t. The source's normalized probability is\nprob (2*t) (Tset t k) and prob (8*t) (Tset t k); Tset gives the singleton at\nk=0 and the left-open, right-closed rational interval for 1 <= k <= t. The\nconclusion is the equality in claim; result proves exactly that proposition.","title":"Hanaoka–Konno Conjecture 4.2: self-similarity of the Riesz walk","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanaoka-konno-2021-riesz-walk-self-similarity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c88e45b6b0288eac045b82159057b55b31cd0e375b3b1b5f0f88a281b50cd00.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hance-2026-twenty-four-cell-preparation-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jonte R. Hance, *Why three? A two-level system with four mutually unbiased\nquestions*, arXiv:2609.10078v1, “Polytopes, quasiprobabilities and contextuality”,\nafter Eq. (sbound), source main.tex line 182:\n\n> The 24-cell subtheory is therefore preparation contextual at least for $\\lambda>2/3$; whether its further equivalences lower the threshold, and whether it is measurement contextual, we leave open, and both are linear programs.\n\nOnly the further-preparation-equivalences threshold clause is addressed.\nLet $V=\\{\\pm e_i:1\\leq i\\leq4\\}\\cup\\{s/2:s\\in\\{\\pm1\\}^4\\}$.\nTake one direction from each of its twelve antipodal pairs as a sharp binary\nquestion, with both outcomes. For every real $\\lambda\\in[0,1]$ the statistics are\n$p(b\\mid n,x)=(1+b\\lambda n\\cdot x)/2$. All classical convex mixtures are included.\n\nA model has an arbitrary measurable ontic space, countably additive probability\npreparations and arbitrary measurable nonnegative normalized binary responses.\nEvery pair of operationally equal convex preparation mixtures has equal ontic\nmixture measures. There is no measurement-noncontextuality or determinism premise.","title":"Exact preparation-noncontextuality threshold of the twenty-four-cell","url":"https://the-omega-institute.github.io/trureturing-pages/research/hance-2026-twenty-four-cell-preparation-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04a56a851a25c9f848d832338653f950c73ab144873bda34e4c5530b821ec53d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a107099-half-iterate-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A107099 comment states: \"Coefficients [x^n] A(x) = 0 (mod 3) except at n = 3^k (conjecture).\" The Name states: \"G.f. satisfies A(A(x)) = x + 4*x^3, where A(x) = Sum_{n>=0} a(n)*x^(2*n+1).\" The displayed expansion begins with $x$, fixing the linear coefficient to one.\n\nThe exact binding claim of [#18858](https://github.com/the-omega-institute/trureturing/issues/18858) includes existence and integrality: there exists an odd $A\\in\\mathbb Q[[x]]$ with linear coefficient one and $A\\circ A=x+4x^3$; for every such $A$ and every $n\\in\\mathbb N$, its coefficient equals an integer $m$, and $3\\mid m$ whenever $n$ is not a power of three. No nonvanishing assertion at powers of three is imposed.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Odd generating function | `IsGF`: every even coefficient vanishes | The literal preregistered condition, including degree zero. |\n| Expansion begins with $x$ | `IsGF`: linear coefficient one | The source normalization. |\n| $A(A(x))=x+4x^3$ | `IsGF`: formal self-substitution equation | Substitution is Mathlib formal power-series substitution. |\n| Coefficients vanish modulo three outside $3^k$ | `claim`: every natural degree has an integer coefficient representative and the off-power implication | The rational coefficient is compared with the cast of that integer. |\n| The entry defines a generating function | `claim`: an `IsGF` solution exists | Prevents vacuity. Universal solution quantification introduces no additional hypothesis. |","title":"hanna a107099 half iterate mod three","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a107099-half-iterate-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f3771897d56d9f9e184b33b7242123264d772a5a117ab6cfce990a502599640.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a174470","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A174470 states: “G.F.: exp( Sum_{n>=1} A014578(n)*(3x)^n/n ), where A014578 is the binary expansion of Thue constant.” Its COMMENT field says: “Conjectured to consist entirely of integers.” The source is revision 2; A014578 is revision 68. The source has no DOI or primary arXiv version.\n\nWrite $b(0)=0$ and, for $n>0$, let $b(n)=1$ when $v_3(n)$ is even and $b(n)=0$ otherwise. Set $c(n)=b(n)3^n$ and\n\n$$\nF(x)=\\exp\\left(\\sum_{n\\ge1}\\frac{c(n)}n x^n\\right)\\in\\mathbb Q[[x]].\n$$\n\nThe assertion includes existence of this rational formal series and integrality of each ordinary coefficient at every $N\\in\\mathbb N$, including $N=0$.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| A014578's zero case and valuation-parity implementation | `b` | Integer-valued bits with explicit zero case. |\n| Weight A014578(n) times $3^n$ | `c` | Literal weight, with natural exponent. |\n| Formal exponential of the logarithmic sum | `IsSourceSeries` | Equality to `PowerSeries.subst (logInput c) (PowerSeries.exp ℚ)`; `logInput` is reused from `DoldExponentialIntegrality`. |\n| “Conjectured to consist entirely of integers” | `claim`, proved by `result` | Existence and all-series/all-degree coefficient integrality; no additional hypotheses. |\n\nThe declarations are in `D5.S1.Recurrence.Algebraic.HannaThueRothExponentialIntegrality`. The claim spells out coefficient integrality and defines no second `IntegralSeries` predicate.","title":"hanna a174470","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a174470/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/722ad31b411e035bc65667b7720cdec37d125d33ab1f567b01a387d2a4d2ac24.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a246056-a246423-legendre-central-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna, [OEIS A246056](https://oeis.org/A246056), Name and Comments; [preregistration #18887](https://github.com/the-omega-institute/trureturing/issues/18887).\n\n> Conjecture: a(n) == 1 (mod 3) when n = 2*A005836(k) for k >= 0, and a(n) == 0 (mod 3) otherwise, where A005836 gives numbers whose base-3 representation contains no 2.\n\nIts generating function is $\\sum_{n\\ge0}x^n(1-2x)^{-2n-1}Q_n(2x)Q_n(3x)$, with offset zero. The conjecture is proved.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The Name generating function | `term` and coefficientwise `A` | Literal preregistered rational expressions; the coefficient at $m$ uses $n\\le m$. |\n| Every natural index | Outer universal quantifier in `claim` | Includes zero. |\n| Integer coefficient | Existential integer $z$ equal to the rational coefficient | Explicit cast to $\\mathbb Q$; integrality is retained. |\n| $n=2\\,A005836(k)$ | An $m$ with $n=2m$ and $2\\notin\\operatorname{Nat.digits}(3,m)$ | Zero has the empty digit list and qualifies. |\n| Residue one on that support, zero otherwise | Both implications in `claim` | Neither support nor complement is dropped. |\n\nThe public `result` proves exactly `claim`, without additional hypotheses. The definitions are mirrored in [the Library note](../Library/RecurrenceComposition/hanna2014a246056a246423.md).","title":"Hanna’s ternary-digit residue conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246056-a246423-legendre-central-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c976d951ed650dc91016c73bc4e4d2cea29077f3c028a0131f98972ae1789f8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a246423-legendre-central-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna, [OEIS A246423](https://oeis.org/A246423), Name and Comments; [preregistration #18888](https://github.com/the-omega-institute/trureturing/issues/18888).\n\n> a(n) == 2 (mod 4) iff n = 2^k for k>=2, and a(n) == 0 (mod 4) elsewhere except at a(0)=1 (conjecture).\n\nIts generating function is $\\sum_{n\\ge0}x^n(1-3x)^{-2n-1}Q_n(2x)Q_n(3x)$, with offset zero. The conjecture is proved.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The Name generating function | `term` and coefficientwise `A` | Literal preregistered rational expressions; the coefficient at $m$ uses $n\\le m$. |\n| $a(0)=1$ | First conjunct of `claim` | The constant term is explicit. |\n| Every positive index, integral coefficients | Universal $n>0$ and integer witness equal to its coefficient | Integrality at zero follows from the first conjunct. |\n| Residue two iff $n=2^k$, $k\\ge2$ | The biconditional in `claim` | The exponent bound is retained. |\n| Zero modulo four elsewhere | Complement implication in `claim` | No complementary case is omitted. |\n\nThe public `result` proves exactly `claim`, without additional hypotheses. The definitions are mirrored in [the Library note](../Library/RecurrenceComposition/hanna2014a246056a246423.md).","title":"Hanna’s power-of-two residue conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246423-legendre-central-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a62f5f8813fc110456d422ee2870308ec43aeef7049305bf231b560ca0f3037.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a271960-dyadic-vanishing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna’s [OEIS A271960](https://oeis.org/A271960), COMMENTS (unpaginated), states:\n\n> Conjecture: [x^(2^n-2^k)] (x/A(x))^(2^n) = 0 for k = 0..n-2 and n >= 2. - _Paul D. Hanna_, Jan 16 2026\n\nIts NAME is:\n\n> G.f. A(x) satisfies: A(x)^2 = A( (x + 2*A(x)^2)^2 ).\n\nThe OFFSET is 1 and the EXAMPLE begins $A(x)=x+2x^2+9x^3+50x^4+312x^5+\\cdots$.\nThe [Library note](../Library/RecurrenceComposition/hanna2016a271960.md) records the source.\nPreregistration: [#19138](https://github.com/the-omega-institute/trureturing/issues/19138).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The defining equation | `IsGF A` | Literal rational power-series equation, with `PowerSeries.subst u f` meaning $f(u)$. |\n| Offset and leading coefficient | First two conjuncts of `IsGF` | Constant coefficient zero and linear coefficient one. |\n| $x/A(x)$ | `xOverA A` | Inverse of the series with coefficient $[x^{n+1}]A$ at index $n$. |\n| $n\\ge2$, $0\\le k\\le n-2$ | Natural binders and the two bounds in `claim` | Natural subtraction is exact under the bounds. |\n| Coefficient and power | Universal conjunct of `claim` | $[x^{2^n-2^k}](x/A(x))^{2^n}=0$. |\n| Existence | Existential conjunct of `claim` | Additional conclusion; it does not restrict the universally quantified solutions. |\n| Every normalized solution | Universal $A$ with `IsGF A` | No selected solution replaces the source’s equation. |","title":"Hanna’s OEIS A271960 dyadic vanishing conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4a874afc320197d94dfc2fe55e8b27e1d18df106740d5cc0a2bc069f5f44b1b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a274479-a377100-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, [OEIS A274479](https://oeis.org/A274479), Name and Comments (unpaginated), as quoted in [the source note](../Library/RecurrenceComposition/hanna2016a274479.md).\n\n> G.f. satisfies: A(x)^2 = A( x^2/(1 - 2*x - 4*x^2) ).\n\n> a(n) = 1 (mod 3) for n>=1 (conjecture).\n\nFor $A(x)=x+\\sum_{n\\ge2}a(n)x^n$, the equation is\n$A(x)^2=A\\!\\left(\\frac{x^2}{1-2x-4x^2}\\right)$.\nThe resolution is **Proved**: a normalized rational power-series solution exists, and every normalized solution has integral coefficients $a(n)\\equiv1\\pmod3$ for all $n\\ge1$.\nPreregistration: [#18850](https://github.com/the-omega-institute/trureturing/issues/18850). The other Hanna problem is separately preregistered and has its own resolution claim.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The displayed generating equation | `HannaGeometricSquareSubstitutionModThree.IsGF` | Literal preregistered equation over rational power series, with the denominator inverse interpreted in that ring. |\n| The example begins $x+x^2+\\cdots$ | Constant coefficient zero and coefficient one at $X$ in `IsGF` | The normalization is explicit. |\n| $a(n)$ for $n\\ge1$ | `coeff n A`, with the hypothesis `1 ≤ n` | Every positive natural index. |\n| Integral coefficients congruent to one modulo three | Integer witness equal to `coeff n A` and `Int.ModEq 3 m 1` in `claim` | Coercion into the rationals is explicit. |\n| A generating function is defined by the equation | Existence conjunct in `claim` | Excludes a vacuous universal statement. |","title":"Hanna's A274479 coefficient congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/352633ffafe267aa1a7334fcbc2f4df1e95e0604404f5f5dd735a3a8817be265.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a338632-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A338632, revision 13, states:\n\n> G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - ...))))), a continued fraction relation.\n\nIts formula line is:\n\n> a(n) = 2 (mod 4) for n > 1 (conjecture).\n\nThe offset is zero. The source note is `Library/RecurrenceComposition/hanna2026a338632.md`.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| The displayed continued fraction | `tail` uses the literal rational-series finite truncations, ending at $A$ | The numerator at level $j$ is $(2j+1)x$. |\n| The equation equals one | `IsGF` | For each coefficient index, every sufficiently deep truncation has that coefficient equal to the constant series one. |\n| The generating series exists | The first conjunct of `claim`, proved by `result` | This makes the universal coefficient assertion nonvacuous. |\n| $a(n)$, offset zero | The degree-$n$ coefficient of any rational series satisfying `IsGF` | The source convention is unchanged. |\n| $n>1$ | $2\\le n$ for natural $n$ | The two ranges coincide. |\n| $a(n)=2\\pmod4$ | An integer representative $m$ of that rational coefficient with $m\\bmod4=2$ | Integrality at the stated indices is part of the conclusion. |\n\nThe quantification over all series satisfying the relation and explicit existence are stronger than selecting one unnamed generating series; they do not weaken the source. The private `gf_unique` identifies every such series with the constructed solution.","title":"Hanna's OEIS A338632 congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a338632-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a146e548d29a3f33cd1ec7905f137f1d48aafa47e5ccafacb6e10704f494530b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a372527-square-composition-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A372527](https://oeis.org/A372527) defines its generating function by:\n\n> Expansion of g.f. A(x) satisfying A( A(x)^2 ) = x*A(x)/(1 - 2*A(x)).\n\nIts conjecture reads:\n\n> Conjecture: a(2^k + 1) == 2 (mod 4) for k >= 0, and a(n) == 0 (mod 4) for n > 0 when n is not of the form 2^k + 1.\n\nThe offset is one and the data start with $a(1)=1$. Accordingly, the zero-residue clause is asserted for $n>1$, as specified in [#18859](https://github.com/the-omega-institute/trureturing/issues/18859). The literal clause for every $n>0$ is false at $n=1$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The displayed generating equation | `IsGF` | The literal rational formal-power-series equation, with the formal inverse of $1-2A$. |\n| Offset one, first coefficient one | `IsGF` requires constant coefficient zero and degree-one coefficient one | These are the source's normalization. |\n| Existence of the generating series | First conjunct of `claim`, proved by `result` | Explicit existence excludes vacuity. |\n| Every $k\\ge0$, coefficient at $2^k+1$ congruent to two | First universal residue clause in `claim` | An integer witness equals the rational coefficient and is congruent to two modulo four. |\n| Other indices | Second residue clause in `claim`, with $n>1$ and no shifted-power representation | An integer witness equals that coefficient and is congruent to zero modulo four. |\n| A selected generating series | Universal quantification over all series satisfying `IsGF` | Private `exists_integral_solution` also proves uniqueness; universality strengthens the assertion. |\n\nThe two public definitions are the expressions registered in #18859. Coefficient agreement and the power-of-two series use existing owners rather than new forwarding definitions.","title":"Hanna's square-composition congruence for OEIS A372527","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a372527-square-composition-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/893e51bb11480ada605a1e005d50360da6d6b230d9f5060cf4f6be9c7f48a9c8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a377100-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, [OEIS A377100](https://oeis.org/A377100), Name and Comments (unpaginated), as quoted in [the source note](../Library/RecurrenceComposition/hanna2024a377100.md).\n\n> G.f. A(x) satisfies A(x) = A(x^3)/A(x^2) + A(x)^2.\n\n> Conjecture: a(n) == 1 (mod 3) for n >= 1, which has been verified for the initial 2305 terms.\n\nFor $A(x)=x+\\sum_{n\\ge2}a(n)x^n$, the equation is\n$A(x)=\\frac{A(x^3)}{A(x^2)}+A(x)^2$.\nThe resolution is **Proved**: a normalized rational power-series solution exists, and every normalized solution has integral coefficients $a(n)\\equiv1\\pmod3$ for all $n\\ge1$.\nPreregistration: [#18851](https://github.com/the-omega-institute/trureturing/issues/18851). The other Hanna problem is separately preregistered and has its own resolution claim.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The displayed generating equation | `HannaCubeSquareQuotientModThree.IsGF` | Literal preregistered equation over rational power series; the A377100 quotient is in Laurent series. |\n| The example begins $x+x^2+\\cdots$ | Constant coefficient zero and coefficient one at $X$ in `IsGF` | The normalization is explicit. |\n| $a(n)$ for $n\\ge1$ | `coeff n A`, with the hypothesis `1 ≤ n` | Every positive natural index. |\n| Integral coefficients congruent to one modulo three | Integer witness equal to `coeff n A` and `Int.ModEq 3 m 1` in `claim` | Coercion into the rationals is explicit. |\n| A generating function is defined by the equation | Existence conjunct in `claim` | Excludes a vacuous universal statement. |","title":"Hanna's A377100 coefficient congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/96cc7040a2be227f29397346e818a87f127a51c6d0dcdeb02f7ba52852ae37bf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a380060-bilateral-theta-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A380060 defines the generating function by:\n\n> G.f. A(x) satisfies A(x) = Sum_{n=-oo..+oo} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.\n\nThe Comments state:\n\n> The term x^(n^2) * (A - x^n)^n/(1 - x^n*A)^n is invariant under sign reversal of n.\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.\n\nFormula (2) states:\n\n> A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * (A(x) - x^n)^n/(1 - x^n*A(x))^n.\n\nThe source is [OEIS A380060](https://oeis.org/A380060), with its citation in\n[the Library note](../Library/RecurrenceComposition/hanna2025a380060.md).\nPreregistration: [#18872](https://github.com/the-omega-institute/trureturing/issues/18872).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Bilateral generating equation | `IsGF` uses the source's formula (2) over rational power series | The zero term is one and opposite indices coincide; no separate Lean Laurent-series folding theorem is asserted. |\n| Infinite sum | Degree $m$ is read from indices $1\\le n\\le m$ | Each summand contains $X^{n^2}$, so omitted indices contribute zero. |\n| Coefficients $a(n)$ | `coeff n A` | Every rational solution of `IsGF` is quantified. |\n| Positive index | Natural $n$ with $0 Conjectures: a(3*n) == A382319(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 1.\n\nThe source equations are $A=X+L(A)^3$ and $B=X(1-X)^{-1}+L(B^3)$,\nwhere $L(F)=\\sum_{k\\ge1}F(X^k)$ and both series have zero constant term.\nAt degree $m$, $L(F)$ has coefficient $\\sum_{1\\le k\\le m,\\ k\\mid m}[X^{m/k}]F$.\nThe empty degree-zero sum is zero. The cube in the second equation is taken\nbefore Lambert summation. The source is recorded in the\n[Library note](../Library/RecurrenceComposition/hanna2025a382318.md).\nPreregistration: [#18861](https://github.com/the-omega-institute/trureturing/issues/18861).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Two generating equations, with offset one | `IsA382318 A`, `IsA382319 B` in `HannaLambertCubeModThree` | Literal rational equations and zero constant coefficients. |\n| Infinite substitution sum | `lambertSum` | Positive-divisor coefficient sum; natural-number division. |\n| Sequence values | `PowerSeries.coeff` | Integer witnesses are cast into rational coefficient equalities. |\n| Nonnegative index | Natural-number binder | Includes zero. |\n| First congruence | First universal conjunct of `claim` | $a(3n)\\equiv b(n)\\pmod3$. |\n| Two other residue classes, $n\\ge1$ | Second universal conjunct of `claim` | Both integer residues are zero. |\n| Existence | Existential conjunct of `claim` | Additional conclusion; no restriction of the source's solution class. |\n| Every solution pair | Universal quantifiers and the two predicate hypotheses | No selected solution replaces the source's generating functions. |\n\nThe first congruence is $a(3n)\\equiv b(n)\\pmod3$ for every natural $n$. The other two are $a(3n+1)\\equiv a(3n+2)\\equiv0\\pmod3$ for $n\\ge1$. No source clause is dropped. The complete proposition is displayed\nat both `claim` and `result` in the Blueprint.","title":"Hanna's A382318 congruence modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff3a0414179794505873503241f21ad2fb4b146aee8fbe9989a873a910f0f93b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-a382319-a382318-lambert-cube-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A382319](https://oeis.org/A382319), Comments (unpaginated), states:\n\n> Conjecture: a(n) == A382318(3*n) (mod 3) for n >= 0.\n\nThe source equations are $A=X+L(A)^3$ and $B=X(1-X)^{-1}+L(B^3)$,\nwhere $L(F)=\\sum_{k\\ge1}F(X^k)$ and both series have zero constant term.\nAt degree $m$, $L(F)$ has coefficient $\\sum_{1\\le k\\le m,\\ k\\mid m}[X^{m/k}]F$.\nThe empty degree-zero sum is zero. The cube in the second equation is taken\nbefore Lambert summation. The source is recorded in the\n[Library note](../Library/RecurrenceComposition/hanna2025a382319.md).\nPreregistration: [#18862](https://github.com/the-omega-institute/trureturing/issues/18862).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Two generating equations, with offset one | `IsA382318 A`, `IsA382319 B` in `HannaLambertCubeModThree` | Literal rational equations and zero constant coefficients. |\n| Infinite substitution sum | `lambertSum` | Positive-divisor coefficient sum; natural-number division. |\n| Sequence values | `PowerSeries.coeff` | Integer witnesses are cast into rational coefficient equalities. |\n| Nonnegative index | Natural-number binder | Includes zero. |\n| Reciprocal congruence | Universal conjunct of `claim` | $b(n)\\equiv a(3n)\\pmod3$. |\n| Existence | Existential conjunct of `claim` | Additional conclusion; no restriction of the source's solution class. |\n| Every solution pair | Universal quantifiers and the two predicate hypotheses | No selected solution replaces the source's generating functions. |\n\nThe congruence is $b(n)\\equiv a(3n)\\pmod3$ for every natural $n$, including zero. No source clause is dropped. The complete proposition is displayed\nat both `claim` and `result` in the Blueprint.","title":"Hanna's A382319 congruence modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1437d20b0df275fd396d9877698de37d37451e3fd339db9b53634d6b5278cbd2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-bilateral-reversion-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna's [OEIS A379202](https://oeis.org/A379202), Name section, states:\n\n> G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 2)^(n+1).\n\nIts Comments section states:\n\n> Conjecture: a(n) is even for n > 1.\n\nThe offset is one and the first coefficient is one. The formal solution is\n$A\\in\\mathbb Q[[X]]$ with $A(0)=0$ and $[X]A=1$. Integer powers and inversion\nare taken in the field $\\mathbb Q((X))$. For each natural $j$, the coefficient\nat degree $j-1$ is the sum over $-j-2\\le n\\le j+2$; all omitted summands\nhave zero coefficient at that degree. The source is recorded in the\n[Library note](../Library/RecurrenceComposition/hanna2024a379202.md).\nPreregistration: [#18876](https://github.com/the-omega-institute/trureturing/issues/18876).\n\n### Statement echo\n\nAll Lean names below belong to\n`D5.S1.Recurrence.Invariants.HannaBilateralReversionParity`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A^n(A^n+2)^{n+1}$ for integer $n$ | `term` | Literal Laurent-series integer powers. |\n| Sum over all integer indices | `bilateralSum` | Coefficientwise finite window; `outside_window` and `literal_finite_window` prove that omitted coefficients vanish. |\n| $1/X$ equals the bilateral sum | `IsGF` | Literal Laurent inverse of $X$. |\n| Offset one, first entry one | Two normalization conjuncts of `IsGF` | Zero constant coefficient and linear coefficient one. |\n| Coefficient $a(n)$ for every $n>1$ | Natural binder and strict inequality in `claim` | Index one is excluded. |\n| Integer, even coefficient | Integer witness and `Even` in `claim` | Witness is cast to the rational coefficient. |\n| Generating series exists | Existential conjunct of `claim` | An additional conclusion, with no added hypothesis. |\n| Uniqueness in the preregistered settlement statement | Private `solution_unique` | Every normalized solution equals the same rational image of the integral inverse. |\n| Any generating series satisfying the equation | Universal series binder and `IsGF` hypothesis | No selected solution replaces the source's solution class. |\n\nNo source clause is dropped. The full proposition appears at both the\n`claim` and `result` Blueprint nodes.","title":"Hanna's bilateral reversion parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-bilateral-reversion-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ee66b030aa517da7613b4d4a1b5f68b0b713e2c5549ca6ede4254446bcdd43e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-nested-square-odd-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392210, by Paul D. Hanna, Feb 02 2026:\n\n> G.f.: x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + sq(x/(1-4*x) + ...)))) where sq(x) = x^2, an\n> infinitely nested square.\n\nThe comment carrying the question:\n\n> It appears that a(2*n-1) is even for n > 1.","title":"The Nested-Square Generating Function Has Even Coefficients at Odd Indices","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-nested-square-odd-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e07ad70b1e67d66bc6ffbeb70d5839d42b5f80221cd0a8b7b6df1da07fe5df23.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna-shift-square-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392203, by Paul D. Hanna, Jan 03 2026:\n\n> G.f. A(x) satisfies A(x - A(x)) = x^2 + x*A(x).\n\nThe comment carrying the question:\n\n> It appears that a(2*n) == 1 (mod 8) and a(2*n+1) == 3 (mod 8) for n >= 1.","title":"Hanna's Series A(x - A(x)) = x^2 + x A(x) Is Rational Modulo Eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-shift-square-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8b486bd1403beeb96a39354bfb9cc1dc66d96632c4b87c2f40435c7c3228d47.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna323-gaussparity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A323689](https://oeis.org/A323689) defines $\\sum_{m\\ge0}(x^m+i)^m/(1+ix^m)^{m+1}$, with constant coefficient replaced by $1$. Its COMMENTS conjecture says odd coefficients occur only at $N=r^2$; its FORMULA also asserts oddness at every such index. The preregistration is [#21417](https://github.com/the-omega-institute/trureturing/issues/21417). The source is OEIS-only, with no identified DOI or arXiv id.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Literal rational summand in NAME and FORMULA | `HannaGaussianSquareParity.summand` | Same powers, imaginary unit and formal inverse; no leading factor. |\n| Locally finite coefficient series | `HannaGaussianSquareParity.A` | Degree zero is $1$; at positive degree $N$, the sum has $1\\le m\\le N$. |\n| Real coefficients and integer parity | `HannaGaussianSquareParity.claim` | Every natural degree has an ordinary integer representative embedded in the complex numbers. |\n| Off-support conjecture and on-support FORMULA | `HannaGaussianSquareParity.result` | Oddness is equivalent to $N=r^2$, including degree zero. |\n\nThe settling result has no hypotheses. The public definitions retain the binding source conventions. The auxiliary recursive array and rational kernel are used within the proofs.","title":"Hanna's A323689 Gaussian coefficient parity","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4a3c7ffd779c57a7b0687286bafad3a5f371af27b2840c3240c2cffd11d656b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna324-gaussparity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A324300](https://oeis.org/A324300) defines $\\sum_{m\\ge0}x^m(x^{m+1}+i)^m/(1+ix^{m+1})^{m+1}$. Its COMMENTS conjecture says odd coefficients occur only at $N=k(k+2)$; its FORMULA also asserts oddness at every such index. The preregistration is [#21418](https://github.com/the-omega-institute/trureturing/issues/21418). The source is OEIS-only, with no identified DOI or arXiv id.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Literal rational summand in NAME and FORMULA | `HannaGaussianShiftedSquareParity.summand` | Same powers, imaginary unit, formal inverse and leading factor $X^m$. |\n| Locally finite coefficient series | `HannaGaussianShiftedSquareParity.A` | Degree $N$ sums $0\\le m\\le N$; no constant replacement. |\n| Real coefficients and integer parity | `HannaGaussianShiftedSquareParity.claim` | Every natural degree has an ordinary integer representative embedded in the complex numbers. |\n| Off-support conjecture and on-support FORMULA | `HannaGaussianShiftedSquareParity.result` | Oddness is equivalent to $N=k(k+2)$, including degree zero. |\n\nThe settling result has no hypotheses. The public definitions retain the binding source conventions. The auxiliary recursive array and rational kernel are used within the proofs.","title":"Hanna's A324300 Gaussian coefficient parity","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1f41429fba2bc18dee59d1ca9297b9a038739c6f84962ed87f366a2726a6b8c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna365771-mod5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A365771, revision 23, defines\n$a(n)=\\binom{2n+1}{n}/(2n+1)\\cdot\\binom{3n-1}{n}$ for $n\\geq0$ and states:\n\n> Conjectures: given A033042 is the sums of distinct powers of 5, then\n> (1) a(5*A033042(n)) == 4 (mod 5) for n > 0,\n> (2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,\n> (3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.\n\nThe problem is preregistered in [#18656](https://github.com/the-omega-institute/trureturing/issues/18656).\nThe three settlements are Proved. For every positive sum $m$ of distinct powers of five,\n$a(5m)\\equiv4\\pmod5$ and $a(5m+1)\\equiv2\\pmod5$. For every positive $n$ outside\n$n=5m$ or $n-1=5m$ with $m$ a sum of distinct powers of five (including zero), $a(n)\\equiv0\\pmod5$.\nThe companion statements are preregistered in [#18657](https://github.com/the-omega-institute/trureturing/issues/18657)\nand [#18658](https://github.com/the-omega-institute/trureturing/issues/18658).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| The formula for $a(n)$, including $n=0$ | `HannaA365771.a` | Literal natural-number binomial quotient and product; subtraction is truncated. |\n| “Sums of distinct powers of 5.” | `HannaA365771.InA033042` | Every digit of `Nat.digits 5 m` is at most one, including the empty digit list at zero. |\n| “for n > 0” in the enumeration | The positive-value hypothesis in `HannaA365771.claim` | A033042 starts at zero, so the positive enumeration indices select exactly its positive values. |\n| Residue four at five times the selected value | `HannaA365771.result` | Proves the full `claim`, with no extra hypothesis. |\n\nUnqualified formal names in this dossier belong to `D5.S1.Recurrence.Residue.HannaA365771`.\nThe companion settling declarations are `D5.S1.Recurrence.Residue.HannaA365771Two.result`\nand `D5.S1.Recurrence.Residue.HannaA365771Zero.result`; their claims retain the positive-$m$\nrestriction for (2) and the negated existential, including $m=0$, for (3).\nNatural division agrees with the displayed quotient because the first factor is an integer Catalan number.","title":"Hanna's three congruences for A365771","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3e4089bf877c56aeb259209a639c20dc94bd8b36e01f02003e4b406fa1798f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna365771-mod5-two","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A365771 revision 23, Comments, states:\n\n> Conjectures: given A033042 is the sums of distinct powers of 5, then\n> (2) a(5*A033042(n) + 1) == 2 (mod 5) for n > 0,\n\nPreregistration: [#18657](https://github.com/the-omega-institute/trureturing/issues/18657).\n\n### Statement echo\n\nEvery positive $m$ in A033042 satisfies $a(5m+1)\\bmod5=2$.\nThe sequence and digit predicate are exactly `D5.S1.Recurrence.Residue.HannaA365771.a`\nand `D5.S1.Recurrence.Residue.HannaA365771.InA033042`. The claim uses natural-number arithmetic.","title":"Hanna's second congruence for A365771","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/619cd4fa4610bbcfccb0c7f2560b6405c12a46ece305cfdfbd3841a5ae22e7cc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna365771-mod5-zero","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A365771 revision 23, Comments, states:\n\n> Conjectures: given A033042 is the sums of distinct powers of 5, then\n> (3) a(n) == 0 (mod 5) for n > 0 except when n or n-1 equals 5*A033042(k) for some k >= 0.\n\nPreregistration: [#18658](https://github.com/the-omega-institute/trureturing/issues/18658).\n\n### Statement echo\n\nEvery $n>0$ with no $m\\geq0$ in A033042 satisfying $n=5m$ or $n-1=5m$ satisfies $a(n)\\bmod5=0$.\nThe sequence and digit predicate are exactly `D5.S1.Recurrence.Residue.HannaA365771.a`\nand `D5.S1.Recurrence.Residue.HannaA365771.InA033042`. The claim uses natural-number arithmetic.","title":"Hanna's third congruence for A365771","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-zero/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff3e615570273283f413c0a05de490d135de2a91803a21a0bfe71b0b011429bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanna397594","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A397594, revision 15, states:\n\n> G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 4^(n-1) for n >= 1.\n\nIts FORMULA gives:\n\n> (1) 4^(n-1) = Sum_{k=0..n-1} [x^k] A(x/n)^n for n >= 1.\n> (2) (4*n)^(n-1) = [x^(n-1)] A(x)^n / (1 - n*x) for n >= 1.\n\nThe conjecture is:\n\n> Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).\n\nThe series is $A(X)=\\sum_{n\\geq0}a(n)X^n\\in\\mathbb Z[[X]]$, with\n$a(0)=1$. Rational rescaling means coefficientwise mapping to\n$\\mathbb Q[[X]]$ followed by $X\\mapsto X/m$. Powers are ordinary formal\nproducts. There is no analytic convergence assumption.\n\n### Statement echo\n\n| Source or binding convention | Lean counterpart | Scope |\n| --- | --- | --- |\n| $a(0)=1$, integer coefficients, offset zero | `IsSourceSeries`, its constant-coefficient conjunct and `PowerSeries ℤ` argument | Literal normalization. |\n| Formula (1), every $m\\geq1$, indices $0,\\ldots,m-1$ | `IsSourceSeries`, rational rescaling and `Finset.range m` | The source's $n,k$ are the binders $m,j$. |\n| Formula (2) | Local `hname_row` equivalence in `result`; the derivation below | The denominator is the constant-one formal inverse. |\n| $n>3$, odd iff $n=2^k-1$ or $n=2^k+1$, $k>1$ | Universal conjunct of `claim`, proved by `result` | Both directions and both strict bounds are retained. |\n| Integer-source existence and quantification over every source | Existential and universal conjuncts of `claim` | Required by preregistration #22490; existence excludes vacuity. |","title":"Integer existence and dyadic parity for OEIS A397594","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna397594/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/289faa02e1f0ba3259d98b081b667af05221af1ad1ebc40ce6b97e1468898424.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanotel-loubenets-2025-high-spin-chsh-nonviolation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Hanotel and E. R. Loubenets, *Nonviolation of the CHSH inequality under local spin-1 measurements on two spin qutrits*, Phys. Scr. 100, 055115 (2025), arXiv:2503.05172v2, Introduction, `Final_Revision_Hanotel_Loubenets.tex`, line 201:\n\n> However, up to the moment the problem whether or not the CHSH inequality is violated under Alice and Bob high spin $s\\geq1$ measurements is still open -- though specifically this Bell inequality has been used for proving \\cite{1, Lou.Kuz.Han:24} nonlocality of every pure nonseparable two-qudit state.\n\nThe same source's Conjecture section, lines 913–914, states:\n\n> Under local Alice and Bob spin-$1$ measurements in an arbitrary nonseparable pure two-qutrit state, the\n> CHSH inequality is not violated.\n\nFor spin $s$, the measurements are the linear projections $S_u=u\\cdot S$ for unit $u\\in\\mathbb R^3$. The companion paper, E. R. Loubenets and L. Hanotel, *High-spin measurements in an arbitrary two-qudit state*, J. Phys. A: Math. Theor. 58, 265304 (2025), arXiv:2412.03470v2, `Revision_Loubenets_Hanotel.tex`, lines 210–237 and 253–261, specifies the spin matrices and the CHSH operator\n$\\mathcal B=S_a\\otimes S_b+S_a\\otimes S_d+S_c\\otimes S_b-S_c\\otimes S_d$.\nIts lines 348–363 give the local-hidden-variable bound $2s^2$.\n\nIssue [#24190](https://github.com/the-omega-institute/trureturing/issues/24190) fixes the reading: for every integer $n=2s\\ge2$, every positive semidefinite trace-one complex matrix $\\rho$ on $\\mathbb C^{n+1}\\otimes\\mathbb C^{n+1}$, and every four unit real directions $a,c,b,d$,\n$|\\operatorname{tr}(\\rho\\mathcal B)|\\le n^2/2=2s^2$.\nThis is proved, including integer and half-integer spins $s\\ge1$ and both pure and mixed states.\n\nThe source definitions and bibliographic locators are also recorded in [the literature note](../Library/Observer/hanotelloubenets2025chshspinone.md).","title":"Linear spin measurements never violate CHSH for any spin $s\\ge1$","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanotel-loubenets-2025-high-spin-chsh-nonviolation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cbbaebda06955403bec61bc4f06265a9386723582a76c45e45d16cd0851a3789.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hanson-2020-jensen-shannon-ball-majorization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. P. Hanson, *Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels* (PhD thesis, Cambridge, 2020; arXiv:2010.02408), derives continuity bounds for Schur-concave functions from the majorization structure of total-variation balls.\n- Every total-variation ball in the probability simplex contains a majorization-minimizer r∗, with r∗ ≺ q for every q in the ball (Eq. (3.1)). Every Schur-concave function, such as an entropy, attains its maximum over the ball at r∗.\n- For p-norm balls with 1 < p < ∞ no such element exists (Section 3.7).\n- Chapter 8, Question 1 asks: \"Do balls in Hellinger distance, Bures distance, the (square-root) Shannon-Jensen divergence, Wasserstein distances, or other metrics on P admit majorization-minimizers?\"\n\nThe verbatim statements are in [the literature note](../Library/Divergence/hanson2020entropiccontinuity.md).\n\nIssue [#18189](https://github.com/the-omega-institute/trureturing/issues/18189) fixes the reading of the Jensen–Shannon clause: for every probability vector p and every positive radius, the square-root Jensen–Shannon ball around p has a majorization-minimizer. This clause is answered in the negative.","title":"Square-root Jensen–Shannon balls need not admit majorization-minimizers","url":"https://the-omega-institute.github.io/trureturing-pages/research/hanson-2020-jensen-shannon-ball-majorization-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a446c689c797d47f19244eb7cd5a745e02306fc94bd04ec2b862df8babcace6c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hardin-barker-a209723-edge-differences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. H. Hardin's OEIS A209723, revision 11, NAME states:\n\n> 1/4 the number of (n+1) X 5 0..2 arrays with every 2 X 2 subblock having distinct clockwise edge differences.\n\nThe FORMULA field states the empirical recurrence and Colin Barker's conjectures:\n\n> Empirical: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3).\n\n> G.f.: x*(6 + x - 11*x^2) / ((1 - x)*(1 - 2*x^2)).\n\n> a(n) = 3*2^(n/2 - 1) + 4 for n even.\n\n> a(n) = 2^((n + 1)/2) + 4 for n odd.\n\nThe offset is $1,1$. Entries are labelled; no symmetry quotient is taken. For a square with top row $a,b$ and bottom row $c,d$, the differences are the signed integers $b-a,d-b,c-d,a-c$. Let $a(n)$ be one quarter of the number of valid $(n+1)$-by-$5$ arrays in the rationals.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Domain and fidelity |\n| --- | --- | --- |\n| Distinct clockwise differences | `DistinctClockwise` | Four integer differences, pairwise unequal; entries have type `Fin 3`. |\n| Every consecutive square | `IsSourceArray` | Every `Fin n` row gap and `Fin 4` column gap is tested. |\n| One quarter of the array count | `a` | `Nat.card` of the entire labelled-array subtype, coerced to rationals and divided by four. |\n| Positive even formula | First conjunct of `claim` | $a(2(m+1))=3\\cdot2^m+4$ for every natural $m$. |\n| Positive odd formula | Second conjunct of `claim` | $a(2m+1)=2^{m+1}+4$ for every natural $m$. |\n| Recurrence | Third conjunct of `claim` | $a(m+4)=a(m+3)+2a(m+2)-2a(m+1)$ for every natural $m$. |\n| Ordinary generating function | `IsSourceSeries`, fourth and fifth conjuncts of `claim` | Existence and the identity for every rational formal series with zero constant coefficient and coefficient $a(n)$ for $n\\ge1$. |\n\nThe natural-index substitutions exactly cover the source domains. The extra series-existence conjunct prevents a vacuous universal assertion. The generating identity is cross-multiplied by $(1-X)(1-2X^2)$, whose constant coefficient is one. The auxiliary value $a(0)$ is outside the source formulas and does not supply the series constant term.","title":"Hardin–Barker five-column clockwise-edge-difference formulas","url":"https://the-omega-institute.github.io/trureturing-pages/research/hardin-barker-a209723-edge-differences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be084b3a6cadf4e41bbb789dbe08d418631db6c9aad203826492ea65d979aca8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:haritha-chithra-2022-distance-seidel-energy-problem-5-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Haritha and A. V. Chithra, *Distance Seidel matrix of a connected graph*, arXiv:2210.05940v2, page 13, Problem 5.1:\n\n> Let G = K_{a,b}, with a, b > 1. Is E_{D^S}(K_{a,b} − e) > E_{D^S}(K_{a,b}), for any edge e of K_{a,b}?\n\nThe answer is affirmative for the source's matrix equation $D^S(G)=J-I-2D$, for every $a,b\\ge2$ and every edge. The source's casewise expression instead gives diagonal one. The universal statement here concerns the zero-diagonal matrix equation; the diagonal-one alternative has only the finite evidence below.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $D^S(G)=J-I-2D$ | `distSeidel` | Literal entrywise expression with shortest-path distance and zero diagonal. |\n| $E_{D^S}(G)=\\sum_r|\\partial_r^S|$ | `energy` | Absolute real characteristic roots, counted with multiplicity; the matrices used by the theorem are real symmetric. |\n| $G=K_{a,b}$ | `completeBipartiteGraph (Fin a) (Fin b)` | Carrier `Fin a ⊕ Fin b`; graph constructions are Mathlib's. |\n| $a,b>1$ | Both natural-number hypotheses in `claim` | Exactly $2\\le a$ and $2\\le b$. |\n| Any edge $e$ | Universal quantifier over `edgeSet` in `claim` | Unordered edges; no particular edge is selected in the public conclusion. |\n| Delete $e$ and increase energy strictly | `deleteEdges` and strict inequality in `claim`, proved by `result` | The singleton edge set is removed; all source clauses are retained. |\n| Finite decidable vertex carrier | The instance parameters of `distSeidel` and `energy` | Supplies finite matrices and sums, without an additional graph restriction. |","title":"Distance Seidel energy after deleting an edge","url":"https://the-omega-institute.github.io/trureturing-pages/research/haritha-chithra-2022-distance-seidel-energy-problem-5-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f0ffbb9ff132077cafc1bab095b9606555590fe64bb8b52ae3bc788be8e25194.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:harkness-2025-maxkcut-qubo-penalty","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Harkness et al. (arXiv:2511.01108, quant-ph) study the penalty (QUBO)\nreformulations of max k-cut used with QAOA, quantum annealers and Ising\nmachines. With edge weights of either sign, `d⁺_v` and `d⁻_v` the sums of the\npositive and of the negative weights at `v`, the one-hot QUBO objective is\n`q(x) = Σ_{uv∈E} w_uv (1 − Σ_j x_uj x_vj) − Σ_v c_v (Σ_j x_vj − 1)²`, and the\nreduced R-QUBO objective on `k − 1` columns is\n`q̄(x) = Σ_{uv∈E} w_uv (1 − Σ_j x_uj x_vj − (1 − Σ_j x_uj)(1 − Σ_j x_vj)) − Σ_v c_v Σ_{i max(d⁺_v/k, −(3/2) d⁻_v)` (Theorem 1), respectively\n`c_v > d⁺_v − 2 d⁻_v` (Theorem 2), and conjectures:\n\n> Conjecture 1. Let G(V, E) be a graph with edge weights w_uv for all\n> {u,v} ∈ E. Let x̂ ∈ {0,1}^{n×k} be an optimal solution of the QUBO\n> formulation with c_v > max{d⁺_v/k, −d⁻_v/2} for every vertex v ∈ V. Then,\n> x̂ is an optimal solution of the max k-cut problem.\n\n> Conjecture 2. Let G(V,E) be a graph with edge weights w_uv for all\n> {u,v} ∈ E. Let x̂ ∈ {0,1}^{n×(k−1)} be an optimal solution for the R-QUBO\n> model with c_v > d⁺_v − d⁻_v for every vertex v ∈ V. Then, x̂ is an optimal\n> solution of the max k-cut problem.\n\nIssue #11239 fixes the reading. Vertices are `Fin n` and parts `Fin k`. The\nweights are a symmetric real function, with weight 0 on non-edges, and the\nedges are the pairs `u < v`. Matrices are Boolean, with entries read as 0 or 1\nby the frozen `bit`.\n\"An optimal solution of the max k-cut problem\" means feasible for the BQO\nformulation (one-hot rows) and maximal for the BQO objective among feasible\npoints; for Conjecture 2 the same holds for the R-BQO formulation (rows with at\nmost one entry). The claim follows the paper's scope `k ≥ 3`.","title":"Tight penalty coefficients for the QUBO reformulations of max k-cut","url":"https://the-omega-institute.github.io/trureturing-pages/research/harkness-2025-maxkcut-qubo-penalty/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/911c3ed6bac51693940a2c92555d0322134fbb20f09f8fca33f6bb2d32418a9e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hart-rowley-greedy-prefix-suffix-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sarah B. Hart and Peter J. Rowley, *A note on involution prefixes in Coxeter\ngroups*, arXiv:2502.00777v1, section 3, printed page 5, immediately before\nReferences, write:\n\n> We suspect that the involution length of any element always equals its\n> ‘suffix’ involution length, but have not tested this extensively enough to\n> make a firm conjecture.\n\nThis is one explicitly tentative, unnumbered published assertion. The quoted\ntext is licensed CC BY 4.0, https://creativecommons.org/licenses/by/4.0/ ; PDF\nline wrapping and typographic ligatures are normalized. It is not Conjecture\n1.2 or 1.3.\n\nThe domain is every Coxeter system whose every nonidentity element has a unique\nmaximum-Coxeter-length involution prefix, and every element of such a system.\nAn involution has order exactly two. A prefix u of w satisfies w = uv and\nlength(w) = length(u) + length(v). The unique ancestor maximizes length over all\ninvolution prefixes. Remove it and recurse on v, with zero factors at identity.\nThe suffix procedure removes the analogous right factor. The assertion equates\nthese two actual greedy counts. No finiteness, rank, or Coxeter-element\nrestriction is imposed, and neither count minimizes arbitrary factorizations.","title":"The tentative Hart–Rowley prefix/suffix equality","url":"https://the-omega-institute.github.io/trureturing-pages/research/hart-rowley-greedy-prefix-suffix-equality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12efa1a43ae1ce0b9fb4154de6205c3efcb15ee416ef81588c5b0abbf12efcb8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hasheminejad-ramazi-2023-mixed-imitation-best-response-nonequilibration","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Hasheminejad and P. Ramazi, *Equilibration of Coordinating Imitation and Best-Response Dynamics*, arXiv:2306.03243v3 [math.DS], IEEE Control Systems Letters 7 (2023) 3078-3083, study finite undirected networks whose agents use either imitation or best response in two-strategy coordination games. The source defines persistence as every agent becoming active infinitely often. Its abstract asks:\n\n> It is therefore, unknown, whether a potential function also exists for mixed populations or if there actually exists a counter example where an activation sequence prevents equilibration.\n\n(`main.tex`, abstract, l. 71.) In Section 2 it states:\n\n> However, it remains open whether the dynamics equilibrate under an arbitrary persistent activation sequence.\n\n(`main.tex`, l. 225.) The conclusion specifies the general-graph question:\n\n> Whether dense trees or general graphs equilibrate under every activation sequence remains an open problem.\n\n(`main.tex`, l. 668.)\n\nThe module's `claim` is the general-graph statement: for every `n : ℕ`, every symmetric and irreflexive adjacency relation on `Fin n`, every imitator assignment and real payoff map satisfying $\\max\\{\\mathrm{pay}_i(B,A),\\mathrm{pay}_i(A,B)\\}<\\min\\{\\mathrm{pay}_i(A,A),\\mathrm{pay}_i(B,B)\\}$ for every agent, every initial state, and every persistent activation sequence, some trajectory state is an `IsEquilibrium`. Here `false` denotes $A$ and `true` denotes $B$; `IsEquilibrium` quantifies over every activation sequence.","title":"Mixed imitation and best-response dynamics need not equilibrate on general graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/hasheminejad-ramazi-2023-mixed-imitation-best-response-nonequilibration/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd3f2631af72a76ee075dfaee12dc247d5f060fd726bbd0ce96ffb340a7e0565.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hastings-2017-small-majorana-fourteen-mode-distance-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. B. Hastings, *Small Majorana Fermion Codes*, Quantum Inf. Comput. 17(13–14), 1191–1205 (2017), arXiv:1703.00612v1, Section II.B, PDF p. 5 (`c.tex`, line 272):\n\n> For $N_{\\rm maj}=14$, we performed a numerical search (described in the next section) for a code with $N_{\\rm stab}=6$ and did not succeed. Hence, we believe that no such code exists, i.e., we believe that $N_{\\rm maj}=16$ is the minimum number of modes to have $d=4,K>0$.\n\nThe paper uses $N_{\\rm maj}$ for the number of Majorana modes, $N_{\\rm stab}$ for the number of independent stabilizers, and $K=N_{\\rm maj}/2-N_{\\rm stab}$ for the number of logical qubits. Its stabilizer bit strings form a self-orthogonal binary subspace $C$, meaning $x\\cdot y=0$ for every $x,y\\in C$, including $x=y$ (`c.tex`, lines 85–105). Logical operators correspond to $C^\\perp\\setminus C$, and distance is their minimum Hamming weight (lines 108–113). The paper requires the all-ones fermion-parity vector $\\mathbf1$ to belong to $C$ (lines 115–118). These definitions and the quotation are also recorded in [the literature note](../Library/Hamming/hastings2017smallmajorana.md).\n\nIssue [#23984](https://github.com/the-omega-institute/trureturing/issues/23984) fixes the statement: for every natural number $n\\le14$ and every binary linear subspace $C\\le\\mathbb F_2^n$, if $C$ is self-orthogonal, $\\mathbf1\\in C$, and $2\\dim C CONJECTURE 1. Let A be a positive semidefinite matrix and B be a Hermitian matrix. Then for 1/2 ⩽ s ⩽ t ⩽ 1 and for every unitarily invariant norm, we have |||A^s B + BA^{1−s}||| ⩽ |||A^t B + BA^{1−t}|||.\n\nThe formal statement retains every quantifier: every natural dimension $n$, all $A,B\\in M_n(\\mathbb C)$ with $A$ positive semidefinite and $B$ Hermitian, all real $1/2\\le s\\le t\\le1$, and every unitarily invariant norm on $M_n(\\mathbb C)$. Real powers are interpreted by continuous functional calculus. No lower spectral bound $A\\ge I$ is part of the conjecture.","title":"Hayajneh–Hayajneh–Kittaneh: mixed-power monotonicity for unitarily invariant norms","url":"https://the-omega-institute.github.io/trureturing-pages/research/hayajneh-2018-unitarily-invariant-norm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e789cbb1e6197d4008dba7cad141523fe269a50f63d5675ecc9714abc5e347f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hedemann-2020-diagonal-correlance-normalization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Samuel R. Hedemann, *Correlance and Discordance: Computable Measures of Nonlocal Correlation*, Quantum Information Processing 19, 189 (2020), arXiv:2001.03453v2, Section II.C, defines the normalization as the maximum of raw correlance over all diagonal states in Eq. (16), and states:\n\n> Instead, one simple example that maximizes $\\mathcal{X}_D$ in all systems is\n> $\\rho_{D_{\\max}}=\\frac{1}{2}(|1\\rangle\\langle1|+|n\\rangle\\langle n|),$\n> yielding the normalization factor, valid for all systems,\n> $\\mathcal{N}_{\\mathcal{X}_D}=\\widetilde{\\mathcal{X}}(\\rho_{D_{\\text{max}}})=\\frac{1}{2}-\\frac{1}{2^N}.$\n\nThis quotation is from `CorrelanceAndDiscordance.tex`, lines 456–465, Eqs. (17)–(18); mathematical macros are expanded and layout commands omitted. Here $N$ is the number of modes and $n$ is their total Hilbert-space dimension. Appendix L, Step 9, line 2385, acknowledges:\n\n> However, a rigorous proof of this is still lacking.\n\nFor $N$ qubits, write a diagonal state as $\\rho=\\operatorname{diag}(p)$ with $p:\\{0,1\\}^N\\to\\mathbb{R}$, $p(x)\\ge0$ and $\\sum_xp(x)=1$. Put $m_i(b)=\\sum_{x:x_i=b}p(x)$ and $q(x)=\\prod_i m_i(x_i)$. The source's raw correlance and reduction product, Eqs. (10)–(11), become\n$R_N(p)=\\operatorname{tr}[(\\operatorname{diag}(p)-\\operatorname{diag}(q))^2]=\\sum_x(p(x)-q(x))^2$.\n\nIssue [#22752](https://github.com/the-omega-institute/trureturing/issues/22752) fixes the qubit consequence of the source's assertion: for every natural $N\\ge2$ and every such probability table, $R_N(p)\\le\\frac12-2^{-N}$. This universal bound is refuted.","title":"Hedemann's proposed diagonal-correlance normalization is exceeded by seven qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/hedemann-2020-diagonal-correlance-normalization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59a7d4f6b4aa28ad32c6d66e2c9b26febbb2c3c3f4349174f2b83fbf5f741a85.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hefg-a232686","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A232686 COMMENTS conjectures concern integer exponential coefficients $a_n=n![x^n]A$ of rational formal power series with $A(0)=1$. The source equation and its nominated congruences are:\n\n> %N A232686 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x))^2.\n> %C A232686 CONJECTURES.\n> %C A232686 a(n) == 0 (mod 2) for n>=2.\n> %C A232686 a(n) == 0 (mod 2^2) for n>=4.\n\n| Record | Formal equation | Nominated congruences | Settling declaration |\n| --- | --- | --- | --- |\n| A232686 | $A'(x)=A(xA'(x))^2$ | $2\\mid a_n$ for $n\\ge2$; $4\\mid a_n$ for $n\\ge4$ | `HannaA232686Congruences.result` |\n\nThe settling declaration belongs to `D5.S1.Recurrence.Residue`. It universally quantifies over rational formal solutions and concludes all-index integrality as well as the two displayed congruences. The source note records the OEIS record and its literal source text.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A'(x)=A(xA'(x))^2$ and $A(0)=1$ | `IsSolution` in `HannaA232686Congruences` | The formal derivative, substitution and constant coefficient are literal. |\n| $a_n=n![x^n]A$ is integral | The existential `a : ℕ → ℤ` with `eCoeff A n = (a n : ℚ)` | Integrality is a conclusion at every index. |\n| $2\\mid a_n$ for $n\\ge2$ | The first divisibility conjunct in `claim` | The threshold and modulus are literal. |\n| $4\\mid a_n$ for $n\\ge4$ | The second divisibility conjunct in `claim` | The threshold and modulus are literal. |","title":"Hanna's derivative-substitution congruences for A232686","url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232686/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c26869da21bd5fe4e16bbd3011c7a181ca815ad122bb21d79627271bd503c7de.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hefg-a232694","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A232694 COMMENTS conjectures concern integer exponential coefficients $a_n=n![x^n]A$ of rational formal power series with $A(0)=1$. The source equation and its nominated congruences are:\n\n> %N A232694 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^4) with A(0)=1.\n> %C A232694 CONJECTURES.\n> %C A232694 a(n) == 0 (mod 3) for n>=3.\n> %C A232694 a(n) == 1 (mod 8) for n>=0.\n\n| Record | Formal equation | Nominated congruences | Settling declaration |\n| --- | --- | --- | --- |\n| A232694 | $A'(x)=A(x(A'(x))^4)$ | $3\\mid a_n$ for $n\\ge3$; $8\\mid a_n-1$ for every $n\\ge0$ | `HannaA232694Congruences.result` |\n\nThe settling declaration belongs to `D5.S1.Recurrence.Residue`. It universally quantifies over rational formal solutions and concludes all-index integrality as well as the two displayed congruences. The source note records the OEIS record and its literal source text.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A'(x)=A(x(A'(x))^4)$ and $A(0)=1$ | `IsSolution` in `HannaA232694Congruences` | The formal derivative, substitution and constant coefficient are literal. |\n| $a_n=n![x^n]A$ is integral | The existential `a : ℕ → ℤ` with `eCoeff A n = (a n : ℚ)` | Integrality is a conclusion at every index. |\n| $3\\mid a_n$ for $n\\ge3$ | The first divisibility conjunct in `claim` | The threshold and modulus are literal. |\n| $8\\mid a_n-1$ for all $n\\ge0$ | The second divisibility conjunct in `claim` | The all-index residue is literal. |","title":"Hanna's derivative-substitution congruences for A232694","url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232694/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5dc6931198940dffe0370dbb468a3b9f15f963f843a165e4663a46d12cfe220.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hefg-a232695","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A232695 COMMENTS conjectures concern integer exponential coefficients $a_n=n![x^n]A$ of rational formal power series with $A(0)=1$. The source equation and its nominated congruences are:\n\n> %N A232695 E.g.f. A(x) satisfies: A'(x) = A(x*A'(x)^5) with A(0)=1.\n> %C A232695 CONJECTURES.\n> %C A232695 a(n) == 1 (mod 5) for n>=0.\n> %C A232695 a(n) == 0 (mod 2) for n>=4.\n\n| Record | Formal equation | Nominated congruences | Settling declaration |\n| --- | --- | --- | --- |\n| A232695 | $A'(x)=A(x(A'(x))^5)$ | $5\\mid a_n-1$ for every $n\\ge0$; $2\\mid a_n$ for $n\\ge4$ | `HannaA232695Congruences.result` |\n\nThe settling declaration belongs to `D5.S1.Recurrence.Residue`. It universally quantifies over rational formal solutions and concludes all-index integrality as well as the two displayed congruences. The source note records the OEIS record and its literal source text.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A'(x)=A(x(A'(x))^5)$ and $A(0)=1$ | `IsSolution` in `HannaA232695Congruences` | The formal derivative, substitution and constant coefficient are literal. |\n| $a_n=n![x^n]A$ is integral | The existential `a : ℕ → ℤ` with `eCoeff A n = (a n : ℚ)` | Integrality is a conclusion at every index. |\n| $5\\mid a_n-1$ for all $n\\ge0$ | The first divisibility conjunct in `claim` | The all-index residue is literal. |\n| $2\\mid a_n$ for $n\\ge4$ | The second divisibility conjunct in `claim` | The threshold and modulus are literal. |","title":"Hanna's derivative-substitution congruences for A232695","url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232695/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4f2b8a5ab6c5ac0517eabc0b7251d1e9d9979cf9f5d640ffaa2a4ce71537340.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hefg-a232696","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A232696 COMMENTS conjectures concern integer exponential coefficients $a_n=n![x^n]A$ of rational formal power series with $A(0)=1$. The source equation and its nominated congruences are:\n\n> %N A232696 E.g.f. A(x) satisfies: A'(x) = A(x/A'(x)) with A(0)=1.\n> %C A232696 CONJECTURES.\n> %C A232696 a(n) == 0 (mod 2) for n>=4.\n> %C A232696 a(n) == 0 (mod 2^2) for n>=10.\n\n| Record | Formal equation | Nominated congruences | Settling declaration |\n| --- | --- | --- | --- |\n| A232696 | $A'(x)=A(x/A'(x))$ | $2\\mid a_n$ for $n\\ge4$; $4\\mid a_n$ for $n\\ge10$ | `HannaA232696Congruences.result` |\n\nThe settling declaration belongs to `D5.S1.Recurrence.Residue`. It universally quantifies over rational formal solutions and concludes all-index integrality as well as the two displayed congruences. The source note records the OEIS record and its literal source text.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A'(x)=A(x/A'(x))$ and $A(0)=1$ | `IsSolution` in `HannaA232696Congruences` | The formal derivative, substitution and constant coefficient are literal. |\n| $a_n=n![x^n]A$ is integral | The existential `a : ℕ → ℤ` with `eCoeff A n = (a n : ℚ)` | Integrality is a conclusion at every index. |\n| $2\\mid a_n$ for $n\\ge4$ | The first divisibility conjunct in `claim` | The threshold and modulus are literal. |\n| $4\\mid a_n$ for $n\\ge10$ | The second divisibility conjunct in `claim` | The threshold and modulus are literal. |","title":"Hanna's derivative-substitution congruences for A232696","url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232696/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f74bdba840c571616deced03ae7cbdd8efc2a5699bceee47be225eeb6cbee6e8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:heib-bruschi-2026-conjecture-80-hidden-center-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Heib and D. E. Bruschi, *On the structural properties of Lie algebras via associated\nlabeled directed graphs*, arXiv:2601.16161v1, attach to a minimal-graph-admissible basis of a\nLie algebra (every bracket of two basis vectors is a multiple of a basis vector) the directed\ngraph with an edge from $x_j$ to $x_\\ell$, labelled $x_k$, whenever $[x_j,x_k]$ is a nonzero\nmultiple of $x_\\ell$. A vertex subset has the ideal-graph-property when no edge leaves it.\nConjecture 80 (§IV.D, p. 46) states that if such an algebra has a nonzero center, then some proper\nnon-empty vertex subset has the ideal-graph-property and spans a solvable ideal. The verbatim\nstatements are in [the literature note](../Library/LieTheory/heib2026structural.md).\n\nIssue [#15636](https://github.com/the-omega-institute/trureturing/issues/15636) fixes the\nreading over every field, as the paper's convention \"We denote any field with the symbol\n$\\mathbb F$\" requires, for the given minimal graph $G(V,E)$ of the conjecture.","title":"A hidden central direction refutes Heib–Bruschi Conjecture 80","url":"https://the-omega-institute.github.io/trureturing-pages/research/heib-bruschi-2026-conjecture-80-hidden-center-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e03b1c42c252ab3eef9fd5626ea204a1b5231c1a47c111b6d8e71f71585d30e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:heib-bruschi-2026-conjecture-83-affine-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Heib and D. E. Bruschi, *On the structural properties of Lie algebras via associated\nlabeled directed graphs*, arXiv:2601.16161v1, attach to a basis of a Lie algebra in which\nevery bracket of two basis vectors is a multiple of a basis vector (a minimal-graph-admissible\nbasis) the directed graph with an edge from $x_j$ to $x_\\ell$, labelled $x_k$, whenever\n$[x_j,x_k]$ is a nonzero multiple of $x_\\ell$. A vertex subset has the\nideal-graph-property when no edge leaves it. Conjecture 83 states that a\nminimal-graph-admissible Lie algebra with trivial center either (i) has no proper non-empty\nvertex subset with the ideal-graph-property, or (ii) is a direct sum of components with\ntrivial center whose every minimal graph has no such subset. The verbatim statements are in\n[the literature note](../Library/LieTheory/heib2026structural.md).\n\nIssue [#13676](https://github.com/the-omega-institute/trureturing/issues/13676) fixes the\nreading over every field, as the paper's convention \"We denote any field with the symbol\n$\\mathbb F$\" requires, with the direct sum read as an internal direct sum of ideals over an\narbitrary index type.","title":"The affine Lie algebra refutes Heib–Bruschi Conjecture 83","url":"https://the-omega-institute.github.io/trureturing-pages/research/heib-bruschi-2026-conjecture-83-affine-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cc3c23562daa78898915830a1cfab931dd1b2caeccc9ad32c9758ef2a46c509.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:heimendahl-heinrich-gross-2022-affine-partition-flag","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Heimendahl, M. Heinrich and D. Gross, *The axiomatic and the operational approaches to resource theories of magic do not coincide*, arXiv:2011.11651v3 (J. Math. Phys. 63, 112201, 2022), conjecture in the remark following Corollary `cor:SO-cap-AD`. The checked TeX member `paper_new.tex`, lines 1219–1227, reads:\n\n> We conjecture that such a projective measurement which maximizes \\eqref{eq:estimation-of-overlapII}\n> has the following form:\n>\n> Measure the first qudit in the computational basis.\n>\n> If $ x \\neq 0 $ is measured, do nothing.\n> If $ 0 $ is measured, measure the second qudit in the computational basis.\n> Continue in this fashion until all qubits are measured.\n>\n> If $ 0 $ has been measured on every qudit, apply a Hadamard gate to every qudit.\n\nWrite $p$ for the source's prime dimension $d$, and $V=\\mathbb F_p^n$, represented by `Fin n → ZMod p`. An affine partition (`isAffinePartition`) is a finite set of nonempty affine subspaces, with distinct parts disjoint, such that a vector belongs to a part if and only if it is nonzero. Its objective is `sqSum`, namely $\\sum_{K\\in\\mathcal K}|K|^2$.\n\nThe cells $K_{i,x}=\\{v\\in V:(\\forall j Let $f$ be a non-negative function that is operator monotone on $[0, \\infty)$. Then $E_{f}(c) \\leq Cf(c)$ holds for all $c > 0$ when $C = 1$.\n\nThe source defines\n\n$$\nE_f(c)=\\inf_{g'\\in\\widehat{\\mathcal M}_+}\n\\left(\\sup_{x\\ge0}(f-g)(x)-\\inf_{x\\ge0}(f-g)(x)+cg'(0)\\right),\n$$\n\nwhere $\\widehat{\\mathcal M}_+$ consists of Fourier transforms of finite positive Borel measures on $\\mathbb R$. Section 5, equation (28), requires $g\\in C^1(\\mathbb R)$. The conjecture is refuted by $f(x)=x/(1+x)$ and $c=1$. Preregistration: [#15279](https://github.com/the-omega-institute/trureturing/issues/15279).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| “$g$ is $C^1(\\mathbb R)$ and $g'$ is the Fourier transform of a finite positive Borel measure” | `IsAdmissible` | `ContDiff` and a finite positive measure, with the complex exponential sign convention. |\n| The displayed oscillation plus $cg'(0)$ | `approxCost` | Supremum and infimum on the subtype of nonnegative reals. |\n| Infimum over admissible $g$ | `approxError` | Real infimum over finite oscillation costs. |\n| “a non-negative function” | The nonnegativity premise of `claim` | Every nonnegative real argument. |\n| “operator monotone on $[0,\\infty)$” | `OperatorMonotoneOn` | Loewner order for positive semidefinite complex matrices at every finite dimension. |\n| “for all $c>0$ when $C=1$” | The positive-$c$ conclusion of `claim` | The constant is exactly one. |\n| Failure of the conjecture | `result : ¬ claim` | The witness satisfies all premises; the lower bound exceeds $f(1)$. |\n\n`approxError` is a real infimum; Lean's real empty-set infimum convention is zero. The refutation instance has a nonempty finite cost set, witnessed by $g=0$, and a uniform finite lower bound. Discarding infinite-cost candidates preserves the source infimum in this domain. No extended-real equivalence is claimed for targets without a finite-cost approximant. Matrix monotonicity at all finite dimensions is the classical Löwner formulation used here; no separate infinite-dimensional transport theorem is claimed.","title":"Herrera's function-approximation conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/herrera-2023-function-approximation-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/79f041147e371ffeb02fcaf3813ae9ddc7c87608a54a5e996d92c1c6d4c8a644.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hertz-lupu-gladstein-heshami-goldberg-2025-loss-purity-log-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Hertz, N. Lupu-Gladstein, K. Heshami and A. Z. Goldberg, *Equalities and inequalities from entanglement, loss, and beam splitters*, arXiv:2501.02047v1, Quantum Stud. Math. Found. **13**, 34 (2026), study the vacuum-environment loss channel ρ_T = Tr₂[B(T)(ρ₁ ⊗ |0⟩⟨0|)B(T)†], with B(T) = exp(arccos(√T)(a₁a₂† − a₁†a₂)).\n- They prove that the purity Tr[ρ_T²] is convex in the transmission T.\n- Conjecture 7 asserts the stronger \"The purity of a state ρ₁ subject to loss is log-convex in T ∈ [0, 1]; ∂² log Tr[ρ_T²]/∂T² ≥ 0\", equivalently (26) (P′)² ≤ P·P″ with P(T) = Tr[ρ_T²].\n- Appendix B extends it to mixtures of identical two-copy states (Conjecture 19, (B4)).\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/hertz2025lossbeamsplitters.md).\n\nIssue [#16345](https://github.com/the-omega-institute/trureturing/issues/16345) fixes the reading: for every single-mode state ρ₁, the function T ↦ log Tr[ρ_T²] is convex on [0, 1]. It is refuted.","title":"The purity of a state under loss is not always log-convex in the transmission","url":"https://the-omega-institute.github.io/trureturing-pages/research/hertz-lupu-gladstein-heshami-goldberg-2025-loss-purity-log-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90fa6d287161cf6fa8fcfc5aaca37791538276f85dd8d6d57d28194e0827b59c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hiai-2026-spectral-geometric-log-euclidean-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fumio Hiai, *Log-majorizations between quasi-geometric type means for matrices*,\nLinear Algebra Appl. 735 (2026), 123–174; arXiv:2510.04691v3.\n\n§2, equation (2.3), verbatim:\n\n> Fα(A, B) := (A^{−1}#B)^α A(A^{−1}#B)^α.\n\n§2, equation (2.4), verbatim:\n\n> SGα,p(A, B) := Fα(A^p, B^p)^{1/p}.\n\n§2, Log-Euclidean and weighted geometric mean definitions, verbatim:\n\n> LEα(A, B) := exp((1 − α) log A + α log B)\n\n> A#αB := A^{1/2}(A^{−1/2}BA^{−1/2})^α A^{1/2}\n\nHere # = #_{1/2}.\n\nProblem 3.20, verbatim:\n\n> Problem 3.20. The case remaining open is whether SGα,p ≺log LEα for α > 1. Note that this is affirmative if SGα,p ≺log SGα,q if α > 1 and p ≥ q (see Problem 3.12).\n\nThe summary table after Problem 3.20 has row “SGα,p ≺log LEα”, with entries\n“none” for 0 < α < 1 and “?” for α > 1.","title":"Hiai's spectral geometric Log-Euclidean comparison fails","url":"https://the-omega-institute.github.io/trureturing-pages/research/hiai-2026-spectral-geometric-log-euclidean-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e16157769bae865fe41cc6a2f4f7b478f7e834fdfcfcddadc3d10961f502dd65.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hiai-mosonyi-hayashi-2009-group-invariant-psi-differentiability-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Hiai, M. Mosonyi and M. Hayashi, *Quantum hypothesis testing with group\nsymmetry*, J. Math. Phys. 50 (2009) 103304, Section 7, verbatim excerpt\nwith the indicated omissions:\n\n> To get a complete solution of the problems of the Chernoff and the Hoeffding bounds and of Stein's lemma, … the key point is to show that (4.5) holds with equality … This can be done, for instance, … [by showing] that the function ψ exists and is differentiable on the interval (0, 1). As the examples of Section 6 suggest, one can expect the differentiability of ψ to hold in our setting of group-invariant state discrimination. The main open question of the present work is to show that this is indeed the case.\n\n| Source object or quantifier | Formal reading |\n|---|---|\n| Finite-dimensional Hilbert space | Complex coordinate space of arbitrary dimension d |\n| Compact group and unitary representation | Finite group G and a homomorphism into unitary complex matrices; finite groups with the discrete topology are compact |\n| Faithful states | Positive definite matrices with trace one |\n| Equation (2.1), tensor conditional expectation | `Defs.twirlPower`, the uniform finite-group Haar average of tensor conjugates, indexed by words `Fin n → Fin d` |\n| Finite logarithmic overlap | `Defs.psiN`, the real logarithm of the real part of the trace of the product of CFC powers |\n| Equation (3.1), asymptotic overlap | `Defs.psi`, the selected limit of `psiN / n`, with convergence at every s in (0,1) an explicit premise |\n| Differentiability assertion | `Defs.claim`, universal differentiability on (0,1) under these premises |\n\nRefuting this finite-group restriction with faithful states refutes the\ncompact-group assertion on its faithful domain.","title":"A corner in the group-invariant asymptotic overlap","url":"https://the-omega-institute.github.io/trureturing-pages/research/hiai-mosonyi-hayashi-2009-group-invariant-psi-differentiability-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72d066033fce35f63cc532e6336839af4c4d9801066cf173db497d094d9d170d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hiai-ruskai-2015-cq-contraction-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hiai and Ruskai, *Contraction coefficients for noisy quantum channels*,\narXiv:1508.03551v1, Section 6, Theorem 6.2 and Conjecture 6.3 (PDF p. 19), state:\n\n> Although the bounds in the above theorem are sufficient to disprove two conjectures as remarked below, we believe that they are optimal, i.e.,\n>\n> Conjecture 6.3. Equality holds in (45c) through (45e) above.\n\nThe map on complex qubit matrices is\n$\\Phi_{\\alpha,\\tau}(w_0I+\\mathbf w\\cdot\\boldsymbol\\sigma)\n=w_0I+\\alpha w_1\\sigma_1+\\tau w_0\\sigma_3$.\nFor every $\\alpha\\geq0$, $0<|\\tau|<1$ and $\\alpha^2+\\tau^2\\leq1$,\nthe conjecture asks for the three equalities\n\n$$\n\\begin{aligned}\n\\eta_{\\widehat{\\mathrm{WY}}}^{\\mathrm{Riem}}(\\Phi_{\\alpha,\\tau})\n &=\\frac{\\alpha^2(1+\\sqrt{1-\\tau^2})}{2(1-\\tau^2)},\\\\\n\\eta_{x^{-1/2}}^{\\mathrm{Riem}}(\\Phi_{\\alpha,\\tau})\n &=\\frac{\\alpha^2}{\\sqrt{1-\\tau^2}},\\\\\n\\eta_{\\mathrm{BKM}}^{\\mathrm{Riem}}(\\Phi_{\\alpha,\\tau})\n &=\\frac{\\alpha^2}{2\\tau}\\log\\frac{1+\\tau}{1-\\tau}.\n\\end{aligned}\n$$\n\nThe kernels are $(1+\\sqrt{x})^2/(4x)$, $x^{-1/2}$ and\n$\\log(x)/(x-1)$, with the BKM value $1$ at $x=1$.\nThe contraction coefficient is the literal supremum of\n$\\langle\\Phi(A),\\Omega_{\\Phi(\\rho)}^\\kappa(\\Phi(A))\\rangle/\n\\langle A,\\Omega_\\rho^\\kappa(A)\\rangle$ over all positive definite\ntrace-one $2\\times2$ complex matrices $\\rho$ and all nonzero traceless\nHermitian matrices $A$. The spectral action is\n$(\\Omega_\\rho^\\kappa(X))_{ij}=\\kappa(\\lambda_i/\\lambda_j)X_{ij}/\\lambda_j$.","title":"Hiai–Ruskai Conjecture 6.3: exact qubit CQ contraction coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/hiai-ruskai-2015-cq-contraction-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a93916f56b7510d9f7d3f3c0d8b770b0e9ea4a052006e25793f2764ed8a2525c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hieu-2026-weak-pareto-connectedness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vu Trung Hieu, *A connectedness result for convex polynomial multi-objective optimization without boundedness*, arXiv:2610.01029v1 (2026-10-01), §4.3 “Scope of the hypotheses”, asks:\n\n> whether $\\mathrm{Sol}^w(K,f)$ is still connected in the general convex case, with $K$ closed convex and the $f_i$ convex and continuous but not necessarily polynomial\n\nThe paper states:\n\n> no explicit example of a disconnected weak Pareto solution set is known\n\nand that Theorem 1\n\n> shows that no such example can have convex polynomial data.\n\nThe theorem in the paper proves connectedness for convex polynomial data without boundedness or a constraint qualification. The question is whether the conclusion survives for general continuous convex objectives.","title":"Connectedness of weak Pareto solutions for continuous convex data","url":"https://the-omega-institute.github.io/trureturing-pages/research/hieu-2026-weak-pareto-connectedness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e17cce2f2952ea80b8fdcadefa7416795bb0cfb8e8cf0a0853a154685c5d95c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:higuchi-sudbery-2000-four-qubit-average-entropy-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every normalized pure state $\\psi\\in(\\mathbb C^2)^{\\otimes4}$, the average two-qubit marginal von Neumann entropy in bits satisfies\n\n$$\\frac{S(\\rho_{AB})+S(\\rho_{AC})+S(\\rho_{AD})}{3}\\le 1+\\frac12\\log_2 3.$$\n\nEquality is attained at the explicit Higuchi–Sudbery state\n\n$$|M_4\\rangle=\\frac1{\\sqrt6}\\bigl[|0011\\rangle+|1100\\rangle+\\omega(|1010\\rangle+|0101\\rangle)+\\omega^2(|1001\\rangle+|0110\\rangle)\\bigr],\\qquad\\omega=e^{2\\pi i/3}.$$\n\nHiguchi–Sudbery, Section 3: “Given that a four-qubit state cannot have maximal entropy of entanglement for every two-qubit subset, we now ask what is the greatest possible average for such entropies, i.e. we seek to maximise”. Their average over the six pairs equals the displayed average over the three complementary cuts.","title":"higuchi sudbery 2000 four qubit average entropy maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/higuchi-sudbery-2000-four-qubit-average-entropy-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e66fa5d32e9e34cacf59a3af4a5249ed4a2991d679a96a95afa4dfc0e9f2637.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hirai-nieuwboer-walter-2023-coth-constant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hirai–Nieuwboer–Walter, arXiv:2303.04771v2, §5, Lemma 5.13 and Remark 5.14:\n\n> \"Lemma 5.13. The constant C = √2 is such that for all a, b ∈ ℝ, one has |coth((a − b)/2) (a coth(a/2) − b coth(b/2))| ≤ C √(ab(a − b) coth(a/2) coth(b/2) coth((a − b)/2)).\"\n\n> \"Remark 5.14. We conjecture, based on numerical evidence, that the optimal constant in the above inequality is C = 1/√2.\"\n\nThe source proves the coefficient $\\sqrt2$ and conjectures the optimal\ncoefficient $1/\\sqrt2$. The repository resolves that conjecture: the optimal\nconstant in\n\n$$\n\\left|\\coth\\frac{a-b}{2}\n\\left(a\\coth\\frac a2-b\\coth\\frac b2\\right)\\right|\n\\le C\\sqrt{ab(a-b)\\coth\\frac a2\\coth\\frac b2\\coth\\frac{a-b}{2}}\n$$\n\nis $C=1/\\sqrt2$. The quantifiers range over all real $a,b$ with\n$a\\ne0$, $b\\ne0$, and $a\\ne b$. Optimality means that for every real\n$C<1/\\sqrt2$, some such pair violates the estimate strictly.\nHere $\\coth x=\\cosh x/\\sinh x$ and square roots are nonnegative real roots.","title":"Optimal hyperbolic cotangent constant","url":"https://the-omega-institute.github.io/trureturing-pages/research/hirai-nieuwboer-walter-2023-coth-constant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/139c35a9ad9bec3b30b2873c3dc2402a7e9cc1438050c674fe561e825c740a28.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hirche-2020-arimoto-information-combining-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Hirche, Rényi Bounds on Information Combining, ISIT 2020,\narXiv:2004.14408v1, §IV-A, Conjecture IV.6:\n\n> There exists a value 1 < α̂ < 1.5783, such that k_α^A(k_α^A⁻¹(x) ∗ k_α^A⁻¹(y)) (47) is convex for 0 < α < 1 and concave for 1 < α ≤ α̂.\n\nEquation (23) defines\n\n$$\nK_\\alpha^A(X\\mid Y)=e^{((1-\\alpha)/\\alpha)H_\\alpha^A(X\\mid Y)}.\n$$\n\nFor binary $X\\sim\\{p,1-p\\}$, $k_\\alpha^A(p)=K_\\alpha^A(X)$, hence\n\n$$\nk_\\alpha^A(p)=(p^\\alpha+(1-p)^\\alpha)^{1/\\alpha},\\qquad\np\\star q=p+q-2pq.\n$$\n\nWrite $k_\\alpha=k_\\alpha^A$ below. Lemma IV.3, cited from\nHsu–Asoodeh–Salamatian–Calmon, gives convexity for $\\alpha\\ge2$;\nLemma IV.4 shows neither convexity nor concavity for\n$\\alpha\\in(1.58,1.97)$.\n\nThe inverse is the restriction to $[0,1/2]$, with range\n$I_\\alpha=[\\min(1,\\delta_\\alpha),\\max(1,\\delta_\\alpha)]$,\nwhere $\\delta_\\alpha=2^{(1-\\alpha)/\\alpha}$. Orders and exponents are real.\nFor each permitted order and every fixed $y\\in I_\\alpha$, the claim is the\nfull two-point `ConvexOn` or `ConcaveOn` inequality on the closed interval\n$I_\\alpha$. The specified witness is $\\hat\\alpha=3/2$.","title":"Arimoto information combining convexity and concavity","url":"https://the-omega-institute.github.io/trureturing-pages/research/hirche-2020-arimoto-information-combining-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/238b81d555a5b27e83a72edf8c1350ee556175bebc2ddfd2af53a833edc3d927.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hirche-2020-cachin-renyi-information-combining-convexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The target is the whole of C. Hirche, “Rényi Bounds on Information\nCombining”, ISIT 2020, arXiv:2004.14408v1, §IV-D, Conjecture IV.18, Eq. (88):\n\n> There exists a value $1 < \\hat\\alpha < 1.3863$, such that\n> $h_\\alpha(h_\\alpha^{-1}(x) \\ast h_\\alpha^{-1}(y))$ is convex for\n> $0 < \\alpha < \\hat\\alpha$ and concave for $\\alpha \\ge 2$.\n\nThe formal claim supplies the witness $\\hat\\alpha=5/4$ and quantifies over\nevery fixed $y\\in[0,\\log2]$, with convexity or concavity in $x$ on the whole\nclosed entropy interval. Put\n\n$$\nh_\\alpha(p)=\n\\begin{cases}\n-p\\log p-(1-p)\\log(1-p),&\\alpha=1,\\\\\n\\log(p^\\alpha+(1-p)^\\alpha)/(1-\\alpha),&\\alpha\\ne1,\n\\end{cases}\n\\qquad p\\star q=p+q-2pq.\n$$\n\nThe inverse branch has probability domain $[0,1/2]$. Logarithms are natural,\nall orders are real, and the Shannon value is Mathlib's `Real.binEntropy`.","title":"Cachin Rényi information combining convexity and concavity","url":"https://the-omega-institute.github.io/trureturing-pages/research/hirche-2020-cachin-renyi-information-combining-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ecd2f8aead603e56383fccdb6a8986797d5988c4166a00c0ba4c704acebfe4b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hirche-george-nuradha-wilde-2026-conditional-hockey-stick-orthogonal-reduction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Hirche, I. George, T. Nuradha and M. M. Wilde, *Conditional contraction coefficients and their applications to quantum networks*, arXiv:2608.27171v1, study contraction coefficients with reference systems.\n- **The coefficient.** η^Δ_D(N) := sup over |R| and ρ_RA ≠ σ_RA of [D((id ⊗ N)(ρ_RA)‖(id ⊗ N)(σ_RA)) − D(ρ_R‖σ_R)] / [D(ρ_RA‖σ_RA) − D(ρ_R‖σ_R)] (Eq. (2.14)), over pairs with a strictly positive finite denominator (Remark 2).\n- **The divergence.** E_γ(ρ‖σ) := Tr(ρ − γσ)₊ (Eq. (5.1)). For γ = 1 the coefficients η^c_Tr and η^Δ_Tr are optimized by orthogonal states (Proposition 8).\n- **The open problem.** \"When considering the hockey-stick divergence, can the optimization in the contraction coefficient be restricted to orthogonal states, similar to Proposition 8? This would directly imply that the associated contraction coefficient is monotone in γ\" (§9, p. 72).\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/hirchegeorgenuradhawilde2026conditional.md).\n\nIssue [#17637](https://github.com/the-omega-institute/trureturing/issues/17637) fixes the reading for the conditional coefficient η^Δ_{E_γ}, γ ≥ 1:\n- S_γ(N) is the set of ratios over every finite reference system R and all states ρ, σ on R ⊗ A with a positive denominator;\n- S^⊥_γ(N) is its subset from pairs with ρσ = 0;\n- (i), the question asked: S_γ(N) = S^⊥_γ(N);\n- (ii), a strengthening: S_γ(N) = S_δ(N) for all γ, δ ≥ 1.\n\nBoth are proved in this reading. The complete coefficient η^c_{E_γ} is not addressed.","title":"Conditional hockey-stick contraction coefficients reduce to orthogonal states and do not depend on γ","url":"https://the-omega-institute.github.io/trureturing-pages/research/hirche-george-nuradha-wilde-2026-conditional-hockey-stick-orthogonal-reduction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9123136b893d1296d5283093ab6221d3bbca375ef1b6de5841a05384ac9e82b4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hirche-guan-tomamichel-2023-renyi-order-three-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Hirche, X. Guan and M. Tomamichel, “Chain Rules for Rényi Information\nCombining”, arXiv:2305.02589v1, Section V, Conjecture V.5, state:\n\n> For $\\alpha\\in[2,3]$ the same holds with $\\geq$ exchanged by $\\leq$ and\n> for $\\alpha\\in\\{2,3\\}$ the above holds with equality.\n\nThe settlement is the $\\alpha=3$ equality clause for two independent,\nequiprobable binary classical-quantum inputs. Each input has two\npositive semidefinite, trace-one complex matrices in an arbitrary finite\ndimension. Put $H_i=\\widetilde H_3^\\downarrow(X_i|B_i)_{\\rho_i}$ and\n$H_{\\rm out}=\\widetilde H_3^\\downarrow(X_1+X_2|B_1B_2)_\\tau$, with $X_2$\ntraced out of the CNOT state. Then\n\n$$\n\\begin{aligned}\nH_1+H_2\\leq\\log2&\\Longrightarrow\nH_{\\rm out}=h_3(h_3^{-1}(H_1)\\ast h_3^{-1}(H_2)),\\\\\n\\log2\\leq H_1+H_2&\\Longrightarrow\nH_{\\rm out}=H_1+H_2-\\log2+\n h_3(h_3^{-1}(\\log2-H_1)\\ast h_3^{-1}(\\log2-H_2)).\n\\end{aligned}\n$$\n\nHere $p\\ast q=p(1-q)+(1-p)q$, and $h_3^{-1}$ is the inverse on probabilities\nin $[0,1/2]$. Both guards hold at the common boundary. The binding matrix,\ninverse and natural-logarithm conventions are in\n[#15115](https://github.com/the-omega-institute/trureturing/issues/15115).","title":"Order-three Rényi information-combining equality","url":"https://the-omega-institute.github.io/trureturing-pages/research/hirche-guan-tomamichel-2023-renyi-order-three-equality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38eb45faf6c5a3df3b900fa799415495e51fe8028d175dbc92d52e120108e6de.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hivert-scott-2026-okada-discriminant-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Florent Hivert and Jeanne Scott, *Diagrammatic Okada monoid and cellularity of the Okada algebra*, arXiv:2609.01440v1, Conjecture 7.9, page 60:\n\n> det G_N = ∏_{S ∈ 𝕐𝔽𝕊_N} [det G^S_N]^{2 dim(Fib(S))}.\n\nThe preamble defines G_N(σ,τ) as tr[ϱ^reg(E_σ)ϱ^reg(E_τ)], where ϱ^reg is the left-regular representation of the Okada algebra. The parameters are a positive integer N, an arbitrary field K, X = (x₁,…,x_{N−1}) and Y = (y₁,…,y_{N−2}). The algebra has relations E_i² = x_i E_i, distant commutation, and E_{i+1}E_iE_{i+1} = y_i E_{i+1}. The conjecture concerns exact equality with this regular trace.","title":"Refutation of the Okada regular-trace discriminant identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/hivert-scott-2026-okada-discriminant-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60cb0fc2ee23efa72b785b33dda2538cb4b38dba558cbcac7f80a17810020b43.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:holevo-shirokov-2026-pure-state-formation-supporting-functional","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. S. Holevo and M. E. Shirokov, *Log-Sobolev inequality, von Neumann entropy and Entanglement of Formation*, arXiv:2609.12667v3, Introduction:\n\n> It is somewhat surprising that the above problem is nontrivial even in the case when the state $\\rho$ is pure. We and GPT-5.6 have found neither a general proof nor counterexamples which would confirm or refute the hypothesis of the existence of a supporting affine functional for the EoF at any pure bipartite state.\n\nSection 3.1 restates the finite-dimensional question. The verbatim passages are in [the literature note](../Library/Entropy/holevo2026logsobolevformation.md).\n\nIssue [#19308](https://github.com/the-omega-institute/trureturing/issues/19308) fixes the reading: for all finite a, b and every unit ψ ∈ ℂ^a ⊗ ℂ^b, there is a Hermitian W with Tr(W P_ψ) = E_F(P_ψ) and Tr(Wσ) ≤ E_F(σ) for every density matrix σ. This is proved.","title":"The entanglement of formation has a supporting affine functional at every pure state","url":"https://the-omega-institute.github.io/trureturing-pages/research/holevo-shirokov-2026-pure-state-formation-supporting-functional/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d55a47fe00523e324d847feb5b1b72e41e7bca9e134372ab602de92dbce2e90f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:horodecki-2022-rank-of-contextuality-non-power-of-two","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Horodecki et al., *The rank of contextuality*, arXiv:2205.10307v1,\nNew Journal of Physics 25 (2023), 073003, Section IX (Section 9 in the journal):\n\n> Finally, it would be interesting to construct a quantum behavior having the rank of contextuality different from the power of 2 (the power of 2 is achieved by a tensor product of the Peres-Mermin behaviors).\n\nThe question asks for an example. The quantum behavior $Q=P\\otimes P$ formed\nfrom two Peres–Mermin squares has rank $3$.\n\n### Statement echo\n\n| Source clause or representation choice | Lean counterpart | Scope |\n| --- | --- | --- |\n| An ordered list of observable outcomes | `restrict C x g` | Boolean outcomes on finite ordered contexts |\n| A family of conditional distributions | `Behavior C`, constrained by `IsQuantum C P` | `Behavior` is the repository's unrestricted real-table carrier; the quantum predicate imposes the Born rule |\n| Mixtures of deterministic assignments, equation (2) | `NonContextual C N` | Nonnegative real weights with total mass one and every restriction-fiber sum |\n| Minimum number of distinct noncontextual behaviors covering every context, equation (3) | `coverSizes` and `IsRankOfContextuality` | One member agrees on all outcomes at each context; the member can depend on that context |\n| Quantum observables and state | `IsQuantum` | Positive semidefinite trace-one state; Hermitian involutions; within-context commutation; ordered spectral-projection products |\n| Construct a quantum behavior | `claim`, proved by `result` | Existential observable count, context count, context lists, table and rank |\n| Rank different from every power of two | Final conjunct of `claim` | All natural exponents, including zero |\n\nThe witness has $18$ observable labels, $36$ contexts, state $I_{16}/16$ and\nrank $3$. Finite labels, ordered contexts and binary outcomes restrict the\nconstruction to a subclass of quantum behaviors; existence in that subclass\nanswers the source's existence question. No hypothesis is added to the question.","title":"A quantum behavior with rank of contextuality three","url":"https://the-omega-institute.github.io/trureturing-pages/research/horodecki-2022-rank-of-contextuality-non-power-of-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22f6a14045a085441c9280d125a7f752728de34ef05134c0b5ecda2c59e34079.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hoster-stump-2025-chow-polynomials-simplicial-posets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Elena Hoster and Christian Stump, *Chow polynomials of simplicial posets\nwith positive h-vector are real-rooted*, arXiv:2508.15538v1, Section 1,\nConjecture 1.5, states:\n\n> Let P be a simplicial poset. Then H_P̂(x), H_P̂*(x) and H^aug_P̂(x) are real-rooted. Moreover, the roots of both H_P̂(x) and H_P̂*(x) interlace the roots of H^aug_P̂(x) = H^aug_P̂*(x).\n\nThis dossier anchors that single conjecture. The settlement refutes its first\nassertion, and therefore the conjunction, without negating its other\nreal-rootedness assertions separately. The definitions are in Section 1,\npage 1, equations (1.1)–(1.2); the conjecture is on page 3 and on page 4\nof the FPSAC 2026 abstract.","title":"Hoster–Stump Conjecture 1.5: simplicial-poset Chow polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/research/hoster-stump-2025-chow-polynomials-simplicial-posets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b087fb19d2565d4f782c3ea145fd8234640dac1c0b12a8cc186721ce3738767.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hou-jiang-miao-2023-root-of-unity-infinity-count-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Hou, Y. Jiang and Y. Miao, *Rational Q-systems at Root of Unity I. Closed Chains*, arXiv:2310.14966v3, SciPost Phys. 16, 129 (2024), Appendix C, conjecture (C.2), page 35:\n\n> Before introducing the algorithm, we make the following conjecture for the number of primitive states with infinite Bethe root(s) by observing the numerical results:\n> N^{pri}_{±∞}(L, M, n_±) = binom(L, M − n_±) − sum_{x=0}^{M−n_±−1} binom(L,x), (C.2),\n> when n_± = n_+ = n_− is a solution to (3.15). When there is no solution to (3.15), N^{pri}_{±∞}(L, M, n_±) = 0.\n\nAppendix C specializes to no twist κ = 1, η = iπ/3, even L and M ≤ L/2. Equations (3.15)–(3.17) imply L ≡ 2(M − n) (mod 6) and 0 ≤ n ≤ 2: η = iπ/3 gives ℓ₁ = 1 and ℓ₂ = 3. For states with infinite roots, n ≥ 1; n ≤ M records that these roots are among the M Bethe roots.","title":"Refutation of the rational Q-system infinite-root primitive count","url":"https://the-omega-institute.github.io/trureturing-pages/research/hou-jiang-miao-2023-root-of-unity-infinity-count-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cd9d74aa97f1819ab23855b0a48f8b50804f2cbbd08eced05d7b6e13f314c46.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hou-zhao-2026-sum-free-code-dimension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Xiang-Dong Hou and Shujun Zhao, “Further Results on Sum-Freedom of Binary\nand q-ary Functions”, arXiv:2609.31489v1, §4.2, printed p. 14,\nimmediately after Theorem 4.6:\n\n> There are some open questions. In Theorem 4.1, the dimension and minimum\n> weight of the code C(f ) are determined and they are independent of f as long as\n> f is APN, i.e., 2nd order sum-free. Do we have the same conclusion for the code\n> Cs (f ) in Theorems 4.5 and 4.6, where f is an sth order sum-free function? If the\n> dimension k and the minimum weight d of Cs (f ) depend on the sum-free function f ,\n> what are the ranges for k and d? Likely, these questions will lead the investigation\n> of sum-free functions to new directions.\n\nDefinition (1.1) requires a nonzero function sum on every s-dimensional\naffine subspace. Equation (4.7) defines C_s(f) as the kernel of the\nparity-check matrix consisting of generator rows of\nR_q(s(q−1)−1,n), followed by the n coordinate rows of f.\n\nThe dimension-only universal assertion, specialized to q = 3, is\n\n$$\n\\forall n,s\\in\\mathbb N,\\quad\nn\\ge2,\\ 1\\le s\\le n-1\\ \\Longrightarrow\\\n\\forall f,g:\\mathbb F_3^n\\to\\mathbb F_3^n,\\quad\n\\operatorname{SumFree}_s(f)\\land\\operatorname{SumFree}_s(g)\n\\ \\Longrightarrow\\ \\dim C_s(f)=\\dim C_s(g).\n$$\n\nThe closed Lean definitions `claim : Prop` and `result : ¬ claim`\nretain exactly these quantifiers. They settle dimension independence\nnegatively, without settling minimum-distance independence or either full range.","title":"Hou–Zhao: the dimension of a sum-free function's code","url":"https://the-omega-institute.github.io/trureturing-pages/research/hou-zhao-2026-sum-free-code-dimension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bd39ea2a8de7b63f594dfc1d1e02676ac13b674436911eb14ede814f69f4896.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hou-zhao-2026-sum-free-code-minimum-weight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Xiang-dong Hou and Shujun Zhao, *Further Results on Sum-Freedom of Binary\nand q-ary Functions*, arXiv:2609.31489v1 (2026-09-25), printed p.14,\nimmediately after Theorem 4.6:\n\n> There are some open questions. In Theorem 4.1, the dimension and minimum\n> weight of the code C(f) are determined and they are independent of f as long\n> as f is APN, i.e., 2nd order sum-free. Do we have the same conclusion for\n> the code C_s(f) in Theorems 4.5 and 4.6, where f is an sth order sum-free\n> function? If the dimension k and the minimum weight d of C_s(f) depend on\n> the sum-free function f, what are the ranges for k and d? Likely, these\n> questions will lead the investigation of sum-free functions to new directions.\n\nThe component considered here is minimum-weight independence alone.\nFor every finite field K of cardinality q, every n ≥ 2, every\n1 ≤ s ≤ n−1 and every f,g:K^n→K^n whose vector sums on **every** affine\ns-plane are nonzero, assert d(C_s(f)) = d(C_s(g)). There is no additional\nplanarity, permutation, non-degeneracy, rank or equal-dimension hypothesis.\n\nThe original definition (1.1) uses affine subspaces. A basis of the direction\nsubspace parametrizes such a subspace bijectively by K^s; independence of\nthe directions prevents repetitions. `SumFree` uses exactly these sums.\nDefinitions (4.2) and (4.7) make C_s(h) the kernel of the evaluations of\nall reduced monomials of total degree at most s(q−1)−1, together with all\nn coordinate rows of h. Each exponent is less than q. These monomials\nspan the source Reed–Muller space, so redundant rows leave the kernel unchanged.\nAll q^n points, including zero, are present.\n\n`claim : Prop` retains the full universal assertion over small finite-field\ncarriers. Every finite field has a small isomorphic carrier. Hamming weight\ncounts nonzero entries; `minimumWeight` takes the infimum over **nonzero**\nkernel words in the extended naturals. The zero code has minimum infinity.\nThis convention adds no premise and is irrelevant to the displayed nonzero codes.\nThe conclusion is literally `result : ¬ claim`.\n\nThe separate dimension result, issue14810 and PR14846, neither proves nor\nrefutes this minimum-weight component. The present result does not classify\nthe complete ranges of k and d and does not contradict Theorem 4.6's bounds.","title":"Hou–Zhao: minimum-weight independence of sum-free codes","url":"https://the-omega-institute.github.io/trureturing-pages/research/hou-zhao-2026-sum-free-code-minimum-weight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18557c7bea8b82333bef2917e98be75c37c3ce88ed50ba8ceb4b3e510b081c11.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:howard-brennan-vala-2013-chsh-graph-regularity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Howard, E. Brennan and J. Vala, *Quantum Contextuality with Stabilizer States*, Entropy 15 (2013) 2340–2362, arXiv:1501.04342v1, Section 5:\n\n> Observe that for all $d \\in \\{2,3,5,7\\}$ that we have discussed, we see some structure in the graphs $\\Gamma_{\\textrm{CHSH}}$ obtained from qudit CHSH inequalities. Whereas it is quite easy to see that all graphs will have order $|\\Gamma_{\\textrm{CHSH}}|=d^3$, we conjecture based on the examples here that $\\Gamma_{\\textrm{CHSH}}$ is a $(2d-1)(d-1)$-regular graph.\n\nThe operators A_j and B_k, the spectral projectors and their rank-one decomposition, and the graph convention are quoted in [the literature note](../Library/QuantumContext/howard2013stabilizercontextuality.md).\n\nIssue [#19492](https://github.com/the-omega-institute/trureturing/issues/19492) fixes the reading:\n- the vertices are the d³ rank-one projectors Π^A_j[a] ⊗ Π^B_k[−jk − a] of the decomposition of the CHSH Bell operator;\n- two vertices are adjacent when the product of their projectors vanishes.\n\nThe conjecture is proved.","title":"The qudit CHSH orthogonality graph is (2d − 1)(d − 1)-regular for every prime d","url":"https://the-omega-institute.github.io/trureturing-pages/research/howard-brennan-vala-2013-chsh-graph-regularity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0388bb9e09df20aaba62fc168dd4282a411e218195024b4bf4cb277629e989f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hsu-et-al-2026-symmetric-satwap-maximal-violation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H.-Y. Hsu, G. N. M. Tabia, K.-S. Chen, M.-E. Liu, T. Vértesi, N. Brunner and Y.-C. Liang, *Trading symmetry for Hilbert-space dimension in Bell-inequality violation*, arXiv:2601.02893v4; Quantum Science and Technology 11(3), 035003 (2026), conjecture following Eq. `Eq:symSATWAP`:\n\n> Our computation results show that for the symmetric inequalities of~\\cref{Eq:symSATWAP}, the measurement of \\cref{Eq:CGLMP-Opt}, in conjunction with a symmetric maximally entangled two-qudit state, indeed reproduces the maximal quantum violation~\\cite{SAT+17} of $2(d-1)$ for $d\\le 100$. Together with the fact~\\cite{Sarkar:2021aa} that this maximal quantum violation self-tests a maximally entangled two-qudit state, we know that the symmetric family of SATWAP inequalities, \\cref{Eq:symSATWAP}, also cannot exhibit a trade-off for $d\\le 100$. Moreover, we conjecture that this holds true for any higher $d$ as well.\n\nThis quotation is from line 532 of the downloaded v4 TeX member `SymmetricBIV-arXiv_v2.tex` (2736 lines, md5 `6116dc142244f252c16a835d291626cf`). The symmetric-strategy definition is at lines 285–297, the measurements at lines 488–497, and the correlators and symmetric SATWAP expression at lines 513–530. See also [the literature note](../Library/Fourier/hsuetal2026tradingsymmetry.md).\n\nThe module uses computational indices $0,\\ldots,d-1$ and real-exponent phases $\\omega^t=e^{2\\pi i t/d}$. Its measurement matrices are\n\n$M_{a|0}=|a\\rangle\\langle a|$, $M_{a|1}=U|a\\rangle\\langle a|U^\\dagger$, $U=TW$,\n\nwhere $T=\\operatorname{diag}(-1,1,\\ldots,1)$ and $W_{ij}=1/[d\\sin((i-j-1/2)\\pi/d)]$. Both parties use this same measurement function. Module settings $0,1$ correspond respectively to settings $1,2$ in the source's symmetric SATWAP expression.\n\nFor a vector $\\psi$, put $P_\\psi(a,b|x,y)=\\langle\\psi|(M_{a|x}\\otimes M_{b|y})|\\psi\\rangle$ and $C_{xy}^{(k)}=\\sum_{a,b=0}^{d-1}\\omega^{-k(a+b)}P_\\psi(a,b|x,y)$. With $a_k=(1-i)\\omega^{k/4}/2$, `satwapValue` is the complex-valued finite sum\n\n$\\tilde I_d(\\psi)=\\sum_{k=1}^{d-1}\\left[\\omega^{-2k}\\left(a_k C_{00}^{(k)}+a_k^*C_{01}^{(k)}+a_k^*C_{10}^{(k)}\\right)+\\omega^{-3k}a_k C_{11}^{(k)}\\right]$.\n\nIssue [#22009](https://github.com/the-omega-institute/trureturing/issues/22009) fixes the attainment statement: for every natural $d\\ge2$, there exists $\\psi\\in\\mathbb C^d\\otimes\\mathbb C^d$ with $\\|\\psi\\|=1$, $S\\psi=\\psi$, $\\operatorname{Tr}_B|\\psi\\rangle\\langle\\psi|=I/d$, and $\\tilde I_d(\\psi)=2(d-1)$. This is proved. The universal quantum upper bound and the self-testing statement used by the source to infer minimal dimension are separate literature inputs.","title":"Identical TW measurements attain $2(d-1)$ for every $d\\ge2$","url":"https://the-omega-institute.github.io/trureturing-pages/research/hsu-et-al-2026-symmetric-satwap-maximal-violation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95790c9f0db1972e3faf9f9f1cf066a7ea2ffab45077fd7c88a2aa40109a0c12.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ht2508","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David He and Daniel Tubbenhauer, *Tensor powers of representations of (diagram) monoids*, arXiv:2508.04054v1, Conjecture 4.2, page 17:\n\n> Let V be a kS-module on which no element acts invertibly, then there is some m ≥ 1 such that Z is a summand of V^{⊗m}, and consequently a(n)=(dim V)ⁿ.\n\nThe field is algebraically closed, the semigroup is nonempty and finite with no two-sided identity, and the representation is finite dimensional and nonunital. The one-dimensional null representation has zero action. The source defines $a(n)$ as an asymptotic equivalent of the indecomposable-summand count $b(n)$, with multiplicity. Thus the final clause means $b(n)/(\\dim V)^n\\to1$, rather than exact equality at finite exponents. Preregistration: [#22531](https://github.com/the-omega-institute/trureturing/issues/22531).\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Algebraically closed field of arbitrary characteristic | `claim`: field and algebraic-closure instances, with no characteristic restriction |\n| Nonempty finite semigroup without identity | `claim`: semigroup, finiteness and nonemptiness instances, and the negated two-sided-identity existence hypothesis |\n| Finite-dimensional nonunital module | `claim`: finite-dimensional vector space and a multiplicative map into endomorphisms |\n| Every element acts noninvertibly | `claim`: nonunit hypothesis for every action |\n| Some positive tensor power contains the null representation as an actual direct summand | `HasNullSummand` and the first conjunct of `claim`; complementary subspace is invariant |\n| Indecomposable summands counted with multiplicity | `IsIndecomposableSummand`, `IsSummandCount`, internal direct sum indexed by `Fin b` |\n| The asymptotic equivalent is the tensor dimension | Count-sequence existence and the ratio limit for every such sequence, the second and third conjuncts of `claim` |\n\nNonemptiness and singularity exclude the zero space; there is no additional nonzero-space hypothesis. The public definitions retain the literal preregistered expressions. The tensor action uses the upstream `PiTensorProduct.map` construction.","title":"A null direct summand in a tensor power of a singular semigroup module","url":"https://the-omega-institute.github.io/trureturing-pages/research/ht2508/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a3260876b52581d1b28e333dfe612e3f8c0fcb67cdc941bc1d69634059d344d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:huang-kosugi-nishi-matsushita-2024-reflection-diagonal-cnot-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Huang, T. Kosugi, H. Nishi and Y.-i. Matsushita, *Optimized Synthesis of Circuits for Diagonal Unitary Matrices with Reflection Symmetry*, J. Phys. Soc. Jpn. 93, 054002 (2024), arXiv:2310.06676v2, state in Sect. 6, Conclusion (`main_arXiv2.tex`, line 847):\n\n> Although we conjecture that the number of entanglement gates is at best $2^{2n-1}+2n-2$ in the $\\{\\mathrm{CNOT}, R_z\\}$-based circuit for the worst scenario of diagonal unitary matrices that satisfy Eq. \\eqref{intro:eq-theta}, […]\n\nThe quoted TeX label denotes Eq. (1): for $n\\ge1$, the real phases of a $2^{2n}\\times2^{2n}$ diagonal unitary satisfy $\\theta_{j,j'}=\\theta_{2^n-1-j,\\,2^n-1-j'}$ for every $0\\le j,j'<2^n$ (`main_arXiv2.tex`, lines 207–227). Equivalently, writing $m=2n$ and $x\\in\\{0,1\\}^m$, $\\theta(\\bar x)=\\theta(x)$, where $\\bar x$ complements every bit. The source specifies ancilla-free $\\{\\mathrm{CNOT},R_z\\}$ circuits at lines 194 and 1225 and realization up to a global phase at lines 429–445.\n\nIssue [#24527](https://github.com/the-omega-institute/trureturing/issues/24527) fixes the reading: for every $n\\ge1$ there is such a phase function $\\theta$ for which every finite circuit realizing $\\operatorname{diag}(e^{\\mathrm i\\theta(x)})$ up to a global phase has at least $2^{2n-1}+2n-2$ CNOT gates. This lower-bound statement is proved. The source's matching upper bound from Algorithm 1 is separate from `claim`.","title":"The worst-case CNOT count for reflection-symmetric diagonal unitaries","url":"https://the-omega-institute.github.io/trureturing-pages/research/huang-kosugi-nishi-matsushita-2024-reflection-diagonal-cnot-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ceb405ac914ebf393513713d24ed1dd87a11924da252f8baa2b957dda98ef87.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:huang-ma-2026-binary-delta-invariant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Min Huang and Qiling Ma, *A mutation invariant for skew-symmetrizable matrices*, arXiv:2602.03194v1, Remark 2.7; Journal of Algebra 712 (2027), 546–558. The [source note](../Library/Eigenstructure/huangma2026mutation.md) gives the verified locator and verbatim definitions.\n\n> We also conjecture that for any fixed n and any choice of skew-symmetrizer d₁, …, dₙ, the invariant δ is binary. This is supported by several explicit computations.\n\nFor an integer matrix $B$, fix positive integer weights $d_i$ with $d_i b_{ij}=-d_j b_{ji}$. The real symmetric matrix $S(B)$ has diagonal $2$, upper entries $\\operatorname{sgn}(b_{ij})\\sqrt{|b_{ij}b_{ji}|}$ and lower entries $\\operatorname{sgn}(b_{ji})\\sqrt{|b_{ij}b_{ji}|}$. The invariant is $\\delta(B)=\\det S(B)\\pmod4$. Binary means at most two residues for fixed size and weights, without asserting that both are attained.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Fixed matrix size and fixed positive integer skew-symmetrizer | `claim` quantifies the natural size and positive vector before choosing the two residues | Same quantifier order; the empty matrix is included as an extension |\n| The diagonal skew-symmetrizer makes the matrix skew-symmetric | `IsSkewSymmetrizer` is positivity and the literal entrywise skew relation | Diagonal zero follows from the relation and positivity |\n| The three cases defining the real symmetrization | `symmetrization` uses diagonal 2, integer sign, absolute value and real square root | Same ordered-index expression |\n| The determinant is an integer and its residue is binary | `claim` chooses an integer determinant and one of two residues for every admissible matrix; `result` proves it | Integrality is included explicitly; the two residues may coincide |","title":"Huang–Ma Remark 2.7: the binary delta invariant","url":"https://the-omega-institute.github.io/trureturing-pages/research/huang-ma-2026-binary-delta-invariant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7c234bf3acd1de22d1cdcbaf9219618b482b8c64dbfc9374329d2b5f926f144d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:huang-stoica-xia-zhong-2020-tree-equations-nonconstant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Huang, B. Stoica, X. Xia and X. Zhong, *Bounds on the Ricci curvature and solutions to the Einstein equations for weighted graphs*, arXiv:2006.06716v1 (2020), define the tree equation of motion using the shortest-path lengths $P_{ij}$ and the vertex sums $c_i$ and $d_i$. For an edge $ij$ they write\n$$\n\\mathrm{tEoM}_{ij}\n=\\frac{1}{P_{ij}}\\left(\\frac{c_i^2}{d_i^2}+\\frac{c_j^2}{d_j^2}\\right)\n-\\frac{c_i}{d_i}-\\frac{c_j}{d_j}=0.\n$$\nFor a tree, they identify $P_{ij}$ on an edge with its edge length, and use $d_i=\\sum_{i'\\sim i}P_{ii'}^{-2}$ and $c_i=\\sum_{i'\\sim i}P_{ii'}^{-1}$. They define $T_q$ as the infinite tree whose vertices have degree $q+1$, and call the setting with all edge lengths equal the constant solution. Section 5.2's second conjecture (ms.tex lines 1931–1934) asks:\n\n> There exists no non-constant solution to the tEoM for $T_2$ with all edge lengths positive.\n\nThe Lean definition `claim` quantifies this assertion over every locally finite simple graph with decidable adjacency that is a tree, has degree $3$ at every vertex, and carries a symmetric positive edge-length function whose `tEoM` vanishes on every edge. It concludes that one real constant equals every edge length.","title":"The Huang–Stoica–Xia–Zhong conjecture on $T_2$ is refuted","url":"https://the-omega-institute.github.io/trureturing-pages/research/huang-stoica-xia-zhong-2020-tree-equations-nonconstant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b669b7edce540dcdde291bf2042cf4cc377ca24c212681b765acaf07da117d49.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hughes-iteration-depth-no-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For a finite alphabet and arbitrary languages `A` and `B`, fix a positive\ninsertion degree `k` throughout each iteration history. A word has iteration\ndepth `m` when `m` is the least stage at which the word occurs for some such\ndegree. Hughes's Section 10 conjecture states that if depth `r` occurs, every\ndepth from zero through `r` occurs. The formalization also proves this statement\nover arbitrary alphabets, with no finiteness or countability assumption.\n\nThis is the arbitrary-language iteration-depth conjecture on page 8. It is\nnot the finite-language Theorem 8 and not the insertion-degree conjecture in\nSection 11.","title":"Hughes iteration-depth no-gap conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/hughes-iteration-depth-no-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa3453166879e88085248d0ec1f8140043a692e052559678180b6edc8dc6a532.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hutton-mehta-vukovic-2025-cq-output-overlap-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hutton, Mehta and Vukovic, *Deciding whether a C–Q channel preserves a bit is QCMA-complete*, arXiv:2508.10664v1, Conjecture 1.5, page 3, ask whether, for quantum states $\\sigma_1,\\ldots,\\sigma_n\\in M_d(\\mathbb C)$ and $k\\le n$ pairwise orthogonal pure states $u_1,\\ldots,u_k\\in\\mathbb C^n$,\n\n$$\n\\frac{1}{k(k-1)}\\sum_{r\\ne s}\\operatorname{Tr}\\!\\left(\\Bigl(\\sum_i |(u_r)_i|^2\\sigma_i\\Bigr)\\Bigl(\\sum_j |(u_s)_j|^2\\sigma_j\\Bigr)\\right)\n\\ge \\min_{i_1,\\ldots,i_k\\text{ distinct}}\\frac{1}{k(k-1)}\\sum_{r\\ne s}\\operatorname{Tr}(\\sigma_{i_r}\\sigma_{i_s}).\n$$\n\nThe denominator requires $2\\le k$. A pure state is a unit vector, and a quantum state is positive semidefinite with trace one. Definition 1.3 specifies the channel on a pure input by $\\Phi(|u\\rangle\\langle u|)=\\sum_i|u_i|^2\\sigma_i$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Arbitrary finite dimensions $n,d$ | Natural-number parameters of `claim` | All dimensions; empty impossible state/frame cases are retained |\n| Arbitrary quantum states $\\sigma_i$ | `CoPRelativeQuantumnessRefutation.IsDensity`: positive semidefinite and trace one | Literal conjunction supplied directly by the existing public owner |\n| $k\\le n$ orthogonal pure inputs | `k ≤ n`, `IsOrthonormalFamily u` | Coordinate inner products are the Kronecker delta, including unit norms |\n| Defined average over unequal ordered pairs | `averageOutputOverlap` | Real reciprocal $1/(k(k-1))$, zero diagonal, real part of each trace |\n| Minimum over $k$ distinct indices | `minDistinctOverlap` | Infimum over injective `Fin k → Fin n`; finite and nonempty under $k\\le n$ |\n| Lower bound | `result : claim` | Same inequality direction |\n| Denominator defined | `2 ≤ k` | Makes the source normalization positive |\n\nNo source clause is dropped. The five local public definitions and the directly reused public `CoPRelativeQuantumnessRefutation.IsDensity` retain the literal conventions of [#20116](https://github.com/the-omega-institute/trureturing/issues/20116).","title":"The minimum output overlap of orthogonal C–Q inputs","url":"https://the-omega-institute.github.io/trureturing-pages/research/hutton-mehta-vukovic-2025-cq-output-overlap-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/edb62fcdb2a1dab176a63f07b0c39af6b419ca55730e677ae7aa3b1f8a9aeb53.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:huynh-vu-zaw-scarani-2023-precession-separable-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Khoi-Nguyen Huynh-Vu, Lin Htoo Zaw and Valerio Scarani,\n*Certification of genuine multipartite entanglement in spin ensembles with\nmeasurements of total angular momentum*, arXiv:2311.00806v2, §III,\nConjecture 2, Eq. (32), PDF p. 8:\n\n> The separable bound for {ȷ̃, ȷ̃′} = {1, K/2} with K ≥ 7 is\n> Psep_K({1, K/2}) = ½ [1 + 2^{−(K−1)} binom(K−1, (K−1)/2) (K−1)/(K+1)].\n\nThe protocol takes odd K, J_k = cos(2πk/K)J_x + sin(2πk/K)J_y,\nand Q_K = (1/K) Σ_k pos(J_k), where 2 pos acts on an eigenvalue m\nas 1+sgn(m), including half weight at zero. Eqs. (7) and (10) express\nthe separable bound as the maximum over unit product pure states.\nThe two factors here have dimensions 3 and K+1. Preregistration:\nhttps://github.com/the-omega-institute/trureturing/issues/11873.","title":"Spin-1 tensor spin-K/2 precession separable bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/huynh-vu-zaw-scarani-2023-precession-separable-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60d6f116c21922a2b657904f17190e86cc7d9ab9603e2feb9b37a24015e2f036.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:huynh-vu-zaw-scarani-2023-universal-gme-threshold-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Khoi-Nguyen Huynh-Vu, Lin Htoo Zaw and Valerio Scarani,\n*Certification of genuine multipartite entanglement in spin ensembles with\nmeasurements of total angular momentum*, Phys. Rev. A 109, 042402,\narXiv:2311.00806v2, Conjecture 3, PDF p. 9:\n\n> Consider a spin ensemble. Perform the precession protocol with odd $K \\geq 3$ on the total angular momentum of the system. If the score $P_K > \\mathbf{P}_K^{\\mathrm{conj}}$ is obtained, then the spin ensemble is GME.\n\nHere $J_k=\\cos(2\\pi k/K)J_x+\\sin(2\\pi k/K)J_y$ is the sum of the\nparticle observables, $Q_K=K^{-1}\\sum_k\\operatorname{pos}(J_k)$ and\n$P_K=\\operatorname{tr}(\\rho Q_K)$. The spectral weight is\n$(1+\\operatorname{sgn}m)/2$, including one half at zero.\nSeparability across a cut means a normalized finite convex mixture of\nproducts of density matrices. GME excludes convex mixtures over all\nnontrivial cuts. Preregistration: [#13586](https://github.com/the-omega-institute/trureturing/issues/13586).","title":"Singlet padding refutes the universal precession GME threshold","url":"https://the-omega-institute.github.io/trureturing-pages/research/huynh-vu-zaw-scarani-2023-universal-gme-threshold-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea9280ff5bf3d517ec8e8841b633161e7ef73b73b0f7fc1b11b594c4ddc48103.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:hypercube-inequality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let `d>=1`, let `x_v>=0` for every vertex `v` of `{0,1}^d`, and let\n`A_d` be the integer matrix whose column at `v` is `(v,1)`. For each\nunordered `(d+1)`-vertex subset `S`, let `A_S` be the square submatrix of\nthose actual columns. Prove\n\n```text\n(sum_{|S|=d+1} |det A_S| * product_{v in S} x_v)\n * (sum_v x_v)^(d-1)\n<= product_{i in Fin d}\n (sum_{v_i=0} x_v) * (sum_{v_i=1} x_v).\n```\n\nEach subset occurs once and there is no factorial factor. The formal\ndefinitions are `augmentedCube` and `hypercubeWeight`; `result` proves the\ndisplayed assertion for every positive natural dimension, including zero\nweights, zero facets and singular determinant data.","title":"The all-dimensional Hypercube Inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/hypercube-inequality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/834a26b20c0b41c2919e592b99b14c82593989a6c3b7b1c8c79a5422928674dd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ikhlef-morin-duchesne-2023-conjectures-c1-c2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Yacine Ikhlef and Alexi Morin-Duchesne, *Fusion of irreducible modules in the\nperiodic Temperley–Lieb algebra*, Appendix C, Conjectures C.1 and C.2,\narXiv:2312.14837v2; SciPost Phys. 17, 132, DOI 10.21468/SciPostPhys.17.5.132.\nThe source note is `D5/L/QuantumContext/ikhlefmorinduchesne2024fusion`.\nPreregistration: [#18569](https://github.com/the-omega-institute/trureturing/issues/18569).\n\nWrite $[k]=(q^k-q^{-k})/(q-q^{-1})$, $\\{k\\}=q^k+q^{-k}$ and\n$\\alpha_z=z+z^{-1}$. Proposition C.1 defines $X^{(p)}_m$ by (C.3)–(C.4):\n\n$$\nX^{(p)}_p=1,\\qquad\nX^{(p)}_{p+1}=\\alpha_x+\\frac{\\alpha_y\\alpha_w}{\\{p+1\\}},\n$$\n\n$$\nX^{(p)}_{k+1}=\\left(\\alpha_x+\n\\frac{\\{p\\}\\alpha_y\\alpha_w}{\\{k\\}\\{k+1\\}}\\right)X^{(p)}_k+\n\\frac{(\\{p\\}^2-\\{k\\}^2)(\\alpha_y^2-\\{k\\}^2)(\\alpha_w^2-\\{k\\}^2)}\n{(q-q^{-1})^2[2k-1][2k+1]\\{k\\}^2}X^{(p)}_{k-1}.\n$$\n\nConjecture C.1 states: “For p ⩾ 1, the function X^{(p)}_m(ω, x, y)\nspecialised to y = εq^m evaluates to” (C.5):\n\n$$\nX^{(p)}_m(w,x,\\varepsilon q^m)=(wx)^{p-m}\n\\prod_{r=-(m-p-1)/2}^{(m-p-1)/2}\n(xq^{2r}+\\varepsilon w)(\\varepsilon q^{2r}+xw).\n$$\n\nProposition C.2 defines $\\sigma^{(0)}_{0,m}$ by (C.24)–(C.25):\n\n$$\n\\sigma_0=1,\\qquad \\sigma_1=\\alpha_x+\n\\frac{\\alpha_y\\alpha_w}{\\{1\\}},\n$$\n\n$$\n\\sigma_{k+1}=\\left(\\alpha_x+\n\\frac{2\\alpha_y\\alpha_w}{\\{k\\}\\{k+1\\}}\\right)\\sigma_k-\n\\frac{[k]^2(\\alpha_y^2-\\{k\\}^2)(\\alpha_w^2-\\{k\\}^2)}\n{[2k+1][2k-1]\\{k\\}^2}\\sigma_{k-1}.\n$$\n\nConjecture C.2 states: “The intermediate seed states σ^{(0)}_{0,m}(w, x, y)\nspecialised to y = εq^m evaluate to” (C.26), the same product with $p=0$.\n\n### Statement echo\n\nAll Lean names in this dossier have prefix\n`D5.S3.Quantum.Algebra.PeriodicTemperleyLiebSeedProduct`.\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Complex nonzero $q,x,w$ | `claim` quantifies complex parameters and requires each nonzero | Exact pointwise domain |\n| $q$ not a root of unity | Every natural $n\\ge1$ has $q^n\\ne1$ | All recurrence poles in scope are avoided |\n| $\\varepsilon=\\pm1$ | Sign disjunction in `claim` | Both signs |\n| $[k]$, $\\{k\\}$, $\\alpha_z$ | `qbr`, `qcb`, `al` | Literal source expressions |\n| (C.3)–(C.4) | `seedX q x y w p n` denotes $X^{(p)}_{p+n}(w,x,y)$ | Three recursion cases, with $k=p+n+1$ |\n| (C.24)–(C.25) | `seedSigma` | Independently transcribed three recursion cases |\n| Symmetric product | `seedProduct` | $N$ factors, $2r=-N+1+2j$, $0\\le j Unlike the upper bound, we note that the lower bound from (\\ref{eq.A_bounds}) has not yet shown to be tight.\n> We believe that the availability at $q=1$ of a policy that always fails purification ($c_k=d_k=0$, $\\forall k$) constitutes a tight lower bound for any other purification policy.\n> We leave this proof as future work.\n\nThe system has one long-lived storage memory and $n$ short-lived generation memories per node. Write $p_g$ for the generation probability per short-lived memory, $p_c$ for the consumption probability, $q$ for the probability of attempting purification, $\\Gamma$ for the depolarising decay parameter and $F_{\\mathrm{new}}$ for the newly generated fidelity.\n\nThe module's `admissible` requires, for each $k\\in\\mathrm{Fin}\\,n$ and every $f\\in[1/4,1]$,\n$$\n0\\le c_k(f-1/4)+d_k\\le1,\\qquad\n0\\le a_k(f-1/4)+b_k\\le\\frac34\\bigl(c_k(f-1/4)+d_k\\bigr).\n$$\nHere $k$ represents the source's $k.\\mathrm{val}+1$ generated links. These are the probability and unnormalised output-fidelity conditions of Appendix C, `main.tex` lines 1734–1767.\n\nWrite $A((a,b,c,d),q)$ for `availability n pg pc q gam fnew a b c d`, the real-valued closed formula in the source's Theorem 1 (`main.tex` lines 378–415). Issue [#21747](https://github.com/the-omega-institute/trureturing/issues/21747) fixes the reading: for every natural $n\\ge1$, real $p_g,p_c\\in(0,1]$, $\\Gamma\\ge0$, $F_{\\mathrm{new}}\\in(1/4,1]$, admissible real coefficient families $a,b,c,d:\\mathrm{Fin}\\,n\\to\\mathbb R$, and $q\\in[0,1]$,\n$$\nA((0,0,0,0),1)\\le A((a,b,c,d),q).\n$$\nThis is `claim`, and `result : claim` proves it. The identification of the closed formula with the limiting fraction of consumption requests finding a link is the source's Theorem 1; that stochastic identification is not formalized in this module. The definitions and source locators are also in [the literature note](../Library/QuantumContext/inestaetal2026entanglementbuffering.md).","title":"The always-failing purification policy attains the sharp availability lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/inesta-davies-kar-wehner-2026-entanglement-buffer-availability-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65e74ad33b74a65bc5fd003527d5591510198bb8fe7185c91a2cae5eb283c63b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:iqbal-2025-pure-state-ecqc","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Iqbal, *On the CQC Conjecture: A sufficient condition and an extension*,\narXiv:2509.08286v2, Closing Remarks and Future Works, p. 24, asks:\n\n> So it would be interesting to develop a method to prove ECQC for pure states\n> analytically, similar to the case of the original CQC conjecture.\n\nThe ECQC assertion is\n\n$$\nI(A:B)\\geq\\min_{S\\subseteq\\{0,\\ldots,d\\},\\ |S|=d}\n\\sum_{i\\in S}I(M_i^A:M_i^B).\n$$\n\nThe source says: \"The elements of the set $\\mathcal{S}$ are collections of $d$\nmutual information resulting from $d+1$ bipartite MUB measurements.\"\nIssue #12896 preregisters its pure-state specialization at prime dimension five,\nwith Shannon and von Neumann entropy in bits and $0\\log 0=0$.\nThe arXiv PDF labels the conjecture 3.1, §3.1, p. 6, and the dimension-five\nbases §4.4.2, pp. 22–23; the preregistration identifies the same quoted content\nas Conjecture 2.1 and §4.5.2.","title":"Pure-state ECQC fails in dimension five","url":"https://the-omega-institute.github.io/trureturing-pages/research/iqbal-2025-pure-state-ecqc/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00119e0efee547a3635e098743bd45346c3fa1486f8b3600a1189bda08e1cbec.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:irvine-least-maximal-digit-sum-base","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A394431, by Jean-Marc Rebert, Mar 20 2026:\n\n> The smallest base b < n where the sum of the digits for the number n in the base b is the\n> largest, with 1 < b < n and a(1) = a(2) = 1.\n\nThe formula line carrying the question, by Sean A. Irvine, Mar 25 2026:\n\n> Conjecture: a(n) = ceiling((n+1)/2) for n>8.","title":"The Least Base Maximizing the Digit Sum of n Is ceiling((n+1)/2) for n > 8","url":"https://the-omega-institute.github.io/trureturing-pages/research/irvine-least-maximal-digit-sum-base/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70abb8cf764c89150342196c20f85601c62673fe6037962a3f4f587bf229169b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:irwin-a138099","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul Curtz's OEIS A138099, revision 26, NAME and FORMULA blocks,\ncontains Benedict W. J. Irwin's unnumbered conjecture:\n\n> Conjecture: a(n) = HookSum(T(n))-HookSum(T(n-1)), where HookSum(A) is the sum of all hooks across a tableau A, and T(n) are the spiraling sequence of tableaux:\n\nThe complete displayed diagrams and arithmetic rows are reproduced in\n`Library/FiniteGeometry/irwin2016a138099.md`. The arithmetic row is\n$R(k,j)=j+\\lfloor k/2\\rfloor$, for $k\\geq1$ and\n$1\\leq j\\leq\\lceil k/2\\rceil$. Put\n$S(k)=\\sum_{i=1}^{k}\\lceil i/2\\rceil$. A source sequence satisfies\n$a(S(k-1)+j)=R(k,j)$ on that domain, with no restriction on $a(0)$.\n\nThe literal square spiral begins at $(0,0)$, travels west, south, east,\nnorth, and repeats these directions with leg lengths $1,1,2,2,3,3,\\ldots$.\n$T_0=\\varnothing$ and $T_{n+1}$ inserts walk cell $n$ into $T_n$.\nFor occupied integer cells, an ordinary hook contains its own cell and the\noccupied cells strictly right in its row and strictly below in its column.\n$H(D)$ sums these hook lengths; $H(\\varnothing)=0$.\n\n### Statement echo\n\n| Source clause or binding interpretation | Lean counterpart | Scope |\n| --- | --- | --- |\n| Irregular triangle read by rows | `triangleEntry`, `trianglePrefix`, `IsSourceSequence` | Every positive row and admissible column; index zero unused |\n| Spiraling sequence of the displayed tableaux | `legDirection`, `WalkState`, `advance`, `walk`, `IsSourceSpiral` | Actual occupied positions; no Ferrers replacement |\n| Sum of all ordinary hooks | `hookLength`, `hookSum` | Every finite integer-cell diagram, including holes |\n| Both implicitly specified sequences exist | First two conjuncts of `claim` | Conclusions, not hypotheses |\n| Every source sequence and source spiral | Universal binders and two source predicates in `claim` | No extra shape assumptions |\n| Increment at every positive index | Final conjunct of `claim`, proved by `result` | Both hook sums cast to integers before subtraction |\n\nThe finite cell carrier is the existing type $\\mathbb Z\\times\\mathbb Z$.\nThe counters implement the literal walk; they add no hypothesis to the\nsource identity. Natural division has positive denominator two, remainder\nhas positive modulus four, and the predecessor expressions are guarded\nby positive indices.","title":"Irwin's hook-sum conjecture for OEIS A138099","url":"https://the-omega-institute.github.io/trureturing-pages/research/irwin-a138099/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8bf4d7fb7bfcd1d3f80075d0de9cdbefb0a7032989a96415ffb1f1fdaa932b6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ishida-2026-cyclic-stable-marriage-blocking-upper-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Yoshiteru Ishida, *A Quadratic Lower Bound for the Shield Number of the Stable Marriage\nProblem: Rearrangement, Extremal Construction, and Biclique Realizability*,\narXiv:2609.17418v1, Section 7.1, p. 8:\n\n> This statement does not assert that µ∗ is maximum-blocking; that is exactly the open\n> arc-lemma upper-bound problem.\n\nConclusion, p. 10:\n\n> It does not claim the exact Shield-Core equality for general n, a universal Hall-feasible\n> biclique theorem, or the general upper bound β(C_n) ≤ ⌊(n − 1)²/4⌋.\n\nThe cyclic profile (1) on labels 0 through n−1 is\nr_M(g,h) = (h−g) mod n + 1 and r_W(h,g) = n − (h−g) mod n.\nSmaller ranks are preferred. A complete matching is a bijection μ from men to women;\n(g,h) blocks it when both r_M(g,h) < r_M(g,μ(g)) and\nr_W(h,g) < r_W(h,μ⁻¹(h)). Write B(μ) for the number of such ordered pairs and\nβ(C_n) for its maximum over matchings.\n\nThe quantified target is: for every natural n≥1 and every permutation μ of Fin n,\nB(μ) ≤ (n−1)²/4, where the division is natural division, hence the floor.","title":"Ishida's cyclic stable-marriage arc-lemma upper bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/ishida-2026-cyclic-stable-marriage-blocking-upper-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c17341b2b6341fcd7048a58f9ec1f6e4ba499a7ebbbc5afda8db395a4f2c6782.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ishikawa-koutschan-2012-motzkin-triangle-pfaffian-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ishikawa and Koutschan write `𝓜^{(k)}_i` for the number of Motzkin paths\nfrom `(0,0)` to `(i−1,k−1)`, with steps `U = (1,1)`, `H = (1,0)`,\n`D = (1,−1)` that never run below the horizontal axis, and define the\nPfaffian of a skew-symmetric `2n × 2n` matrix as the signed sum over the\npartitions of `[2n]` into two-element subsets. Part (i) of their Conjecture\n`conj.gen` states, for positive integers `n` and `k`:\n\n> Pf((j−i)𝓜^{(k)}_{i+j−2})_{1≤i,j≤2n} equals ∏_{i=0}^{m−1}∏_{j=0}^{k−1}(4ki+2j+k)\n> if m = n/k is an integer, and it equals\n> (∏_{j=1}^{⌊k/2⌋} 1/(2j−k))(∏_{i=0}^{m−1}∏_{j=1}^{k}(4ki+2j−k)) if k is odd and\n> m = (n+⌊k/2⌋)/k is an integer. The Pfaffian is zero in all other cases.\n\nAt `k = 1` this is the paper's Theorem `thm.pfMotz`. Issue #10039 fixes the\nreadings: the Pfaffian is the source's matching sum over the rationals; the\nvalue `𝓜^{(k)}_0` enters only the diagonal entry at `i = j = 1`, where it is\nmultiplied by `j − i = 0`; the first case is read as `k ∣ n` and the second as\n`k` odd with `k ∣ n + ⌊k/2⌋`, the first taking precedence (both hold only at\n`k = 1`, where the two products agree); part (ii) is not settled.","title":"Ishikawa and Koutschan's Pfaffian conjecture for the Motzkin triangle, part (i)","url":"https://the-omega-institute.github.io/trureturing-pages/research/ishikawa-koutschan-2012-motzkin-triangle-pfaffian-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30330aa065638fc1897e751082fe14f26615d2aab98c0f7ce3b8cb442d843279.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jack-2024-finite-otto-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Trevor Jack, *Answering Five Open Problems Involving Semigroup Conjugacy*,\narXiv:2411.16284v1, Section 9, Problem 9.4, p. 14; Semigroup Forum 112\n(2026), 450–466, DOI 10.1007/s00233-026-10614-w:\n\n> Is there a finite semigroup for which ∼o is a congruence and S/∼o is not cancelative?\n\nThe answer is negative. Section 2, p. 2 defines the relation by:\n\n> ∀a, b ∈ S (a ∼o b ⇔ ag = gb and bh = ha for some g, h ∈ S).\n\nBoth witnesses lie in $S$; no identity is adjoined.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| finite semigroup | `claim` quantifies over `S : Type` with `Semigroup S` and `Finite S` | arbitrary finite carrier, with no chosen enumeration |\n| Otto conjugacy | `Otto` | exactly the two existential witness equations in Section 2 |\n| is a congruence | `OttoIsCongruence` | multiplication compatibility; `otto_refl`, `otto_symm`, `otto_trans` prove equivalence |\n| quotient is cancelative | `QuotientCancelative` | both representative cancellation implications |\n| no such finite semigroup | `result : claim` | universal negative answer; empty carriers satisfy the cancellation statement vacuously |","title":"Finite Otto congruence quotients are cancellative","url":"https://the-omega-institute.github.io/trureturing-pages/research/jack-2024-finite-otto-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b5efe392174d57c75bd73bc665bce7e919351101fc88f8a36b8ea07a8f643e52.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jacq-lardizabal-2017-oqw-gambler-mean-hitting-time-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. S. Jacq and C. F. Lardizabal, *Open Quantum Random Walks on the Half-Line: The Karlin–McGregor Formula, Path Counting and Foster's Theorem* (J. Stat. Phys. 169, 547–594, 2017; arXiv:1701.08400v3), state the following in Remark 5.3 of the source TeX `JL_rev2.tex`, lines 1291–1293:\n\n> Numerical experiments allows us to recognize a behavior that such OQW has in common with Example 5.1, namely, that the mean hitting time has the form $E_{k,\\rho}(\\tau)=1+h(M,M-k)\\rho_{11}+h(M,k)\\rho_{22}+j(M,k)Re(\\rho_{12}),$ where $h, j$ are functions of $M$ and $k=1,\\dots, M-1$, with $j$ being a function satisfying $j(M,k)=-j(M,M-k)$. We conjecture that this holds for the gambler's ruin problem associated to *every* pair or matrices $L$, $R$ with $L^*L+R^*R=I$ inducing an OQW on vertices $\\{0,\\dots, M\\}$. A proof of this statement is, up to our knowledge, unknown.\n\nThe formal `claim` fixes two-dimensional complex matrices $l,r$ with $l^*l+r^*r=1$, every natural $m$ with $3\\le m$, and functions $h,j:\\mathbb N\\to\\mathbb N\\to\\mathbb R$. It requires the stated antisymmetry for every $k$ with $1\\le k\\le m-1$ and, for every such $k$ and every `IsDensity` matrix $\\rho$, requires `meanHittingTime` to be different from $\\top$ and its `toReal` value to equal $1+h(m,m-k)(\\rho\\ 0\\ 0).re+h(m,k)(\\rho\\ 1\\ 1).re+j(m,k)(\\rho\\ 0\\ 1).re$. Here `IsDensity` is positive semidefiniteness together with trace one, and `meanHittingTime` has codomain `ℝ≥0∞`, the extended nonnegative reals.","title":"The Jacq–Lardizabal mean-hitting-time form is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/jacq-lardizabal-2017-oqw-gambler-mean-hitting-time-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/abf49dd5e6e2a537efb005254b317567d937329f59bc71dec9500a6f3a605387.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jaffe-primitive-frobenius-successor-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The target is exactly Conjecture 7.3 in arXiv:2610.17267v1:\n\n> For all a,b >= 3 relatively prime and for c(a,b) the minimum integer satisfying both c(a,b)>b and gcd(c(a,b),a)=1, we have g(a,c(a,b))/g(a,b) <= 5.\n\nSection 3 defines\n\\[\nR(a,b)=\\{ax+by:x,y\\in\\mathbb Z_{\\geq0},\\ \\gcd(x,y)=1\\},\n\\quad\ng(a,b)=\\max(\\mathbb Z\\setminus R(a,b)),\n\\]\nwith infinity when a greatest missing integer does not exist. Zero\ncoefficients are allowed. There is no \\(a We conjecture that the multiplicity of every nonzero eigenvalue of |C|∘r∘ (except the top eigenvalue) equals two.\n\nFor every natural m at least one, put n=2m+1 and\nC_ab=cos((a-b)pi/n), with a,b from zero through n-1. The conjecture\nconcerns the matrix A(r) whose entries are |C_ab|^r for every real\nr at least zero. Every nonzero characteristic-polynomial root other\nthan the top Fourier value must have algebraic multiplicity exactly\ntwo. The formal definition `Defs.claim` retains all these quantifiers\nand both exceptions.\n\nThe sentence is attributed to the publisher's indexed full text. The\njournal PDF was inaccessible, so its page number is unverified.\nThe first version of arXiv:1803.06803 omits this sentence; the journal\narticle is the source of the conjecture.","title":"Jain: multiplicity two for absolute-cosine Hadamard powers","url":"https://the-omega-institute.github.io/trureturing-pages/research/jain-2017-abs-cosine-hadamard-power-multiplicity-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4488f8f7eddea9d084d5857d12b5c56339a5ea7dcbc9167fb0b310f926b3c166.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jakobczyk-2016-trace-norm-discord-mutual-correlation-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Jakóbczyk, *Maximal mutual correlation: an algebraic measure of correlations in bipartite quantum systems*, arXiv:1607.06585v1, abstract:\n\n> In the states containing some quantum correlations, maximal mutual correlation is always greater or equal to geometric measure of quantum discord. This relation is proved to be valid in the class of Bell - diagonal states and it is conjectured that is also true for general two - qubit states.\n\nThe quantities are defined as follows:\n- **Trace-norm geometric discord:** D₁(ρ) = min over projective measurements 𝒫 on A of ‖ρ − 𝒫(ρ)‖₁, where 𝒫(ρ) = Σ_k (P_k ⊗ I) ρ (P_k ⊗ I).\n- **Maximal mutual correlation:** C_M(ρ) is the supremum of |ω(ab) − ω(a)ω(b)| over local a, b of operator norm 1.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/jakobczyk2016maximalmutual.md); C_M is the quantity of the companion settlement `D5/S3/Quantum/Information/GeometricDiscordMutualCorrelationBound`.\n\nIssue [#19706](https://github.com/the-omega-institute/trureturing/issues/19706) fixes the reading: for every two-qubit density matrix ρ, D₁(ρ) ≤ C_M(ρ). This is proved.","title":"Maximal mutual correlation dominates the trace-norm geometric discord of two qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/jakobczyk-2016-trace-norm-discord-mutual-correlation-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c2fc1b11b3098a0c3211d05e16015257fed9e0c6581b0b8d42fef9599bfbc70.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jana-mahato-sivasubramanian-2025-steiner-path-peak","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Jana, I. Mahato and S. Sivasubramanian, *Unimodality and peak location of the\ncharacteristic polynomials of two distance matrices of trees*, Computational\nand Applied Mathematics 44, 223 (2025), arXiv:2407.03309v1, Conjecture 29,\npage 18:\n\n> For a path P_n on n > 5 vertices, if CharPoly_{𝔇_{P_n}}(x) = Σ_{i=0}^{2n−3} a_i x^i and |a_ℓ| = max{|a_0|, |a_1|, …, |a_{2n−4}|}, then ℓ = n − 1.\n\nThe [Library note](../Library/GraphInvariants/janamahatosivasubramanian2025steinerpeak.md)\nquotes the distance, basis and principal-submatrix definitions.\nPreregistration: [#19152](https://github.com/the-omega-institute/trureturing/issues/19152).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Minimum edges of a connected subtree containing the four vertices | `steiner n S` | A connected subtree of a path is an interval with $b-a$ edges; the natural infimum ranges over exactly the containing intervals |\n| Ordered basis $(e_1,f_1,…,e_{n-2},f_{n-2},e_{n-1})$ | `basis n t` | Even zero-based indices give $e_i=\\{i,i+1\\}$ and odd indices give $f_j=\\{j,j+2\\}$ |\n| Principal matrix $\\mathfrak D_2(P_n)[B,B]$ | `D n` | Each integer entry is the Steiner distance of the union of its row and column pairs |\n| Every path with $n>5$ | `claim` quantifies over natural $n$ with $5 In fact, a limited numerical exploration with some other pdfs (like the exponential $\\exp({-}w)\\,\\cax_{[0,\\infty)}(w)$ and $\\cax_{[0,1)}(w)$) shows that there may hold more generally\n> 2\\,\\frac{\\mu_3}{\\mu_4}\\,{\\rm ln}(q-1)2~,\n\nHere 𝓕₀ and 𝒦 are defined in Section 1. Under the standing assumption that 𝓕₀'' is first positive and then negative, t_c is the unique positive zero of 𝒦. The verbatim definitions are in [the literature note](../Library/StatisticalMechanics/janssen2025annealedpotts.md).\n\nIssue [#18836](https://github.com/the-omega-institute/trureturing/issues/18836) fixes the reading: the lower inequality, for every q > 2 and every probability density on (0, ∞) with finite fourth moment satisfying the standing assumption and the uniqueness of the positive zero of 𝒦. It is refuted.","title":"A weight density below Janssen's moment lower bound for the annealed Potts critical parameter","url":"https://the-omega-institute.github.io/trureturing-pages/research/janssen-2025-annealed-potts-moment-lower-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4fa66aed1f2f7430df508f7d7aa13443dba8ca98bd8a89fec4d9f58eab9a02b7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:javaheri-2026-cyclic-arrangement-conjecture-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mohammad Javaheri, *A new arrangement inequality (revised and extended\nversion)*, arXiv:2610.33653v1, Section 1, Conjecture 1.3, states that for all\nreal a,b ≥ 0 with a+b ≠ 1, the largest dimension N(a,b) for which\n\n (Σᵢ xᵢ)(Σᵢ xᵢᵃ xᵢ₊₁ᵇ) ≤ (Σᵢ xᵢ^((a+b+1)/2))²\n\nholds for every nonnegative cyclic tuple equals\n\n N*(a,b) = max { n ≥ 1 : n = 1 or\n (a+b−1)² ≤ 8ab sin²(π/n) }.\n\nThe cyclic successor is xₙ₊₁ = x₁. The source writes the largest valid\ndimension and the threshold maximum as named quantities. The Lean encoding\nmakes those implicit extrema explicit: `IsN` is the extended-natural\nleast-upper-bound predicate for the set of valid positive dimensions, and\n`IsNStar` is the greatest-element predicate for the threshold set. It then\nstates the full universal claim together with existence of the two specified\nobjects, so the refutation does not depend on an underspecified maximum.\n\nThe claim was preregistered in GitHub issue [#22620](https://github.com/the-omega-institute/trureturing/issues/22620)\nbefore the Lean probe and proof.","title":"Refutation of Javaheri's cyclic arrangement dimension conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/javaheri-2026-cyclic-arrangement-conjecture-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1f6b6daac23ec4712ea1e7f0747f4e7f64ed74f4f6c22cdc3d944344512abe6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jean-seo-2022-cubic-error-detecting-identifying-code-density","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jean and Seo, *Optimal Error-detection system for Identifying Codes*, arXiv:2208.06052v1, Section 4.2.2, Conjecture 1:\n\n> For a cubic graph G, DET:IC%(G) ≥ 3/4.\n\nThe source explicitly defines density for finite graphs. Its Theorem 2.1 characterizes a DET:IC by two-domination and asymmetric two-separation. The source statements and locators are in [the Library note](../Library/GraphInvariants/jeanseo2022errordetection.md). Issue [#19618](https://github.com/the-omega-institute/trureturing/issues/19618) preregisters the finite connected reading.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| $N[u]=N(u)\\cup\\{u\\}$ and $N_S[u]=N[u]\\cap S$ | `closedDominators` | Closed neighborhood, intersected with the finite detector set. |\n| Every vertex is at least 2-dominated | First conjunct of `IsDETIC` | At least two closed detectors at every vertex. |\n| Distinct vertices are 2#-distinguished | Second conjunct of `IsDETIC` | Either directed neighborhood difference has cardinality at least two. |\n| Cubic graph | `SimpleGraph` and `IsRegularOfDegree 3` in `claim` | Finite simple graph; degree three everywhere. |\n| Lowest density is at least $3/4$ | Every code has rational density at least $3/4$ in `claim` | Equivalent universal reading of a lower bound on the minimum. |\n| Connectedness | `Connected` in `claim` | Additional restriction; a connected counterexample also refutes the unrestricted conjecture. |\n| Finite enumerability and decidability | `Fintype`, `DecidableEq`, `DecidableRel` | Computational presentation of finite graphs, not an extra mathematical restriction. |","title":"A connected cubic counterexample to Jean–Seo Conjecture 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/jean-seo-2022-cubic-error-detecting-identifying-code-density/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7025c29f07e5922c1881b49ee957b3a0e26836691fe69f1ae30ebd3f9f3d396e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jen2610","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nik Jenzer, *Star-Ordered Rotor Loops in Clifford Algebras*, arXiv:2610.21593v1,\nTheorem 5.5 and the following scope remark, TeX Remark 5.7, rendered Remark 5.5,\npage 19, asks whether the permuted loop formula remains true without the star condition:\n\n> Nevertheless, numerical evidence suggests that the formula also holds without the star condition in that case: tested on the triangle family from Remark 4.4, which violates the star condition, under all six permutations of {1, 2, 3} with random parameter sets; the largest deviation encountered was in the range of double machine precision. A general proof would have to capture the cancellation of the odd off-diagonal terms in coarser groups than pairs; we have not determined such a grouping.\n\nFor an anticommuting unit bivector family $B_1,\\ldots,B_k$ in a finite-dimensional\nnondegenerate real Clifford algebra, $k\\geq1$, put\n$c_i=\\cos(\\theta_i/2)$, $s_i=\\sin(\\theta_i/2)$, $w_i=s_i^2$,\n$U_i=c_i+s_iB_i$, and\n$\\Theta_k(\\tau)=(U_1\\cdots U_k)(\\widetilde U_{\\tau(1)}\\cdots\\widetilde U_{\\tau(k)})$.\nThen the proved statement is\n\n$$\n\\langle\\Theta_k(\\tau)\\rangle_0\n=\\sum_{S\\subseteq\\{1,\\ldots,k\\}}\n\\eta(|S|)(-1)^{\\iota(\\tau,S)}\n\\prod_{i\\in S}w_i\\prod_{i\\notin S}(1-w_i),\n\\qquad\n\\eta(m)=(-1)^m(-1)^{m(m+1)/2}.\n$$\n\nHere $\\iota(\\tau,S)$ counts pairs $i\\tau^{-1}(j)$. No star condition is assumed.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Real finite-dimensional space and nondegenerate quadratic form | `claim` binders `V`, real module and finite-dimensional instances, `q`, `q.Nondegenerate` | All source signatures |\n| Positive family size | `k`, `1 ≤ k` | Every positive natural size |\n| Homogeneous bivectors, square $-1$, pairwise anticommutation | `IsBivector`, the three conjuncts of `AnticommutingUnitFamily` | No independence assumption |\n| Native scalar coefficient and reversion | `scalarProjection`, `CliffordAlgebra.reverse` in `returnProduct` | The exterior equivalence reads grades only; multiplication remains Clifford multiplication |\n| All real angles and all return permutations | `θ`, `τ` in `claim` | Zero sine and cosine values included |\n| Ordered products, subset weights and restricted inverse inversions | `subsetProduct`, `rotor`, `chain`, `returnProduct`, `loop`, `a`, `c`, `s`, `w`, `inversionCount`, `eta` | `Fin k` is the increasing relabelling of the one-based source indices |\n| Removal of the star condition | `StarCondition` is defined, but is absent from `claim` | No replacement hypothesis |\n\nThe canonical names in the table belong to\n`D5.S3.QuadraticForms.JenzerPermutedRotorLoopAlgebra`.\nThe settling theorem is `D5.S3.QuadraticForms.JenzerPermutedRotorLoop.result`.\nThe public definitions retain the source expressions and the binding conventions in #22266.","title":"Jenzer's permuted rotor-loop formula without the star condition","url":"https://the-omega-institute.github.io/trureturing-pages/research/jen2610/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b2977a6592a3e37e055562b1bbc10c0c063ec38fc566714ca3be61a4829a20ea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jeudy-rouleux-global-permutation-lie","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Jeudy and M. Rouleux, *Generalized exchange operators for a system of spin-1 particles*, arXiv:2411.03952v1, Section 3.1, pp. 6–7; DOI 10.1088/1742-6596/2912/1/012043. The source is recorded in [jeudyrouleux2024exchange](../Library/LieTheory/jeudyrouleux2024exchange.md). The named expectation is preregistered in [#19130](https://github.com/the-omega-institute/trureturing/issues/19130).\n\n> Moreover since that the CSCO of the P- and Q-reps are the same in this case (see [Jin], p.70), we expect that Schrödinger Hamiltonian P_S, as well as other Hamiltonians, belongs to L.\n\nThe settlement refutes the membership assertion for the Schrödinger Hamiltonian at $(S,N)=(1,3)$. It uses all six global permutations of the three spin labels, hence also refutes membership in the smaller Lie algebra generated by the transpositions alone. It asserts nothing about other Hamiltonians.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Spin one, three particles, $\\mathcal H=\\mathbb C^{3^3}$ | Matrix indices `Fin 3 → Fin 3`, coefficients `ℂ` | Exactly three labels at each of three particle positions |\n| $Q(ij)=\\widetilde Q(ij)^{\\otimes3}$ | Global-action entry in `L` is one iff $x=\\alpha\\circ y$ | A single label permutation acts on every particle |\n| The $Q(\\alpha)$ generate $L$ | `L`, using `LieSubalgebra.lieSpan` and all six label permutations | A larger generator set strengthens the exclusion |\n| $P_{ij}$ exchanges particles $i,j$ | Entries in `schrodingerHamiltonian` are one iff $x=y\\circ\\operatorname{swap}(i,j)$ | Zero-based indexing |\n| $H_S(1,3)=\\sum_{\\langle i,j\\rangle}P_{ij}$ | `schrodingerHamiltonian`, with pairs $(0,1),(0,2),(1,2)$ | Each unordered pair occurs once |\n| The Hamiltonian belongs to $L$ | `claim` | Literal membership assertion |\n| Refutation | `result : ¬ claim` | No additional hypotheses |\n\nThe public definitions display the literal global-action and exchange entries; the generic matrix and Lie constructions come directly from Mathlib. The equality projector and all three auxiliary theorems are private.","title":"The Jeudy–Rouleux global-permutation Lie-algebra expectation","url":"https://the-omega-institute.github.io/trureturing-pages/research/jeudy-rouleux-global-permutation-lie/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/249215beb20f0b145597d391d0f1d63ef7390bb552946cde490e27f51778fcd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jia-capel-2023-generic-lie-length","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Yifan Jia and Angela Capel, *A generic quantum Wielandt's inequality*, arXiv:2301.08241v3, Quantum 8 (2024) 1331, Conjecture 6, p. 19:\n\n> Let $S$ be a random Lie-generating set of $\\mathfrak{su}(n)$, then $\\mathtt{Lie}\\ell(S)=\\Theta(\\log n)$ almost surely.\n\nDefinition 3, p. 17, counts generator letters in right-nested brackets. It takes $U_1=\\mathrm{span}(S)$, $U_{k+1}=\\mathrm{span}[S,U_k]$, and the least positive $L$ for which the cumulative layers span the Lie algebra. The number $g\\ge2$ of generators is fixed independently of $n$. The random law is Lebesgue measure in real linear coordinates on $\\mathfrak{su}(n)^g$; the almost-everywhere assertion also applies to probability laws absolutely continuous with respect to this measure (by the definition of absolute continuity).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| $\\mathfrak{su}(n)$ | `GenericLieLengthFiltration.suSet` | Real submodule with $X^*=-X$ and $\\operatorname{tr}X=0$. |\n| Right-nested brackets and cumulative layers | `layer`, `lieLength`, `rightWordEval`, `rightSpan` | `layer` at zero is bottom; at one it is the real span of the generators. |\n| Fixed generator count | `GenericLieLength.claim` | For every $g\\ge2$, positive constants precede the dimension quantifier. |\n| Dimensions | `claim` | Every $n:\\mathbb N$ with $2\\le n$; dimension and Lie-length are cast to $\\mathbb R$ in the estimates. |\n| Random, almost surely | `suCoordinateMap`, `suPairMeasure`, `claim` | `Measure.map` of coordinate Lebesgue measure; a null exceptional set at each dimension. |\n| $\\Theta(\\log n)$ | `GenericLieLength.result` | $c_g=1/\\log(g+1)$ and $C=32/\\log 2$ satisfy both inequalities. |\n| Lie-generating | `GenericLieLength.ae_span_top_of_witness` with the explicit spanning pair | Generation follows almost everywhere; it is not an extra hypothesis that narrows the random law. |\n\nThe layer recurrence advances from $k+1$ to $k+2$, so the successor rule is asserted for $1\\le k$. Matrix units are Mathlib's `Matrix.single`. The dimension quantifier and real casts are as stated in the table. These conventions avoid applying a positive-layer recurrence to the zero layer.","title":"Generic logarithmic Lie-length of su(n)","url":"https://the-omega-institute.github.io/trureturing-pages/research/jia-capel-2023-generic-lie-length/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec0bb085c5e99c6b3fb04aa4ddd4cd18f36b6289eafdb7423fa788ffc7a47bd0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jiang-klco-di-matteo-2025-cartesian-control-angle-matrix-invertibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Jiang, N. Klco and O. Di Matteo, *Non-Abelian dynamics on a cube: Improving quantum compilation through qudit-based simulations*, arXiv:2506.10945v1 (Phys. Rev. D 112, 074512, 2025), state the following conjecture in `subsec_3c_restricted-rotations.tex`, lines 38–46 (the display labelled `eq:conjecture`). The quotation expands the source macro `\\ctrlseq` as $\\boldsymbol C$ and renders its displayed equation inline:\n\n> By keeping $\\boldsymbol C$ general, we sacrifice the guarantee that the matrix $M$ is invertible, although we conjecture our particular implementation yields an invertible $M$ when $\\{\\boldsymbol C\\} = \\mathbb{Z}_{d_1} \\times \\mathbb{Z}_{d_2} \\times \\cdots \\times \\mathbb{Z}_{d_k}$, i.e., uniform control but non-uniform qudit dimension.\n\nEquation (5), `subsec_3a_uniform-rotations.tex`, lines 38–49, defines $\\boldsymbol\\theta=M\\boldsymbol\\beta$ and $M_{ij}=\\prod_l(-1)^{f(b_l^{(i)},g_l^{(j)})}$, where $f(b,g)=1$ when $0 For $s=3$, direct computation (Propositions 3.22 and 3.23) shows that only $d=2$ is Gorenstein in even dimensions $d\\le6$; we conjecture this extends to all $d\\ge4$. Characterize all $(s,r)$ with $s\\ge2$ for which $\\mathcal P_{2r}^{(s)}$ is Gorenstein.\n\nTheir set is\n$\\mathcal P_d^{(s)}=\\{x\\in\\mathbb R_{\\ge0}^d:\n x_i+x_{i+1}\\le s+\\delta_i,\\ 1\\le i\\le d-1\\}$,\nwhere $\\delta_i=0$ for odd $i$ and $\\delta_i=1$ for even $i$.\nThe Gorenstein characterization used here is the source's interior-lattice\ntranslation condition: some integer $q\\ge1$ and $c\\in\\mathbb Z^d$ satisfy\n$\\operatorname{int}((n+q)P)\\cap\\mathbb Z^d\n=c+(nP\\cap\\mathbb Z^d)$ for every integer $n\\ge0$.","title":"The Gorenstein pairs for alternating adjacent-sum polytopes","url":"https://the-omega-institute.github.io/trureturing-pages/research/jiang-wen-zhong-2026-alternating-polytope-gorenstein-pairs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60b161f1d24fb844440e7bb3b69a82cd398d4e467b032ad8047a9506cce50ef1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jivulescu-lancien-nechita-2022-werner-tester-incompleteness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. A. Jivulescu, C. Lancien and I. Nechita, *Multipartite Entanglement Detection Via Projective Tensor Norms*, arXiv:2010.06365v1 (Annales Henri Poincaré 23 (2022), 3791–3838), Section `sec:conclusions`, posed the following question in `main.tex`, line 1365:\n\n> The main question that remains unanswered at this point is whether our family of entanglement criteria is complete, without allowing for permutation of indices before applying the testers. Concretely, we would like to know if, for any entangled state $\\rho$ on $\\mathbb C^{d_1}\\otimes\\cdots\\otimes\\mathbb C^{d_m}$, there exist testers $\\mathcal E_i:S_1^{d_i}\\to\\ell_2^{n_i}$, $1\\leq i\\leq m$, such that $\\|\\mathcal E_1\\otimes\\cdots\\otimes\\mathcal E_m(\\rho)\\|_{\\ell_2^{n_1}\\otimes_\\pi\\cdots\\otimes_\\pi\\ell_2^{n_m}}>1$. Even for bipartite states we are only able to show a weaker version of this statement (see Section \\ref{sec:completeness}).\n\nThe source defines the projective tensor norm by the infimum of the sums of products of factor norms (Definition, `main.tex`, lines 234–238), the trace norm by $\\|X\\|_1=\\operatorname{Tr}\\sqrt{X^*X}$ (`main.tex`, lines 288–290), and a complex tester as a complex-linear map of norm exactly one from the Schatten-1 space to the Euclidean space (`main.tex`, lines 340–352). Its Werner state is\n$\\sigma_\\mu=\\mu\\frac{I+F}{d(d+1)}+(1-\\mu)\\frac{I-F}{d(d-1)},\\qquad 0\\leq\\mu\\leq1$\n(`main.tex`, line 946).\n\nThe Lean declaration `claim : Prop` is the bipartite instance. For every $d_1,d_2:\\mathbb N$ and every $\\rho:\\operatorname{Matrix}(\\operatorname{Fin}d_1\\times\\operatorname{Fin}d_2)(\\operatorname{Fin}d_1\\times\\operatorname{Fin}d_2)\\,\\mathbb C$, its hypotheses are `ρ.PosSemidef`, `ρ.trace = 1`, and `¬ Separable ρ`; its conclusion requires $n_1,n_2:\\mathbb N$ and complex-linear maps $e_i:\\operatorname{Matrix}(\\operatorname{Fin}d_i)(\\operatorname{Fin}d_i)\\,\\mathbb C\\to\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin}n_i)$ satisfying `IsTester e₁ ∧ IsTester e₂ ∧ 1 < ‖testerImage e₁ e₂ ρ‖`.\n\nHere `IsTester e` is exactly the conjunction $\\forall x,\\ \\|e x\\|\\leq\\operatorname{traceNorm}(x)$ and $\\exists x,\\ x\\neq0\\land\\|e x\\|=\\operatorname{traceNorm}(x)$. The formal `Separable` predicate is membership in the real convex hull of matrices `Matrix.kroneckerMap (· * ·) ρ₁ ρ₂`, where each factor is positive semidefinite with trace $1$.\n\nThe output of `testerImage` has type `⨂[ℂ] s : Bool, EuclideanSpace ℂ (Fin (cond s n₂ n₁))`: the `false` factor is the first Euclidean output space and the `true` factor is the second. Its norm $\\|\\cdot\\|$ is Mathlib's bundled `PiTensorProduct.projectiveSeminorm` on this tensor product. This represents the source's $\\ell_2^{n_1}\\otimes_\\pi\\ell_2^{n_2}$ norm from `main.tex`, lines 234–238: the two factors carry their Euclidean norms, and each finite pure-tensor expansion has cost equal to the sum of the products of those factor norms. Mathlib permits explicit complex scalar coefficients in these expansions; absorbing each coefficient into either factor gives the source's convention, with the same cost. The tensor norm notation and application of `PiTensorProduct.projectiveSeminorm` are definitionally equal.\n\nThe settlement is the negation of this bipartite claim. The source's universal multipartite question is therefore already false at two parties.","title":"Complex testers do not detect every entangled Werner state","url":"https://the-omega-institute.github.io/trureturing-pages/research/jivulescu-lancien-nechita-2022-werner-tester-incompleteness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/79f04c281e0c9fd5053ac5f413a4784dab42858325437eee23299d7252ad251f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jmpp-2022-absolute-two-incoherence-purity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johnston, Moein, Pereira and Plosker, *Absolutely k-incoherent quantum states and spectral inequalities for factor width of a matrix*, Phys. Rev. A 106, 052417, arXiv:2205.05110v2, Section 5, printed page 17:\n\n> Theorem 7 gives a sufficient condition for absolute 2-incoherence that is also necessary in dimension n = 3. It is unclear whether or not it is necessary in dimension n = 4 or higher.\n\nThe condition is $\\operatorname{Tr}(\\rho^2)\\le1/(n-1)$. The universal necessity assertion for all $n\\ge4$ is refuted by one density matrix in dimension five. The source and literal definitions are in [the Library note](../Library/Eigenstructure/johnstonmoeinpereiraplosker2022incoherence.md); the external question is preregistered in [#17863](https://github.com/the-omega-institute/trureturing/issues/17863).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| A vector has at most $k$ nonzero entries | `FactorWidthTwoCone.IsKSparse` | The finite support-cardinality expression. |\n| A finite sum $\\sum_jv_jv_j^*$ of such vectors | `FactorWidthTwoCone.IsKIncoherentSum` | Every summand has the required support; individual vectors need not be normalized. |\n| A mixed quantum state | `GHZMeasureBiseparableBound.IsDensity` | Positive semidefinite matrix with complex trace one. |\n| Every unitary conjugate is $k$-incoherent | `AbsoluteTwoIncoherencePurityRefutation.IsAbsolutelyKIncoherent` | All complex unitary matrices are quantified. |\n| Necessity for dimension four or higher | `AbsoluteTwoIncoherencePurityRefutation.claim` | Every natural $n\\ge4$, every density matrix satisfying absolute two-incoherence. |\n| The eigenvalue purity bound | Real part of the trace of the square | For Hermitian matrices, the spectral theorem identifies this with the sum of squared eigenvalues. |\n| Refutation | `AbsoluteTwoIncoherencePurityRefutation.result` | Exact negation of `claim`, witnessed at $n=5$. |\n\n`FactorWidthTwoCone.IsLocally2` is the Hermitian support-two quadratic-form condition in Definition 1(b). It is an auxiliary dual-cone predicate, not an extra hypothesis in `claim` or `result`.","title":"Absolute two-incoherence beyond the purity bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/jmpp-2022-absolute-two-incoherence-purity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ffb40751298a01a5059aec4ab75f4d375587a15b9ac0217ad794d6ed15d1f253.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:johnson-mondal-tsatsomeros-2026-embeddable-pick-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. R. Johnson, S. Mondal, and M. J. Tsatsomeros, *Powers, Roots, and Positivity*, arXiv:2609.06298v1, Table 2, “Behavior of Matrix Classes under Pick Functions and Fractional Powers”, p. 22, row 13 reads verbatim:\n\n> Embeddable Matrix | Open | Yes\n\nThe note below the table states verbatim:\n\n> The entries marked Open indicate cases for which preservation under Pick functions remains an open question.\n\nRemark 3.28, p. 17, states verbatim:\n\n> The embedding problem, introduced in [11], asks whether a given stochastic matrix A can be realized as the transition matrix of a continuous-time Markov process. It was shown in [24] that a nonsingular stochastic matrix A is embeddable if and only if A = e^B for some essentially nonnegative matrix B satisfying Be = 0.","title":"Pick functions do not preserve embeddable matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/johnson-mondal-tsatsomeros-2026-embeddable-pick-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/660827e9f406e14f6d2e4eec5aaa8fa38b27f5a602fa18aad60bd06d9ffe23c8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:johnson-mondal-tsatsomeros-2026-sidm-pick-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. R. Johnson, S. Mondal, and M. J. Tsatsomeros, *Powers, Roots, and Positivity*, arXiv:2609.06298v1, §1, ask verbatim:\n\n> Given A ∈ C and a Pick function f, is f[A] ∈ C?\n\nTable 2, “Behavior of Matrix Classes under Pick Functions and Fractional Powers”, p. 22, row 12 reads verbatim:\n\n> Strongly Infinitely Divisible Matrix (SIDM) | Open | Yes\n\nThe note below the table states verbatim:\n\n> The entries marked Open indicate cases for which preservation under Pick functions remains an open question.\n\nDefinition 2.7 states verbatim:\n\n> Let D := C\\(−∞, 0]. A complex function f analytic on D is called a Pick function if ℑf(z) ℑz ≥ 0, ∀z ∈ D\n\nDefinition 3.25 states verbatim:\n\n> A nonnegative matrix A ∈ Mn(R) is said to be an infinitely divisible matrix (IDM) if, for every positive integer m, there exists a nonnegative matrix Km such that (Km)^m = A. If, in addition, A is invertible, then A is called a strongly infinitely divisible matrix (SIDM).","title":"Pick functions do not preserve strong infinite divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/johnson-mondal-tsatsomeros-2026-sidm-pick-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6bbdae43a021c91c57662005d011e302dd7bafe6c4cd8e03320c25626795b67f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:johnson-wakefield-2022-rank-three-moebius-question-6-19","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"John Johnson and Max Wakefield, *A non-associative incidence near-ring with a generalized Möbius function*, Ars Mathematica Contemporanea 24, #P2.03, Question 6.19, ask:\n\n> Is there a rank 3 non-modular connected matroid $M$ such that $\\mathcal M(M,-1)=0$?\n\nThe answer for finite matroids is negative. The precise source definitions and finite-carrier conventions are fixed by [#19762](https://github.com/the-omega-institute/trureturing/issues/19762). The Library source is `Library/FiniteGeometry/johnsonwakefield2024incidencenearring.md`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Lattice of flats $L(M)$, ordered by inclusion | `Flats M`, the subtype of `Matroid.IsFlat` | Literal binding convention |\n| Rank and top rank | `rk M F`, `M.eRank` | Extended rank converted to a natural number; finite carrier |\n| Definition 5.1, $x\\le a\\le y\\le b\\le z$, with diagonal value one and all other flag values zero | `IsJFunction` | The rectangle equation, rather than the product formula, defines the public predicate |\n| Definition 6.2, the partial-flag sum with exponent $3r(M)-r(x)-r(y)-r(z)$ | `MoebiusPoly` | All sums are finite; exponent subtraction is in $\\mathbb N$ |\n| Rank three, connected, nonmodular | `claim` hypotheses `M.eRank = 3`, `Connected M`, and `¬ Modular M` | Connectedness is retained literally |\n| No such vanishing example | `result : claim` | Every solution of the defining rectangle equation has nonzero evaluation |\n| Finite matroid setting | `Finite α` with `M : Matroid α` | No infinite-carrier assertion |","title":"Johnson–Wakefield Question 6.19: nonvanishing in rank three","url":"https://the-omega-institute.github.io/trureturing-pages/research/johnson-wakefield-2022-rank-three-moebius-question-6-19/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73e2cfa24117e6e2be9236b601bf94db60bcd7e8f89199405341f77ddde4ca00.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:johnston-martin-duro-rumynin-2024-knutson-index-question-2-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johnston, Martín Duro and Rumynin define the Knutson index and the finite-set\nsubindex for a finitely generated commutative ring $R$, a homomorphism\n$\\alpha : R \\to \\mathbb{Z}$ and a regular element $r$. Question 2.5 asks\nwhether there is a finite $S \\subset R$ such that\n$\\mathcal{K}_r(x)=\\mathcal{K}_r^S(x)$ for every $x\\in R$.\n\n### Statement echo\n\nThe Lean definitions `IsRegularElement`, `knutsonIndex`, `eligible`,\n`eligibleIndex` and `knutsonSubindex` mirror the source conventions. The\nsettling declaration `result` proves the negation of the universal `claim`.\nThe ring hypotheses are `[CommRing R]` and `[Algebra.FiniteType ℤ R]`, and the\ncomparison is in $\\mathbb{N}_\\infty$ after casting the ordinary index.","title":"Johnston–Martín Duro–Rumynin Question 2.5","url":"https://the-omega-institute.github.io/trureturing-pages/research/johnston-martin-duro-rumynin-2024-knutson-index-question-2-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3cc8a97415d9ae0ceed771710a38f86ce4d62325d2b9c0f71937168f8bd2117.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:johnston-russo-2026-fourier-ldoi-locc-tightness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johnston and Russo, “Distinguishability of locally diagonal orthogonally invariant\nquantum states,” Example 17, arXiv:2604.12808v1, ask:\n\n> Whether the lower bound in Equation (61) is tight—that is, whether opt_LOCC(E) = 1/2 − (n − 2)/(2n²)—remains an open question.\n\nThe uniform ensemble contains $n^2$ states, with prior $1/n^2$ each. Its diagonal\nvectors are $\\phi_{ii}=\\sum_k U_{ik}|kk\\rangle$, where $U$ is the normalized Fourier\nmatrix. For $i We still do not know whether or not Corollary 5.4 is tight when n ≥ 5.\n\nCorollary 5.4 guarantees antidistinguishability when every distinct-state overlap is at most $c_n=\\sqrt{(n-2)/(2(n-1))}$. Issue #15573 specifies the finite-dimensional unit-state convention and the squared-overlap margin. The settlement proves non-tightness for every $n\\ge10$.\n\n### Statement echo\n\n| Source condition | Formal counterpart | Scope |\n| --- | --- | --- |\n| A set of $n$ pure states | $\\psi:\\operatorname{Fin}(n)\\to\\mathbb C^d$ | Every finite ambient dimension $d$ |\n| Normalized states | $\\|\\psi_i\\|=1$ for every $i$ | No dimension or distinctness restriction |\n| POVM with zero probability for the corresponding state | `IsAntidistinguishable` | Positive semidefinite effects, identity sum, zero Born quadratic form |\n| Tightness of the overlap guarantee | `IsTight` | Every positive squared-overlap margin admits a non-antidistinguishable family |\n| Positive source cap | $c_n>0$ for $n\\ge10$ | Squaring reparametrizes positive margins: $(c_n+\\varepsilon)^2-c_n^2>0$ and $\\sqrt{c_n^2+\\delta}-c_n>0$ |\n| Non-tightness | `claim` and `result` | Every $n\\ge10$; cases $5\\le n\\le9$ remain open |\n\nThe positive margin $\\delta_n$ is additional proved information, rather than a hypothesis strengthening the source question. The displayed finite-dimensional POVM definition is literal. The Gram construction is reused as `Matrix.gram`, and the squared Frobenius norm is reused as `RHLinalg.frobSq`.","title":"Non-tightness of the antidistinguishability overlap bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/johnston-russo-sikora-2025-antidistinguishability-tightness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/89f6d4f5097d819b2bf6342b507be1386c6f2cef1d48bb124930e0b7d36b5005.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:jokinen-et-al-2024-k-measurement-pseudobroadcasting","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Jokinen, M. Weilenmann, M. Plávala, J.-P. Pellonpää, J. Kiukas and R. Uola, *No-Broadcasting Characterizes Operational Contextuality*, Phys. Rev. Lett. 133, 240201 (2024), arXiv:2406.07305v1, Appendix D, state:\n\n> One can also see that $\\mathcal S(\\mathcal A)$ is a Cartesian product of $k$ simplexes if $\\mathcal A$ is generated by $k$ independent measurements. This leads to a conjecture that if $\\mathcal A$ is generated by $k$ measurements, then $1\\mapsto k$-pseudobroadcasting is equivalent to non-contextuality. This is only a conjecture since even if $\\mathcal A$ is generated by $k$ measurements, $\\mathcal S(\\mathcal A)$ in general is only a subset of a Cartesian product of $k$ simplexes. We leave resolving this issue and proving or disproving the conjecture for future work.\n\nThis quotation is from `Arxiv.tex`, line 482, in the arXiv v1 source archive (534 lines; md5 `cc2fa845c0b0f4155ed18dcb76e4c92e`). The source macros `\\hS`, `\\hA` and `\\btn{k}` are expanded to $\\mathcal S$, $\\mathcal A$ and $1\\mapsto k$; the wording is unchanged. The main-text conjecture is on line 348. The definitions and further quotations are in [the literature note](../Library/QuantumContext/jokinenetal2024nobroadcasting.md).\n\nIssue [#21862](https://github.com/the-omega-institute/trureturing/issues/21862) fixes the reading of generation: $\\mathcal A$ is the real convex hull of the binary classical post-processings $\\sum_o c_oM_j(o)$, with $0\\le c_o\\le1$, of $k$ finite POVMs. A subtheory has a convex set $\\mathcal T$ of density matrices and a convex set $\\mathcal A$ of effects, contains $I$ among its effects, and is closed under $E\\mapsto I-E$. The conjectured equivalence is refuted by a qubit subtheory generated by one four-outcome POVM.","title":"One generating measurement does not make pseudo-broadcasting equivalent to non-contextuality","url":"https://the-omega-institute.github.io/trureturing-pages/research/jokinen-et-al-2024-k-measurement-pseudobroadcasting/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2eadc5310972556ae295f6b20413f75099084fc6fd391b28cc8c9cfabcdf926c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:joshi-2025-sturmian-map-cumulative-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gandhar Joshi, *Automatic sequences from algebraic, dynamical, and computational\nperspectives*, Open University PhD thesis, §6.1, p. 118:\n\n> “The data generated by the A(d) formulae in Corollary 4.3.13 heuristically directs\n> to a conjecture that for every n ∈ N, d ≤ qₙ ⟹ A(d) ≤ A(qₙ) (Proposition 4.4.4)\n> could be true for all g(d). As done for the specific case in [32, Corollary 4.17],\n> this possibility for all Sturmians will be assessed in the author’s future work.”\n\nThe literal conjecture quantifies over irrational $0<\\alpha<1/2$, positive\n$n$, the convergent denominator $q_n$, and $1\\le d\\le q_n$.\nPreregistration: https://github.com/the-omega-institute/trureturing/issues/17837.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Definition 2.2.3: the letter at $n$ uses $\\{(n+1)\\alpha\\}$ and intervals $[0,1-\\alpha)$, $[1-\\alpha,1)$ | `lowerMechanicalWord α α`; private `sturmian_letter_iff` | Same indexing and intercept; true is the second letter. |\n| Definitions 2.6.1–2.6.2: all starts, either letter, positive finite lengths, infinity if unbounded | `mapLength` | Full nested supremum in $\\mathbb N\\cup\\{\\infty\\}$. |\n| Irrational slope in Chapter 4's $0<\\alpha<1/2$ | `claim` slope hypotheses | Literal domain. |\n| Every positive $n$, with $q_0=1$ and $q_1=a_1$ | `1 ≤ n`, `(GenContFract.of α).dens n` | Natural index and real denominator, identified with natural $q$. |\n| Every positive $d\\le q_n$ has $A(d)\\le A(q_n)$ | The final quantified implication of `claim` | No additional condition on $g(d)$. |\n| Refutation | `result : ¬ claim` | Witness $\\alpha=2-\\sqrt3$, $n=1$, $q=3$, $d=2$. |","title":"A stronger cumulative Sturmian progression bound is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-2025-sturmian-map-cumulative-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/47efee48dc460fa0bda6c57caf20b28c8e19714bb5016d20387b3482e5b7beda.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:joshi-rust-2025-thue-morse-first-longest-start","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joshi and Rust, *Monochromatic arithmetic progressions in the Fibonacci,\nThue-Morse, and Rudin-Shapiro words*, Theoretical Computer Science 1050\n(2025), 115391, arXiv:2501.05830v2, Section 3.2.2, Conjecture 3.8:\n\n> We have i(2^n+1)=3·2^(2n)-2^n-1,\n> i(2^(2n)-1)=3·2^(4n)-2^(2n)+1,\n> i(2^(2n+1)-1)=2^(2n+1)-1.\n\nThe display omits ranges. The contextual interpretation, from the preceding\nmaximum-length formulas, is `n>=2` in the first equation and `n>=1` in the\nother two. Equivalently, for every natural `e>=2`, with `q=2^e`,\n`i(q+1)=3q^2-q-1`, `i(q-1)=3q^2-q+1` for even `e`, and\n`i(q-1)=q-1` for odd `e`. This range is an explicitly disclosed\ninterpretation, not a printed quantifier. The previously known `i(3)=45`\nexcludes the plus-family extension to `e=1`; it is not a new refutation.\n\nThe word is the zero-indexed Thue-Morse word `0110100110010110...`, with\n`t(0)=0`, `t(2u)=t(u)` and `t(2u+1)=1-t(u)`. Definitions 2.2, 2.3 and\n2.5 make `A(d)` the global maximum length over every nonnegative start\nand either letter, and `i(d)` the least start attaining it.","title":"First longest Thue-Morse progressions at power-of-two differences","url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-rust-2025-thue-morse-first-longest-start/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d94af8f14644890ceedca254086e0e82ed2c861e48ab2e187399575ff932a83.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:joshi-rust-2025-thue-morse-infinite-odd-fibers","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joshi and Rust, arXiv:2501.05830v2, Section 3.2.1, Question 3.7,\nsecond clause asks: “Are there infinitely many values of n for which\nO_max(n)=∞?” The first and third clauses are outside this target.\n\nExact question locator: https://arxiv.org/html/2501.05830v2#S3.Thmtheorem7.\n\nThe staged claim is that the set of natural lengths n for which the set\nof positive odd differences d satisfying ExactMax(d,n) is infinite is\nitself infinite. ExactMax requires a start attaining length n and a\nuniversal bound on every monochromatic AP length at every natural start.\nMAP and the actual zero-indexed thueMorse are reused from the existing\nWords sources. This covers both colors and the entire infinite word.","title":"Infinite odd fibers of Thue-Morse global maxima","url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-rust-2025-thue-morse-infinite-odd-fibers/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0acecf0d32335c35dc05b880b62ce1859d1b3875dcbb6029bdc47f0b5436c27.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:joshi-rust-fibonacci-map-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joshi and Rust, *Monochromatic arithmetic progressions in the Fibonacci,\nThue-Morse, and Rudin-Shapiro words*, arXiv:2501.05830v2, Theoretical\nComputer Science 1050 (2025), 115391, Conjecture 4.24:\n\n> For all d >= 1, (A(d)-1)/d < sqrt(5)/tau.\n\nHere `tau=(1+sqrt(5))/2` and `A(d)` is the global maximum positive length\nof a monochromatic arithmetic progression of difference `d` in the\nzero-indexed infinite Fibonacci fixed point `0 -> 01, 1 -> 0`. Both symbols\nand every nonnegative starting position are included.","title":"Fibonacci-word MAP maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-rust-fibonacci-map-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7092498a1e0aa8b7ab08c72bf61a2ea2fd540b7eda8be0a361f57c9fc749d198.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:joshi-rust-fibonacci-map-first-start","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joshi and Rust, *Monochromatic arithmetic progressions in the Fibonacci,\nThue-Morse, and Rudin-Shapiro words*, Theoretical Computer Science 1050\n(2025), 115391, arXiv:2501.05830v2, Conjecture 3.19:\n\n> i(F_(2n+1)) = F_(2n+3) - 2, and i(F_(2n)) = F_(4n) - 1.\n\nThe quantifier is every natural `n >= 1`, as independently clarified in\nthe author's A364648 comment. Here `F_0=0`, `F_1=1`. The word is the\nzero-indexed fixed point `010010100100101...` of `0 -> 01, 1 -> 0`.\nFor a positive difference `d`, `A(d)` is the global maximum of positive\nlengths over every nonnegative start and both symbols; `i(d)` is the least\nstart attaining that maximum, again allowing either symbol. Definitions\n2.2, 2.3 and 2.5 supply these conventions.","title":"First longest progressions at Fibonacci differences","url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-rust-fibonacci-map-first-start/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/49c442456c4c6ac4060dde6bd7baa8578975dcff6a6b95be72c9281965220903.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kade-2025-box-six-vertex-determinant-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kade (arXiv:2509.03416v1, section 2.3) puts the six-vertex model in a square\nbox with arrow-reflecting walls: `2M` horizontal spectral lines, in pairs\n`x_i, 1/x_i`, run from the right wall to the left wall, and `2M` vertical lines,\nin pairs `y_j, 1/y_j`, run from the top wall to the bottom wall. The bulk\nweights are `a = p t − 1/(p t)`, `b = t − 1/t`, `c = p − 1/p` with `t = x/y`, and\nthe walls carry the diagonal K-matrices\n`K_L(x) = diag(b(xξ_L), b(x/(qξ_L)))`, `K_R(x) = diag(b(xξ_R), b(xq/ξ_R))`,\n`K_U(y) = diag(b(yξ_U), b(yq/ξ_U))`, `K_D(y) = diag(b(yξ_D), b(y/(qξ_D)))`.\nThe thesis proposes\n\n> Z_M({x_i}|{y_i}) = ∏_{i,j} (x_i/y_j − y_j/x_i) W(x_i, y_j) / ∏_{i · det[ c² a(x_i y_j) a(1/(x_i y_j)) F^LU(x_i) F^DR(y_j) / ((x_j/y_i − y_i/x_j) W(x_i, y_j)) ]\n\nwith `W(x, y) = a(xy) a(1/(xy)) a(x/y) a(y/x)`,\n`F^LU(x) = b(xξ_L) b(xq/ξ_U) + b(x/(qξ_L)) b(xξ_U)` and\n`F^DR(y) = b(yξ_D) b(yq/ξ_R) + b(y/(qξ_D)) b(yξ_R)`, and its concluding chapter\nsays that the conjecture \"has to be proven or discarded\".\n\nIssue #10444 fixes the readings: the absolute arrow conventions of the six\nvertices and of the diagonal wall components are those of the thesis's figures\n(`6Vvertices`, `KLeft_mat`, `KRight_mat`, `KUp_mat`, `KDown_mat`); rows carry\n`x_1, 1/x_1, x_2, 1/x_2, …` from the top and columns `y_1, 1/y_1, …` from the\nleft; `q` is the crossing parameter `p`, the only value among those tried at\nwhich the `2 × 1` box partition function is symmetric in its two horizontal\nspectral parameters, as integrability requires; the parameters are complex and\nthe formula is asserted wherever none of its printed denominators vanishes.","title":"The box partition function of the six-vertex model","url":"https://the-omega-institute.github.io/trureturing-pages/research/kade-2025-box-six-vertex-determinant-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7fb2563482b5d87e52adc5f7ea6073dd430c7b286ea9ffd286ee3131de28e9d9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kag329479-binarynull","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Kagey's OEIS A329479, revision 25, final FORMULA line, states:\n\n> Conjectured recurrence: a(n) = 2a(n-1) + 2a(n-2) - 5a(n-3) - 2a(n-4) + 10a(n-5) - 4a(n-6) - 4a(n-7) + 8a(n-8).\n\nThe source NAME is: “Number of degree n polynomials f with all nonzero coefficients equal to 1 such that f(k) is divisible by 3 for all integers k.” Thus $a(n)$ counts distinct integer polynomials of degree exactly $n$, with leading coefficient one, every coefficient in $\\{0,1\\}$, and $3\\mid f(k)$ for every integer $k$. The source offset is one. The conjecture is interpreted for every natural $n\\ge9$, using integer arithmetic for the counts and natural subtraction for the indices.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Degree $n$ integer polynomials | `a`: subtype of `Polynomial Int` with `natDegree = n` | Exact degree; distinct polynomials remain distinct |\n| All nonzero coefficients equal one | `a`: every coefficient is zero or one, with `Monic` | For the positive degrees used here, monicness expresses the forced leading one |\n| Divisible by three for all integers | `a`: divisibility of every integer evaluation | Full universal condition |\n| Eight-term recurrence | `claim`; `result` proves it | All eight coefficients and signs retained |\n| Offset one | Guard $9\\le n$ | Every preceding index is at least one |\n\nThe source COMMENT describes strings of length $n$, while the NAME and six degree-seven examples use polynomial degree $n$. The formal count follows the NAME and examples, so the coefficient string has length $n+1$. No recurrence-defined sequence or quotient by polynomial functions replaces this count.","title":"Kagey's order-eight recurrence for binary null polynomials modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/kag329479-binarynull/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07206cd7b0264e167463f284852da6a9717de7909e2e7934e234c5446d406d40.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kagey-2020-a331859-colliding-blocks-records","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A331859 (Peter Kagey, 2020) counts collisions in the colliding-blocks\nsystem of the 3Blue1Brown videos:\n\n> Suppose there is a block A of mass n sliding left toward a stationary block B\n> of mass 1, to the left of which is a wall. Assuming the sliding is\n> frictionless and the collisions are elastic, a(n) is the number of\n> collisions between A and B plus the number of collisions between B and the\n> wall.\n\nIts FORMULA section gives `a(n) = ceiling(Pi/arctan(sqrt(1/n))) - 1`, and its\nCOMMENTS state\n\n> Conjecture: The values of n for which a(n) != A121854(n) is a subset of\n> A331903.\n\nwhere A121854 is `floor(Pi*(sqrt(n)))` and A331903 is the positions of records\nin A331859.\n\nIssue #10098 fixes the readings: `a(n)` is the FORMULA closed form for\n`n ≥ 1`, whose identification with the collision count is the published count\nof Galperin (2003) and is not formalized here; A121854 is `⌊π√n⌋`; A331903 is\nthe set of `n ≥ 1` with `a(k) < a(n)` for all `1 ≤ k < n` (strict records, as\nthe entry's data `1, 2, 4, 5, 6, 8, …` shows).","title":"Colliding blocks: the collision count leaves floor(π√n) only at records","url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb5dc4afbe702cce2ce86cc461998171df9865d7d42176156d79c1eb785ff1e8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kagey-2021-inversion-first-letter-conjecture-5-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Kagey, “Expected value of letters of permutations with a given number of k-cycles”, arXiv:2112.05281v1, Section 5.2, page 20, Conjecture 5.1 states:\n\n> For fixed k and n > k, the expected value of the first letter of a permutation with a given number of inversions satisfies a rational function in n given by E[π(1) | π ∈ S_n, inv(π) = k] = M(n + 1, k)/M(n, k), where M(n, k), as above, is the number of permutations w ∈ S_n such that inv(w) = k.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Fixed $k$ and $n>k$ | `claim` universally quantifies both natural parameters and assumes $k ZMod m -> T` with literal periodic lift `F : Z x Z -> T`, every h in `{r,r^3,rf,r^3f}` belonging to R, every allowed translation representative u before h, and every integral affine center. It asks for a genuine H on the full product `ZMod gcd(n,m) x ZMod gcd(n,m)` whose literal plane lift is pointwise identical to F. Equal and unequal dimensions, unit dimensions, nonprimitive periods, all quotient representatives, and nontrivial design actions are included. No minimality claim and no D8 action on an unequal finite rectangle is part of the problem.\n\nThe operational source maps are `r : (x,y) -> (-y,x)`, `r^3 : (x,y) -> (y,-x)`, `rf : (x,y) -> (y,x)`, and `r^3f : (x,y) -> (-y,-x)`, with labels `d -> d·h`. The source's inverse conjugation coordinate map is kept distinct from these operational maps.","title":"Rectangular-torus rotation descent","url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-keehn-2023-rectangular-torus/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc06d25e2f10ea18b9b7953686989f41a507c4b9ba733e4086a43bd25dd1d803.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kagey350532-perfectpowers","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Kagey's OEIS A350532 COMMENT states verbatim:\n\n> Conjecture: T(2*n,2) = n.\n\nThe settled statement is $T(2n,2)=n$ for every $n\\in\\mathbb N$, including zero.\n\nThe source's NAME defines the count verbatim:\n\n> Triangle read by rows: T(n,k) is the number of degree-n polynomials over Z/2Z of the form f(x)^m for some m > 1 with exactly k nonzero terms; 1 <= k <= n + 1.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Degree-n polynomials over Z/2Z, exactly k nonzero terms, some exponent m > 1 | `KageyA350532Binomials.T` | The literal cardinality subtype; distinct nonzero polynomials, with witnesses for base and exponent. |\n| Conjecture: T(2*n,2) = n. | `KageyA350532Binomials.claim` and `KageyA350532Binomials.result` | Literal universal assertion under the preregistered conventions. |\n| No restriction to positive degree in the definition | `p ≠ 0` together with `p.natDegree` | The preregistered nonzero convention prevents zero from entering the degree-zero count. |\n| Natural-number variables | The natural-number binders and arithmetic | Subtraction and division in the trinomial assertion are natural-number operations. |","title":"Binomial perfect-power counts over GF(2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey350532-perfectpowers/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ce3d029a70c6c54b70382bea57ce69b9554a9f7ae2a001ccac5916ca67074f5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kagey350532-trinomials","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Kagey's OEIS A350532 COMMENT states verbatim:\n\n> Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1.\n\nThe settled statement is $T(2n,3)=(n^2-n)/2$ for every $n\\in\\mathbb N$ with $1\\leq n$.\n\nThe source's NAME defines the count verbatim:\n\n> Triangle read by rows: T(n,k) is the number of degree-n polynomials over Z/2Z of the form f(x)^m for some m > 1 with exactly k nonzero terms; 1 <= k <= n + 1.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Degree-n polynomials over Z/2Z, exactly k nonzero terms, some exponent m > 1 | `KageyA350532Binomials.T` | The literal cardinality subtype; distinct nonzero polynomials, with witnesses for base and exponent. |\n| Conjecture: T(2*n,3) = (n^2 - n)/2 for n >= 1. | `KageyA350532Trinomials.claim` and `KageyA350532Trinomials.result` | Literal universal assertion under the preregistered conventions. |\n| No restriction to positive degree in the definition | `p ≠ 0` together with `p.natDegree` | The preregistered nonzero convention prevents zero from entering the degree-zero count. |\n| Natural-number variables | The natural-number binders and arithmetic | Subtraction and division in the trinomial assertion are natural-number operations. |","title":"Trinomial perfect-power counts over GF(2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey350532-trinomials/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/481a0f77078dc41b78b6f5aee35e9d7ad22db296f5f4b22b5a7f87612a869508.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kalmykov-karp-2013-reciprocal-gamma-turanian-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. I. Kalmykov and D. B. Karp, *Log-concavity for series in reciprocal\ngamma functions and applications*, arXiv:1206.4814v1, Conjecture 1:\n\n> Under hypotheses of Theorem 2 the function µ → g(µ, x) is log-concave on (0, ∞) for each fixed x ≥ 0. Moreover, the function x → g(µ + α, x)g(µ + β, x) − g(µ, x)g(µ + α + β, x) has positive power series coefficients for µ ≥ −1 and µ + α ≥ 0, µ + β ≥ 0, where α, β > 0.\n\nTheorem 2's sequence hypotheses are nontriviality, nonnegativity,\nlog-concavity $g_{n-1}g_{n+1}\\le g_n^2$ for $n\\ge1$, and no internal\nzeros. Equation (20) is\n\n$$g(\\mu,x)=\\sum_{n=0}^{\\infty}\\frac{g_n x^n}{\\Gamma(n+\\mu)},$$\n\nwhere reciprocal Gamma is zero at its poles. Only the “Moreover” clause\nis refuted; the first assertion is not addressed.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.ReciprocalGammaTuran`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $1/\\Gamma(s)$ | `Defs.rgamma` | Inverse real Gamma, zero at nonpositive integers. |\n| Theorem 2's sequence hypotheses | `Defs.AdmissibleSeq` | Nontrivial, nonnegative, log-concave, without internal zeros. |\n| Equation (20)'s formal series | `Series.formalSeries` | Degree $n$ coefficient $g_n/\\Gamma(n+\\mu)$, with no factorial weight. |\n| Degree $m$ Turanian coefficient | `Defs.turanCoeff`, `Series.coeff_mul` | Exact finite Cauchy sum over $0\\le k\\le m$. |\n| Moreover clause | `Defs.claim` | Every admissible sequence and all real shifts in the printed range, at every natural degree. |\n| Boundary failure | `Boundary.d_two_neg` | For every $0 All conclusions of Theorem 3 are valid for all α > 0, while conditions on other parameters remain intact. In particular, µ → g(µ; x) is log-concave for each fixed 0 < x < Rg.\n\nTheorem 3, printed p. 11:\n\n> For a real sequence {gₙ}ₙ≥₀ and fixed 0 < q < 1 define\n>\n> $g(\\mu;x)=\\displaystyle\\sum_{n=0}^{\\infty}\\frac{g_nx^n}{\\Gamma_q(\\mu+n)}.$ (16)\n>\n> Suppose {gₙ}∞ₙ₌₀ is doubly positive. Then (16) has a positive radius of convergence Rg and the generalized Turánian\n>\n> $\\Delta_g(\\alpha,\\beta;x)=g(\\mu+\\alpha;x)g(\\mu+\\beta;x)-g(\\mu;x)g(\\mu+\\alpha+\\beta;x)$\n>\n> is positive for 0 < x < Rg if α ∈ N, β > 0 and µ ≥ max(−β, −1). If, in addition, α ≤ β + 1, then Δg(α, β; x) has positive power series coefficients at all powers of x.\n\nThe sequence is nontrivial, nonnegative, log-concave and has no\ninternal zeros, the definition of doubly positive on printed p. 3.\nEquation (6), printed p. 4, defines\n\n$$\\Gamma_q(z)=(1-q)^{1-z}\\frac{(q;q)_\\infty}{(q^z;q)_\\infty}.$$\n\nIts reciprocal is extended by zero at the poles $z\\in-\\mathbb N_0$.\nOnly the coefficient-positivity conclusion is refuted here.\n\n### Formal reading\n\nThe new Lean names have prefix `D5.S3.Analytic.QGammaTuran.Result`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Reciprocal of (6) | `rqgamma` | Real infinite-product quotient, extended by zero at nonpositive integers. |\n| Doubly positive sequence | `D5.S3.Analytic.ReciprocalGammaTuran.Defs.AdmissibleSeq` | Existing nontriviality, nonnegativity, log-concavity and no-internal-zero hypotheses. |\n| Degree $m$ coefficient | `qturanCoeff` | Exact finite Cauchy sum over $0\\le k\\le m$. |\n| Coefficient clause of Conjecture 1 | `claim` | Every $0 All conclusions of Theorem 3 are valid for all α > 0, while conditions on other parameters remain intact. In particular, µ → g(µ; x) is log-concave for each fixed 0 < x < Rg.\n\nTheorem 3, printed p. 11:\n\n> For a real sequence {gₙ}ₙ≥₀ and fixed 0 < q < 1 define\n>\n> $g(\\mu;x)=\\displaystyle\\sum_{n=0}^{\\infty}\\frac{g_nx^n}{\\Gamma_q(\\mu+n)}.$ (16)\n>\n> Suppose {gₙ}∞ₙ₌₀ is doubly positive. Then (16) has a positive radius of convergence Rg and the generalized Turánian\n>\n> $\\Delta_g(\\alpha,\\beta;x)=g(\\mu+\\alpha;x)g(\\mu+\\beta;x)-g(\\mu;x)g(\\mu+\\alpha+\\beta;x)$\n>\n> is positive for 0 < x < Rg if α ∈ N, β > 0 and µ ≥ max(−β, −1). If, in addition, α ≤ β + 1, then Δg(α, β; x) has positive power series coefficients at all powers of x.\n\nEquation (6), printed p. 4:\n\n> The q-gamma function is defined by\n>\n> $\\Gamma_q(z)=(1-q)^{1-z}\\displaystyle\\frac{(q;q)_\\infty}{(q^z;q)_\\infty}$ (6)\n>\n> for |q| < 1 and all complex z such that qᶻ⁺ᵏ ≠ 1 for k ∈ N₀.\n\nThe reciprocal is extended by zero at nonpositive integers. Doubly\npositive means nontrivial, nonnegative, log-concave and without internal\nzeros, as on printed p. 3. The function assertion has no\n$\\alpha\\le\\beta+1$ restriction.\n\n### Formal reading\n\nThe names below belong to `D5.S3.Analytic.QGammaTuran.FunctionResult`,\nexcept for the reused reciprocal and sequence condition.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Reciprocal of (6) | `Result.rqgamma` | The same real infinite-product quotient as in the coefficient settlement. |\n| Doubly positive sequence | `D5.S3.Analytic.ReciprocalGammaTuran.Defs.AdmissibleSeq` | Nontriviality, nonnegativity, log-concavity and no internal zeros. |\n| Series (16) | `qseries` | The real infinite sum, in the printed order of factors. |\n| Radius of (16) at the given $\\mu$ | `radius` | Mathlib scalar-series radius in $[0,\\infty]$, for coefficients $g_n\\,\\texttt{rqgamma}(q,\\mu+n)$. |\n| Function clause | `claim` | All permitted parameters and every $0 0 for the radius\nof convergence in (16), we claim that the next conjecture is true.”\nMathlib's `FormalMultilinearSeries.ofScalars` represents its coefficient\nsequence, and `FormalMultilinearSeries.radius` supplies the standard\nextended nonnegative radius. For this scalar series the coefficient-map\nnorm is exactly the absolute value of the coefficient.\n\nAt the example, absolute summability at $x=3/2$ gives $3/2\\le R_g$.\nThus $0<93/100<3/2\\le R_g$, so the refutation holds strictly inside\nthe disk of convergence of (16). The same geometric comparison proves\nsummability of all four shifted series at $x=93/100$.","title":"Refutation of reciprocal q-gamma function positivity","url":"https://the-omega-institute.github.io/trureturing-pages/research/kalmykov-karp-2018-qgamma-turanian-function/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b02fd97032d21d139172f3853a322b16fcbe503cd3d67dd9adff61a19642f4b8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kalmykov-karp-pfq-conjecture-one-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. I. Kalmykov and D. B. Karp, *Log-concavity and Turán-type inequalities\nfor the generalized hypergeometric function*, arXiv:1611.06724v1, Section 3,\nConjecture 1:\n\n> All conclusions of Theorem 3 hold for all α, β ≥ 0 if µ ≥ 1.\n\nFor positive parameter vectors of lengths $p$ and $q$, put\n\n$$\nc(t)=\\frac{\\prod_{i=1}^p\\Gamma(a_i+t)}{\\prod_{j=1}^q\\Gamma(b_j+t)},\n\\qquad\nf(\\mu;x)=\\sum_{n=0}^{\\infty}c(\\mu+n)\\frac{x^n}{n!}.\n$$\n\nEquation (9) defines\n\n$$\n\\Delta_f(\\alpha,\\beta;x)=\nf(\\mu+\\alpha;x)f(\\mu+\\beta;x)\n-f(\\mu;x)f(\\mu+\\alpha+\\beta;x),\n$$\n\nand the coefficient of degree $m$ is\n\n$$\n\\delta_m=\\sum_{k=0}^m\n\\frac{c(\\mu+\\alpha+k)c(\\mu+\\beta+m-k)\n-c(\\mu+k)c(\\mu+\\alpha+\\beta+m-k)}{k!(m-k)!}.\n$$\n\nFor $p\\le q$, condition (13) consists of\n\n$$\n\\frac{e_{q-j}(\\mathbf b)}{e_{p-j}(\\mathbf a)}\n\\le\\frac{e_{q-j-1}(\\mathbf b)}{e_{p-j-1}(\\mathbf a)}\n\\qquad(0\\le j Conjecture 1. All conclusions of Theorem 3 hold for all α, β ≥ 0 if µ ≥ 1.\n\nThe relevant branch of Theorem 3, page 5, states verbatim (line breaks joined):\n\n> Theorem 3. If p ≤ q and conditions (13) are satisfied, then ∆f (α, β; x) ≥ 0 for x ≥ 0, µ, β ≥ 0 and α ∈ N, where ∆f (α, β; x) is defined in (9) with f from (10). Moreover, if α ≤ β + 1, then δm ≥ 0 for all m ∈ N0.\n\nFor positive parameter vectors with $p\\le q$, equations (9) and (10) are\n\n$$\nc(t)=\\frac{\\prod_i\\Gamma(a_i+t)}{\\prod_j\\Gamma(b_j+t)},\\qquad\nf(\\mu;x)=\\sum_{n=0}^\\infty c(\\mu+n)\\frac{x^n}{n!},\n$$\n\n$$\n\\Delta_f(\\alpha,\\beta;x)=f(\\mu+\\alpha;x)f(\\mu+\\beta;x)\n-f(\\mu;x)f(\\mu+\\alpha+\\beta;x).\n$$\n\nCondition (13) is\n\n$$\n\\frac{e_{q-j}(\\mathbf b)}{e_{p-j}(\\mathbf a)}\n\\le\\frac{e_{q-j-1}(\\mathbf b)}{e_{p-j-1}(\\mathbf a)}\\quad(0\\le j We will always assume that a₂ is not empty (i.e. p₂ ≥ 1), while all other subvectors are allowed to be empty.\n\nIt then says:\n\n> In this section we will consider log-convexity of the function f (µ; x) defined in (3) under these assumptions.\n\nSection 3 sets $a_1,b_1$ empty, identifying $a=a_2$ in (10). The first\nsentence admits a standing-scope reading; the next admits a section-local\nreading. The source-scope ambiguity is unresolved. The encoded claim imposes\n$1\\le p$.\nIts nonempty-numerator counterexample refutes both that restricted universal\nclaim and the broader claim allowing $p=0$.\n\n| Source object | Lean name in `D5.S3.Analytic.HypergeometricShiftTuran` |\n| --- | --- |\n| Gamma product ratio | frozen `Defs.coeff` |\n| Condition (13) | frozen `Defs.cond13`, using `Defs.esymm` |\n| Equation (10), real infinite sum | `FunctionResult.fval` |\n| Equation (9) | `FunctionResult.turanian` |\n| Universal function clause | `FunctionResult.claim` |\n| Its negation | `FunctionResult.result` |","title":"The function clause of the shifted Gamma Turanian conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/kalmykov-karp-pfq-conjecture-one-function/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f543ce9a82d7c36a0c0f0358a239d3d2febe6f04bfed954390b1a1f2f25fc784.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kalmykov-karp-pfq-conjecture-two-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. I. Kalmykov and D. B. Karp, *Log-concavity and Turán-type inequalities\nfor the generalized hypergeometric function*, arXiv:1611.06724v1, Section 3,\npage 6:\n\n> Conjecture 2. Suppose p1 ≤ q1 , p2 ≤ q2 and conditions (13) hold for a1 , b1 and a2 , b2 . Then the function µ → f (µ; x) defined in (3) is log-concave on [1, ∞) and the corresponding generalized Turánian ∆f (α, β; x) has nonnegative coefficients at all powers of x.\n\nThe standing assumption on page 2 is:\n\n> We will always assume that a2 is not empty (i.e. p2 ≥ 1), while all other subvectors are allowed to be empty.\n\nFor positive parameter vectors in two blocks, equation (3) defines\n\n$$\nf(\\mu;x)=\\frac{\\Gamma(\\mathbf a_2+\\mu)}{\\Gamma(\\mathbf b_2+\\mu)}\n\\,{}_pF_q(\\mathbf a_1,\\mathbf a_2+\\mu;\n\\mathbf b_1,\\mathbf b_2+\\mu;x).\n$$\n\nHere a Gamma value at a vector means the product over its entries. Writing\n$(a)_n$ for the rising factorial and $c(t)$ for the shifted Gamma product\nratio, the Taylor coefficient is\n\n$$\nF_n(\\mu)=\n\\frac{\\prod_i(a_{1,i})_n}{\\prod_j(b_{1,j})_n}\n\\frac{c(\\mu+n)}{n!},\\qquad\nc(t)=\\frac{\\prod_i\\Gamma(a_{2,i}+t)}{\\prod_j\\Gamma(b_{2,j}+t)}.\n$$\n\nEquation (2) defines\n\n$$\n\\Delta_f(\\alpha,\\beta;x)=\nf(\\mu+\\alpha;x)f(\\mu+\\beta;x)\n-f(\\mu;x)f(\\mu+\\alpha+\\beta;x),\n$$\n\nwhose degree-$m$ coefficient is\n\n$$\n\\delta_m=\\sum_{k=0}^m\n\\bigl[F_k(\\mu+\\alpha)F_{m-k}(\\mu+\\beta)\n-F_k(\\mu)F_{m-k}(\\mu+\\alpha+\\beta)\\bigr].\n$$\n\nFor each block separately, condition (13) is\n\n$$\n\\frac{e_{q-j}(\\mathbf b)}{e_{p-j}(\\mathbf a)}\n\\le\\frac{e_{q-j-1}(\\mathbf b)}{e_{p-j-1}(\\mathbf a)}\n\\qquad(0\\le j Our conjecture is that the statement is valid for all $n \\geqslant 2$, but in this work we consider repeaters with up to eight segments only, and for such $n$ we have verified this statement directly.\n\nThe statement is the displayed equality at line 1080:\n\n$\\tilde{D}_n(N_1, \\ldots, N_n) = D^\\star_n(N_1, \\ldots, N_n)$.\n\nHere $N_i$ is the positive integer ready time of segment $i$. Swapping adjacent segments adds $|N_i-N_{i+1}|$ and replaces their ready times by $\\max(N_i,N_{i+1})$. The source defines $\\tilde D_n$ by repeatedly choosing an adjacent pair with least maximum, allowing arbitrary ties (`skr.tex`, lines 1033–1044). It defines $D^\\star_n$ by minimizing the swap cost plus the recursively optimal cost of the shortened list over every adjacent first pair, with $D^\\star_2(N_1,N_2)=|N_1-N_2|$ (lines 1049–1065).\n\nIssue [#25201](https://github.com/the-omega-institute/trureturing/issues/25201) fixes the reading: for every list of $n\\ge2$ positive integers and every complete greedy history, its accumulated cost equals $D^\\star_n$. This is proved. The formal statement also includes zero ready times and terminal lists.","title":"Swapping an earliest adjacent pair minimizes dephasing for every number of segments","url":"https://the-omega-institute.github.io/trureturing-pages/research/kamin-shchukin-schmidt-van-loock-2023-swap-earliest-dephasing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/940534c1b09994a2e45c5cb567473aac44917a0bcd642f73a39315ed1b6bf11c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kang-jager-2026-universal-permutative-lift-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The cited v1 asks whether a given minimal/strictly ergodic (Toeplitz)\nsubshift admits a permutative sliding block code of length ℓ≥2 whose entire\ninverse image is minimal/strictly ergodic. This dossier includes the delivered\ngeneral minimal refutation and its strengthening to a regular Toeplitz,\nstrictly ergodic base. Both concern the same numbered external question.\n\nThe original `UniversalRefutation.claim` quantifies over all finite discrete\nalphabets A with at least two letters and all nonempty minimal two-sided\nsubshifts. Its conclusion asks for some k≥0 and h:A^(k+2)→A, bijective in\nboth endpoints with all remaining letters fixed, whose whole inverse image\nis minimal. `UniversalRefutation.result : Not claim` is the general-branch\nrefutation retained as a preliminary result.\n\nThe strengthened `BalancedRegularRefutation.claim` already restricts the\ninput to binary subshifts that are nonempty and minimal, have exactly one\ninvariant probability measure, and are generated by one aperiodic Toeplitz\npoint with a genuine essential period structure. The structure has positive\nnonempty essential periods, strict growth, divisibility into the next period,\nand coverage of every signed coordinate; its actual normalized periodic\nskeleton densities tend to one. The conclusion still allows every length\nk+2 and every bipermutative local rule, and asks for minimality of the whole\ninverse image. `BalancedRegularRefutation.result : ¬ claim` is its literal\nnegation. Since strict ergodicity implies minimality, it also rules out a\nstrictly ergodic whole lift. A binary counterexample suffices for the original\nfinite-alphabet question.","title":"Question 1.10: regular Toeplitz and strictly ergodic strengthening","url":"https://the-omega-institute.github.io/trureturing-pages/research/kang-jager-2026-universal-permutative-lift-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a494176631a319088fc347526c29fdf015f13bf246bd4fcb1069cb8e3bb926b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kania-kakol-2026-problem-8-2-ch-talagrand","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kania and Kąkol, *Talagrand compacta, 2DCP, and pointwise quotients*,\narXiv:2607.06808v2, revised 10 July 2026, PDF p.19, ask:\n\n> Problem 8.2. Can the prediction principle in Theorem A be weakened from ♢(S) to CH?\n\n### Exact statement correspondence\n\nThere is one mathematical input, `hCH : Cardinal.continuum = ℵ₁`, and one\nliteral carrier `K := (SignedOrdinalRealization.finalPrefix hCH).LimitSpace`.\nThe public theorem is\n`TalagrandTopologicalConsequences.actual_CH_Talagrand_scheme_and_endpoints`.\nAll clauses below occur together in its statement and are consumed by its\nproof.\n\n| Original requirement | Lean statement and derivation |\n| --- | --- |\n| Actual closed nonempty coordinate compacta and onto restrictions | The first conjunction retains all `H.carrier`, `H.map` and `H.projection` readbacks and surjectivity. |\n| Initial omega cube | Every carrier through omega is the whole cube; the final restriction onto the omega cube is onto. |\n| Closed-cover successor split for omega ≤ alpha < omega-one | Closed Y,Z cover `H.Space a`; `H.carrier (nextStage a)` is literally the range of Boolean insertion from `splitCarrier Y Z`. |\n| Proper limit and final thread equations | Each proper limit above omega is exactly its proper-slice limit carrier; the final carrier is exactly all compatible restrictions. |\n| Countable-stage and final topology | Every proper stage is compact Hausdorff, zero-dimensional and metrizable. K is nonempty compact Hausdorff and zero-dimensional, with its actual limiting cone. |\n| Whole original enumerations | U, B and nu are onto; B and V are the original whole births. Every signed measure has regular Jordan parts, variation mass one, support concentration and positive mass at least one half. |\n| Original A–F and every signed selection | Every proper cutoff has the original `P.Laws D`, including every-choice signed F, with every original-label measure readback. No final-stage F is postulated. |\n| G and cofinal bookkeeping | One eta gives the same support launches and signed source measures. The fixed G clopen is the new-coordinate-true cylinder. Its fixed increasing column sequence works for arbitrary complete signed norm-one lifts, with infinitely many evaluations at least one sixth and infinitely many exact zeros. Every row/label pair is treated cofinally above any base. |\n| Theorem A closed-copy prohibition | For every closed A and continuous f : A → K with disjoint range, the range is finite. Every two disjoint homeomorphic subspaces, one closed, are consequently finite. This is stronger than excluding disjoint nonmetrizable closed copies. |\n| No 2DCP | No sequence of nonempty compact T(n) has two disjoint copies of T(n+1) in T(n). No nesting or infinitude premise is added. |\n| Grothendieck | Every sequence in the full real continuous dual converging pointwise to FInf converges under every continuous real bidual test. There is no input norm bound. |\n| Entire signed dual unit ball | Every continuous map from all of `StoneCech ℕ` into `WeakDual ℝ C(K,ℝ)` bounded by one is noninjective, including principal and free points. |\n| Point-space consequences | Every convergent sequence is eventually its actual limit; no continuous StoneCech injection exists; K is infinite, not first countable and not metrizable. |","title":"Kania–Kąkol Problem 8.2: the full Talagrand construction under CH","url":"https://the-omega-institute.github.io/trureturing-pages/research/kania-kakol-2026-problem-8-2-ch-talagrand/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e88e35ec0cd63dd42c6a738774680a55030355bafacc49f99b279cd41d5e95ec.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karlsson-2016-bccb-order-nine-affine-suborbits","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. R. Karlsson, *BCCB complex Hadamard matrices of order 9, and MUBs*, arXiv:1604.03770v1, Linear Algebra Appl. 504 (2016), Conjecture 3:\n\n> $BC_{9A}^{(1)}$ and $BC{}_{9B}^{(1)}$ are the only affine suborbits of $BC_{9}^{(2)}$.\n\nThe definition of an affine orbit, the matrix, the unitarity locus x + y + u + w = 1 and the two families are quoted in [the literature note](../Library/QuantumContext/karlsson2016bccbhadamard.md).\n\nIssue [#19395](https://github.com/the-omega-institute/trureturing/issues/19395) fixes the reading. Consider:\n- a nonconstant family of four unit parameters on a nonempty open subset of ℝ^p with sum 1;\n- whose canonically dephased matrix has entries of the form c e^{iL(t)} with L linear.\n\nUp to a fixed permutation of the parameters, such a family is (μ, −μ, −ω, −ω²) or (1, ξ, ωξ, ω²ξ). The conjecture is proved.","title":"The two Karlsson families are the only affine suborbits of BC_9^(2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/karlsson-2016-bccb-order-nine-affine-suborbits/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a9b842bc301b08677c5f5e3445cfe82e97394e36494088e102f463e634542b30.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-2016-incomplete-beta-coefficient-log-concavity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Karp, *Normalized incomplete beta function: log-concavity in parameters\nand other properties*, §3, Conjecture 1:\n\n> Conjecture 1 The coefficients φ_k are positive for all a, b, α, β > 0 and x ∈ (0, 1), so that b → x^{−a}(1 − x)^{−b} I_x(a, b) is strictly coefficient-wise Wright log-concave on (0, ∞) for these values of parameters.\n\nHere equations (1), (3) and (4) give\n\n$$c_j(a,b)=\\frac{\\Gamma(a+b+j)}{\\Gamma(b)\\Gamma(a+j+1)},\\qquad\n\\varphi_n=\\sum_{k=0}^n\n\\bigl[c_k(a,b+\\alpha)c_{n-k}(a,b+\\beta)\n-c_k(a,b)c_{n-k}(a,b+\\alpha+\\beta)\\bigr].$$\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.IncompleteBeta`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $c_j(a,b)$ in (1) | `coeff` in `Defs` | Exactly the displayed Gamma quotient, with natural index $j$. |\n| $\\varphi_n$ in (3), (4) | `phi` in `Defs` | The finite convolution difference over $0\\le k\\le n$. |\n| $C_n(s,t)$ | `convolution` in `Moment` | $\\sum_{k=0}^n c_k(a,s)c_{n-k}(a,t)$, with $a$ explicit. |\n| The symmetric beta kernel | `W` in `Moment` | The finite powered kernel used in the moment representation below. |\n| Conjecture 1 | `claim`, `Result.result` | Every real $a,b,\\alpha,\\beta>0$ and every natural $n$, with strict positivity. |\n| $x\\in(0,1)$ | No additional coefficient hypothesis | $\\varphi_n$ does not depend on $x$, so this source quantifier is vacuous for the coefficients. |","title":"Coefficient-wise Wright log-concavity of normalized incomplete beta functions","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-2016-incomplete-beta-coefficient-log-concavity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/751fcb974e12db419612e6def27ccf94f466720f5c671c6a4569099e69ef031c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-2016-incomplete-beta-first-parameter-coefficient-signs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Karp, *Normalized incomplete beta function: log-concavity in parameters\nand other properties*, §3, Conjecture 2:\n\n> Conjecture 2 The coefficients ψ_k are negative for all a, α, β > 0 and x ∈ (0, 1) if 0 < b < 1 and positive if b > 1, so that a → x^{−a}(1 − x)^{−b} I_x(a, b) is strictly coefficient-wise Wright log-convex on (0, ∞) for α, β > 0, x ∈ (0, 1) and 0 < b < 1; and a → x^{−a}(1 − x)^{−b} I_x(a, b) is strictly coefficient-wise Wright log-concave on (0, ∞) for α, β > 0, x ∈ (0, 1) and b > 1.\n\nEquations (1) and (6) give\n\n$$c_j(t,b)=\\frac{\\Gamma(t+b+j)}{\\Gamma(b)\\Gamma(t+j+1)},\\qquad\n\\psi_n=\\sum_{k=0}^n\\bigl[c_k(a+\\alpha,b)c_{n-k}(a+\\beta,b)\n-c_k(a,b)c_{n-k}(a+\\alpha+\\beta,b)\\bigr],$$\n\n$$\\psi_{a,b,\\alpha,\\beta}(x)\n=x^{2a+\\alpha+\\beta}(1-x)^{2b}\\sum_{n\\ge0}\\psi_nx^n.$$\n\n### Formal reading\n\nUnless otherwise indicated, Lean names have prefix\n`D5.S3.Analytic.IncompleteBetaShape`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $c_j(t,b)$ in (1) | `D5.S3.Analytic.IncompleteBeta.coeff` | The same Gamma quotient; no new coefficient definition. |\n| $\\psi_n$ in (6) | `Defs.psi` | Exactly the displayed finite convolution difference, with natural index $n$. |\n| $S_n(p,q;b)$ | `Mixture.convolution` | $\\sum_{k=0}^n c_k(p,b)c_{n-k}(q,b)$. |\n| Uniform shifted beta mixture | `Mixture.mixtureDensity` | The average of Beta$(p+k+1,q+n-k+1)$ densities for $0\\le k\\le n$. |\n| Sliding kernel | `Kernel.F`, `Kernel.H` | The uniform affine average and its beta average used below. |\n| Conjecture 2 | `Defs.claim`, `Result.result` | Both strict signs for all real $a,b,\\alpha,\\beta>0$ and all natural $n$. |\n| $x\\in(0,1)$ | No additional coefficient hypothesis | The coefficients do not depend on $x$, so this quantifier is vacuous for them. |","title":"First-parameter coefficient signs of the normalized incomplete beta function","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-2016-incomplete-beta-first-parameter-coefficient-signs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebafde4d2445e5082f67c3abac4c5990ac2f8292d2115c68c764c2b1f53244d4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-prilepkina-2015-meijer-zero-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Karp and Prilepkina, *Completely monotonic gamma ratio and infinitely\ndivisible H-function of Fox*, arXiv:1501.05388v2, Section 2, Conjecture 1:\n\n> If aᵢ ≥ 0, i = 1, …, p and Σᵢ₌₁ᵖ (bᵢ − aᵢ) > 0 then # zeros of G^{p,0}_{p,p}(x | b/a) on (0, 1) ≤ # zeros of v(t) on (0, 1).\n\nHere $p\\ge1$, $a_i,b_i$ are real,\n\n$$v(t)=\\sum_{k=1}^p(t^{a_k}-t^{b_k}),\\qquad\n\\psi=\\sum_{i=1}^p(b_i-a_i),$$\n\nand the contour-integral definition is\n\n$$G^{p,0}_{p,p}\\!\\left(x\\;\\middle|\\;\\begin{matrix}b\\\\a\\end{matrix}\\right)\n=\\frac{1}{2\\pi i}\\int_L\n\\frac{\\prod_{i=1}^p\\Gamma(a_i+s)}\n{\\prod_{i=1}^p\\Gamma(b_i+s)}x^{-s}\\,ds.$$\n\nThe contour separates the numerator poles in the usual Mellin–Barnes\nrepresentation. The zero counts below count distinct zeros.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.MeijerZeroCount`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $R(s)=\\prod_i\\Gamma(a_i+s)/\\prod_i\\Gamma(b_i+s)$ | `Defs.gammaQuot` | Complex Gamma quotient of the real parameter vectors. |\n| $G^{p,0}_{p,p}(x\\mid b/a)$ | `Defs.meijerG` | `mellinInv 1 (gammaQuot a b) x`, with normalization $1/(2\\pi)$ in the real integral. |\n| $v(t)=\\sum_k(t^{a_k}-t^{b_k})$ | `Defs.control` | Real powers of real arguments; the comparison uses $00$ | `Defs.claim` | The formal claim restricts to $\\psi>1$. It quantifies over natural $p$; the case $p=0$ cannot satisfy $\\psi>1$. |\n| Number of zeros on $(0,1)$ | `Set.encard` of each displayed zero set | Extended natural cardinality, including infinite zero sets. |\n| $a=(n+c,n+c)$, $b=(c,2n+d+c)$ | `Result.family` | All natural $n,d\\ge2$ and real $c>0$; $p=2$ and $\\psi=d$. |\n| Refutation of the comparison | `Result.result` | The unconditional theorem has type `¬ Defs.claim`. |\n\nThe formal claim restricts to $\\psi>1$, where the contour may be taken\nas the vertical line $\\Re s=1$. Since $a_i\\ge0$, this line is to the\nright of every numerator Gamma pole. The Gamma quotient decays as\n$O(|\\Im s|^{-\\psi})$ on this line, so the integral is absolutely\nconvergent. A counterexample in this restricted range is also a\ncounterexample to the source statement with $\\psi>0$.","title":"The Karp–Prilepkina Meijer zero-count conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-prilepkina-2015-meijer-zero-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf2b0c988a0d22cc1d46c39a210c0d21f1bf1bf6f928ef7ef7eba54543cb179b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-sitnik-2009-luke-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Karp and S. M. Sitnik, *Inequalities and monotonicity of ratios for\ngeneralized hypergeometric function*, J. Approx. Theory 161 (2009),\n337–352, DOI 10.1016/j.jat.2008.10.002, arXiv:math/0703084v2,\nConjecture 1, printed p. 14:\n\n> Theorem 3 is true for all σ > 0 and ∑ᵢ₌₁ᵠ (bᵢ − aᵢ) > 0.\n\nTheorem 3, printed p. 6, equation (23):\n\n> For bₖ > aₖ > 0, k = 1, . . . , q, x > −1 and σ ≥ 1 the inequality\n>\n> $\\displaystyle\\frac{1}{(1+x\\prod_{i=1}^q(a_i/b_i))^\\sigma}<{}_{q+1}F_q(\\sigma,(a_q);(b_q);-x)$\n>\n> holds true.\n\n### Formal reading\n\nLean names below have prefix `D5.S3.Analytic.KarpSitnikLuke.Result`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Rising-factorial product ratio | `D5.S3.Analytic.HypergeometricShiftTuran.TwoBlockResult.pochRatio` | Existing quotient of upper and lower rising-factorial products. |\n| Summand of (1) | `hyperTerm` | Extra upper parameter σ, factorial denominator and power zⁿ. |\n| Function inside the convergence disk | `hyper` | Real infinite sum of these summands. |\n| Restricted Conjecture 1 | `claim` | All positive vectors with positive sum(b−a), σ>0, −1−1. Its refutation refutes the conjecture.\nFor q=0 the positive-sum assumption is impossible.","title":"Refutation of the sum-condition Luke lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-sitnik-2009-luke-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a6378f915d8b1f31627e65f25e4f57a29b59224c32dd3879c3680ceb5cf3740.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-sitnik-2009-luke-upper-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Karp and S. M. Sitnik, *Inequalities and monotonicity of ratios for\ngeneralized hypergeometric function*, J. Approx. Theory 161 (2009),\n337–352, DOI 10.1016/j.jat.2008.10.002, arXiv:math/0703084v2,\nConjecture 2, printed p. 14, equation (38):\n\n> For 0 < σ ≤ min(a₁, a₂, . . . , a_q), ∑ᵢ₌₁ᵠ (bᵢ − aᵢ) > 0 and x > 0:\n>\n> $\\displaystyle {}_{q+1}F_q(\\sigma,(a_q);(b_q);-x)<\\frac{1}{\\left[1+x\\prod_{i=1}^q\\frac{\\Gamma(a_i)\\Gamma(b_i-\\sigma)}{\\Gamma(b_i)\\Gamma(a_i-\\sigma)}\\right]^\\sigma}.\\qquad (38)$\n\nThe preceding sentence reads:\n\n> The asymptotic formula (16) and numerical experiments suggest the following\n\nFormula (16), printed p. 5, is preceded by:\n\n> It follows that for σ < min(a₁, a₂, . . . , a_q) we will have\n\n$$ {}_{q+1}F_q(\\sigma,(a_q);(b_q);-z)\n=\\frac{\\prod_i\\Gamma(a_i-\\sigma)\\Gamma(b_i)}\n{\\prod_i\\Gamma(a_i)\\Gamma(b_i-\\sigma)}z^{-\\sigma}\n+o(z^{-\\sigma}),\\qquad z\\to\\infty. $$\n\nThe quotation transcribes the extracted display, which places a $\\sigma$\nnext to the Gamma product. Write\n$K=\\prod_{i=1}^q\\Gamma(a_i)\\Gamma(b_i-\\sigma)/(\\Gamma(b_i)\\Gamma(a_i-\\sigma))$.\nSeparately, the asymptotically precise reading via (16) is\n$1/(1+xK^{1/\\sigma})^\\sigma$: the inner exponent $1/\\sigma$\nmakes its leading asymptotic coefficient agree with (16).\nThe counterexample has $\\sigma=1$, where both readings coincide.\n\n### Formal reading\n\nLean names below have prefix `D5.S3.Analytic.KarpSitnikLuke.UpperResult`,\nexcept where their frozen owner is specified.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Rising-factorial ratio | `D5.S3.Analytic.HypergeometricShiftTuran.TwoBlockResult.pochRatio` | Quotient of the products for a and b. |\n| Summand of (1) | `D5.S3.Analytic.KarpSitnikLuke.Result.hyperTerm` | Extra upper parameter σ, factorial denominator and zⁿ. |\n| Series value | `D5.S3.Analytic.KarpSitnikLuke.Result.hyper` | Real infinite sum of these summands. |\n| Gamma product in (38) | `gammaProduct` | Exactly K defined above, with finite index type Fin q. |\n| Restricted Conjecture 2 | `claim` | Positive parameters, positive sum(b−a), 0<σ≤aᵢ, 00 and the strict upper bound. |\n| Refutation | `result` | Exactly ¬ claim. |\n\nEach additional condition weakens the universal assertion: $x<1$\nrestricts to the power-series disk, summability restricts to convergent\nseries, and $K>0$ restricts the parameter set. The latter, with $x>0$,\nmakes the real-power base positive. Refuting the weaker assertion\nrefutes the conjecture. The $q=0$ positive-sum assumption is impossible.","title":"Refutation of the sum-condition Luke upper bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-sitnik-2009-luke-upper-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d15d2b8d10bb6f47b9841d11600f9f3e8d82ab46b69e7864d7e2ea0f396a72f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-sitnik-2010-kummer-ratio-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Karp and Sitnik, *Log-convexity and log-concavity of hypergeometric-like\nfunctions*, arXiv:0902.3073v1, the unnumbered Conjecture after Theorem 4:\n\n> Conjecture. The ratio in the middle of (10) is monotone decreasing from (0, ∞) onto (1, A), where A is the left-hand side of (10); the ratio in the middle of (11) is monotone increasing from (−∞, 0) onto (B, 1), where B is the left-hand side of (11).\n\nWrite $M(s;c;x)={}_1F_1(s;c;x)$ and\n\n$$R(x)=\\frac{M(b+\\delta;c;x)M(a;c;x)}\n {M(a+\\delta;c;x)M(b;c;x)}.$$\n\nTheorem 4 assumes $\\delta>0$ throughout. Its inequality (10), for\n$b>a>0$ and $c,x>0$, is\n\n$$\\frac{\\Gamma(a+\\delta)\\Gamma(b)}\n {\\Gamma(b+\\delta)\\Gamma(a)}0>x$, is\n\n$$\\frac{\\Gamma(c-a-\\delta)\\Gamma(c-b)}\n {\\Gamma(c-b-\\delta)\\Gamma(c-a)}0$ and $00$ and $a Conjecture 1. Suppose b ≥ 0, ν1 ≥ ν2 > −1 and 0 < a1 ≤ a2. Then the ratio x → Iν1(a1 x)/Iν2(a2 x) is unimodal on (0, ∞). Moreover, if ν1 ≥ ν2 > 0 it is log-concave.\n\nThe modified Bessel function of the first kind has the standard series\n\n$$\nI_\\nu(z)=\\sum_{k=0}^{\\infty}\\frac{(z/2)^{2k+\\nu}}{k!\\Gamma(k+\\nu+1)}.\n$$\n\nThe encoded claim is\n\n$$\n\\forall\\nu_1\\ge\\nu_2>0\\ \\forall\\,0 Assume that $\\{f_n\\}_{n=0}^{\\infty}$ is log-concave and independent of $\\mu$. Then the function\n>\n> $$\n> \\mu\\to f(\\mu;x)=\\sum_{n=1}^{\\infty} f_n\\frac{(\\mu)_{3n}}{(3n-1)!}x^n\n> $$\n>\n> is coefficient-wise log-concave.\n\nDefinition 1.1, page 2, uses nonnegative, nontrivial real sequences,\n$f_k^2\\ge f_{k-1}f_{k+1}$ for positive natural $k$, and interval support.\nIn particular, internal zeros are excluded, but zero endpoints and finite\nsupports are allowed. Define\n\n$$\na_0(t)=0,\\qquad a_n(t)=\\frac{(t)_{3n}}{(3n-1)!}\\quad(n\\ge1),\n$$\n\nand, for every natural $m$ and nonnegative real $\\mu,\\alpha,\\beta$,\n\n$$\n\\delta_m=\\sum_{k=0}^{m} f_kf_{m-k}\n\\left[a_k(\\mu+\\alpha)a_{m-k}(\\mu+\\beta)\n-a_k(\\mu)a_{m-k}(\\mu+\\alpha+\\beta)\\right].\n$$\n\nThe assertion is $\\delta_m\\ge0$ for every such sequence, parameter triple\nand degree. The endpoint summands vanish. These are finite formal Cauchy\ncoefficients, without convergence or growth assumptions.\n\nThe coefficient-wise convention on printed page 1 reads:\n\n> if the coefficients $\\delta_k$ at all powers of $x$ in (2) are non-negative (non-positive) for all $\\alpha,\\beta\\ge0$ we will say that $\\mu\\to f(\\mu;x)$ is coefficient-wise log-concave (log-convex ).\n\nThe Lean namespace prefix is `D5.S3.Analytic`. The coefficient is\n`KarpZhangNinh.Result.Kernel.a`, the finite Turanian is `KarpZhangNinh.Result.delta`,\nthe sequence predicate is the reused\n`KarpZhangReciprocal.Result.IsLogConcave`, and the universal proposition is\n`KarpZhangNinh.Result.claim`.","title":"Coefficientwise log-concavity of the triple rising-factorial series","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-zhang-2023-conjecture-3-8/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70d8ddc822cc68274e06ddee73d66161b5a78062ea06662e32764400468c01dc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-zhang-2023-conjecture-4-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dmitrii Karp and Yi Zhang, *Log-concavity and log-convexity of series containing\nmultiple Pochhammer symbols*, arXiv:2305.09029v3, page 8, Conjecture 4.2:\n\n> Assume that $\\{h_k\\}_{k=0}^{\\infty}$ is log-concave (log-convex ) and independent of $\\mu$. Then the function $\\mu\\to h(\\mu;x)$ defined in (15) is coefficient-wise log-concave (log-convex ).\n\nFormula (15) defines\n\n$$\nh(t;x)=\\sum_{k\\ge0}h_k a_k(t)x^k,\\qquad\na_k(t)=\\frac{(t)_{2k}}{(t+1)_k k!}.\n$$\n\nDefinition 1.1, page 2, starts with a nonnegative nontrivial real sequence.\nThe log-concave class has $h_k^2\\ge h_{k-1}h_{k+1}$ for every positive\ninteger $k$ and interval support, with the precise printed zero-support\ndisjunction. The log-convex clause reads:\n\n> If $f_k^2\\le f_{k-1}f_{k+1}$ for all $k\\in\\mathbb N$, the sequence is called log-convex (this inequality implies that a nontrivial sequence is strictly positive).\n\nHere $\\mathbb N=\\{1,2,\\ldots\\}$. The literal inequalities permit the\nexception supported on $\\{0\\}$; no extra strict-positivity assumption is\nimposed. For $\\mu,\\alpha,\\beta\\ge0$, set\n\n$$\n\\delta_m=\\sum_{k=0}^{m}h_kh_{m-k}\n\\left[a_k(\\mu+\\alpha)a_{m-k}(\\mu+\\beta)\n-a_k(\\mu)a_{m-k}(\\mu+\\alpha+\\beta)\\right].\n$$\n\nThe assertion is $\\delta_m\\ge0$ for log-concave $h$ and $\\delta_m\\le0$\nfor log-convex $h$, for every natural degree $m$ and every nonnegative\nreal base parameter and shifts. It concerns finite formal coefficients,\nwithout convergence or growth assumptions.\n\nAll Lean names in the table have prefix `D5.S3.Analytic`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Formula (15) | `KarpZhangCatalanPower.Coeff.a` | Literal rising-factorial quotient, including $k!$. |\n| Definition 1.1, log-concave class | `KarpZhangReciprocal.Result.IsLogConcave` | Existing predicate, including nontriviality and exact interval support. |\n| Definition 1.1, log-convex class | `KarpZhangCatalanPower.Result.IsLogConvex` | Nonnegativity, nontriviality and adjacent inequalities at positive indices; no support restriction. |\n| Turanian coefficient | `KarpZhangCatalanPower.Result.delta` | Exactly the finite Cauchy sum above. |\n| Conjecture 4.2 | `KarpZhangCatalanPower.Result.claim` | Both implications for the same $h,\\mu,\\alpha,\\beta,m$. |\n| Settlement | `KarpZhangCatalanPower.Result.result` | A proof of exactly `claim`. |","title":"Coefficientwise log-concavity and log-convexity of Catalan-power series","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-zhang-2023-conjecture-4-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/abed66ec5a92fac9176d28a44cd2abcf2ffb5619c4ff0d0b67c5fd284ea597ce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:karp-zhang-2023-conjecture-5-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dmitrii Karp and Yi Zhang, *Log-concavity and log-convexity of series containing\nmultiple Pochhammer symbols*, arXiv:2305.09029v3, page 12, Conjecture 5.2:\n\n> Suppose that $\\{y_k\\}_{k=0}^{\\infty}$ is a log-concave sequence. Then the functions $\\mu\\to\\widetilde y(\\mu;x)$ and $\\mu\\to\\widehat y(\\mu;x)$ are coefficient-wise log-concave on $\\mu>0$.\n\nThe two series are\n\n$$\n\\widetilde y(\\mu;x)=\\sum_{k=0}^{\\infty}y_k\\frac{(2\\mu)_k}{(\\mu)_k k!}x^k,\n\\qquad\n\\widehat y(\\mu;x)=\\sum_{k=0}^{\\infty}y_k\\frac{\\Gamma(\\mu+k)}{\\Gamma(2\\mu+k)k!}x^k.\n$$\n\nHere $(a)_k$ is the rising factorial. Definition 1.1, page 2:\n\n> Let $\\{f_k\\}_{k=0}^{\\infty}$ be a non-negative nontrivial real sequence. We call $\\{f_k\\}_{k=0}^{\\infty}$ log-concave (or Pólya frequency two – $PF_2$, or doubly positive) if $f_k^2\\ge f_{k-1}f_{k+1}$ for each $k\\in\\mathbb N$, and $\\{f_k\\}_{k=0}^{\\infty}$ is supported on an interval of integers (i.e., $f_N=0$ implies either $f_{N+i}=0$ for all $i\\in\\mathbb N_0$ or $f_{N-i}=0$ for $i=0,\\ldots,N$).\n\nThe Lean predicate retains nonnegativity, nontriviality, adjacent inequalities\nfor positive natural indices, and the printed zero-support disjunction.\nNatural subtraction is used only with the stated bounds.\n\nFor either series $f$, coefficientwise log-concavity means that every\ncoefficient of\n\n$$\nf(\\mu+\\alpha;x)f(\\mu+\\beta;x)-f(\\mu;x)f(\\mu+\\alpha+\\beta;x)\n$$\n\nis nonnegative for every $\\mu>0$ and every real $\\alpha,\\beta\\ge0$.\nThis is an assertion about finite Cauchy sums in each natural degree, so it\nrequires no convergence or growth assumption on $y$.\n\nFormal correspondence. All Lean names below have prefix `D5.S3.Analytic`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Definition 1.1 | `KarpZhangReciprocal.Result.IsLogConcave` | All four sequence conditions, including nontriviality and the exact zero convention. |\n| Reciprocal-series coefficient before multiplication by $y_k$ | `KarpZhangReciprocal.Coeff.c` | $(2\\mu)_k/((\\mu)_k k!)$. |\n| Reciprocal Turanian coefficient | `KarpZhangReciprocal.Result.delta` | The degree-$m$ finite Cauchy coefficient with both factorials retained. |\n| Gamma-series coefficient before multiplication by $y_k$ | `KarpZhangGamma.Series.gammaCoeff` | $\\Gamma(\\mu+k)/(\\Gamma(2\\mu+k)k!)$. |\n| Gamma Turanian coefficient | `KarpZhangGamma.Series.gammaDelta` | The degree-$m$ finite Cauchy coefficient of the gamma-series Turanian. |\n| Conjecture 5.2 | `KarpZhangConjecture.Result.claim` | Both coefficient inequalities for the same $y,\\mu,\\alpha,\\beta,m$, universally quantified. |\n| Settlement | `KarpZhangConjecture.Result.result` | A proof of exactly `claim`. |","title":"Coefficientwise log-concavity of the two Karp--Zhang series","url":"https://the-omega-institute.github.io/trureturing-pages/research/karp-zhang-2023-conjecture-5-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e34a0bfb47f8d9c9f0f87f14bd0d36293136b786f47c9a925833ac62bc50ef7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kasel-2026-displacement-ladder","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kasel, arXiv:2609.02939, Appendix B.4, states: “The conjectured growth L(m) = m − 2”. The exact target was preregistered in [#14885](https://github.com/the-omega-institute/trureturing/issues/14885).\n\nLet $b(v)=\\lceil\\log_2 v\\rceil$ for $v\\ge2$, implemented by the total natural-number function `Nat.clog 2 v`. Set\n\n$$S_A(m)=\\{v\\in[1,4^m]:b(v)\\ge2,\\ b(v)\\text{ even}\\},\\qquad\nD=\\{3,4,9,10,11,12,13,14,15,16\\}.$$\n\nA scheme consists of natural-valued stages $s$ and fibre positions $r$. It orders values by stage, then by fibre position. Validity requires injectivity within each stage and excludes both increasing and decreasing three-term arithmetic progressions in that concatenation. Normalization requires $s(v)\\ge\\lfloor b(v)/2\\rfloor$ on $S_A(m)$. Write $\\delta(v)=s(v)-\\lfloor b(v)/2\\rfloor$ and let $L(m)$ be the minimum, over valid normalized schemes, of $\\max_{v\\in D}\\delta(v)$. The proved statement is $L(m)=m-2$ for every natural $m\\ge2$.","title":"kasel 2026 displacement ladder","url":"https://the-omega-institute.github.io/trureturing-pages/research/kasel-2026-displacement-ladder/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/34fb7b83835a5b448dead5d076b7d2a4136a06fca7ac4dd72810e2159b433b01.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kauffman-mukherjee-vojtechovsky-2025-maximal-r-clique-divisibility-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Printed page 21 of arXiv:2504.09368v1, section 5.5, states:\n\n> Problem 5.24. Does there exist a finite connected rack Q and a maximal R-clique C of Q such that |C| does not divide |Q|?\n\nThe printed question is existential. The preregistered claim24 is its negation: every inclusion-maximal R-clique in every finite connected rack has cardinality dividing the carrier cardinality. The result refutes this universal divisibility assertion; it answers the printed existence question Yes.","title":"Kauffman–Mukherjee–Vojtěchovský Problem 5.24","url":"https://the-omega-institute.github.io/trureturing-pages/research/kauffman-mukherjee-vojtechovsky-2025-maximal-r-clique-divisibility-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6360e129fbcee539fa4271c51c4a44557e1b7da64c455a2afe8e3c2373f3f9ed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kauffman-mukherjee-vojtechovsky-2025-maximal-r-clique-equal-size-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Printed page 21 of arXiv:2504.09368v1, section 5.5, states:\n\n> Problem 5.22. Do all maximal R-cliques in a finite connected rack have the same size?\n\nThe affirmative reading says that any two inclusion-maximal R-cliques C and D have equal cardinalities. The formal claim22 quantifies over every Q : Type with Fintype Q and DecidableEq Q, every binary operation mul, and both cliques, with IsRack mul and Connected mul as premises.","title":"Kauffman–Mukherjee–Vojtěchovský Problem 5.22","url":"https://the-omega-institute.github.io/trureturing-pages/research/kauffman-mukherjee-vojtechovsky-2025-maximal-r-clique-equal-size-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c551be0d682127708c5c73a4ad6635bfc53f39422cf0157bf5f05d72a767985.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kay-2010-transfer-rate-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Kay, *A Review of Perfect State Transfer and its Application as a\nConstructive Tool*, arXiv:0903.4274v3 (Int. J. Quantum Inf. 8, 641 (2010)),\nsubsection \"Transfer Rate\". For the ordered eigenvalues\n$\\lambda_1 < \\dots < \\lambda_N$ of a mirror-symmetric chain the perfect state\ntransfer condition is\n\n> $\\lambda_n-\\lambda_{n-1}=(2m_n+1)\\pi/t_0$ where $t_0$ is the state transfer time, and $m_n$ is a positive integer (which can vary with $n$).\n\nand $B'(\\lambda_n)=\\prod_{m=1\\neq n}^N(\\lambda_n-\\lambda_m)$. The lemma of\nthat subsection (the sixth lemma of the source) reads:\n\n> If a set of eigenvalues is chosen to fulfill the perfect state transfer condition of Eqn.~(\\ref{eqn:st_cond}), then a necessary and sufficient condition to perfectly achieve the rate $M/2t_0$ for integer $M$ is that all the $R_k$ for $k=0\\ldots M-1$ should be equal, where\n> $R_k=\\sum_{n=1}^N\\frac{(-1)^n}{B'(\\lambda_n)}$,\n> and the sum is restricted to those terms satisfying the condition\n> $\\frac{t_0}{\\pi}(\\lambda_n-\\lambda_1)\\mod M=k.$\n\nIt is followed by the conjecture:\n\n> We conjecture that it is impossible to fulfill the condition of Lemma \\ref{lemma:rate} for any $M>2$, although we only have a proof for $M>N/2$.\n\nIssue #14475 fixes the reading. A spectrum is a function `lam : Fin N → ℝ`;\nthe condition is `0 < t₀` together with positive integers `m` such that every\ngap between consecutive levels is `(2 m + 1) π / t₀`; the level `n` lies in\nthe class `k` modulo `M` when the real number `(t₀/π)(λ_n − λ_1)` is an integer\ncongruent to `k`; the sign `(−1)^n` uses the source's index, which starts at\none. The conjecture is the statement that for every `N ≥ 1`, every such\nspectrum and transfer time, and every integer `M > 2`, the sums `R_0, …,\nR_{M−1}` are not all equal. The case `N = 0` is excluded because all sums of\nthe empty spectrum are empty and therefore equal.","title":"Kay's transfer-rate conjecture for perfect state transfer chains","url":"https://the-omega-institute.github.io/trureturing-pages/research/kay-2010-transfer-rate-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b6fa6e46a85930b5a6568bbd38f820a2b05e494daad5ce5c1ba1e82b380c08a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kaygorodov-martin-gonzalez-viruel-2026-evolution-algebra-derivations-conjecture-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of results and a list of open problems*, arXiv:2610.22248v1, state Conjecture 5: if an evolution algebra is finite-dimensional over an algebraically closed field of characteristic zero and its derivation algebra is semisimple, then the derivation algebra is a finite direct sum of orthogonal Lie algebras $\\mathfrak{so}_{m_i}$ with $m_i\\geq 3$. The empty sum is allowed.\n\nThe source's Section 1.1 defines an evolution algebra by a natural basis whose distinct basis vectors multiply to zero. Proposition 14 realizes the orthogonal summands; its converse realization assertion is not separately formalized here.\n\n### Statement echo\n\nThe public `claim` in `D5/S3/QuadraticForms/EvolutionAlgebras/SemisimpleDerivationClassification.lean` preserves the source's field, algebraic-closedness, characteristic-zero, finite-dimensionality, natural-basis, semisimplicity and finite direct-sum clauses. The public `result` proves that literal proposition.","title":"Kaygorodov–Martín González–Viruel Conjecture 5","url":"https://the-omega-institute.github.io/trureturing-pages/research/kaygorodov-martin-gonzalez-viruel-2026-evolution-algebra-derivations-conjecture-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ed47cd36fbfca4b953ccf26df4781cb5232216fbf63f740b4c39cbc3d7628964.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kaygorodov-martin-gonzalez-viruel-2026-evolution-automorphism-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, section 2.3,\nConjecture 3, pages 27–28, state:\n\n> Over C, the maximal order of the automorphism group of a perfect\n> n-dimensional evolution algebra is\n> max { product_l (l(2^l−1))^(m_l) m_l! : sum_l l m_l=n },\n> that is 1, 6, 21, 72, 155, 1296 for n=1,...,6 and n! for n>=7.\n\nThe dimensions are positive. Multiplication is bilinear and need not be\nassociative. A natural basis has zero mixed products. Perfectness means\nthat the span of all pair products is the whole algebra. Automorphisms\nare all multiplication-preserving complex linear equivalences.","title":"The largest automorphism groups of perfect evolution algebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/kaygorodov-martin-gonzalez-viruel-2026-evolution-automorphism-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ae19b28437b466149ce6961c404c2b96e0fbd74bf3ccf0348b3fa0c5af8d0d7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kaygorodov-martin-gonzalez-viruel-2026-finite-aut-infinite-natural-bases","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martin Gonzalez and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, section 2.2.2,\nQuestion 14 asks:\n\n> Describe the set of natural bases (up to permutation and scaling) of a\n> non-degenerate evolution algebra. When is it finite? Can it be infinite\n> for an algebra whose automorphism group is finite?\n\nThe endpoint is the entire final interrogative only. Its positive answer\nrequires a permitted field and a finite-dimensional commutative bilinear,\npossibly nonassociative evolution algebra with zero annihilator, finite\nfull linear multiplication-automorphism group, and infinitely many natural\nbases modulo all permutations and independent nonzero scalings. It does\nnot ask for every field, a group-scheme classification, or algebra-automorphism\norbits of bases.","title":"Infinite natural-basis quotient with finite full automorphisms","url":"https://the-omega-institute.github.io/trureturing-pages/research/kaygorodov-martin-gonzalez-viruel-2026-finite-aut-infinite-natural-bases/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f0edae4996a66098956e20fa2529ca8d4d007c062d82da10520d4ec513742ae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kaygorodov-martin-gonzalez-viruel-2026-hilbert-natural-basis-uniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martin Gonzalez and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, section 2.2.8,\nQuestion 46 (page 26), ask:\n\n> If the square of a Hilbert evolution algebra is dense, is its natural\n> orthonormal basis unique up to permutation and unimodular scalars?\n\nThe assertion quantifies over real or complex complete separable Hilbert\nspaces, bilinear products with bounded left multiplication for every fixed\nvector, and two complete natural orthonormal bases indexed by the positive\nintegers. The algebraic square is the linear span of all pair products;\ndensity means norm density. Completeness of a basis means closed linear\nspan equal to the Hilbert space. The following sentence about derivations,\nautomorphisms and spectral theory is excluded.","title":"Natural orthonormal bases of a dense-square Hilbert evolution algebra","url":"https://the-omega-institute.github.io/trureturing-pages/research/kaygorodov-martin-gonzalez-viruel-2026-hilbert-natural-basis-uniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd21720a362c8c50f1f14368bbb4f2acd34a32924b0664af27f2894fff40b61c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kaygorodov-martin-gonzalez-viruel-2026-nilpotency-spectrum-conjecture-9","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, section 2.3,\nConjecture 9 (Spectrum of nilpotency indices), state:\n\n> Let $\\Bbbk$ be an infinite field. An $n$-dimensional nilpotent evolution\n> $\\Bbbk$-algebra of nilpotency index $k+1$ exists if and only if either\n> $k=2^a$ with $0\\leq a\\leq n-1$, or $k$ has at least two non-zero binary\n> digits and $\\lfloor\\log_2 k\\rfloor+s_2(k)\\leq n-1$, where $s_2(k)$ is\n> the number of non-zero binary digits of $k$.\n\nThe whole assertion quantifies over every infinite field $K$, each integer\n$n\\geq0$ and each integer $k\\geq1$. The algebra is a vector space of\ndimension $n$ with a $K$-bilinear product and a natural basis satisfying\n$e_i e_j=0$ for distinct indices. No associativity, power-associativity,\nunit, nondegeneracy, perfectness, nonzero-square or condition on bases of\nlower-power subspaces is imposed. The source's section 1.1 defines the\nordinary lower powers by\n\n$$\n A^1=A,\\qquad A^m=\\sum_{i=1}^{m-1}A^iA^{m-i}\\quad(m\\geq2),\n$$\n\nwhere a product of subspaces is the linear span of all pair products and\nthe sum is a sum of subspaces. Thus every positive split and every\nparenthesization is included. Index $k+1$ means $A^{k+1}=0$ and $A^k\\ne0$.\nThe logarithm is its integer floor on positive $k$; all bounds, including\n$n-1$ at zero dimension, are integer bounds. Neither $k=0$ nor a zeroth\npower convention can settle this assertion. Both directions and both\narithmetic alternatives belong to this one target.","title":"The ordinary nilpotency spectrum of evolution algebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/kaygorodov-martin-gonzalez-viruel-2026-nilpotency-spectrum-conjecture-9/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0c4016c4bed23c284759e5253e909e0b7f8d466fa2fbe092d00c2bda3f6121b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kechrimparis-bae-2020-isolated-qubit-ensembles","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Spiros Kechrimparis and Joonwoo Bae, *Optimal measurement preserving qubit\nchannels*, New Journal of Physics 22, 123024, DOI 10.1088/1367-2630/abcc9b,\narXiv:2006.16366v2, Conclusions:\n\n> We expect that no isolated ensembles exist but we have not managed to establish the result in general.\n\nTheir Definition 1 calls a channel optimal measurement preserving (OMP) when\none POVM is optimal for both the original and output ensembles. Isolation means\nthat only the identity CPTP map shares any optimal POVM. Preregistration: #15684.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite qubit ensemble | Natural label count, real priors, complex two-by-two matrices in `claim` | Literal preregistered convention |\n| Probabilities of physical states | Strictly positive priors with sum one; frozen `IsDensity` | No extra hypothesis |\n| Minimum-error optimum | `IsOptimalPOVM`: frozen `IsPOVM` and score comparison with every POVM | The complete defining expression |\n| One optimum before and after the channel | `IsOMP`: an existential shared POVM | Does not require all optima to be preserved |\n| Only the identity preserves an optimum | `IsIsolated`: every frozen `IsCPTP` map satisfying `IsOMP` equals `LinearMap.id` | The source's trivial OMP set |\n| No isolated ensembles | Universal `claim` | Refuted by `result : ¬ claim` at four states |\n\nThe real part of the trace reads the real discrimination score; complete\npositivity and trace preservation are those of the frozen channel owner.\nA unique original optimum makes sharing any optimum equivalent to preserving\nthat optimum. The journal title differs from the separate JSAC paper *Channel\nCoding of a Quantum Measurement*.","title":"kechrimparis bae 2020 isolated qubit ensembles","url":"https://the-omega-institute.github.io/trureturing-pages/research/kechrimparis-bae-2020-isolated-qubit-ensembles/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c43baf8faaeb7575c6252cc186181ad9891e79794081ef1a7a3c55f42ddb6d45.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kemp-ramgoolam-degeneracy-graph-determinant-and-monomial-basis","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kemp and Ramgoolam's arXiv:2603.05259v2 defines a layered source tree with recursive threshold sets in (3.1)–(3.4). Its ordinary source columns are the disjoint union in (3.5)–(3.6). The leaf-path matrix is (6.21). The determinant conjecture is the signed sibling-only product (6.22): at the last layer (i=L), the exponent is one, while at every earlier layer (i Let the base dimension $d\\geqslant 2$ and differential order $k\\geqslant 1$ be arbitrary. Put the complete generalised Wronskian $W_d^k=\\Bone\\wedge \\partial_\\bx\\wedges \\partial^k_{\\bx...\\bx}$ as the $N$\\nobreakdash-ary bracket, $N={d+k\\choose d}$, on the $N$\\nobreakdash-ary SH-Lie polynomial algebra $\\Bbbk[x^1,...,x^d]$. Does there exist an homogeneous polynomial $p(\\bx)\\in \\Bbbk[\\bx]$ of degree $k+1$ such that the $N$\\nobreakdash-ary SH-Lie polynomial subalgebra $\\EuA=\\Bbbk_k[\\bx]\\oplus\\langle p(\\bx)\\rangle$ is \\emph{perfect} w.r.t\\ the Wronskian, i.e.\\ $\\Span_\\Bbbk W_d^k(\\EuA,...,\\EuA)=\\EuA$?\n\nThe source macros are retained in the quotation: `\\Bone` is the identity operator, `\\bx` denotes $\\boldsymbol x$, `\\wedges` denotes $\\wedge\\cdots\\wedge$, `\\EuA` denotes $\\mathcal A$, and `\\Span` denotes linear span. The source takes $\\Bbbk=\\mathbb R$ or $\\mathbb C$ and writes $\\Bbbk_k[\\boldsymbol x]$ for polynomials of degree at most $k$ (TeX lines 538–542 and 613–623).\n\nIssue [#22115](https://github.com/the-omega-institute/trureturing/issues/22115) fixes the statement over a characteristic-zero field $K$. Put\n\n$I=\\{\\alpha\\in\\mathbb N^d:|\\alpha|\\le k\\},\\qquad W(f)=\\det(\\partial^\\alpha f_\\beta)_{\\alpha,\\beta\\in I},$\n\n$A_p=K_{\\le k}[\\boldsymbol x]+Kp,\\qquad D_p=\\operatorname{span}_K\\{W(f):f:I\\to A_p\\}.$\n\nHere $\\partial^\\alpha$ is ordinary iterated partial differentiation, without divided-power normalization. The settled statement is: for every characteristic-zero field $K$, every $d\\ge2$ and every $k\\ge1$,\n\n$\\bigl(\\exists p\\in K[\\boldsymbol x],\\ p\\ne0,\\ p\\text{ homogeneous of degree }k+1,\\ D_p=A_p\\bigr)\\ \\Longleftrightarrow\\ k=1.$\n\nThis classification is proved.","title":"Perfect homogeneous Wronskian extensions exist exactly at order one","url":"https://the-omega-institute.github.io/trureturing-pages/research/kenins-kiselev-2026-wronskian-perfect-lonely-algebra/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3e1032fb19c0aaeb8b0302092d5175789e017ccdf29c972b82f8d0198436a2a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kerenidis-2026-butterfly-lambda-two-relative-design-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Iordanis Kerenidis, *Scalable Quantum Machine Learning: Trainability,\nExpressivity and Efficiency*, arXiv:2607.24014v2, Conjecture 20\n(`quantum_v2.tex`, `conj:lambda2_design`), asserts:\n\n> The unitary butterfly W_n at uniformly random parameters is an\n> ε-approximate unitary 2-design on U(n) in the antisymmetric 2-particle\n> representation Λ² ℂ^n, in the relative (multiplicative) sense, with\n> ε = O(1/n).\n\nThe displayed assertion is\n`(1−ε) Φ₂^(Λ²,Haar) ⪯ Φ₂^(Λ²,W_n) ⪯ (1+ε) Φ₂^(Λ²,Haar)`\nas completely positive maps on End(Λ²ℂ^n ⊗ Λ²ℂ^n).\nThe Library note quotes the complete conjecture, including its separate\nfixed-point-space sentence, and the literal circuit definition.\nIssue #11543 preregisters the quantified reading: there are a real c and\nnatural N₀ such that every n=2^K ≥ N₀ admits 0 ≤ ε ≤ c/n satisfying\nboth inequalities. The circuit has one pass through increasing strides,\nwith the full phase layer before each disjoint RBS layer.","title":"The single-pass butterfly fails the relative exterior-square two-design bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/kerenidis-2026-butterfly-lambda-two-relative-design-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc3458ab82b93c4a98cd98d56aaafa878bfead47c6accafd90ffe6c1063fbe7e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kharel-limbu-khatri-khanal-adhikari-2026-periodic-ssh-excess-pseudo-entropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. K. Kharel, M. Limbu, N. Khatri, A. Khanal and K. Adhikari, *Pseudo entropy and topological phases of matter*, arXiv:2606.29311v1, test the conjectured sign of the averaged excess pseudo entropy in the Su–Schrieffer–Heeger model.\n- **The quantity.** For pure states ψ, φ with ⟨φ|ψ⟩ ≠ 0, T = |ψ⟩⟨φ|/⟨φ|ψ⟩, T_A = Tr_B T and S(T_A) = −Tr(T_A log T_A) (Eqs. (1)–(3)). The averaged excess is ∆S = Re S(T_A) − (S(ρ_A^ψ) + S(ρ_A^φ))/2 (Eq. (21)).\n- **The conjecture.** \"In [25, 26], it was conjectured that the averaged excess entropy is non-positive, ∆S ≤ 0, when the states |ψ⟩ and |φ⟩ are in the same phase\" (§IV, p. 5).\n- **The assertion.** \"Under periodic boundary conditions, the conjecture is fully satisfied with no exceptions at all system sizes and across all three phases: topological, trivial, and crossing\" (§VI, p. 9), for half-filled ground states of the periodic Hamiltonian of Eq. (15) at w = 1.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/khareletal2026pseudoentropy.md).\n\nIssue [#17503](https://github.com/the-omega-institute/trureturing/issues/17503) fixes the reading:\n- The Fock space of the 2N modes (1,A), (1,B), …, (N,A), (N,B) carries the Jordan–Wigner annihilation operators in this order.\n- A half-filled ground state is a nonzero N-particle vector of minimal energy in the N-particle sector.\n- A is the first L unit cells, 1 ≤ L < N, and Tr_B is the partial trace over the remaining modes.\n- The entropy of a matrix is −Σ λ log λ over its eigenvalues with multiplicity.\n- The assertion: for all such N, L, all v₁, v₂ in the same phase, and all such ground states, ∆S ≤ 0.\n\nThe assertion is refuted in this reading.","title":"Same-phase periodic SSH ground states can have positive averaged excess pseudo entropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/kharel-limbu-khatri-khanal-adhikari-2026-periodic-ssh-excess-pseudo-entropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e5c654f6bfebe36d4acdd915ea8c64c91704806f815aabcc2bb74ed78f073670.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:khavinson-pereira-putinar-saff-shimorin-2011-nonunivalence-radius","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Khavinson, R. Pereira, M. Putinar, E. B. Saff and S. Shimorin, *Borcea's variance conjectures on the critical points of polynomials*, arXiv:1010.5167v2, §5, Theorem 5.1 and Conjecture 4. See [khavinson2011borcea](../Library/Zeros/khavinson2011borcea.md).\n\n> Theorem 5.1 (Alexander-Kakeya). If a polynomial F of degree n has no critical points in a closed disk of radius R, then it is univalent in the concentric disk of radius R sin(π/n).\n\n> If F is an nth degree polynomial with F(0) = 0, then F is not univalent in any closed disk of radius larger than σ_p(F) sin(π/n), p ≥ 1, centered at zero.\n\n> Of course, if the conjecture is true for p = 1, then it follows for all p ≥ 1. By the Alexander-Kakeya Theorem, the truth of this conjecture implies the truth of Conjecture 1 for the same value of p. It can be easily seen that the strongest case p = 1 of this conjecture is true for n = 2.\n\nFor every $n\\ge1$, every complex polynomial $F$ of degree $n$ vanishing at zero, every real $p\\ge1$ and every real $r>\\sigma_p(F)\\sin(\\pi/n)$, the assertion forbids injectivity on the closed disk centered at zero with radius $r$. Zeros are counted with algebraic multiplicity, and $\\sigma_p$ is the infimum over all complex centers of the normalized root moment. The formal definition includes every stated quantifier and imposes no simple-root or geometric restrictions.","title":"Khavinson–Pereira–Putinar–Saff–Shimorin: nonunivalence radius","url":"https://the-omega-institute.github.io/trureturing-pages/research/khavinson-pereira-putinar-saff-shimorin-2011-nonunivalence-radius/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23b66ca58338a93574f2550c0fbe5967033360c3578dae29cc620a23cd24db63.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:khesin-li-2024-bipartite-encoder-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. B. Khesin and A. M. Li, *Equivalence Classes of Quantum Error-Correcting Codes* (2024), arXiv:2406.12083v1, Section 8, Conjecture 8.3 (`main.tex`, lines 1768–1771):\n\n> In an equivalence class with bipartite forms, all bipartite forms in the class can be transformed from one to another using only output permutations, input permutations, and row operations on the input-output adjacency matrix.\n\nThe source introduces these forms as graphs with “no edges among output nodes” (`main.tex`, line 1631). Its Section 7 class construction uses the operations of Conjecture 6.3 (`main.tex`, lines 1557 and 1582–1584). The operations and their locators are recorded in [the literature note](../Library/QuantumContext/khesinli2024equivalence.md).\n\nIssue [#26139](https://github.com/the-omega-institute/trureturing/issues/26139) fixes the operation-generated reading. A graph has vertex set $\\mathrm{Fin}(k)\\sqcup\\mathrm{Fin}(n)$, with inputs on the left and outputs on the right. A bipartite form has only input–output edges: neither input–input nor output–output edges. Equivalence is `Relation.EqvGen Step`, generated by local complementation, output permutations, input permutations, binary additions of distinct input rows, and input-edge removal. Local Clifford decorations are absent.\n\nFor all natural $k,n$ and all such bipartite forms $G,H$, `claim` says that operation-generated equivalence implies `Relation.ReflTransGen BipartiteStep G H`: a finite, possibly empty, sequence of output permutations, input permutations and binary additions of distinct input rows. This is proved. There is no positivity, connectedness, full-rank or normalization hypothesis.","title":"Bipartite encoder forms in an operation-generated class differ only by permutations and row operations","url":"https://the-omega-institute.github.io/trureturing-pages/research/khesin-li-2024-bipartite-encoder-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/26c0e6c0bfadadc847125c1966241ed26009da853b82fd5bd7dbbefb3de1e634.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:khesin-li-2024-equivalence-class-sizes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Khesin and Li, *Equivalence Classes of Quantum Error-Correcting Codes*, arXiv:2406.12083v1, Conjecture 7.1, state:\n\n> For positive integers n > k, the equivalence classes of [n,k] codes with a single connected component have sizes divisible by 3^k.\n\nThe proposition concerns the labelled normalized encoders counted in Section 7. Every normalized member of the class is required to be connected, as in the figures' restriction to classes in which every graph is prime. Completeness of the operations as a characterization of physical code equivalence is a separate question, Conjecture 6.3.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive integers $n>k$ | Natural parameters in `claim`, with `0 < k` and `k < n` | Both positivity and strict inequality are retained. |\n| Encoder graph with $k$ inputs and $n$ outputs | `SimpleGraph (Fin k ⊕ Fin n)` | Inputs and outputs have distinct labels. |\n| Operations 1–5 of Conjecture 6.3 | `Step` | Local complementation, separate input/output permutations, modulo-two row addition and input-edge removal are explicit constructors. |\n| Removed output Clifford decorations | Decoration-free carrier | Operation 6 has no graph action; this is the source's normalized convention. |\n| Claim 6.4 normalized adjacency matrices | `Normalized` | No input-input or pivot-pivot edges; the first $k$ output columns form the identity block. |\n| One operation followed by renormalization | `NormalizedStep`, using `Step` and `Relation.ReflTransGen RenormStep` | Normalization is required at both endpoints; an empty renormalization sequence is allowed. |\n| Equivalence classes of distinct normalized encoders | `Relation.EqvGen NormalizedStep` and `classSize` | The subtype is counted by `Nat.card`; no quotient by graph isomorphism is taken. |\n| A single connected component | The hypothesis that every normalized class member is `Connected` | This is the preregistered interpretation of the source's connected-class restriction. |\n| Size divisible by $3^k$ | `result` proves `claim` | No additional conclusion is assumed. |\n\nThe source's XOR expression for local complementation is written literally in `Step.complement`. The consumed private correspondence `lc_source` identifies it with the existing graph operation `GraphStateMonogamyForbiddenSubgraph.lc` used in the proof.","title":"Divisibility of connected encoder equivalence classes","url":"https://the-omega-institute.github.io/trureturing-pages/research/khesin-li-2024-equivalence-class-sizes/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ba78c4fd122cc99cc5de9d64aa0d3067fe155d5e20e47fdc12694d00d9dd3e3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kiem-kye-2026-local-distinguishability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Young-Hoon Kiem and Seung-Hyeok Kye, *Local geometry for Schmidt number witnesses*, arXiv:2608.14199v1, Section 5, states:\n\n> Nevertheless, we conjecture that the inclusion BP_ℓ ⊋ BP_{ℓ+1} is locally distinguishable around ϱ_{E⊥} for ℓ ≥ k, whenever ϱ_{E⊥} is on the boundary of the convex set BP_k.\n\nFor a nonzero proper complex subspace $E\\subset\\mathbb C^m\\otimes\\mathbb C^n$, write $\\rho=P_{E^\\perp}/\\dim E^\\perp$. The result says that if $\\rho\\in\\partial_H BP_k$, then every neighbourhood of $\\rho$ meets $BP_\\ell\\setminus BP_{\\ell+1}$ whenever $1\\le k\\le\\ell<\\min(m,n)$. Here $H$ is the trace-one Hermitian affine space, and $BP_j$ tests nonnegativity against all vectors of Schmidt rank at most $j$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Trace-one Hermitian affine space $H$ | `ambient m n` | Both Hermiticity and complex trace one. |\n| $j$-blockpositivity | `BP m n j`; imported `schmidtRank` | The real quadratic form is nonnegative on every vector whose coefficient matrix has rank at most $j$. |\n| Normalized projection onto $E^\\perp$ | `rhoPerp E` | `Submodule.starProjection` converted by `Matrix.toEuclideanLin.symm` and divided by the complex dimension. |\n| Nonzero proper subspace | `E ≠ ⊥`, `E ≠ ⊤` in `claim` | The entire source range is retained. |\n| Boundary in $H$ | `OnBoundary` | Membership in the closed set $BP_k$ and in the closure of $H\\setminus BP_k$. |\n| Adjacent inclusions in the finite chain | `1 ≤ k`, `k ≤ ℓ`, `ℓ < min m n` | The range in Sections 3 and 5; it implies $m,n\\ge2$. |\n| Every neighbourhood meets the difference | `LocallyDistinguishable` | Ambient and relative neighbourhoods give the same test because the difference lies in $H$. |\n| The conjecture for every subspace | `result : claim` | No genericity hypothesis. |\n\nAll public definition bodies are the literal conventions of [#17787](https://github.com/the-omega-institute/trureturing/issues/17787), with the matrix abbreviation expanded. The imported Schmidt rank is the rank of the coefficient matrix.","title":"Local distinguishability at every boundary projection state","url":"https://the-omega-institute.github.io/trureturing-pages/research/kiem-kye-2026-local-distinguishability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3d219461e80be72044e29b3ab68297cb29812ee3401b4a98fa431abb5f24421.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kimb341249-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Clark Kimberling's OEIS A341249, NAME and COMMENTS, revision 10, states:\n\n> a(n) = floor(r*floor(s*n)), where r = 2 + sqrt(2) and s = sqrt(2).\n\n> Conjecture: 1 < r*s*n - a(n) < 2 for n >= 1.\n\nThis is the tier-1 external conjecture preregistered in\nhttps://github.com/the-omega-institute/trureturing/issues/18673.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| r = 2 + sqrt(2) | r | Real constant $2+\\sqrt 2$. |\n| s = sqrt(2) | s | Real constant $\\sqrt 2$. |\n| a(n) = floor(r*floor(s*n)) | a | Natural index; both floors integer valued; products real. |\n| for n >= 1 | claim | All natural $n$ satisfying $1\\le n$. The formula-defined value at zero imposes no extra condition. |\n| 1 < r*s*n - a(n) < 2 | claim | Conjunction of both strict inequalities, with integer values cast to real. |\n| Refutation | result | Negation of that complete universal conjunction. |\n\nThe namespace is `D5.S0.Certificates.KimberlingA341249BoundRefutation`.","title":"Kimberling's A341249 strict bound is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/kimb341249-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2e6ff115a9cb849afee7a6b95df86bd4d4611167f48787dce1a64370612bc50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kimberling-array-first-occurrence-l4v5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alex Ashburn, arXiv:2610.05776v1, Section 5, Conjecture 15:\n\n> For every v ≥ 2, the least N with a(N) = v is L_{4v−5}.\n\nPut φ = (1 + √5)/2 and R_i = {⌊kφ^i⌋ : k ≥ 1}, for positive natural\nrow indices i. The occurrence count a(N) is the number of rows containing\nN. The Lucas sequence has L₀ = 2, L₁ = 1 and L_{n+2} = L_{n+1} + L_n.\n\nThe Lean `claim` says, for every natural v ≥ 2, both a(L_{4v−5}) = v and\nthat every positive N < L_{4v−5} has a(N) ≠ v. The count uses the finite\nindex interval 1 ≤ i ≤ N; the proved row-index bound shows that this loses\nno containing row.","title":"First occurrences in Kimberling's array","url":"https://the-omega-institute.github.io/trureturing-pages/research/kimberling-array-first-occurrence-l4v5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a7f58a39605c7498eb278bf04ec736c5b57809218fced98a94e1389e8d52f23.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kimoto-wakayama-2019-even-normalized-apery-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Kimoto and M. Wakayama define normalized Apéry-like rational numbers $\\widetilde J_k(n)$ by a three-term inhomogeneous recurrence in Section 3.4 of *Apéry-like numbers for non-commutative harmonic oscillators and automorphic integrals*. The source TeX `Eichler20191217.tex`, lines 1608–1617, states the even part of Conjecture 3.6 as follows (the source macro `\\tJ` is rendered as $\\widetilde J$):\n\n> For positive integers $m,n,s$ such that $mp\\ge s$ and $mp^{n-1}\\ge s$, we have\n> $$\\frac{p^{2sn}\\widetilde J_{2s+2}(mp^n)}{p^{2s}\\widetilde J_{2s+2}(mp)}\\equiv\\frac{p^{2s(n-1)}\\widetilde J_{2s+2}(mp^{n-1})}{p^{2s}\\widetilde J_{2s+2}(mp)}\\ (\\not\\equiv0)\\pmod{p^n}.$$\n\nThe compiled proposition `claim` is the odd-prime restriction of this assertion. It quantifies over natural $p$ with `p.Prime` and $p\\ne2$, then over positive natural $m,n,s$ satisfying $s\\le mp$ and $s\\le mp^{n-1}$, and requires both the displayed rational congruence and noncongruence of the right ratio to zero. The declaration `result` has type $\\neg\\,\\texttt{claim}$.","title":"The even Kimoto–Wakayama normalized Apéry congruence is refuted","url":"https://the-omega-institute.github.io/trureturing-pages/research/kimoto-wakayama-2019-even-normalized-apery-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f4fe185870624806564069fdcb3ecc0bb58db67521251e4a41aa6f0dff30526.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kirkland-2019-interior-pst-converse-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Kirkland, D. McLaren, R. Pereira, S. Plosker and X. Zhang, *Perfect quantum state transfer in weighted paths with potentials (loops) using orthogonal polynomials*, arXiv:1708.03283v2 (Linear and Multilinear Algebra 67, 2019), study perfect state transfer (PST) on weighted paths with real edge weights and real diagonal potentials. Their Proposition `prop:internal` states (source TeX `main.tex`, lines 309–312):\n\n> For a weighted path with or without potentials, PST between vertices $1$ and $n$ implies PST between vertices $j$ and $n+1-j$, for each $j=2, \\ldots, n-1$. If for some $j$ with $2 \\le j \\le n-1$ there is PST between vertices $j$ and $n+1-j$, and if in addition none of the eigenvectors of the Hamiltonian has a zero entry in the $j$--th position, then the converse holds.\n\nThe question immediately following it is (source TeX `main.tex`, line 315):\n\n> It is an open question if the converse holds even if there are eigenvectors with zeros in the jth place.\n\nThe formal reading settled here uses the repository's vertices $0,\\ldots,m$. For every $m : \\mathbb N$ with $3 \\le m$, every $r : \\mathrm{Fin}\\,m \\to \\mathbb R$ with $0 < r(k)$ for every $k$, every $q : \\mathrm{Fin}(m+1) \\to \\mathbb R$, and every $j : \\mathrm{Fin}(m+1)$ satisfying\n$1 \\le j.val$, $j.val \\le m-1$, and $j.val \\ne m-j.val$, it asserts\n\n$$\n\\left(\\exists t : \\mathbb R,\\ 0 Another open question is whether the hypotheses of Theorem 1.1 are in fact necessary conditions for generalized action graphs to exist.\n\nThis dossier anchors exactly that question, with the strict positivity in\nTheorem 1.1. The theorem assumes positive integers $s_n$ for every $n\\ge0$\nand $z_i$ for every $i\\ge1$, with $s_0=1$ and\n\n$$\ns_n=\\sum_{i=1}^{n}z_i s_{n-i}\\qquad(n\\ge1).\n$$\n\nThe necessity assertion quantifies every positive integer sequence and the\nexistence of one entire family of generalized action graphs, and concludes\nthe normalization and existence of coefficients satisfying all these\nconditions. The value $z_0$ is irrelevant and unrestricted. The earlier\narXiv:2507.22861v1, Section 6, is an alias of this question: its Definition\n4.1 and Theorem 2.1 correspond to journal Definition 1.1 and Theorem 1.1.\nIt is not a second target. Preregistration is\n[#23113](https://github.com/the-omega-institute/trureturing/issues/23113).\n\nDefinition 1.1, pages 158 and 172, gives the full source model:\n\n> The sequence Gn of generalized action graphs for a particular sequence sn of positive integers is a sequence of directed, labeled, rooted trees such that:\n> Axiom 1 Define G0 as the graph with s0 vertices labeled 0 and no edges. Construct Gn from Gn−1 by adding sn new vertices, which are each labeled n.\n> Axiom 2 For vertex v in Gn, the subtree of Gn with root v is isomorphic to some Gk such that k ≤ n.\n> Axiom 3 All leaves in the graph Gn have label n.\n\nThe paragraph following the repeated definition on page 172 specifies:\n\n> Note that subtree of G with root v here means the subgraph of G consisting of all descendants of v and edges between them in G.\n\nSection 3.1, page 172, specifies the label convention:\n\n> Note that when we consider isomorphism in generalized action graphs, the labels on the vertices may be shifted, but the underlying structure of the graphs are the same.","title":"Strictly positive coefficients are not necessary for generalized action graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/klanderman-2026-positive-action-graph-necessity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3add19f20a68ebc8beddf1b84eafc89f93b9dc8b0bc7f8233870c79ea15f85ed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:klausen-2025-lattice-lindbladian-zero-spectrum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Frederik Ravn Klausen, *Spectra of generators of Markovian evolution in the thermodynamic limit: From non-Hermitian to full evolution via tridiagonal Laurent matrices*, arXiv:2206.09879v3 (2025), asks in “Further questions” (`section04.tex`, line 574; verbatim TeX):\n\n> Prove that if $\\Li$ is \\eqref{eq:finite-range}, \\eqref{eq:translation-invariance}, and \\eqref{eq:Rank1} then $0 \\in \\sigma(\\Li)$, thereby strengthening \\Cref{thm:no_gap}.\n\nHere the source macro `\\Li` denotes $\\mathcal L$, the three assumption labels mean (Finite Range), (TransInv), and (Rank 1), and the theorem label is `thm:no_gap`. The source assumes bounded self-adjoint $H$ and $G>0$; (Rank 1) means $\\operatorname{rank}(L_k)=1$ for every $k\\in\\mathbb Z$ (`section02.tex`, lines 12 and 50). It takes $\\mathcal L$ on $\\mathrm{HS}(\\ell^2(\\mathbb Z))$ (`section02.tex`, lines 6–13 and 54–57). Its theorem (`section03.tex`, lines 506–509) reads:\n\n> If $\\Li$ is \\eqref{eq:finite-range}, \\eqref{eq:translation-invariance}, \\eqref{eq:Rank1} and $0 \\in \\sigma(\\Li)$.\n> Then $\\Li$ is gapless or $\\dim(\\ker(\\Li))=\\infty.$\n\nIssue [#23482](https://github.com/the-omega-institute/trureturing/issues/23482) fixes the matrix-kernel reading. Put $E=\\ell^2(\\mathbb Z\\times\\mathbb Z;\\mathbb C)$, the module's `ObserverHilbertSpace (ℤ × ℤ)`. For every $r\\in\\mathbb N$, every $g\\in\\mathbb R$, every $h:\\mathbb Z^2\\to\\mathbb C$, and every $\\ell:\\mathbb Z^3\\to\\mathbb C$, assume:\n\n- `IsFiniteRange r h ℓ`: $h(x,y)=0$ whenever $|x-y|>r$, and $\\ell(k,x,y)=0$ whenever $\\max\\{|x-k|,|y-k|\\}>r$;\n- `IsTranslationInvariant h ℓ`: $h(x+1,y+1)=h(x,y)$ and $\\ell(k+1,x+1,y+1)=\\ell(k,x,y)$ for all integer indices.\n\nThe conclusion of `result` is that there exists a continuous complex linear operator $T:E\\to E$ such that, for every $\\rho\\in E$ and every $x,y\\in\\mathbb Z$, $(T\\rho)(x,y)=\\operatorname{lindbladEntry}(g,h,\\ell,\\rho,x,y)$, and $0\\in\\sigma_{\\mathbb C}(T)$. Explicitly, the entry is\n\n$$\n\\begin{aligned}\n&-i\\left(\\sum_z h(x,z)\\rho(z,y)-\\sum_z\\rho(x,z)h(z,y)\\right)\\\\\n&\\quad+g\\sum_k\\left[\n\\sum_u\\sum_v\\ell(k,x,u)\\rho(u,v)\\overline{\\ell(k,y,v)}\n-\\frac12\\sum_z\\left(\\sum_w\\overline{\\ell(k,w,x)}\\ell(k,w,z)\\right)\\rho(z,y)\n-\\frac12\\sum_z\\rho(x,z)\\left(\\sum_w\\overline{\\ell(k,w,z)}\\ell(k,w,y)\\right)\n\\right].\n\\end{aligned}\n$$\n\nAll indices in these sums range over $\\mathbb Z$; the Lean definition uses unconditional sums, whose nonzero terms are finite under `IsFiniteRange`. The theorem imposes no rank condition on the jumps, no self-adjointness condition on $h$, and no sign condition on $g$. Its specialization to $g>0$ and self-adjoint $h$ answers the registered question, including its rank-one case.","title":"Zero belongs to the spectrum of a finite-range translation-invariant lattice Lindbladian","url":"https://the-omega-institute.github.io/trureturing-pages/research/klausen-2025-lattice-lindbladian-zero-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8997689e30e37b844b43fc86cf20d69b3564d78428a7606db1165415c433c274.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:klavzar-et-al-2025-mobile-general-position-cylinders","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Klavžar, Krishnakumar, Kuziak, Shallcross, Tuite and Yero, “Moving through\nCartesian products, coronas and joins in general position”, arXiv:2505.00535v2,\nDiscrete Applied Mathematics 379 (2026), 768–780,\nDOI 10.1016/j.dam.2025.10.041, Section 3, after Proposition 3.7:\n\n> Hence, the only unknown values are Mobgp(C₇ □ Pₛ) and Mobgp(C₁₀ □ Pₛ) for s ≥ 5. We conjecture that the answer is four in these cases.\n\nThe conclusion is $\\operatorname{Mob}_{gp}(C_7\\square P_s)=\n\\operatorname{Mob}_{gp}(C_{10}\\square P_s)=4$ for every integer $s\\ge5$.\nIt is the tier-1 published conjecture preregistered in\n[issue #17705](https://github.com/the-omega-institute/trureturing/issues/17705).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Shortest-path membership | `OnGeodesic` | Sum of the two distances equals the endpoint distance, on the connected cylinders. |\n| No three occupied vertices on a common shortest path | `IsGeneralPosition` | All triples of pairwise distinct occupied vertices are quantified. |\n| One move along an edge to an unoccupied vertex, preserving general position | `LegalMove` | Exactly one occupied vertex is erased and the new vertex inserted. |\n| A sequence of legal moves from the initial set visits every vertex | `IsMobile` | A finite sequence with the stated initial configuration, every consecutive move legal, and a visit to every vertex. |\n| Largest cardinality of a mobile general position set | `mobGP` | Natural-number supremum; the upper bound and a size-four witness establish attainment here. |\n| Cartesian product of cycle and path | `cylinder` | Mathlib `cycleGraph r` and `pathGraph s`, on `Fin r × Fin s`; path height counts vertices. |\n| Both exceptional circumferences, all $s\\ge5$, answer four | `claim`, `result` | Both equalities, the natural-height quantifier and its guard occur literally. |\n\nZero-based labels are a relabelling of the source vertices. The pairwise\ninequalities in `IsGeneralPosition` express three distinct vertices; the\nsequence length and configuration indices express the source's finite moves.\nThey impose no extra hypothesis on the cylinder claim.","title":"Mobile general position on the seven-cycle and ten-cycle cylinders","url":"https://the-omega-institute.github.io/trureturing-pages/research/klavzar-et-al-2025-mobile-general-position-cylinders/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/499919675b10e930a84da43bf8a12e8d769bb8f65546547053882168f44cd94e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:klimov-2023-w-state-ti-mps-bond-dimension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Klimov, R. Sengupta and J. Biamonte, *On Translation-Invariant Matrix\nProduct States and advances in MPS representations of the $W$-state*,\narXiv:2306.16456v2, Section 3, conjecture after their Theorem 1:\n\n> These experiments lead us to conjecture that the representation obtained in Theorem \\ref{W state representation} could be optimal both in terms of asymptotics and the constant factor or in other words it is impossible to come up with a TI MPS representation with PBC with bond dimension smaller than $\\floor*{\\frac{n}{2}}+1$ for the $W$-state of order $n$.\n\nA TI MPS representation with PBC of bond dimension $d$ of the normalized\n$W$-state of order $n$ is a pair of complex $d\\times d$ matrices $A_0$, $A_1$\nwith $\\operatorname{Tr}(A_{i_1}\\cdots A_{i_n})=1/\\sqrt n$ when exactly one\n$i_j$ is $1$ and $0$ otherwise (Section 2). Problem 4 of Section 4 asks\nwhether $d(n)=\\lfloor n/2\\rfloor+1$, and the table of Section 4 lists\n$d(7)=4$.","title":"The minimal translation-invariant MPS bond dimension of the W-state","url":"https://the-omega-institute.github.io/trureturing-pages/research/klimov-2023-w-state-ti-mps-bond-dimension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1d0619e63fbcce800e735f0951618c127a2e0dc25701da8318f9516748b0ac3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:klobus-2016-chained-monogamy-realizability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"W. Kłobus, M. Oszmaniec, R. Augusiak, A. Grudka, *Communication strength\nof correlations violating monogamy relations*, Found. Phys. 46 (2016)\n620–634, arXiv:1408.1223v2, Section 5:\n\n> Although we cannot prove it as in the case $M=2$, we conjecture that all\n> possible values of the correlators in $S_{A\\to BE}^i$ and\n> $S_{B\\to AE}^i$ that satisfy inequalities (\\ref{ElPrat}) can always be\n> realized with some signaling probability distribution $\\vec{p}$ for which\n> $R_M(\\vec{p})=2M+\\Delta$.\n\nA box is a family $p(a,b,e|A_i,B_j)$ of distributions on $a,b,e=\\pm1$,\n$i,j=0,\\dots,M-1$, with one setting $E$ for the third party, all one- and\nthree-party expectation values zero, and a common value of\n$\\langle B_0E\\rangle_{A_i}$. $R_M=I^M_{AB}+2\\langle B_0E\\rangle$ with the\nchained Bell expression $I^M_{AB}=\\sum_k(\\langle A_kB_k\\rangle+\\langle\nA_{k+1}B_k\\rangle)$, $A_M=-A_0$, and $\\Delta\\in[0,2]$. The coordinates are\n$x_A^i=\\langle B_iE\\rangle_{A_i}$, $y_A^i=\\langle B_iE\\rangle_{A_{i+1}}$,\n$x_B^i=\\langle A_iE\\rangle_{B_{i-1}}$, $y_B^i=\\langle A_iE\\rangle_{B_i}$\n($1\\le i\\le M-1$), $x_B^0=\\langle A_0E\\rangle_{B_0}$,\n$y_B^0=\\langle A_0E\\rangle_{B_{M-1}}$, and (ElPrat) is\n$$\\sum_{i=1}^{M-1}(-1)^{a_i}(x_A^i-y_B^i)+\\sum_{i=1}^{M-2}(-1)^{b_i}\n(x_B^{i+1}-y_A^i)+(-1)^c(y_A^{M-1}+y_B^0)+x_B^1+x_B^0\\ge\\Delta$$\nfor all $a_i,b_i,c\\in\\{0,1\\}$.","title":"Realizability of the correlators of boxes violating chained monogamy","url":"https://the-omega-institute.github.io/trureturing-pages/research/klobus-2016-chained-monogamy-realizability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d8fea2accb323f298a0d87d7ff85c36cb0ab1134c743dcb537fd84d550cca68.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:klostermeyer-self-conjugate-even-polynomial-conjecture-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The sole target is Section 2, Conjecture 2 of the versioned source identified\nby the literature note, PDF page 4 immediately after Theorem 9:\n\n> Let p(x) be an even polynomial over GF(2). Then p(x) is self-conjugate if and only if p(x)=x^{2^{k_1}}+x^{2^{k_2}}+...+x^{2^{k_m}}, where m is odd if x^0 is a term and m is even otherwise.\n\nThe source defines self-conjugacy by f(x)=f(x+1), as equality of formal\npolynomials. For each of the six explicit conventional readings e of even,\nlet Domain(e,p) mean, respectively: every supported exponent is even; the\nnatural degree is even; the support cardinality is even; the constant\ncoefficient is zero; X divides p; or p(-X)=p(X). For each such e the complete\npublished assertion is\n\\[\n \\forall p\\in\\mathbb F_2[X],\\quad\n \\operatorname{Domain}(e,p)\\Longrightarrow\n \\left[p(X+1)=p(X)\\ \\Longleftrightarrow\\\n \\exists m\\in\\mathbb N,\\ k:\\operatorname{Fin}(m)\\to\\mathbb N,\n \\ p=\\sum_{i\\in\\operatorname{Fin}(m)}X^{2^{k(i)}}\n \\ \\land\\ \\operatorname{SourceParity}(p,m)\\right],\n\\]\nwhere SourceParity(p,m) is the conjunction of the implications from nonzero\nconstant coefficient to odd m and from zero constant coefficient to even m.\nThe indices may repeat; m=0 and k(i)=0 remain allowed. The optional-constant\nvariant separately adds an arbitrary c in the same field as C(c) to the sum.\nIt is robustness of the witnesses and does not repair the literal display.\n\nThe Lean claim is the existential disjunction, over these six readings, of\nthe complete published iff or its optional-constant counterpart. Thus its\nliteral negation refutes every complete published iff and every separately\nlabelled robustness counterpart. There is no auxiliary family condition in\nthe claim whose failure could stand in for a source refutation.","title":"Self-conjugate even polynomial Conjecture 2","url":"https://the-omega-institute.github.io/trureturing-pages/research/klostermeyer-self-conjugate-even-polynomial-conjecture-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da2045b612559a3f4652969646889f34adf364d4ed7ca18f5adade43db582dfa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kmv27-evobalance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, Question 27,\npage 24, ask:\n\n> Is the balance independent of the natural basis for arbitrary (possibly degenerate) evolution algebras?\n\nThe source's section 1.1 states: “Unless stated otherwise, algebras are\nfinite-dimensional.” The assertion concerns every field, bilinear evolution\nmultiplication and pair of natural bases. It adds no nondegeneracy,\nperfectness, characteristic or connectedness assumption. The other two\nquestions about a complete system of graph invariants are outside this\nsettlement. Preregistration: [#21625](https://github.com/the-omega-institute/trureturing/issues/21625).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A field and a finite-dimensional algebra | `claim` quantifies `k`, `V`, field, additive-group, module and finite-dimensional instances, and bilinear `μ` | No associativity or unit assumption |\n| Evolution algebra | Existing `DerivationAnnihilatorObstruction.IsEvolutionAlgebra` | The arbitrary-field natural-basis predicate is reused |\n| Every two natural bases | `ι`, `κ`, their `Fintype` instances, `B`, `F`, and both zero mixed-product laws in `claim` | Different index types are allowed |\n| Row square coefficient gives an arrow | `associatedGraph` | Basis coordinates of the square, with the source's row order |\n| A path retains actual arrows traversed forwards or backwards | `Path.vertex`, `Path.edge`, `Path.isEdge`, `Path.traverses` | Opposing arrows remain distinct choices |\n| A cycle has equal first and last vertices | `IsCycle` | All finite lengths, including zero |\n| Forward cardinality minus backward cardinality | `pathBalance` | Both counts are cast to integers; loops enter both sets |\n| Gcd of all absolute cycle balances | `graphBalance` | Absolute integer ideal generator; no length cutoff |\n| The balances agree | Equality in `claim`; `result` proves its negation | The rational four-dimensional instance refutes the universal assertion |\n\nThe graph definitions also accept infinite vertex types and the height\nlemma arbitrary moduli; these stronger helper domains do not restrict the\nfinite-basis assertion. Closing universes at `Type` supplies a closed\nproposition and includes the rational counterexample. The zero gcd is zero.","title":"Natural-basis dependence of balance in degenerate evolution algebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/kmv27-evobalance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bdf77e96c11fcc86c9171c17475ead4c96fa8f7072f741691d1a2dbd6b134133.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kmvc4-simpleaut","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of\nresults and a list of open problems*, arXiv:2610.22248v1, Conjecture 4,\nsection 2.3, page 28:\n\n> Over ℂ, the maximal order of the automorphism group of a simple\n> n-dimensional evolution algebra is max{n!, n(2ⁿ−1)}.\n\nThe dimension is positive. The source's natural-basis condition is zero\nmixed products. The product is complex bilinear and commutative, without\nassociativity or a multiplicative unit. Automorphisms are the entire group\nof complex linear multiplication-preserving equivalences.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Every positive dimension | `claim` quantifies over $n\\in\\mathbb N$ with $1\\le n$ |\n| Every complex vector space of dimension $n$ | Arbitrary carrier, complex module, finite-dimensional instance and finrank equality |\n| Evolution product and natural basis | Bilinear product, arbitrary finite basis, zero mixed products and commutativity |\n| Simple algebra | `IsSimple`: $A^2\\ne0$ and every complex-linear two-sided ideal is zero or the whole space |\n| Full finite automorphism group and upper bound | `Finite (Aut μ)` together with the bound on `Nat.card` |\n| Maximum is attained | A product on $\\mathbb C^n$ with the dimension, commutativity, natural basis, simplicity, finiteness and exact cardinality |\n\nThe simplicity definition is literal: “We recall that an algebra A is\nsimple if A² ≠ 0 and 0 is the only proper ideal.” This sentence is from\nthe survey's cited arXiv:1601.01832v2, section 4, page 18. All two-sided\ncomplex-linear ideals are quantified, including those not spanned by\nnatural-basis vectors. The coordinate representative in the attainment\nclause does not restrict the universal clause. `Finite` prevents the\nnatural cardinality of an infinite group from satisfying a bound through\nits value zero. The extra dimension equality in the attaining clause\nrecords that the chosen representative has the required dimension.","title":"The simple evolution-algebra automorphism maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/kmvc4-simpleaut/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66a4507bc23c8dc5f43e51cb6128c7bcc737dfe95c4280d2555bffd1a104cc68.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kmvq31","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kaygorodov, Martín González and Viruel, *Evolution algebras: a survey of results and a list of open problems*, arXiv:2610.22248v1, section 2.2.5, page 24, ask: “Let G be a linear algebraic group over 𝕜. Is there a finite-dimensional evolution 𝕜-algebra A with Aut(A) ≅ G(𝕜) as abstract groups? with Aut(A) ≅ G as algebraic groups? with 𝐀𝐮𝐭(A) ≅ G as group schemes?”\n\nThe second and third interrogatives have a negative answer. The counterexample is $G=\\mathrm{SL}_3$ over $\\mathbb C$. The public `claim` is the affirmative universal statement for polynomial-closed matrix groups and full natural automorphism functors over every commutative unital complex coefficient algebra. It includes coefficient algebras with nilpotents. The abstract-group interrogative is outside this assertion.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Linear algebraic group $G$ over a field. | `LinearAlgebraicGroup` with a matrix degree, polynomial equations, points, zero-locus characterization and change-of-algebra stability. | The field is specialized to $\\mathbb C$, sufficient to refute the universal question. |\n| Finite-dimensional evolution algebra $A$. | Existential $d\\in\\mathbb N$ and bilinear product on $\\mathrm{Fin}(d)\\to\\mathbb C$, satisfying `IsEvolutionAlgebra`. | An arbitrary natural basis is existential; zero dimension and degenerate algebras are included. |\n| Full automorphism group or group scheme. | `automorphisms` contains all invertible linear maps preserving multiplication; `Realizes` gives multiplicative equivalences at every coefficient algebra and naturality for every algebra homomorphism. | No identity or associativity is imposed on multiplication. |\n| Algebraic groups and group schemes. | The literal functor formulation in `claim`. | The comparison uses characteristic-zero smoothness; no theorem about arbitrary abstract point groups is asserted. |\n| Is there such an $A$ for every $G$? | `result` proves the negation of `claim`. | The witness group is `slGroup 3`. |","title":"Kaygorodov–Martín González–Viruel Question 31","url":"https://the-omega-institute.github.io/trureturing-pages/research/kmvq31/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f1a310b8a709dd521315f58719db81c71253a12f45f3e459e7f3aab8d9021db.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:knight-dialler-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A327692 is defined by\n\n> Number of length-n phone numbers that can be dialed by a chess knight on a 0-9\n> keypad that starts on any number and takes n-1 steps\n\nand carries, at revision #41 of Apr 22 2024, a formula line still labelled a\nconjecture:\n\n> Conjectures from _Colin Barker_, Oct 01 2019: (Start)\n> G.f.: 2*x*(5 + 10*x - 7*x^2 - 8*x^3 + 2*x^4) / (1 - 6*x^2 + 4*x^4).\n> a(n) = 6*a(n-2) - 4*a(n-4) for n>6. (End)\n\nA later comment by Francesca Arici, dated Apr 17 2024, adds that the recurrence\nalso holds at `n = 6` and sketches a route through the eigenvalues of the\nadjacency matrix, ending at \"reduces to checking an algebraic condition on the\nnonzero eigenvalues\". The condition is not checked there and the formula line\nwas not reclassified.","title":"The Knight-Dialler Recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/knight-dialler-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a18e7ec515718266cfa7b7e70b9873ac4f7c530427c8bd2b11ca224419256a63.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koch-narayan-2025-bipartite-unique-domination-edge-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Koch and Narayan, *Maximal bipartite graphs with a unique minimum dominating set*,\narXiv:2511.01719v1, Conjecture 1, assert |E(G)| <= m(n,gamma) for every finite\nsimple bipartite graph without isolated vertices, with a unique minimum\ndominating set, gamma >= 2 and n >= 3gamma. With a=ceil(gamma/2),\nb=floor(gamma/2), the exact printed expression is\n\n Phi = max(0,n-3gamma-2a-b+1),\n m = 2gamma+2ab+min(n-3gamma,2a-b+1)(2a+1)\n + sum(i=1,...,Phi) ((2a+1)+ceil(i/2)).","title":"The Koch-Narayan bipartite edge bound is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/koch-narayan-2025-bipartite-unique-domination-edge-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59eeb45d57fd6155b9bda4204e790bd9d743f43f5176aae8ebdb808a78d0d37c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kok-jaco-addition-law-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kok, Fisher, Wilkens, Mabula and Mukungunugwa,\n*Characteristics of Finite Jaco Graphs, J_n(1), n in N*,\narXiv:1404.0484v1, printed page 7, Theorem 2.10 (Conjectured), states:\n\n> For the Jaco Graphs J_n(1), J_m(1) with n>=3, m>=3, n!=m we have\n> Delta(J_(n+m)(1))=Delta(J_n(1))+Delta(J_m(1)), if J_n(1) or J_m(1)\n> has a unique Jaconian vertex; Delta(J_n(1))+Delta(J_m(1))+1 otherwise.\n\nThis quotation normalizes only mathematical typography. Printed page 11\nexplicitly asks: “Prove Theorem (conjectured) 2.10.”\nDefinitions 1.1--1.4 and 1.7, printed page 2, give positive vertices,\nthe recursion i For any finite linear Jaco graph Jn (x), n ≥ 2 there exists a pair of\n> vertices i.e. v1 , vn for which a minimal (v1 , vn )-path (not necessarily a\n> diam-path) i.e. Pd (Jn (x)) exists such that a γ-set of Pd (Jn (x)) is a\n> γ-set of Jn (x). We call the path Pd (Jn (x)) the primary minimal dom-path.\n\nand states Conjecture 2.9 on printed page 7, verbatim:\n\n> For any linear Jaco graph Jn (x), n ≥ 1 the length of a diam-path and a\n> primary minimal dom-path Pd satisfy |Pd | − |diam(Jn (x))| ≤ 1.\n\nThe paper measures path length in edges. The formal target is the weakest\nselection-independent consequence: for every positive `n`, some dom-path has\nedge length at most the Mathlib diameter of `J_n(x)` plus one. Refuting this\nexistential consequence refutes every possible choice of primary minimal\ndom-path satisfying the source's defining property.","title":"Refutation of Kok's dom-path diameter conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-dom-path-diameter-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ffb3a27d9fa579d751fa5e2c9d9da89591b1f7cd834eddcc19ad128b5e0fbbdc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kok-jaco-exponential-domination-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kok, arXiv:2507.16500v1, section 2.3, Conjecture 2.12, printed page 9,\nstates these two clauses verbatim:\n\n> The vertex subscripts of a γ-set X = ”{v1 }” ∪ {v2 , v7 , v20 , v54 , . . . } of the infinite linear Jaco graph J∞ (x) is given by the sequence A000149: a(t) = ⌊e^t⌋, t = 0, 1, 2, . . . where e ≈ 2.71828 is the Euler number (or Napier’s constant).\n\n> Furthermore, it implies that sequence A000149 is p-graphical where p(G) = {i : j the subscript of vj ∈ X with X some γ-set of G} and F = {G : G = Jn(x), n = 1, 2, 3, . . . }.\n\nThe formal target is only the domination clause implied by the first quotation.\nA gamma-set is a minimum dominating set, so a failure of domination refutes\nthe gamma-set assertion. Gamma-set minimality and the separate p-graphical\nclause are not encoded.","title":"Refutation of Kok's A000149 domination clause","url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ab2be56abc5e36832d2a208a2f2f4a299b6a9d12c07654a90064a6ed339a978.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kok-jaco-unique-maximum-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johan Kok, *Integer sequences with conjectured relation with certain graph\nparameters of the family of linear Jaco graphs*, arXiv:2507.16500v1,\nSection 2.2, Conjecture 2.3, printed page 5\n([primary locator](https://arxiv.org/html/2507.16500v1#S2.Thmtheorem3)), states:\n\n> For $t = 1,2,3,\\dots$ the values given by sequence A003622 i.e. the Wythoff compound sequence AA: $a(t) = \\lfloor t \\times (\\frac{1+\\sqrt{5}}{2})^2\\rfloor -1$ are the unique maximum degrees of corresponding linear Jaco graphs.\n>\n> Furthermore, it implies that sequence A003622 is $p$-graphical where $p(G) = \\Delta(G)_{unique}$ and $\\mathcal{F} \\in \\{G:G=J_n(x), n = 1,2,3,\\dots\\}$.\n\nThe preceding paragraph says the maximum-degree values that appear once\nacross graph orders are intended and explicitly excludes $\\Delta(J_1(x))=0$.\nThis is numerical value uniqueness across orders, not a singleton set of\nmaximizing vertices within one graph; Conjecture 2.5 concerns the latter.\n\nRetain the original recursive graph with $f(x)=x$. For every positive $i$,\nlet $d_i=\\#\\{j:1\\le j0}\\to\\mathbb N_{>0},\\ \\forall t\\ge1,\\\n r(t)\\ge2\\ \\land\\ a(t)\\ge1\\ \\land\\ \\Delta_{r(t)}=a(t)\n \\ \\land\\ \\forall m\\ge1,\\ [\\Delta_m=a(t)\\Rightarrow m=r(t)]\\right).\n$$\n\nThere is one whole infinite indexed family $G_t=J_{r(t)}$, not only separate\nfinite prefixes. Every positive graph order remains in the reference class,\nincluding $m=1$ in the realizing clause. The Lean witness is a function on\nall natural numbers whose assertions are restricted to positive indices.\nIt additionally identifies $r$ with `realizingOrder`; no value at zero is\nclaimed as a source sequence term.\n\nThe source conventions are explicit. Read the printed family/element notation\nmemberwise: each realizing graph belongs to the class of source graphs. The\nintroduction's existential realizing-family definition permits selection of\ncorresponding orders; neither exhaustion of all orders nor $r(t)=t$ is\nrequired. Extend that termwise definition to the positive infinite index set\nprinted in Conjecture 2.3. On\n$D=\\{J_n:n\\ge1,\\ U(\\Delta_n)\\}$, interpret the qualified parameter as\n$p_D(G)=\\Delta(G)$. No value is assigned outside $D$. Read “are the unique\nmaximum degrees” as the full iff classification, retaining both inclusions.\nThese conventions normalize the source's notation; they are not an\nauthor-issued correction or a certificate of uniquely intended semantics.","title":"Kok's unique maximum-degree values and infinite realizing family","url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-unique-maximum-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0e7d7b8731ba62fdf4f7bb8e05dc68c4e8098221c791744525216a212a853c3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kok-linear-jaco-maximum-degree","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johan Kok, *Integer sequences with conjectured relation with certain graph\nparameters of the family of linear Jaco graphs*, arXiv:2507.16500v1,\nSection 2.2, Conjecture 2.2, states:\n\n> For the finite linear Jaco graphs J_{n}(x), n=2,3,4,\\dots the respective\n> maximum degrees \\Delta(J_{n}(x)) are yielded by the sequence A319433 with\n> closed formula \\Delta(J_{n}(x))=\\lfloor\\frac{2(n+2)}{1+\\sqrt{5}}\\rfloor-1,\n> n=2,3,4,\\dots\n>\n> Furthermore, it implies that sequence A319433 is p-graphical where\n> p(G)=\\Delta(G) and \\mathcal{F}=\\{G:G=J_{n}(x),n=1,2,3,\\dots\\}.\n\nThe formal graph is the underlying simple undirected graph of the finite\ndirected linear Jaco graph with `f(x)=x`. For vertices `v_i`, `1 ≤ i ≤ n`,\nwrite `d_i` for the number of earlier vertices `v_k` satisfying\n`2*k-d_k ≥ i`; for `i < j ≤ n`, the directed arc `(v_i,v_j)` exists exactly\nwhen `2*i-d_i ≥ j`. `J_n(x)` forgets orientation and `Delta` is its ordinary\nmaximum degree.\n\nThe sequence term uses its actual offset-zero index. For `n > 0`, write the\ncanonical Zeckendorf representation with weights `Fib_2=1`, `Fib_3=2`, and no\nconsecutive occupied positions. Delete the digit position of weight `Fib_2`,\nwhether its digit is one or zero: remove occupied index 2 if present, lower\nevery surviving occupied index by one, and decode in the same Fibonacci\nsystem. Call the result\n`a(n)=deleteLeast(n)`, with `a(0)=a(1)=0`.\n\nThe complete assertion is\n\n$$\n\\left(\\forall n\\in\\mathbb N,\\;2\\le n\\Rightarrow\n\\left(\\Delta(J_n(x))=\\left\\lfloor\\frac{2(n+2)}{1+\\sqrt5}\\right\\rfloor-1\n\\land\\Delta(J_n(x))=a(n)\\right)\\right)\n\\land\n\\left(\\forall n\\in\\mathbb N,\\;1\\le n\\Rightarrow\n\\Delta(J_n(x))=a(n)\\right).\n$$\n\nThe second conjunct retains the singleton `J_1(x)`, whose degree and deletion\nvalue are both zero. The first conjunct starts at `n=2`, exactly as in the\nsource. The index-zero sequence term is not reassigned to `J_1`; no `J_0`,\noffset shift, order cutoff, subsequence, changed graph, or extra premise is\nincluded.","title":"Kok's maximum degree and literal A319433 deletion","url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-linear-jaco-maximum-degree/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/afa7b3149462f612c195210c5fc45cb51510f40608b46f067ae46057ed54783c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koljancic-oblak-2026-nssp-minimum-arcs-problem-1-8","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sara Koljančić and Polona Oblak, *Combinatorial aspects of the nonsymmetric\nstrong spectral property for graphs*, Problem 1.8, printed page 759:\n\n> Is the minimum number of arcs in a directed graph G on n vertices that requires the nSSP equal to 2n−1?\n\nThe source and preregistration are linked from\n[issue #25108](https://github.com/the-omega-institute/trureturing/issues/25108).\nThe [Library note](../Library/Eigenstructure/koljancicoblak2026nssp.md)\nrecords source versions, attribution and licensing. The endpoint-looped double\npath is the source's Theorem 5.5 and Remark 2.10(1), not a new upper construction.\n\n| Source clause | Exact Lean meaning |\n| --- | --- |\n| n vertices | Fin n, for every natural n with 1 ≤ n |\n| Directed graph, loops allowed | Finset (Fin n × Fin n); each ordered pair appears at most once |\n| Exact matrix support | A i j ≠ 0 if and only if (i,j) belongs to E |\n| nSSP | Every real X with A i j * X i j = 0 and A * X.transpose − X.transpose * A = 0 is zero |\n| Requires nSSP | Every real A of that exact support has nSSP |\n| Minimum equals 2n−1 | All requiring E have at least 2n−1 arcs, and at the same n some requiring E has exactly 2n−1 arcs |\n\nThese definitions occur in\n`D5.S3.Combinatorics.SpectralGraph.KoljancicOblakMinimumArcs`.\n`MinimumArcs` is the entire two-clause assertion; `result` states it directly.\nThere is no additional identity-Hadamard, trace, sign, connectivity, symmetry,\neigenvalue, genericity or finite-order restriction in that assertion.","title":"The minimum number of arcs requiring nSSP","url":"https://the-omega-institute.github.io/trureturing-pages/research/koljancic-oblak-2026-nssp-minimum-arcs-problem-1-8/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f8090d52e38e0751d7186c37e1c70cd28a0779c827e9e07e9cfdcf212e2b994.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koljancic-oblak-2026-question-5-11-even-loop-path","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Koljančić–Oblak Question 5.11 asks whether every positive even order n and every nonempty loop set L on the double path admit a real nSSP matrix with exactly that pattern. Both orientations of every consecutive path edge occur, and only the loops at vertices in L are added. The final witness has no sign, symmetry, trace, spectrum or genericity restriction. The real test matrix X is unrestricted; the defining equations are entrywise A times X equal to zero and A times transpose X minus transpose X times A equal to zero.\n\nThe exact Lean proposition is `D5.S3.Combinatorics.SpectralGraph.KoljancicOblakEvenPath.claim`. Unit #25129, preregistration comment 6554378006, owns this event. The companion `D5/S3/Combinatorics/SpectralGraph/KoljancicOblakEvenPathNSSP.result` has this exact proposition as its type. Both mathematical hosts are frozen. The result proves the full event stated here.\n\nThe source is Sara Koljančić and Polona Oblak, *Combinatorial aspects of the non-symmetric Strong Spectral Property for graphs*, arXiv:2604.21552v1, Question 5.11, with Definitions 1.1, 1.3 and 1.5 fixing nSSP, allowing a pattern and double-path loops. The later Electronic Journal of Linear Algebra publication, volume 42, pages 755–779, DOI 10.13001/ela.2026.10523, states Question 5.11 on PDF page 22, printed page 776. The linked Library note owns the source and license correspondence.\n\nFor every natural n with 0 < n and Even n, and every nonempty finite set L of vertices in Fin n, there exists a real matrix A indexed by Fin n whose nonzero entries are exactly the two orientations of consecutive path edges and the diagonal entries selected by L. For every real matrix X on the same indices, A ∘ X = 0 together with A * Xᵀ − Xᵀ * A = 0 implies X = 0. Here ∘ is the entrywise product. This includes n = 2 and singleton endpoint loops; no restriction on X or extra premise on A is added to the event.","title":"Even double paths with a nonempty loop set allow nSSP","url":"https://the-omega-institute.github.io/trureturing-pages/research/koljancic-oblak-2026-question-5-11-even-loop-path/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1cd90ff3349c843f53e6404b462b62881e4e2290a19ec39404a5ab5a0a501201.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kopel-2026-eb-norm-coefficient","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Kopel, *A sharp norm inequality for entanglement-breaking channels*, arXiv:2609.27906v1, fixes\na Hermitian traceless basis $\\lambda_1,\\dots,\\lambda_{d^2-1}$ of $M_d$ with\n$\\mathrm{Tr}[\\lambda_i\\lambda_j]=2\\delta_{ij}$, writes a channel in Bloch form $r\\mapsto Ar+c$ and\nproves (Theorem 1) that every entanglement-breaking channel satisfies\n$\\|A\\|_*^2+\\tfrac{d(d-1)}{2}|c|^2\\le(d-1)^2$. Item 3 of its open questions asks: \"Is the\ncoefficient $d(d-1)/2$ optimal?\" The verbatim statements are in\n[the literature note](../Library/QuantumChannels/kopel2026sharpnorm.md).\n\nIssue [#14492](https://github.com/the-omega-institute/trureturing/issues/14492) reads the\nquestion as: for every $d\\ge2$, every admissible basis and every real $B>d(d-1)/2$, some\nentanglement-breaking channel has $\\|A\\|_*^2+B|c|^2>(d-1)^2$. The inequality with a coefficient\n$\\kappa$ becomes stronger as $\\kappa$ grows, so this says that $d(d-1)/2$ is the largest admissible\ncoefficient.","title":"Optimality of the coefficient in the entanglement-breaking norm inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/kopel-2026-eb-norm-coefficient/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1f5abc9be955cfcce40318123019a5f632db5cf3c6564ce746ce68a7616b33cb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koprowski-lewis-2026-conjecture-3-1-hypermatrix-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every finite field `F` of cardinality `q`, every positive integer `k`, and\nnonincreasing nonnegative masks `lambda, mu : Fin k -> Nat` satisfying\n`mu j <= lambda j`, `lambda j <= k - j`, and `mu j < k - j`, count the pairs of\nactual `(k+1) x k` matrices `(M1, M2)` that respect the two zero regions and\nare nondegenerate. The source's nondegeneracy condition is the nonvanishing of\nCayley's second hyperdeterminant; equivalently, every nonzero linear\ncombination of the two faces has rank `k` over the algebraic closure.\n\nThe resolved formula is\n\n```text\nq^(k*k) * (q-1)^(2*k)\n * product (q-bracket (k+1-j-lambda j))\n * product (q-bracket (k-j-mu j)),\n```\n\nwhere the products range over `j : Fin k` and\n`[r]_q = sum_{e=0}^{r-1} q^e`. The count is over the literal matrix-pair\nsubtype, with no count, bijection, inverse, normalization, or rational-combination\npremise added.","title":"Koprowski–Lewis Conjecture 3.1: the original masked hypermatrix count","url":"https://the-omega-institute.github.io/trureturing-pages/research/koprowski-lewis-2026-conjecture-3-1-hypermatrix-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3d4377dd3baebf7f52a7385424ade2b9bebed0303dcbf37d4cbe561e6c2649f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:korzekwa-et-al-2019-three-distinguishability-product-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Korzekwa, S. Czachórski, Z. Puchała and K. Życzkowski, *Distinguishing classically indistinguishable states and channels*, J. Phys. A: Math. Theor. 52, 475303 (2019), arXiv:1812.09083v3, conjecture in Section III.B, Eq. (30), `Distinguishability.tex` lines 462–467 (notation macros expanded):\n\n> Moreover, based on numerical evidence, we conjecture that points belonging to $\\mathcal A^3_4$ have the following product structure:\n>\n> $a b \\boldsymbol{f}^1+a (1-b) \\boldsymbol{f}^2+(1-a) b \\boldsymbol{f}^3+(1-a) (1-b) \\boldsymbol{f}^4,$\n>\n> with $a,b\\in[0,1]$.\n\nThe source defines $\\mathcal A_d^M$ by the existence of $M$ perfectly distinguishable quantum states with the same classical version (`Distinguishability.tex`, lines 147–149), defines that version as the diagonal in the preferred basis (lines 236–241), and identifies perfect distinguishability with pairwise orthogonal supports (lines 297–300). The $\\boldsymbol f^i$ are the four face centres of the probability simplex, each a permutation of $(1,1,1,0)/3$ (line 462). The definitions and quotation are also recorded in [the literature note](../Library/HilbertOperators/korzekwaetal2019distinguishing.md).\n\nFor the formal statement, index the four coordinates and face centres by $0,1,2,3$, and put $f^j_k=(1-\\delta_{jk})/3$. `InRegion p` means that $p$ is nonnegative with coordinate sum one and that there exist three complex $4\\times4$ positive semidefinite matrices $\\rho_i$ of trace one satisfying $\\rho_i\\rho_j=0$ for $i\\ne j$ and $\\operatorname{Re}(\\rho_i)_{kk}=p_k$ for every $i,k$. `ProductForm f p` is precisely the displayed combination with $a,b\\in[0,1]$.\n\nIssue [#25867](https://github.com/the-omega-institute/trureturing/issues/25867) fixes `claim`: there exists a permutation $\\sigma$ of the four face centres such that every $p$ satisfying `InRegion p` has `ProductForm` with centres $f^{\\sigma(0)},\\ldots,f^{\\sigma(3)}$. The existential labelling accommodates the different conventions in the source's text and figure. The module proves `result : ¬ claim`.","title":"The three-distinguishability region in dimension four is not of product form","url":"https://the-omega-institute.github.io/trureturing-pages/research/korzekwa-et-al-2019-three-distinguishability-product-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da94c0316e4018960cfb1aeb2e1fbda78fa0b1064cecf9124cbb549be1ced619.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kotdp-weighteddistinct","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A022629 defines $a_1(n)=[x^n]\\prod_{k\\ge1}(1+kx^k)$ and states:\n\n> Conjecture: log(a(n)) ~ sqrt(n/2) * (log(2*n) - 2).\n\nThe coefficient index ranges over every natural number, including zero. The claim means that the quotient tends to one along all natural indices. For a positive natural exponent $r$, write\n$a_r(n)=[x^n]\\prod_{k\\ge1}(1+k^r x^k)$.\nThe related entries A092484, A265840, A265841 and A265842 take $r=2,3,4,5$, respectively. Their denominators are $\\sqrt{2n}(\\log(2n)-2)$ for $r=2$, and $r\\sqrt{n/2}(\\log(2n)-2)$ for $r=3,4,5$.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| A022629: “Expansion of Product_{m>=1} (1 + m*q^m).” | `KotesovecA022629.a` | The coefficient of degree $n$ in factors $1$ through $n$; later factors cannot affect that coefficient. |\n| A022629: the quoted conjecture | `KotesovecA022629.claim`, `KotesovecA022629.result` | Exact source denominator; no assumed positivity or asymptotic hypothesis. |\n| A092484: “Expansion of Product_{m>=1} (1 + m^2*q^m).” and its logarithmic conjecture | `KotesovecA092484.a`, `.claim`, `.result` | The literal $r=2$ normalization. |\n| A265840, A265841, A265842: products with exponents $3,4,5$ and their logarithmic conjectures | The corresponding modules' `a`, `claim`, `result` | Each source denominator is retained. |\n| Natural indices, including zero | Each `a` has domain $\\mathbb N$ | Total real logarithm and division affect only initial exceptional indices. |\n| Every fixed positive natural exponent | `KotesovecA022629.all_exponents` | A consumed family theorem; fixing $r$ introduces no additional hypothesis into the five source claims. |\n\nAll Lean names above have prefix `D5.S0.Asymptotics.GeneratingFunctions`. The coefficient expression and the triangular sum are inline where needed; no higher-stratum triangular-number definition is imported.","title":"Kotesovec's weighted distinct-partition logarithmic limits","url":"https://the-omega-institute.github.io/trureturing-pages/research/kotdp-weighteddistinct/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1623c424d9c1ff132c6dcb1d7f16d38728e3719aa2a7d94e42a4f5b6a8e5bed9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kotesovec-2024-a368959-family","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Václav Kotěšovec, [OEIS A368959](https://oeis.org/A368959), COMMENTS; [preregistration #19242](https://github.com/the-omega-institute/trureturing/issues/19242).\n\n> Conjecture 2: This sequence contains all the terms from A338325, plus 243, 972, 1944, 2401, ...\n\nThe NAME is “Numbers m such that Sum_{k=0..m-1} exp(2*Pi*i*k^s/m) <> 0 for all s>=2.” A001694 denotes powerful numbers. A338325 has NAME “Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.” Membership uses positive natural numbers.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The complete sum over $0\\le k Conjecture 1: This is a subset of A001694.\n\nThe NAME is “Numbers m such that Sum_{k=0..m-1} exp(2*Pi*i*k^s/m) <> 0 for all s>=2.” A001694 denotes powerful numbers. A338325 has NAME “Biquadratefree powerful numbers: numbers whose exponents in their prime factorization are either 2 or 3.” Membership uses positive natural numbers.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The complete sum over $0\\le k Consider the model of simulation with postselection that allows: 1) unitary operations with ancilla; 2) classical randomization; 3) projective measurements; 4) adaptive unitary operations based on measurement results. Can \\textit{every} channel $\\Phi$ be simulated in this way with success probability $\\Omega(1)$ using one additional qubit and $O(1)$ rounds?\n\nThe verbatim passages are in [the literature note](../Library/QuantumContext/kotowski2026smallancilla.md).\n\nIssue [#19196](https://github.com/the-omega-institute/trureturing/issues/19196) fixes the reading: universal constants R₀ and q₀ > 0 such that every channel on M_d(ℂ), for every d, has a randomized adaptive protocol on ℂ^d ⊗ ℂ² with at most R₀ projective-measurement stages and success map qΦ with q ≥ q₀. The answer is no.","title":"One ancilla qubit and bounded rounds cannot simulate every channel with constant success","url":"https://the-omega-institute.github.io/trureturing-pages/research/kotowski-kotowski-2026-one-qubit-simulation-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ecce1f09f1fe9a2ef356402a100948f1d9a068d67d92175d31376d2e6087690.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koutschan-krattenthaler-schlosser-2024-shifted-g-determinants","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Koutschan, Krattenthaler and Schlosser, arXiv:2401.08481v1, Conjecture 23, states: “Moreover, the following identities are conjectured to hold for all n ≥ 3:”. Equation (10.12) is\n\n$G_{2,1,2,0}(n)=2G_{7,4,7,-1}(n-1)$.\n\nThe preregistration is [#19228](https://github.com/the-omega-institute/trureturing/issues/19228). The [Library note](../Library/RecurrenceComposition/koutschan2024determinantevaluations.md) records the source definition and all three equations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (1.2): falling factorial divided by $p!$ for $p\\ge0$, zero for $p<0$ | `ShiftedG.binom` | Rational `Ring.choose` on the nonnegative branch, integer lower argument. |\n| Section 10 determinant | `ShiftedG.G` | Integer parameters, rational entries, integer powers of 2 and zero-based `Fin n` indices. |\n| “for all n ≥ 3” | `ShiftedG.claim` | Natural size and the original lower bound. |\n| Every consecutive equality of (10.12) | `ShiftedG.claim` and `ShiftedG.result` | Original coefficients, signs and size shifts; a chain is a conjunction of adjacent equalities. |\n\nNatural subtraction is truncated at zero; within the source range it agrees with integer subtraction. The rational/integer coercions express the same source entries. There are no additional assumptions on the conjecture.","title":"Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.12)","url":"https://the-omega-institute.github.io/trureturing-pages/research/koutschan-krattenthaler-schlosser-2024-shifted-g-determinants/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5345fb3ea840b119f4e14d92cc5e29d9af6ab4539b4edfbb8aeacde9344658d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koutschan-krattenthaler-schlosser-2024-ten-eleven-determinants","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Koutschan, Krattenthaler and Schlosser, arXiv:2401.08481v1, Conjecture 23, states: “Moreover, the following identities are conjectured to hold for all n ≥ 3:”. Equation (10.11) is\n\n$G_{1,1,0,-2}(n)=-\\frac1{49}G_{2,3,0,-6}(n)=-\\frac27G_{6,4,5,-3}(n-1)=-\\frac4{5577}G_{7,6,5,-7}(n-1)$.\n\nThe preregistration is [#19237](https://github.com/the-omega-institute/trureturing/issues/19237). The [Library note](../Library/RecurrenceComposition/koutschan2024determinantevaluations.md) records the source definition and all three equations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (1.2): falling factorial divided by $p!$ for $p\\ge0$, zero for $p<0$ | `ShiftedG.binom` | Rational `Ring.choose` on the nonnegative branch, integer lower argument. |\n| Section 10 determinant | `ShiftedG.G` | Integer parameters, rational entries, integer powers of 2 and zero-based `Fin n` indices. |\n| “for all n ≥ 3” | `TenEleven.claim` | Natural size and the original lower bound. |\n| Every consecutive equality of (10.11) | `TenEleven.claim` and `TenEleven.result` | Original coefficients, signs and size shifts; a chain is a conjunction of adjacent equalities. |\n\nNatural subtraction is truncated at zero; within the source range it agrees with integer subtraction. The rational/integer coercions express the same source entries. There are no additional assumptions on the conjecture.","title":"Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.11)","url":"https://the-omega-institute.github.io/trureturing-pages/research/koutschan-krattenthaler-schlosser-2024-ten-eleven-determinants/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90035773b33a99e732db3bb3d9dd6d64e903c16cf369c883d263740450e7e4b1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:koutschan-krattenthaler-schlosser-2024-ten-ten-determinants","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Koutschan, Krattenthaler and Schlosser, arXiv:2401.08481v1, Conjecture 23, states: “Moreover, the following identities are conjectured to hold for all n ≥ 3:”. Equation (10.10) is\n\n$G_{3,0,3,3}(n)=\\frac23G_{0,1,-2,-4}(n+1)=-\\frac1{672}G_{1,3,-2,-8}(n+1)=\\frac1{63}G_{5,4,3,-5}(n)=\\frac4{1002001}G_{6,6,3,-9}(n)=-\\frac85G_{9,5,8,-2}(n-1)$.\n\nThe preregistration is [#19236](https://github.com/the-omega-institute/trureturing/issues/19236). The [Library note](../Library/RecurrenceComposition/koutschan2024determinantevaluations.md) records the source definition and all three equations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (1.2): falling factorial divided by $p!$ for $p\\ge0$, zero for $p<0$ | `ShiftedG.binom` | Rational `Ring.choose` on the nonnegative branch, integer lower argument. |\n| Section 10 determinant | `ShiftedG.G` | Integer parameters, rational entries, integer powers of 2 and zero-based `Fin n` indices. |\n| “for all n ≥ 3” | `TenTen.claim` | Natural size and the original lower bound. |\n| Every consecutive equality of (10.10) | `TenTen.claim` and `TenTen.result` | Original coefficients, signs and size shifts; a chain is a conjunction of adjacent equalities. |\n\nNatural subtraction is truncated at zero; within the source range it agrees with integer subtraction. The rational/integer coercions express the same source entries. There are no additional assumptions on the conjecture.","title":"Koutschan–Krattenthaler–Schlosser Conjecture 23, equation (10.10)","url":"https://the-omega-institute.github.io/trureturing-pages/research/koutschan-krattenthaler-schlosser-2024-ten-ten-determinants/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de3748332e331d5a5cee0cebfc9d5ee00c5ed9c87dd95f289356ccc3185d61b6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kovacec-2026-strengthened-marcus-de-oliveira","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Kovačec, *A conjecture more precise and stronger than the one by Marcus and de Oliveira*,\nDMUC Preprint 26-11, v1, Conjecture 7, p. 5:\n\n> Given any two n-multisets of complex numbers A, B such that a Jordan curve J and the bounded component J(A, B) are defined, there holds ∆(A, B) ⊆ closure(J(A, B)).\n\n[The source note](../Library/Eigenstructure/kovacec2026strongermoc.md) gives the\ndefinitions and URL. [Preregistration #17803](https://github.com/the-omega-institute/trureturing/issues/17803)\nspecifies the exact witness and the literal Lean conventions. The resolution is **Refuted**.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity and scope |\n| --- | --- | --- |\n| Any two $n$-multisets of complex numbers | `claim` quantifies over every natural `n` and `a b : Fin n → ℂ` | Functions retain multiplicities; no restriction on the spectra is added. |\n| $z_\\sigma=\\prod_i(a_i+b_{\\sigma(i)})$ | `sigmaPoint` | The product ranges over all `Fin n`; permutations are `Equiv.Perm (Fin n)`. |\n| $\\Delta=\\{\\det(D_A+UD_BU^*):U\\in U(n)\\}$ | `detRange` | `Matrix.diagonal`, `Matrix.unitaryGroup`, `star` and determinant give the literal expression. |\n| Permutations differing by one transposition | `graph` | Union of the real closed segments for `sigma` and `sigma * Equiv.swap i j`, with `i ≠ j`. |\n| Every positive prescribed distance admits an unobstructed segment | `exterior` | All real `M > 0`, a complex endpoint at distance `M`, and disjointness from the whole graph. |\n| Exterior boundary is a Jordan curve | `IsJordanCurve` applied to the exterior frontier | The entire boundary is the range of a continuous injective map from `Circle`. |\n| The bounded component of the complement of the boundary | `connectedComponentIn` at each point outside that frontier, with `Bornology.IsBounded` | Every bounded component is quantified. Under Jordan separation this is the source's unique bounded component; no Jordan curve theorem is assumed in Lean. |\n| $\\Delta\\subseteq\\overline{J(A,B)}$ | The inclusion in `claim`; `result` proves its negation | One bounded component containing zero suffices to refute the universal claim. |\n\nThe endpoint parameter and boundedness hypothesis identify the source's component;\nthey do not restrict the counterexample's determinant range. No source clause is dropped.","title":"The strengthened Marcus–de Oliveira conjecture has a three-dimensional counterexample","url":"https://the-omega-institute.github.io/trureturing-pages/research/kovacec-2026-strengthened-marcus-de-oliveira/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e619a78a034089f2231b436f3ddffd1aeec48d433e69d5121eb1a9532f0dc02b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kraus-simanek-2024-polynomial-content-no-maximizer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Kraus and B. Simanek, *New Perspectives on Torsional Rigidity and Polynomial Approximations of z-bar*, arXiv:2309.16450v1, Conjecture 2.1, p. 3:\n\n> For each N ∈ ℕ, there is an n ∈ ℕ so that among all n-gons with area 1, ρ_N has no maximizer.\n\nThe source characterizes $\\rho_N(\\Omega)$ as the squared distance from $\\bar z$ to complex polynomials of degree at most $N$ in $L^2(\\Omega,dA)$. The settlement is **Proved** by `D5/S3/Analytic/Torsion/PolynomialContentNoMaximizer.result`, with preregistration [#16916](https://github.com/the-omega-institute/trureturing/issues/16916).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| For each $N\\in\\mathbb N$ | `claim` quantifies over all natural degrees, including zero | Covers the source domain and zero. |\n| There is a polygon side count | The existential index in `claim` represents index plus three sides | Every polygon has at least three sides; the constructed count is $2\\max(3,N+2)$. |\n| A simple polygonal Jordan region | `IsSimplePolygon` requires nonadjacent edge disjointness, adjacent-edge intersection exactly at the shared vertex, and consecutive noncollinearity; `IsPolygonRegion` requires a bounded connected open region with frontier equal to Mathlib’s `Polygon.boundary ℝ` of those vertices | The closed path $v_0\\to v_1\\to\\cdots\\to v_{n+2}\\to v_0$ is the union of `Polygon.edgeSet ℝ`, identified with vector-space segments by `affineSegment_eq_segment` and `finRotate_apply`. |\n| Area one | `volume` of the region equals one | Complex planar Lebesgue measure. |\n| Polynomial content | `rhoN` is the infimum of the squared integral errors over `Polynomial.degreeLE` | The source's squared-distance expression. |\n| Has no maximizer | `claim` negates the existence of an admissible polygon dominating all other admissible polygons | No additional assumption on the competing polygon. |","title":"Polynomial content has no maximizer on suitable unit-area polygons","url":"https://the-omega-institute.github.io/trureturing-pages/research/kraus-simanek-2024-polynomial-content-no-maximizer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ac1e0f65efc4510db753318f925fbab2b076eb11eaf354ee468497021eaeed0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krawec-2015-recycled-coin-uniformity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"W. O. Krawec, *History dependent quantum walk on the cycle with an unbalanced coin*, Physica A\n428 (2015), arXiv:1411.6298v1, studies a two-coin walk on the d-cycle.\n- The coin flip applies C(π/4) or C(π(1+φ)/4) to the second coin according to the first coin.\n- The second coin moves the walker one step left or right.\n- The coins are then swapped.\n\nThe memory parameter is φ ∈ [0, 8). §III.B and §IV conjecture, from numerics for d ≤ 300 on the grid φ = m/10, that the time-averaged position distribution is uniform for every φ outside E_d. Here E_d = {0, 2, 4, 6} if 12 ∤ d and E_d = {0, 1, 2, 4, 5, 6} if 12 ∣ d. The source states it for every initial coin state at position 0. The verbatim statements are in [the literature note](../Library/QuantumChannels/krawec2015history.md).\n\nIssue [#15747](https://github.com/the-omega-institute/trureturing/issues/15747) fixes the reading: d ≥ 2, every unit coin α ∈ ℂ⁴ at position 0, and the Cesàro averages over times 1 … T converging to 1/d.","title":"A resonant memory parameter refutes Krawec's recycled-coin uniformity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/krawec-2015-recycled-coin-uniformity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/278b81c7bb54a2f196426ca54d918d92cb1037fdc8fc061b3eecc8997584f661.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kreh-2015-minimal-sets-conjecture-18","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Definition 16 (Unicode transcription, preserving the printed superscripts):\n\n> Definition 16. For a given set M ⊂ N define a sequence δⁿ(M) of sets recursively by δ⁰(M) := M, δ(M) := δ¹(M) := M \\ S(M), δⁿ⁺¹(M) := δ(δⁿ(M)) and let ηⁿ(M) := |S(δⁿ(M))|, η(M) := η¹(M).\n\nConjecture 18 (Unicode transcription, preserving the printed superscripts):\n\n> Conjecture 18. There are only countably many infinite sets M ⊂ N with η(M) ≤ η⁰(M). For all other sets we have ηⁿ(M) → ∞.\n\nKreh writes `N` for the positive integers, writes `x ⊳ y` when the decimal\nstring of `x` is obtained from that of `y` by deleting zero or more digits,\nand writes `S(M)` for the elements of `M` minimal under that order. The Lean\ndefinition `digitSubseq a b` is `a ⊳ b`; `minimal`, `peel`, and `eta` are\nKreh's `S`, `δ`, and `η` constructions.\n\nThe literal refuted statement is the second sentence: for every infinite set\n`M` of positive natural numbers with `eta M 0 < eta M 1`, every natural bound\n`B` eventually satisfies `B ≤ eta M k` at every later layer. This is exactly\n`claim`, including the explicit positivity premise. Lean's `Set.ncard` is zero\non an infinite set, but that convention is never used in the refutation:\nevery displayed minimal layer is proved to be a singleton or a pair. That\ntheorem does not address the first sentence, concerning countably many\ninfinite sets with `η(M) ≤ η⁰(M)`. A companion result for that sentence is\ndescribed below; this dossier's resolution binding remains the divergence\ntheorem.","title":"Refutation of the divergence sentence in Kreh's Conjecture 18","url":"https://the-omega-institute.github.io/trureturing-pages/research/kreh-2015-minimal-sets-conjecture-18/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51c4de617981ecf2f2388f27d346cce39571fa47ada4d9eeb3a6cd02d2c120ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2021-cstar-novak-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, *C*-algebraic Schur product theorem, Pólya-Szegő-Rudin question and Novak's conjecture*, arXiv:2108.06662v1, Conjecture 4.3, p. 10; journal DOI [10.4134/JKMS.j210627](https://doi.org/10.4134/JKMS.j210627), states:\n\n> Let $\\mathcal A$ be a unital C*-algebra. Then the matrix $[\\prod_{l=1}^d(1+\\cos(x_{j,l}-x_{k,l}))/2-1/n]_{1\\le j,k\\le n}$ is positive for all $n,d\\ge2$ and all choices of $x_j=(x_{j,1},\\ldots,x_{j,d})\\in\\mathcal A_{\\rm sa}^d$.\n\nThe source defines $\\cos X=(e^{iX}+e^{-iX})/2$ using the exponential series, and positivity means self-adjointness together with $\\langle Mv,v\\rangle\\ge0$ for every $v\\in\\mathcal A^n$, where the source's left-linear form is $\\sum_j\\sum_k M_{jk}v_kv_j^*$.","title":"A matrix refutation of Krishna's C*-algebraic Novak conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2021-cstar-novak-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f97d8a57b5b691b6041d6ce3ba332926fa554cb1398226081d26ee66093567f3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2022-cstar-de-bruin-sharma","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, \"C*-algebraic Schoenberg Conjecture\", arXiv:2206.06653v1 (14 June 2022; single version), Section 2. The source text is quoted verbatim:\n\n> **Conjecture 2.3 (C*-algebraic de Bruin-Sharma Conjecture).** Let $\\mathcal{A}$ be a C*-algebra, $d\\in \\mathbb{N}\\setminus\\{1\\}$ and let $P(z) \\coloneqq (z-a_1)(z-a_2)\\cdots (z-a_d)$ be a polynomial over $\\mathcal{A}$ with $a_1, a_2, \\dots, a_d \\in \\mathcal{A}$. Assume that $P'$ can be written as $P'(z)\\coloneqq d (z-b_1)\\cdots (z-b_{d-1})$ on $\\mathcal{A}$ with $b_1, b_2, \\dots, b_{d-1} \\in \\mathcal{A}$. If $\\sum_{j=1}^{d}a_j=0,$ then\n> $$\\sum_{k=1}^{d-1}(b_kb_k^*)^2\\leq \\frac{2}{d^2}\\left(\\sum_{j=1}^{d}a_ja_j^*\\right)^2+ \\frac{d-4}{d}\\sum_{j=1}^{d}(a_ja_j^*)^2$$\n> and\n> $$\\sum_{k=1}^{d-1}(b_k^*b_k)^2\\leq \\frac{2}{d^2}\\left(\\sum_{j=1}^{d}a_j^*a_j\\right)^2+ \\frac{d-4}{d}\\sum_{j=1}^{d}(a_j^*a_j)^2.$$\n\nThe ordered derivative is defined in the source immediately before the conjecture. The encoding uses zero-based Fin indices, complex scalar coefficients, star, and the C*-order. It restricts the source to unital C*-algebras in Type with PartialOrder and StarOrderedRing, and d >= 2; this subclass is enough for a counterexample to the source's universal assertion.","title":"The C*-algebraic de Bruin–Sharma Conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2022-cstar-de-bruin-sharma/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68be5f7a1e7537b7cca3da09093aa43535604bc2c20cf5ef69c867bf3a591191.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2022-cstar-dubinin-sugawa-dual-mean-value","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, *C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture*, arXiv:2206.08154v1, Section 3, p. 7, Conjecture DUALSMALE (journal Conjecture 3.1), states:\n\n> Let $\\mathcal A$ be a commutative C*-algebra. Let $P(z)=(z-a_1)\\cdots(z-a_n)$ be a polynomial of degree $n\\geq2$ over $\\mathcal A$, with $a_1,\\ldots,a_n\\in\\mathcal A$. If $z\\in\\mathcal A$ is not a critical point of $P$, then there exists a critical point $w\\in\\mathcal A$ such that\n> $$\\frac{\\|P'(z)\\|}{n}\\leq\\frac{\\|P(z)-P(w)\\|}{\\|z-w\\|}.$$\n\nSection 2 defines $P'$ as the sum of ordered products obtained by omitting one factor. The source's gloss makes noncritical mean $P'(z)\\ne0$; critical means $P'(w)=0$. The Lean definitions preserve this convention, use zero-based `Fin n` root indices, and cast $n$ to $\\mathbb R$ in the displayed quotient.\n\nThe carrier `A : Type` with `[CommCStarAlgebra A]` restricts the universal claim to unital commutative C*-algebras in `Type`. This weakens the source assertion, so its negation refutes that assertion. The witness algebra $\\mathbb C\\times\\mathbb C$ is in this subclass.","title":"The C*-algebraic Dubinin–Sugawa norm mean value conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2022-cstar-dubinin-sugawa-dual-mean-value/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7c6b41bec1223ae3756ce148505835018d77dfedf6a27b4db56b2f96e98ed231.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2022-cstar-higher-order-smale-mean-value","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, *C*-algebraic Smale Mean Value Conjecture and Dubinin-Sugawa Dual Mean Value Conjecture*, arXiv:2206.08154v1, Section 2, Conjecture HIGHERMEAN; journal DOI 10.26117/2079-6641-2026-54-2-38-47, Conjecture 2.3, p. 44:\n\n> Let $\\mathcal A$ be a commutative C*-algebra. Let $P(z)\\coloneqq(z-a_1)\\cdots(z-a_n)$ be a polynomial of degree $n\\ge2$ over $\\mathcal A$, $a_1,\\ldots,a_n\\in\\mathcal A$. If $z\\in\\mathcal A$ is not a critical point of $P$, then there exists a critical point $w\\in\\mathcal A$ of $P$ such that\n>\n> $$\\frac{\\|P^{(k)}(z)\\|}{k!}\\frac{\\|P(z)-P(w)\\|^{k-1}}{\\|P'(z)\\|^k}\\le4^{k-1},\\qquad\\forall\\,2\\le k\\le n.$$\n\nThe binding convention of [#14982](https://github.com/the-omega-institute/trureturing/issues/14982) uses `Polynomial A`, iterates of `Polynomial.derivative`, and `CommCStarAlgebra A`. The factor index `Fin n` retains every root with multiplicity. Noncritical means $P'(z)\\ne0$; the same $w$ must satisfy every inequality.","title":"The Higher Order C*-algebraic Smale Mean Value Conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2022-cstar-higher-order-smale-mean-value/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f0c2e0a12f1ff1225f867d7338ef643f1d09339bd06dd3db112ebeb5300415e1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2022-cstar-kushel-tyaglov","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, \"C*-algebraic Schoenberg Conjecture\", arXiv:2206.06653v1 (14 June 2022; single version), Section 2. The source text is quoted verbatim:\n\n> **Conjecture 2.4 (C*-algebraic Kushel-Tyaglov Conjecture).** Let $\\mathcal{A}$ be a C*-algebra, $n\\in \\mathbb{N}\\setminus\\{1\\}$ and let $P(z) \\coloneqq (z-a_1)(z-a_2)\\cdots (z-a_d)$ be a polynomial over $\\mathcal{A}$ with $a_1, a_2, \\dots, a_d\\in \\mathcal{A}$. Assume that $P'$ can be written as $P'(z)\\coloneqq d (z-b_1)\\cdots (z-b_{d-1})$ on $\\mathcal{A}$ with $b_1, b_2, \\dots, b_{d-1} \\in \\mathcal{A}$. Then\n> $$\\sum_{k=1}^{d-1}(b_kb_k^*)^2\\leq \\frac{d-6}{d}\\sum_{j=1}^{n}(a_ja_j^*)^2+\\frac{1}{d^2}\\left(\\sum_{j=1}^{d}a_ja_j^*\\right)^2+ \\frac{1}{d^2}\\left[\\sum_{j=1}^{d}a_j^2-\\frac{1}{d^2}\\left(\\sum_{k=1}^{d}a_k\\right)^2\\right] \\left[\\sum_{j=1}^{d}a_j^2-\\frac{1}{d^2}\\left(\\sum_{k=1}^{d}a_k\\right)^2\\right]^* +\\frac{2}{d}\\sum_{j=1}^{d}a_j\\left[a_j+\\frac{1}{d}\\sum_{k=1}^{d}a_k\\right]\\left[a_j+\\frac{1}{d}\\sum_{k=1}^{d}a_k\\right]^*a_j^*-\\frac{4}{d^3}\\sum_{j=1}^{d}a_j \\left[\\sum_{k=1}^{d}a_k \\right] \\left[\\sum_{k=1}^{d}a_k \\right]^*a_j^*$$\n> and\n> $$\\sum_{k=1}^{d-1}(b_k^*b_k)^2\\leq \\frac{d-6}{d}\\sum_{j=1}^{d}(a_j^*a_j)^2+\\frac{1}{d^2}\\left(\\sum_{j=1}^{d}a_j^*a_j\\right)^2+ \\frac{1}{d^2}\\left[\\sum_{j=1}^{d}a_j^2-\\frac{1}{d^2}\\left(\\sum_{k=1}^{d}a_k\\right)^2\\right]^* \\left[\\sum_{j=1}^{d}a_j^2-\\frac{1}{d^2}\\left(\\sum_{k=1}^{d}a_k\\right)^2\\right] +\\frac{2}{d}\\sum_{j=1}^{n}a_j^*\\left[a_j+\\frac{1}{d}\\sum_{k=1}^{d}a_k\\right]^*\\left[a_j+\\frac{1}{d}\\sum_{k=1}^{d}a_k\\right]a_j-\\frac{4}{d^3}\\sum_{j=1}^{d}a_j^* \\left[\\sum_{k=1}^{d}a_k \\right]^* \\left[\\sum_{k=1}^{d}a_k \\right]a_j.$$\n\nThe ordered derivative is defined in the source immediately before the conjecture. In Conjecture 2.4, the encoding reads the printed n as d. Indices become zero-based Fin types and the product order is retained. The encoding uses complex scalar coefficients, star, and the C*-order, and restricts the source to unital C*-algebras in Type with PartialOrder and StarOrderedRing and d >= 2; this subclass is enough for a counterexample to the source's universal assertion.","title":"The C*-algebraic Kushel–Tyaglov Conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2022-cstar-kushel-tyaglov/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e5f22483001fd929e5298245ab3a3446c7329fc02d090af280ac8c6239641d2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishna-2022-cstar-schoenberg","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahesh Krishna, \"C*-algebraic Schoenberg Conjecture\", arXiv:2206.06653v1 (14 June 2022; single version), Section 2. The source text is quoted verbatim:\n\n> **Conjecture 2.1 (C*-algebraic Schoenberg Conjecture).** Let $\\mathcal{A}$ be a C*-algebra. Let $d\\in \\mathbb{N}\\setminus\\{1\\}$, $P(z) \\coloneqq (z-a_1)(z-a_2)\\cdots (z-a_d)$ be a polynomial over $\\mathcal{A}$ with $a_1, a_2, \\dots, a_d \\in \\mathcal{A}$. If $P'$ can be written as $P'(z)= d(z-b_1)(z-b_2)\\cdots (z-b_{d-1})$ on $\\mathcal{A}$ with $b_1, b_2, \\dots, b_{d-1} \\in \\mathcal{A}$, then\n> $$\\sum_{k=1}^{d-1}b_kb_k^*\\leq \\frac{1}{d^2}\\left[\\sum_{j=1}^{d}a_j\\right]\\left[\\sum_{j=1}^{d}a_j\\right]^*+ \\frac{d-2}{d}\\sum_{j=1}^{d}a_ja_j^*$$\n> and\n> $$\\sum_{k=1}^{d-1}b_k^*b_k\\leq \\frac{1}{d^2}\\left[\\sum_{j=1}^{d}a_j\\right]^*\\left[\\sum_{j=1}^{d}a_j\\right]+ \\frac{d-2}{d}\\sum_{j=1}^{d}a_j^*a_j.$$\n\nThe ordered derivative is defined in the source immediately before the conjecture. The encoding uses zero-based Fin indices, complex scalar coefficients, star, and the C*-order. It restricts the source to unital C*-algebras in Type with PartialOrder and StarOrderedRing, and d >= 2; this subclass is enough for a counterexample to the source's universal assertion.","title":"The C*-algebraic Schoenberg Conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishna-2022-cstar-schoenberg/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb9ce907ddcd02b50ea5c9330f8d1328e7fe10824a32385a37bf07f8d46d761e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishnan-zero-forcing-generalized-petersen-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Arnav Krishnan, *A correction to the Zero Forcing Number of the Generalized Petersen Graphs P(n,3)*, arXiv:2607.19412v1, Conjecture 5:\n\n> Z(P(n,3)) = 8 for every n ≥ 13.\n\nThe paper proves the upper bound for `n ≥ 9` (Proposition 4) and the lower bound for `13 ≤ n ≤ 20` by exhaustive search, and states that a\nlower-bound proof valid for all `n` is missing.","title":"The Zero Forcing Number of P(n,3) Is Eight for Every n At Least Thirteen","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishnan-zero-forcing-generalized-petersen-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55b1ded078fd1e590581e9cc00e54226fbd309ef082dde3ed0403c48c7417fd5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krishnaswami-rameshan-2026-rotor-right-special-word","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Govind S. Krishnaswami and Anirudh Rameshan, *Symbol sequences from three-rotor coincidences and their word-complexity*, arXiv:2606.25766v1 [nlin.CD] (2026), §5.3.1, “Conjecture on right-special word”, state:\n\n> we conjecture that the $(n-1)$-word that branches (in two ways) coincides with the unique $n$-word starting with 0 read backwards (excluding the 0).\n\nThis is a verbatim excerpt from line 701 of the downloaded v1 TeX `3rotor-symb-dyn.tex`, md5 `5d78bd4b19257909d542bbe5b1469202`. The surrounding scope is UHE quasiperiodic trajectories starting at the origin with irrational slope $\\sigma>1$ (line 663). Discussion question (1), line 917, explicitly asks for a settlement. The source specifies its encoding in §5.3.4, line 835:\n\n> Given a Sturmian sequence $s_1 s_2 s_3 \\cdots$, the corresponding rotor sequence is $02 s_1 2 s_2 2 s_3 2 \\cdots$.\n\nIssue [#20821](https://github.com/the-omega-institute/trureturing/issues/20821) fixes the formal reading. For $m\\ge1$, define $s_m=3$ exactly when $m=\\lfloor k\\sigma\\rfloor_{\\mathbb N}+k$ for some natural $k\\ge1$, and $s_m=1$ otherwise. Here $\\lfloor\\cdot\\rfloor_{\\mathbb N}$ is the natural-valued floor used by `cutting`. Define $R_0=0$, $R_{2j-1}=2$ and $R_{2j}=s_j$ for $j\\ge1$, as in `rotor`.\n\nA finite list $w$ of natural symbols is a factor when $w_t=R_{i+t}$ for some natural starting position $i$ and every $t<|w|$. It is right-special when $wa$ and $wb$ are factors for two distinct natural symbols $a\\ne b$; the occurrences may start at different positions. These are `IsFactor` and `IsRightSpecial`.\n\nThe claim is: for every irrational real $\\sigma>1$, every natural $n\\ge2$, and every finite list $w$ of length $n-1$,\n\n$w\\text{ is right-special in }R\\quad\\Longleftrightarrow\\quad w=\\operatorname{reverse}(R_1R_2\\cdots R_{n-1}).$\n\nThis reading is proved. The [literature note](../Library/Rewriting/krishnaswamirameshan2026rotor.md) gives the source quotations and locators.","title":"The right-special rotor word is the reversed prefix after the initial zero","url":"https://the-omega-institute.github.io/trureturing-pages/research/krishnaswami-rameshan-2026-rotor-right-special-word/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8de695fd8618d95e500122e690b334ffeaf9c65a06ec1f30098f6fb4954698b8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:krop-mittal-wigal-2024-game-cordiality-tree-path","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Krop, Mittal and Wigal, *The Cordiality Game and the Game Cordiality Number*,\narXiv:2403.18060v1 (2024), published as *Graphs and Combinatorics* 40 (2024),\narticle 75, state in Section 3, Conjecture 3.2:\n\n> For any tree $T$ of order $n$, $c_g(T)\\le c_g(P_n)$.\n\nTheir Section 1 defines the cordiality game: Admirable labels a selected\nvertex by $0$, Impish labels one by $1$, edge labels are sums modulo $2$,\nand the discrepancy is $d=|e_1-e_0|$. Admirable starts in $c_g$ and minimizes\nthe terminal discrepancy while Impish maximizes it.","title":"The tree–path conjecture for the game cordiality number","url":"https://the-omega-institute.github.io/trureturing-pages/research/krop-mittal-wigal-2024-game-cordiality-tree-path/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68a6700f2ab71b4a2d523ef61919b7030a2c57d6206559b3420b3d9f8245aec1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kruger-werner-2005-single-polynomial-separability-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"O. Krüger and R. F. Werner, *Some Open Problems in Quantum Information Theory*, arXiv:quant-ph/0504166v1, Problem 3, \"Polynomial entanglement invariants\", the third closely related problem:\n\n> Decide whether the set of separable states can be described in terms of a polynomial invariant $f$, such that $f(\\rho)\\ge 0$ is equivalent to separability. There are many weaker versions of this statement, which may be of interest. For example, we might merely ask for a sufficient or a necessary separability criterion, and we might allow $f$ to depend on the dimensions.\n\nHere a polynomial invariant is a real function on bipartite density operators that is constant on local-unitary orbits $\\rho\\mapsto(U_A\\otimes U_B)\\rho(U_A\\otimes U_B)^*$ and is computed by a polynomial in the matrix elements of $\\rho$. The definitions are quoted in [the literature note](../Library/QuantumContext/krugerwerner2005openproblems.md).\n\nIssue [#20112](https://github.com/the-omega-institute/trureturing/issues/20112) fixes the reading: for all local dimensions $d_A,d_B\\ge2$ there is a real polynomial $f$ in the real and imaginary parts of the entries, invariant under local unitaries on density operators, with $f(\\rho)\\ge0$ iff $\\rho$ is separable for every density operator $\\rho$. This dimension-dependent version, the weakest the source names for a characterization, is refuted.","title":"No single polynomial inequality characterizes separability in every dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/kruger-werner-2005-single-polynomial-separability-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c68fa8baa6468c054df0cdcdf512d06df38a75c893454131015c4a340237f33.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ks2105-2phi1denom","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kim and Stanton, *Three families of q-Lommel polynomials*, arXiv:2105.10096v3, Section 8, Conjecture 8.3.\n\nLet $R=\\mathbb Z[a,b,c,q]$ and $K=\\operatorname{Frac}(R)$. The formal series is\n$F(A,B;C;q,z)=\\sum_{r\\ge0}(A;q)_r(B;q)_r z^r/((C;q)_r(q;q)_r)$, where\n$(u;q)_r=\\prod_{j=0}^{r-1}(1-uq^j)$. Put\n$D_n=\\prod_{k=0}^n(1-cq^k)^{\\lfloor(n+1)/(k+1)\\rfloor}$.\n\nDefine $\\gamma_n=[z^n](F(aq,bq;cq;q,z)/F(a,b;c;q,z))$. The assertion is\n$\\forall n\\in\\mathbb N\\;\\exists P_n\\in R:\\;\\gamma_n/(1-c)=P_n/D_n$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The standard basic hypergeometric series | `KimStantonDenominatorCore.F` and `qShiftedFactorial` | Both lower products, including $(q;q)_r$, are present. |\n| The displayed contiguous quotient | `KimStantonSimultaneousDenominator.gamma` | The upper and lower parameter shifts are literal. |\n| Every natural coefficient index | `KimStantonSimultaneousDenominator.claim` | The quantifier includes zero. |\n| An integer polynomial in all four variables | The existential `MvPolynomial (Fin 4) ℤ` binder | The variables are ordered $a,b,c,q$. |\n| The prescribed floor exponent product | `KimStantonDenominatorCore.D` | Natural division is floor division. |\n| The claimed identity | `KimStantonSimultaneousDenominator.result` | No additional hypotheses. |","title":"Kim–Stanton Conjecture 8.3: integer polynomial denominators","url":"https://the-omega-institute.github.io/trureturing-pages/research/ks2105-2phi1denom/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60233e2492aea12a17134e0b1d444ebd30b4e4bad740ac4e2e93ea7a3197ac24.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ks2105-single-2phi1denom","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kim and Stanton, *Three families of q-Lommel polynomials*, arXiv:2105.10096v3, Section 8, Conjecture 8.4.\n\nLet $R=\\mathbb Z[a,b,c,q]$ and $K=\\operatorname{Frac}(R)$. The formal series is\n$F(A,B;C;q,z)=\\sum_{r\\ge0}(A;q)_r(B;q)_r z^r/((C;q)_r(q;q)_r)$, where\n$(u;q)_r=\\prod_{j=0}^{r-1}(1-uq^j)$. Put\n$D_n=\\prod_{k=0}^n(1-cq^k)^{\\lfloor(n+1)/(k+1)\\rfloor}$.\n\nDefine $\\gamma_n=[z^n](F(aq,b;cq;q,z)/F(a,b;c;q,z))$. The assertion is\n$\\forall n\\in\\mathbb N\\;\\exists P_n\\in R:\\;\\gamma_n/(1-c)=P_n/D_n$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The standard basic hypergeometric series | `KimStantonDenominatorCore.F` and `qShiftedFactorial` | Both lower products, including $(q;q)_r$, are present. |\n| The displayed contiguous quotient | `KimStantonSingleDenominator.gamma` | The upper and lower parameter shifts are literal. |\n| Every natural coefficient index | `KimStantonSingleDenominator.claim` | The quantifier includes zero. |\n| An integer polynomial in all four variables | The existential `MvPolynomial (Fin 4) ℤ` binder | The variables are ordered $a,b,c,q$. |\n| The prescribed floor exponent product | `KimStantonDenominatorCore.D` | Natural division is floor division. |\n| The claimed identity | `KimStantonSingleDenominator.result` | No additional hypotheses. |","title":"Kim–Stanton Conjecture 8.4: integer polynomial denominators","url":"https://the-omega-institute.github.io/trureturing-pages/research/ks2105-single-2phi1denom/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc0e913b87db69f046b6dffad46ad48a0ffda29d16f6284d22d1b4ac0cfd5bea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ku-2026-monochromatic-k-in-a-row-conjecture-5-6","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kuo-Han Ku, *Monochromatic k in a row*, arXiv:2606.12880v1,\nsection 5.1.2, states:\n\n> Conjecture 5.6. For every $k\\in\\mathbb{N}_{\\geq 2}$ and $d\\in\\mathbb{N}$, $D(k,[k]^d)=1-\\frac{2}{k}$.\n\nDefinitions 2.2, 2.4 and 2.6 fix the ordinary board\n$[k]^d=\\{1,\\ldots,k\\}^d$, all full signed integer lines, near-line\navoidance, inclusion-maximal saturation, and the maximum saturated density.\nThis universal assertion is refuted at $k=3,d=3$.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Every integer $k\\geq2$ and natural dimension $d$ | `KuGeometry.claim` quantifies over natural `k`, the hypothesis $2\\leq k$, and every natural `d`. |\n| Ordinary points in $\\{1,\\ldots,k\\}^d$ | `Board k d` is `Fin d → Fin k`; a label is interpreted as its natural value plus one in the integers. |\n| Every full line $p+tv$, $0\\leq t We have also numerically verified that non-zero phases ϑ_m lead to worse result, and we can thus conjecture that this fully characterizes the optimal steady-state general-dyne monitoring.\n\nThe [Library note](../Library/QuantumBounds/kuaserafinigenoni2025daemonic.md) gives the source locators and Eq. (66). Write $r=2\\chi/\\kappa$, $\\nu=2n_{\\rm th}+1$ and $c=\\sqrt{1-r^2}$, with $\\kappa>0$, $0\\le\\chi<\\kappa/2$ and $n_{\\rm th}\\ge0$. For every finite physical general-dyne measurement covariance $M$ and physical stationary conditional covariance $S$, the result proves\n\n$$\\sqrt{\\det S}\\ge v_* = \\frac{(\\nu-1)c+\\sqrt{(\\nu-1)^2c^2+4\\nu}}{2}.$$\n\nThe pure aligned covariance\n\n$$M_*=\\operatorname{diag}\\left(\\frac{1-r}{1+r},\\frac{1+r}{1-r}\\right)$$\n\nhas a physical stationary solution attaining this value. No uniqueness of the minimizing measurement is asserted by the public theorem.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Stable thermal OPO, Eqs. (57)–(58) | `claim` parameters `κ`, `χ`, `nth` and its four hypotheses; `driftA` | All damping, pump and thermal parameters in the stated regime |\n| Thermal input and diffusion | Literal scalar multiples of the identity inside `eMat`, `bMat`, `steadyEq` | $\\sigma_{\\rm in}=\\nu I$ and $D=\\kappa\\nu I$ |\n| Passive coupling in physical q,p coordinates | `coupling` | $C=\\sqrt\\kappa\\,\\Omega$, with $\\Omega=\\begin{pmatrix}0&1\\\\-1&0\\end{pmatrix}$ |\n| Source E and B, Eqs. (21)–(25) | `eMat`, `bMat` | The inverse matrix square root is `CFC.sqrt` of the inverse positive matrix |\n| Stationary Riccati equation, Eq. (55) | `steadyEq` | Every matrix product, transpose and diffusion term is retained |\n| Every physical measurement and stationary covariance | `claim` first conjunct; frozen `IsGaussianMeasurementCov (m := 1)` | Symmetry and the complex uncertainty matrix being positive semidefinite; arbitrary phase, squeezing and impurity |\n| Global optimum | `vStar`; `result` first conjunct | Universal determinant-square-root lower bound |\n| Aligned pure measurement, Eq. (66) | `optMeas`; `result` second conjunct | Existence of a physical stationary attainer with equality |\n\nThe symplectic, thermal-input and diffusion expressions are inlined in the public formulas. Proofs use the frozen `PositiveOneModeWilliamson.physicalJ2` through `omega_sq`, and the frozen physical covariance predicate and public `GaussianThermometryJointMeasurementRefutation.physical_data`. The finite $2\\times2$ real matrix carrier is the source's single-mode covariance representation. Requiring $S$ to be physical restricts the conclusion to conditional covariance solutions; it adds no hypothesis to the source's physical statement. The damping scale cancels without restricting its positive value.","title":"The aligned general-dyne measurement attains the global steady-state optimum","url":"https://the-omega-institute.github.io/trureturing-pages/research/kua-serafini-genoni-2025-general-dyne-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ec34c67d11f854e2cc5de26eb7e1f10f72ddf8b16fe5ff0d4e1c0a1451093b3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kubota-sekido-yoshino-2023-cubic-grover-twice-odd-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Kubota, H. Sekido and K. Yoshino, *Regular graphs to induce even\nperiodic Grover walks*, arXiv:2307.13227v1, Discrete Mathematics 348\n(2025) 114345, ask after Theorem 4.10:\n\n> Question 4.11. Let l be an odd integer that is a multiple of 3. Do 2l-periodic 3-regular graphs exist?\n\nGraphs have the scope of §2.1: finite, simple, connected and undirected.\nAn arc is an ordered adjacent pair, represented by Mathlib's `G.Dart`;\nits origin, terminus and reversal are `a.fst`, `a.snd` and `a.symm`.\nSection 2.2 defines the complex Grover matrix by\n\n$$\nU_{a,b}=\\begin{cases}\n2/\\deg_G(t(b))-1 & a=b^{-1},\\\\\n2/\\deg_G(t(b)) & t(b)=o(a),\\ a\\ne b^{-1},\\\\\n0 & t(b)\\ne o(a).\n\\end{cases}\n$$\n\nThe period is the least positive integer $\\tau$ with $U^\\tau=I$.\nThe precise claim in issue #12338 uses $l\\in\\mathbb N$, `Odd l`,\n`3 ∣ l`, a finite vertex type, connectedness, degree three at every\nvertex, and the negation of period $2l$. Oddness already excludes zero;\nthe natural-number formulation preserves the source's positive periods.","title":"Cubic Grover walks cannot have twice an odd period","url":"https://the-omega-institute.github.io/trureturing-pages/research/kubota-sekido-yoshino-2023-cubic-grover-twice-odd-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bed1e29e0ed7abe3b19d45a447fde3cbefdc94aa8eeb207afc7aa61959dd7069.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kulhandjian-hanzo-2026-flagship-distance-conjecture-25","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kulhandjian and Hanzo, arXiv:2610.02392v1, Section VII, Conjecture 25\n(“Flagship distance, original”), state: “For prime $p$ odd,\n$d_{\\min}(\\mathcal Q(p,2))\\le p+1$.” Issue\n[#13799](https://github.com/the-omega-institute/trureturing/issues/13799)\nrecords the complete statement and its literature checks.\n\nFor every odd prime $p$, take a primitive $\\alpha\\in\\mathbb F_{p^3}$ and\n$n=p^2+p+1$. The source's literal trace definitions are\n$\\tau_H(i)=[\\operatorname{Tr}(\\alpha^i)=0]$ and\n$\\tau_Q(i)=[\\operatorname{Tr}(\\alpha^{2i})=0]$.\nSet $A=\\operatorname{circ}(\\tau_H)$,\n$M_Q=\\operatorname{circ}(\\tau_Q)$, and $H=(A\\mid M_QA)$ over\n$\\mathbb F_p$, using entries $v_{i-j}$ for circulants. Stabilizers are\n$\\operatorname{rowspan}(H)$. The centralizer consists of pairs with zero\nsigned symplectic pairing against every stabilizer, and Pauli weight counts\npositions where either component is nonzero. The result constructs a\ncentralizer vector outside the stabilizers of weight at most $p+1$.\nIt quantifies over every primitive $\\alpha$.","title":"Kulhandjian–Hanzo Conjecture 25: the odd-prime distance bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/kulhandjian-hanzo-2026-flagship-distance-conjecture-25/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e040326add10bbad11a7884a7f7c5843f5c16f266a408e3900ee221ecb908928.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kulhandjian-hanzo-2026-q2-distance-ceiling","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Kulhandjian and L. Hanzo, *Singer-Difference-Set Qudit Stabilizer Codes\nfrom Non-Degenerate Quadrics in PG(d,q)*, arXiv:2610.02392v1, construct for a\nprime power $q$ and $d\\ge2$ a stabilizer code $\\mathcal Q(q,d)$ with\n$H=(A\\mid M_QA)$: $A=\\mathrm{circ}(\\boldsymbol\\tau_H)$ is the Singer incidence\nmatrix of $\\mathrm{PG}(d,q)$ and $M_Q=\\mathrm{circ}(\\boldsymbol\\tau_Q)$ the\nquadric circulant. Proposition 23 (Section VII.E) states for $q=2$ and\n$d\\in\\{2,\\dots,6\\}$ that $u=M_Q\\boldsymbol\\tau_H\\equiv\\mathbf 1$, that every\n$Z_iZ_j$ lies in the centralizer, and, by exhaustive search for $d\\le4$, that\n$Z_0Z_1$ is not a stabilizer, so $d_{\\min}=2$. Conjecture 24 asserts the\nproposition for every $d\\ge2$, and the paper's list of open problems asks to\nsettle it analytically. The verbatim statements are in\n[the literature note](../Library/QuantumStates/kulhandjian2026singer.md).\n\nIssue [#13470](https://github.com/the-omega-institute/trureturing/issues/13470)\nfixes the reading. At $q=2$: $K=\\mathrm{GF}(2^{d+1})$, $n=2^{d+1}-1$,\n$(\\boldsymbol\\tau_H)_i=1$ iff $\\operatorname{Tr}(\\alpha^i)=0$,\n$(\\boldsymbol\\tau_Q)_i=1$ iff $\\operatorname{Tr}(\\alpha^{3i})=0$, circulants\nhave entries $v_{(i-j)\\bmod n}$, the centralizer is\n$\\{(\\mathbf a,\\mathbf b):\\mathbf aH_x^\\top=\\mathbf bH_z^\\top\\}$ and the\nstabilizers are the row space of $H$. `claim` asserts, for every $d\\ge2$ and\nevery primitive $\\alpha$ (the paper fixes one): $u=\\mathbf 1$; for $i\\ne j$,\n$(e_i+e_j,\\mathbf 0)$ is in the centralizer and not a stabilizer; every\ncentralizer element outside the stabilizers has symplectic weight $\\ge2$. The\nwitness has weight 2 also as a Hamming weight, so $d_{\\min}=2$ under either\nreading of the paper's weight.","title":"The q = 2 Singer-quadric qubit codes have minimum distance 2 for every d ≥ 2","url":"https://the-omega-institute.github.io/trureturing-pages/research/kulhandjian-hanzo-2026-q2-distance-ceiling/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3f8420dc0a135b33e4be879c16bda24dcf54b9505ff8b79d683837bb681044f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kumar-2026-rectangular-weak-two-metric-dimension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Aryan Kumar, arXiv:2609.32161v1, §7:\n\n> The rectangular k = 2 values below the threshold m = 2n − 2 remain to be determined.\n\nThe vertices of $K_n\\square K_m$ are pairs in $[n]\\times[m]$, with\n$d((i,j),(a,b))=[i\\ne a]+[j\\ne b]$. For a landmark set $S$, put\n$\\Delta_S(x,y)=\\sum_{w\\in S}|d(x,w)-d(y,w)|$.\nA weak $k$-resolving set has $\\Delta_S(x,y)\\ge k$ for every distinct pair;\n$\\operatorname{wdim}_k$ is its least cardinality.\n\nThe exact rectangular answer is\n\n$$\n\\operatorname{wdim}_2(K_n\\square K_m)\n=\\min\\left\\{\\left\\lceil\\frac{2(n+m)}3\\right\\rceil,\\,2n-2\\right\\}\n\\qquad(4\\le n E(m, 2) is the third most inefficient for odd m.\n\nThe source defines expressions from copies of $x$ using addition, multiplication and parentheses, evaluates them in $\\mathbb{Z}/m\\mathbb{Z}$ at a base $s$, and defines $E(m,s)$ as the maximum expression complexity over residues. The binding proposition reads “third” by individual unit bases: for every odd $m \\ge 3$, at most two units $s$ satisfy $E(m,2) The CGLMP inequality for 2-qutrit states is monogamous, that is, if\n> $\\rho_{ABC}$ is any 3-qutrit state such that $\\rho_{AB}$, $\\rho_{BC}$ and\n> $\\rho_{AC}$ are its three 2-qutrit RDMs, at most one of these violates the\n> CGLMP inequality.\n\nHere $\\mathcal B_{CGLMP}(\\rho)$ is the maximum of the CGLMP expression\n$\\mathcal I_3$ (Eq. (CGLMP1)) over the angles of $A_k=U_{\\text{FT}}U(\\vec\\phi_k)$\nand $B_l=U^*_{\\text{FT}}U(\\vec\\varphi_l)$, with\n$U(\\vec\\phi)=\\mathrm{diag}(e^{-i\\phi(j)})$ and a computational-basis\nmeasurement after each operator (Eq. (CGLMP2)). A state violates the\ninequality when $\\mathcal B_{CGLMP}>2$.","title":"Monogamy of the qutrit CGLMP inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/kumari-2017-cglmp-monogamy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33df16584108bfd5d72267e0e32159fc4a32834242c0ccdaf4bce125fea1ea21.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kumari-adhikari-2022-structured-negativity-coincidence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Kumari and S. Adhikari, “Structured negativity: A physically realizable\nmeasure of entanglement based on structural physical approximation”,\narXiv:2209.03909v1 (2022-09-08, quant-ph), state in the abstract:\n\n> For d ⊗ d dimensional state, we conjecture from the result obtained in this work that negativity coincide with the structured negativity when the number of negative eigenvalues of the partially transposed matrix is equal to d(d−1)/2.\n\nThe conclusion repeats: “Thus, we conjecture that the negativity and\nstructured negativity coincides when q=d(d−1)/2.” Issue\n[#11475](https://github.com/the-omega-institute/trureturing/issues/11475)\npreregisters this quantified reading and the counterexample before any Lean.\nThe claim ranges over natural dimensions d ≥ 2 and positive semidefinite,\ntrace-one matrices on ℂ^d ⊗ ℂ^d. The partial transpose on B sends the entry\nρ_(i,j),(k,l) to ρ_(i,l),(k,j); q counts its strictly negative eigenvalues\nwith multiplicity.","title":"Negativity and structured negativity differ for a pure two-qutrit state","url":"https://the-omega-institute.github.io/trureturing-pages/research/kumari-adhikari-2022-structured-negativity-coincidence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2cd148a1474666f5b3de62494a63affad3abcc7db08e493ed18ce453d8a70abd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kur107667-ballot","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mikhail Kurkov's FORMULA conjecture in Paul D. Hanna's [OEIS A107667, approved revision 25](https://oeis.org/history/view?seq=A107667&v=25) asserts\n$T(n+m,m)=(n+m+1)P(n+1,n+1,m)$ for all $n,m\\in\\mathbb N$.\nThe preregistration is [#21639](https://github.com/the-omega-institute/trureturing/issues/21639).\nThe source is OEIS-only, with no claim-source DOI or arXiv identifier identified.\n\nThe source NAME defines a lower triangular matrix by $T=D+\\operatorname{SHIFT\\_LEFT}(T^2)$ with diagonal $1,2,3,\\ldots$.\nThus $T(a,b)=0$ for $a;\nthe preregistration is .\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The $k\\times k$ first-kind minor with one-based indices | `B` | `Fin k` values plus one preserve both source indices. |\n| Nested ranges $n\\ge j_1\\ge\\cdots\\ge j_m\\ge0$ | Reused `KurkovNestedStirlingIdentity.Chain n m` | Antitone maps into `Fin (n+1)` enumerate each source tuple once. |\n| Sign, squared pair differences, falling factorial, numerator power and denominator factorial | `sourceTerm` | Rational differences and division; natural factorial indices and exponents. |\n| Division by the product of $(n+i)!$ | `sourceRHS` | The entire chain sum retains the full normalization. |\n| Equality for all natural $n,k$ and $m\\ge1$ | `claim`, `result` | The source boundaries $n=0$ and $k=0$ are included. |\n\nThe four public definitions are the literal source determinant, summand,\nnormalized sum and quantified assertion. There is no extra hypothesis on $n$ or $k$.","title":"Kurkov's three-parameter Stirling-minor Vandermonde sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/kurk396563/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/650058e27afd7606294693438bace1a0614646fdaacebcd95fae5032d7d8f85c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kurzynski-2017-eighteen-mode-bosonic-assignment-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Kurzyński, *Contextuality of identical particles*, Phys. Rev. A 95(1), 012133 (2017), arXiv:1609.01250v1, asks in “Contextuality of $N>2$ bosons” (`contextuality_fermions.tex`, line 233):\n\n> The Hardy-like proof shows that there is the state-dependent contextuality in the 18-mode scenario for any $N>0$. However, the lack of the state-independent contextuality was only shown for $N=2$. It is possible that for some $N>2$ the bosonic occupation numbers of 18 modes cannot be assigned in a non-contextual way. This would require further research, since for large numbers of particles there are many ways in which the particles can be distributed between the modes.\n\nThe source's bosonic model assigns a nonnegative integer to each mode, independently of the context, with the occupations in every context summing to the particle number $N$ (lines 70–76, Eq. (1)). A mode $v_{ij}$ belongs to contexts $i$ and $j$ (line 83); the nine contexts are displayed in lines 132–221. The definitions and source locator are also recorded in [the literature note](../Library/Observer/kurzynski2017contextualityidentical.md).\n\nThe formal definition `modes` is the set\n\n$$\nE=\\{(1,2),(1,6),(1,7),(1,8),(2,3),(2,8),(2,9),(3,4),(3,7),\n(3,9),(4,5),(4,7),(4,8),(5,6),(5,8),(5,9),(6,7),(6,9)\\}.\n$$\n\nFor $i\\in\\{1,\\ldots,9\\}$, put $C_i=\\{(a,b)\\in E:a=i\\text{ or }b=i\\}$. The definition `IsNoncontextualAssignment` requires one function $v:\\mathbb N\\times\\mathbb N\\to\\mathbb N$ satisfying $\\sum_{p\\in C_i}v(p)=N$ for every such $i$. Values outside $E$ are unrestricted and do not enter any context sum.\n\nIssue [#24135](https://github.com/the-omega-institute/trureturing/issues/24135) fixes the statement `claim`: for every $N\\in\\mathbb N$, such an assignment exists if and only if $N$ is even. This classification is proved.","title":"Eighteen-mode bosonic occupation assignments exist exactly for even particle numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/kurzynski-2017-eighteen-mode-bosonic-assignment-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab31116354d3a884635593709068e5dc13ff1193fe6a06145641faead403ccb4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:kvashchuk-2026-multicopy-design-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Kvashchuk, P. Chernyshova, L. E. A. Porto, T.-A. Ohst, L. B. Vieira and M. T. Quintino, *The most discriminable quantum states in the multicopy regime*, arXiv:2604.26927v3, study N equiprobable states received in k copies.\n- The discriminability Discr is the largest average success (1/N) Σ_i Tr(M_i ρ_i^{⊗k}) over POVMs.\n- A weighted state k-design (Definition 6) is a probability-weighted family whose weighted k-th tensor moment is Π_sym/Tr Π_sym, with every state of positive weight pure.\n- N′(d,k) is the least size of such a design (§3.3).\n\nFor mixed states with N = N′(d,k) + 1, §11.2 proposes a design on N′(d,k) states together with the maximally mixed state I/d. §12, question 4, asks whether this construction is optimal in every dimension, citing numerical evidence for qubits. The verbatim statements are in [the literature note](../Library/QuantumBounds/kvashchuk2026multicopy.md).\n\nIssue [#15935](https://github.com/the-omega-institute/trureturing/issues/15935) fixes the reading: for all d, k > 0, every weighted k-design on N′(d,k) states, appended with I/d, has Discr at least that of every ensemble of N′(d,k) + 1 states.","title":"A design with the maximally mixed state is not the most discriminable qubit ensemble at N = N′(2,3) + 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/kvashchuk-2026-multicopy-design-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77d79021db49f172132afab38c5fcdfc05d72d436878772f15f7fbd7c9f9f1c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:la-piana-2025-c5-graph-quantum-magic-squares","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. La Piana, *Graph Quantum Magic Squares and Free Spectrahedra*, arXiv:2512.08797v3, Question 1.3 asks:\n\n> Given a graph Γ, does the matrix convex hull of its quantum permutation matrices P^(Γ) generate all graph quantum magic squares? That is, mconv(P^(Γ)) = M^(Γ)?\n\nSection 4.1 names $C_5$ and the Petersen graph among the candidates for strict containment. The settlement is **Proved** for $C_5$, under the complete conventions preregistered in [#19303](https://github.com/the-omega-institute/trureturing/issues/19303):\n\n$\\operatorname{mconv}(\\mathcal P^{(C_5)})_s\\subseteq\\mathcal M^{(C_5)}_s$ at every internal size, and some $X\\in\\mathcal M^{(C_5)}_2$ lies outside that hull.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Definition 2.1: positive semidefinite blocks and identity row/column sums | `IsQMS` | All external and internal natural-number sizes; complex matrices |\n| Definition 3.1: adjacency-block commutation | `IsGraphQMS` | Every pair of vertices, both adjacency-filtered sums |\n| Definition 3.2: graph magic squares | `M` | Set of blocks satisfying `IsGraphQMS` |\n| Definition 3.2: idempotent, Hermitian graph magic squares | `P` | Both projection equations for every block |\n| Definition 2.3: normalized finite blockwise compressions | `IsMatrixConvex` | All finite families, varying input sizes and all output sizes |\n| Smallest containing matrix convex family | `mconv` | Intersection of every containing matrix convex family |\n| Section 4.1, named candidate $C_5$ | `claim`, `result` | All-level containment and existential separation |\n\nThe source uses positive sizes and a Hermitian ambient family. The preregistered encoding uses arbitrary complex block families and includes level zero; `IsQMS` itself enforces Hermitian blocks through positivity. The size-zero matrices are unique. `Fin` indices, the matrix identity `1`, and `SimpleGraph.cycleGraph 5` implement the source's indices, identity and graph. The strictness statement is existential in the internal size, with a size-two witness; it does not assert strictness at every size.","title":"Strict containment for graph quantum magic squares on the five-cycle","url":"https://the-omega-institute.github.io/trureturing-pages/research/la-piana-2025-c5-graph-quantum-magic-squares/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c04308c36dd6f9419fc4b33e0a6d40a3ba4fa0b8db459e12f97ec2ec52e00270.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:labelle-2025-toda-numerator-unimodality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Labelle, \"On a specialization of Toda eigenfunctions\", arXiv:2502.10655v3,\nSection 7, Conjecture 7.3:\n\n> The polynomial $(q)_\\alpha^2\\mathfrak{J}_\\alpha$ is unimodal.\n\nSection 1 fixes a split semisimple Lie algebra over the rationals, its simple\nroots, and its invariant form normalized so short roots have squared length 2.\nDefinition 1.1 defines $\\mathfrak{J}_\\alpha$ in $\\mathbb{Q}(q)$ for every\nnonnegative root-lattice coordinate vector. Unimodality means weak increase\nup to a peak and weak decrease thereafter.","title":"Labelle's Toda numerator unimodality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/labelle-2025-toda-numerator-unimodality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8df72b8fe6b1f87c3b9e975450d8f454c0c62fedaebeb02e88f3b81f4d9a2bff.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lalou-mbarek-skender-togni-conjecture-one-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lalou, Mbarek, Skender, and Togni, *Completely Independent Spanning Trees in\nSplit Graphs: Structural Properties and Complexity*, arXiv:2512.15486v2,\nSection 5, Conjecture 1 (verbatim apart from plain-text typography):\n\n> Every hypergraph H satisfies\n> chi_p^2(H) = chi_p(H) - ceil(alpha_{chi_p(H)}(H)/2).\n\nHere chi_p and chi_p^2 are the maximum feasible panchromatic and\nbipanchromatic color counts. A bipanchromatic coloring requires every global\ncolor class to occur at least twice. The quantity alpha_k is the minimum number\nof globally unique colors over all panchromatic k-colorings.","title":"Refutation of the panchromatic pairing conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/lalou-mbarek-skender-togni-conjecture-one-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/496661385c28904ecb53d323aba7b452686e4c2a9d6fea0eb9efab4636654390.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:landman-robertson-robertson-2023-fibonacci-super-accessibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Landman, Robertson and Robertson, *Ramsey properties for integer sequences with restricted gaps*, Moscow Journal of Combinatorics and Number Theory 12(3), 181–195, Section 5, question 3, ask:\n\n> Is the set of Fibonacci numbers 2-super-accessible? We know it is 2-accessible but that it is not 2-large.\n\nThe [source note](../Library/Certificates/landman2023fibonaccisuperaccessibility.md) gives the definitions and locators. The external named question and literal Lean conventions are in [#16039](https://github.com/the-omega-institute/trureturing/issues/16039).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The Fibonacci set $F=\\{f_n\\}_{n\\ge1}$ | `fibSet` | Membership is a positive-index Fibonacci value; repeated values do not change a set. |\n| A $k$-term $S$-ap has positive start and gap in $S$ | `HasAP S A k` | Every index below $k$ belongs to the color class $A$. |\n| A positive increasing $S$-diffsequence has successive differences in $S$ | `HasDiffSeq S A k` | The positivity, strict inequalities, difference memberships and all $k$ color memberships are retained. |\n| Every $r$-coloring of the positive integers, with literal colors $1,\\ldots,r$ | `SuperAccessible r S` | The total natural-valued function is constrained only on positive inputs. Its value at zero is irrelevant. |\n| Arbitrarily long color-1 $S$-aps, or arbitrarily long color-$j$ $S$-diffsequences for one $2\\le j\\le r$ | The disjunction and existential in `SuperAccessible` | The color $j$ is chosen before the length, exactly as in Definition 14. |\n| The question for $r=2$ and $S=F$ | `claim` | The literal definitions above are used. |\n| A negative answer | `result : ¬ claim` | One coloring violates both alternatives. |","title":"The Fibonacci numbers are not 2-super-accessible","url":"https://the-omega-institute.github.io/trureturing-pages/research/landman-robertson-robertson-2023-fibonacci-super-accessibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5adf1253ae3c069e22cc8ba31699e254c29b6dd36512cefbfa9605880dcfd1aa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:laneve-2024-qsp-collapse-conjecture-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Laneve and S. Wolf, *On multivariate polynomials achievable with quantum signal processing*,\narXiv:2407.20823v2 (Quantum 9, 1641 (2025)), Conjecture 8: if a Protocol C segment\n$A_m\\tilde W\\cdots A_1\\tilde W$ ($\\tilde W=\\operatorname{diag}(1,a,b)$, $A_k\\in SU(3)$) maps a polynomial\nstate of effective dimension $\\le2$ to one of effective dimension $\\le2$ while every intermediate\nstate has effective dimension $>2$, then the segment acts as $a^kb^h\\cdot U$ on some pair of\nsubspaces $H\\to H'$ of dimension $d\\ge2$. The verbatim statements are in\n[the literature note](../Library/QuantumBounds/laneve2024multivariateqsp.md).\n\nIssue [#14369](https://github.com/the-omega-institute/trureturing/issues/14369) reads the\nconclusion permissively — subspaces $H,H'$ of $\\mathbb C^3$ with its Euclidean (Hermitian) norm, any\nlinear isometry $U:H\\to H'$ and any $k,h\\in\\mathbb N$ — so that its refutation covers the printed form,\nwhere $U\\in SU(d)$ is taken in orthonormal bases of $H$ and $H'$.","title":"A three-level QSP segment that collapses without a common monomial factor","url":"https://the-omega-institute.github.io/trureturing-pages/research/laneve-2024-qsp-collapse-conjecture-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a08ff0c538a952bcbbd2ba0beba925c66e023503728d19f4c3c485cef0cad9b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lanier-beguinot-rioul-2025-reverse-pinsker-condition-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Lanier, J. Béguinot and O. Rioul, *From Classical to Quantum: Explicit Classical Distributions Achieving Maximal Quantum f-Divergence*, arXiv:2501.14340v1, §III-C, p. 5:\n\n> We conjecture that (49) is in fact not necessary for (50) to hold.\n\nThe same paragraph states:\n\n> Our bound holds in general for any f-divergence, in particular for the maximal f-divergence.\n\nSection I specifies a convex real generator on the nonnegative half-line with $f(1)=0$. Equation (4) defines\n\n$$\nD_f^{\\max}(\\rho\\Vert\\sigma)\n=\\operatorname{Tr}\\bigl(\\sigma^{1/2}f(\\sigma^{-1/2}\\rho\\sigma^{-1/2})\\sigma^{1/2}\\bigr).\n$$\n\nTheorem 4 assumes $|\\rho-\\sigma|\\leq\\rho+\\sigma$ (49), and asserts\n\n$$\nD_f(\\rho\\Vert\\sigma)\\leq\\frac{\\|\\rho-\\sigma\\|_1}{2}\n\\left(\\frac{f(m)}{1-m}+\\frac{f(M)}{M-1}\\right),\\tag{50}\n$$\n\nwhere $m,M$ are the extreme eigenvalues of $\\sigma^{-1/2}\\rho\\sigma^{-1/2}$. Preregistration: [#19943](https://github.com/the-omega-institute/trureturing/issues/19943).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity and scope |\n| --- | --- | --- |\n| Every finite dimension | `claim`: natural dimension and `Matrix (Fin n) (Fin n) ℂ` | Includes the qubit instance. |\n| Convex real generator for nonnegative arguments, zero at one | `ConvexOn ℝ (Set.Ici 0) f`, `f 1 = 0` | Values outside the half-line do not affect the positive relative matrix. |\n| States; invertible second state | `ρ.PosSemidef`, `ρ.trace = 1`, `σ.PosDef`, `σ.trace = 1` | Positive definiteness expresses an invertible positive state. |\n| Equation (4) | `maxDivergence` | Literal relative-matrix expression and real part of the CFC trace. |\n| Equation (50) | `reversePinskerBound` | Repository trace norm; real spectral infimum and supremum. The witness has nonzero denominators. |\n| Remove condition (49) | `claim` | No absolute-difference order hypothesis. |\n| Conditional printed assertion | private `claimTheorem4` | Adds exactly the matrix-order condition (49). |\n| Refutation | public `result` | Negates the complete binding `claim`. |\n\nThe public formulas inline the source relative-matrix expression. The proof uses `D5.S3.Resource.LogDet.PathSpectralClassical.relativeCongruence`, which has that expression, without introducing a duplicate definition. The library note is `Library/Divergence/lanier2025maximal.md`.","title":"A qubit refutation of the maximal reverse Pinsker bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/lanier-beguinot-rioul-2025-reverse-pinsker-condition-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01431261aca5533c773675dd6f90a25bf7a81d016e56edefb817aa2308907ec0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lanina-sleptsov-2026-vassiliev-one-parameter-finite-generation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Lanina and A. Sleptsov, *Algebraic structures of Vassiliev invariants for knot families*, arXiv:2508.02385v2, Nuclear Physics B 1022 (2026), 117268, conjecture in the abstract (`main.tex`, line 165):\n\n> We conjecture that such 1-parametric algebra of Vassiliev invariants is always finitely generated, while in the case of more parameters, we provide example of the knot family with infinite number of generators.\n\nSection 4.1, “Generic one-parametric family” (`main.tex`, line 1166), asks:\n\n> Can this force such a 1-parametric algebra of Vassiliev invariants to be infinitely generated? Examples and reasonings provide the intuition that a 1-parametric algebra of polynomial Vassiliev invariants always has only a finite number of generators. Still, we currently lack strict mathematical proof.\n\nThe source restricts attention to rational-valued invariants (`main.tex`, line 350), defines primary invariants as generators of the algebra restricted to a family (line 356), and represents the one-parameter restrictions by polynomials with rational coefficients (lines 1114–1117). These passages are recorded in [the literature note](../Library/Topology/laninasleptsov2026vassilievfamilies.md).\n\nIssue [#22008](https://github.com/the-omega-institute/trureturing/issues/22008) fixes the algebraic reading: for every index type $\\iota:\\mathrm{Type}$ and every family $p:\\iota\\to\\mathbb Q[X]$, there is a finite set $s$ of indices such that\n\n$\\mathbb Q[p_i\\mid i\\in s]=\\mathbb Q[p_i\\mid i:\\iota]$.\n\nHere $\\mathbb Q[S]$ denotes the smallest rational subalgebra containing $S$. Thus the generators can be chosen from the original family of restricted invariants. This is the statement of `claim`, proved by `result`.","title":"Finite generation of one-parameter polynomial Vassiliev algebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/lanina-sleptsov-2026-vassiliev-one-parameter-finite-generation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af6cbd1a78a1e2c107ccd5e15830a1233558ef89ae20c27b1595258d8788df1c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:larsson-manabe-2026-rational-bracket-collision-density","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Urban Larsson and Hikaru Manabe, *Additive Subtraction Games*, arXiv:2603.10414v2,\nSection 7, Question 20(iii), define the collision density of an arbitrary bracket\nexpression and separately ask: “Is it always rational?” Section 1 specifies\ninteger coefficients and positive rational moduli.\n\nFor every finite $t$, coefficients $c_i\\in\\mathbb Z$ and positive\n$\\alpha_i\\in\\mathbb Q$, set\n$w_n=\\sum_{i=1}^t c_i\\lfloor n\\alpha_i\\rfloor$ for $n\\geq0$ and\n$W=\\{w_n:n\\geq0\\}$. For every positive integer $k$, set\n$C(k)=\\{x\\in W:x+k\\in W\\}$. The result proves that\n\n$$\\lim_{N\\to\\infty}\\frac{|C(k)\\cap\\{0,\\ldots,N\\}|}{N}$$\n\nexists and is rational. This is the complete rationality subquestion, including\nexistence of the limit, rather than the other requests in Question 20.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Every finite number of terms, including zero | `t : ℕ` and `Fin t` |\n| Integer coefficients of either sign | `c : Fin t → ℤ` |\n| Positive rational moduli | `α : Fin t → ℚ` and `∀ i, 0 < α i` |\n| Natural sequence indices including zero | `bracket c α n`, `n : ℕ` |\n| Actual distinct image values at $x$ and $x+k$ | `collision` has two independent existential indices; the count filters values once |\n| Every positive integer shift | `k : ℕ` with `0 < k` |\n| Inclusive nonnegative interval | `Finset.range (N + 1)` |\n| Denominator $N$ | Real cardinality divided by `(N : ℝ)` |\n| Existence and rationality of the whole limit | `∃ a : ℚ, Tendsto … atTop (𝓝 (a : ℝ))` in `claim` |\n\nNo monotonicity, injectivity, positive drift or nonnegative-coefficient premise\nis imposed. The empty expression, zero coefficients and all drift signs remain\nin scope.","title":"Rational collision density of bracket expressions","url":"https://the-omega-institute.github.io/trureturing-pages/research/larsson-manabe-2026-rational-bracket-collision-density/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/56420900ff2c2be13ead380912d852c129ddb788d6be6925e83a373a51ccfb1d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lau-2026-synchronous-doily-one-tenth","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Tony Lau, *Beyond the Magic Square Game: Widening the Gap for Two Bell States*, arXiv:2603.20748v2, §5, asks whether a value below 31/35 occurs for some p < 1/7 and identifies the one-tenth candidate: if the asymmetric strategy bound in Lemma 4.3 can be strengthened to twelve synchronous wins out of fifteen, then the 1/10-synchronous doily game has classical value 22/25. The settled statement is\n\n`classicalValue (1/10) = 22/25`,\n\nwhere `classicalValue` is the supremum over all deterministic pairs of functions from the fifteen equations to Boolean triples, the nonsynchronous distribution is uniform over the 90 ordered intersecting pairs, and the synchronous distribution is uniform over the fifteen diagonal pairs. The referee predicate uses the fifteen equations in Table 3, their stated right-hand sides, parity validity, and equality of every shared variable.","title":"The one-tenth synchronous doily value","url":"https://the-omega-institute.github.io/trureturing-pages/research/lau-2026-synchronous-doily-one-tenth/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22d8694c0b7a44bad44696ab27f0e338e8dabc5934ddbf1055264f1827ca1320.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lazorec-tarnauceanu-2025-rational-subgroup-conjugacy-attainment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mihai-Silviu Lazorec and Marius Tărnăuceanu, *On the number of conjugacy\nclasses of subgroups of a finite group*, arXiv:2508.14267v2, Section 4,\nProblem 4.1:\n\n> Let a,b be integers such that $1\\leq a any ratio of the form $\\frac{a}{a+1}$ can be achieved by computing $d'$\n> for specific non-nilpotent supersolvable groups having one conjugacy\n> class of non-normal subgroups. We also showed that each ratio of the\n> form $\\frac{a}{b}$ can be achieved as the limit of $(d'(G_n))_{n\\geq1}$\n> where $(G_n)_{n\\geq1}$ is a sequence of nilpotent groups formed as\n> direct products of some modular p-groups. Hence, we pose the following\n> question: Is it true that there is a group G such that\n> $d'(G)=\\frac{a}{b}$?\n\nSection 1 specifies that all groups are finite, defines $L(G)$ as the\nsubgroup lattice and $k'(G)$ as the number of conjugacy classes of\nsubgroups, and sets $d'(G)=k'(G)/|L(G)|$. Bottom and top belong to $L(G)$.\nConjugacy means inner conjugation by elements of $G$.\n\nThe exact settling statement is\n\n```lean\n∀ a b : ℤ, 1 ≤ a → a < b → ∃ W : FiniteGroup,\n b * (classCount W.Carrier : ℤ) = a * (latticeCount W.Carrier : ℤ)\n```\n\nThe source clauses correspond as follows.\n\n| Source clause | Formal meaning |\n| --- | --- |\n| Integers $a,b$ | Universal `a b : ℤ`, with no coprimality or reduced-fraction condition |\n| $1\\leq a Number of n X 1 0..1 arrays with every row and column least squares fitting to a zero slope\n> straight line, with a single point array taken as having zero slope.\n\nThe comment carrying the question, by Gus Wiseman, Jan 07 2023:\n\n> Conjecture: A binary word is counted iff it has the same sum of positions of 1's as its reverse,\n> or, equivalently, the same sum of partial sums as its reverse.\n\nThe companion entry OEIS A359402, which lists the numbers whose binary expansion satisfies the\nposition condition, repeats the same bridge in its own comments:\n\n> Conjecture: Also numbers whose binary expansion has as least squares fit a line of zero slope,\n> counted by A222955.","title":"Zero Slope Fits a Sample Best Exactly When Its Weighted Index Sum Is Balanced","url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e5e577113461bd29146a76afe92a76950056a41b9e6f23d700a5e9f9939768a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-2026-reduction-operator-invertibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eunghyun Lee, *Integrability of multispecies long-range swap models with\nspecies-dependent interpolation*, arXiv:2604.12136v1, Remark 3.1 after\nEquation (28), source label `1122am331`, conjectures:\n\n> These results lead us to conjecture that $\\mathfrak A_k$ is invertible for all parameters $\\mu_i \\in [0,1]$.\n\nLEE-A is preregistered in [#14343](https://github.com/the-omega-institute/trureturing/issues/14343).\nFor every $N\\ge1$, $n\\ge2$, $j\\ge1$, $k\\ge0$ with $j+k+1\\le n$ and\n$\\mu\\in[0,1]^N$, the literal recursion\n$\\mathfrak A_0=I$,\n$\\mathfrak A_k=I-\\mathcal B_{k+1}\\mathfrak A_{k-1}^{-1}\\mathcal B'_k$\nhas invertible matrices on words $\\operatorname{Fin}(n)\\to\\operatorname{Fin}(N)$.","title":"lee 2026 reduction operator invertibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-2026-reduction-operator-invertibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/977ececb35f05c27ddf71a82e324633753887cb58f7a44a13bb998bef5966b6d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-baek-park-kim-nha-2022-output-coherence-tensorization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Lee, K. Baek, J. Park, J. Kim and H. Nha, *Fundamental limits on concentrating and preserving tensorized quantum resources*, Phys. Rev. Research 4, 043070 (2022), arXiv:2104.12307, ask whether correlated inputs preserve more of a resource through parallel noisy channels than product inputs.\n- For coherence they use the maximal coherence η(ρ) = max_{i≠j} |ρ_ij|/√(ρ_ii ρ_jj) of Lami, Regula and Adesso (Eq. (15)). It governs the fidelity of asymptotic coherence distillation under strictly incoherent operations.\n- They define the output maximal coherence η̃(Φ) = max_ρ η(Φ(ρ)) (Eq. (28)).\n- **Conjecture 1** (Eq. (29)): η̃(Φ₁ ⊗ Φ₂) = max{η̃(Φ₁), η̃(Φ₂)}. The authors support it with numerics on random channels in dimensions (2, 2) and (2, 3).\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/leebaekparkkimnha2022tensorizedresources.md).\n\nIssue [#18019](https://github.com/the-omega-institute/trureturing/issues/18019) fixes the reading: for all finite-dimensional channels, given by Kraus families, Eq. (29) holds, with the supremum over all density matrices. This conjecture is refuted.","title":"The output maximal coherence of channels is not tensorization-stable","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-baek-park-kim-nha-2022-output-coherence-tensorization-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa20146c01b9f91e27f683aff26e193546c24912d50c37549f5ad01169c5ac7b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-heinosaari-2026-adaptive-query-four-qubit-states","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Lee and T. Heinosaari, *Perfect Discrimination of Non-Orthogonal Quantum States via Adaptive Post-Measurement Queries*, arXiv:2608.01167v1, study state discrimination with a one-bit query.\n- **Pre-measurement query.** P_pre maximizes, over the binary partitions of the symbols and the measurements of the two subensembles, the success probability after the answer is known (Eq. (3)).\n- **Adaptive post-measurement query.** P^adap_post maximizes over a measurement whose outcome selects the partition and the guesses (Eq. (5)).\n- **The suggestion.** \"A relevant question is whether there exists an ensemble of four qubit states with this property. … These examples suggest that P^adap_post > P_pre may never occur for qubit ensembles with n ≥ 4, though we leave a proof as an open question\" (§VI, p. 6).\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/leeheinosaari2026adaptivequery.md).\n\nIssue [#17780](https://github.com/the-omega-institute/trureturing/issues/17780) fixes the reading: for every n ≥ 4 and every ensemble of n qubit states (priors q_x ≥ 0 with Σ q_x = 1 and 2 × 2 density matrices), P^adap_post ≤ P_pre. This suggestion is refuted, and the §VI question is answered affirmatively, by an ensemble of four qubit states.","title":"A four-state qubit ensemble where an adaptive post-measurement query beats a pre-measurement query","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-heinosaari-2026-adaptive-query-four-qubit-states/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a36f5ed890f1c3a54d36a91bf5fe67a665ad9fac437c6686e466e703eceed27.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-kjoshanssen-2026-quantum-cerny-pure-target","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pui Hang Lee, Pui Yee Lee and Bjørn Kjos-Hanssen, arXiv:2609.40154v1,\n§6 item 5:\n\n> Does a general quadratic-type saving qc_pure(w) ≤ O(√|w|) hold,\n> i.e. a pure-target analogue of Theorem 2.4?\n\nAn instance consists of two quantum channels on d by d density matrices\nand a start state. Letters act from left to right. The reachable set contains\nthe images of the start state under all binary words. A word synchronizes\nwhen its channel is constant on this set. The complexity qc_pure(w) is the\nleast dimension of an instance in which w is the unique shortest\nsynchronizing word and its common output is pure.\n\nThe precise proposed uniform bound is the existence of a real constant C\nand a natural threshold m₀ such that qc_pure(w) ≤ C√|w| for every binary\nword with |w| ≥ m₀.","title":"Pure-target quantum Černý complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-kjoshanssen-2026-quantum-cerny-pure-target/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97078bb96bd67ab11f01c72f5d788db832ea8aec6db0ef8c8815846f3d826224.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-kjoshanssen-2026-quantum-cerny-thue-morse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lee, Lee and Kjos-Hanssen, *Quantum Černý complexity of binary words*,\narXiv:2609.40154v1, define the quantum Černý complexity $\\mathrm{qc}(w)$ as the\nleast $d$ for which two quantum channels on $d\\times d$ density matrices and a\nstart state make $w$ the unique shortest synchronizing word (Definition 1.1).\nOpen problem 1 reads\n\n> (Thue--Morse prefix.) We conjecture $\\qc(01101001)=2$; the bound\n> $\\qc(01101001)\\le3$ follows from Theorem~\\ref{thm:kmp}.\n\nIssue #13185 fixes the reading: channels and density matrices are the frozen\n`QuantumChannel (Fin d) (Fin d)` and `DensityState (Fin d)`, letters act left\nto right, the reachable set is the set of images of the start state, and\n`qc w = sInf {d | 1 ≤ d ∧ some instance of dimension d has w as its unique\nshortest synchronizing word}`.","title":"The Thue–Morse prefix 01101001 has quantum Černý complexity 3, not 2","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-kjoshanssen-2026-quantum-cerny-thue-morse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dae08c5e7060afa8f8561b22da8a929662999e52c61dfe240f215f8404a13a96.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lee-liu-2025-integer-two-domination-three-row","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jia-Ying Lee and Chia-An Liu, *The integer {2}-domination number of grids*,\narXiv:2502.00134v1, Conjecture 2.5:\n\n> For positive integer n, we have γ{2}(G3,n) = ⌊3n/2⌋ + 1.\n\nThe settlement is **Proved** for every positive width. Its preregistration is\n[#17651](https://github.com/the-omega-institute/trureturing/issues/17651).\nHere $G_{m,n}=P_m\\mathbin{\\square}P_n$. An integer {2}-dominating function assigns\neach vertex a value in $\\{0,1,2\\}$ and has sum at least $2$ on each closed\nneighbourhood; $\\gamma_{\\{2\\}}$ is the minimum total weight.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Cartesian product of paths $P_m\\mathbin{\\square}P_n$ | `grid m n`, using `SimpleGraph.pathGraph` and `SimpleGraph.boxProd` | Vertices in `Fin m × Fin n` represent the source's one-based labels. |\n| Values in $\\{0,1,2\\}$ | `IsTwoDominating m n f` bounds every natural value by $2$ | Exactly the abstract and Section 4 domain. |\n| Sum on each closed neighbourhood is at least $2$ | The second conjunct of `IsTwoDominating` adds the vertex value to its open-neighbour sum | Each closed-neighbourhood vertex is counted once. |\n| Minimum total weight | `gamma2 m n` is the natural infimum of attained feasible weights | The constant function $2$ is feasible, so the weight set is nonempty and its infimum is its minimum. |\n| Every positive integer $n$ | `claim` quantifies over every natural $n$ with $1\\le n$ | No upper cutoff or additional hypothesis. |\n| $\\lfloor3n/2\\rfloor+1$ | Natural division in `claim` | Natural division equals the floor for these nonnegative integers. |\n| Equality for three rows | `result : claim` | Both bounds, including width $1$, are proved. |\n\nSection 1 also allows arbitrary nonnegative integer values. Truncating each\nvalue at $2$ preserves feasibility: a neighbourhood containing a value at least\n$2$ still has sum at least $2$, and otherwise its values are unchanged. Weight\ncannot increase. Thus the two minimum-weight formulations coincide. This\ncorrespondence is a prose argument; the Lean predicate uses the bounded domain.\nThe general row parameter in the definitions names the graph family and adds no\nhypothesis to the three-row conclusion.","title":"Lee–Liu integer {2}-domination of three-row grids","url":"https://the-omega-institute.github.io/trureturing-pages/research/lee-liu-2025-integer-two-domination-three-row/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/224ca9f1e36a8682210cce3f12f0792c0b2d597faf6573cc95b244b62f296ce0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lemos-moura-ribas-silva-2024-consecutive-davenport-conjecture-3-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Lemos, A. O. Moura, S. Ribas and A. T. Silva, “A note on weighted consecutive Davenport constant,” arXiv:2404.11312v1, Conjecture 3.4, p. 3, asks whether $C(G)=|G|$ for every finite group $G$. The settlement is **Proved** under the literal conventions preregistered in [#19765](https://github.com/the-omega-institute/trureturing/issues/19765).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Ordered sequences over $G$ | `List G` | The ordered product is `List.prod` in list order. |\n| Ordered subsequences $g_i\\cdots g_j$ with $1\\le i\\le j$ | `T ≠ [] ∧ T <:+: S` | Nonempty contiguous infix is literal. |\n| Consecutive product-one condition for $A=\\{1\\}$ | `HasConsecutiveProductOne S` | `T.prod = 1` keeps the unweighted convention. |\n| Smallest positive threshold | `C G` | `sInf` of the positive natural thresholds with the stated property. |\n| Every finite group has $C(G)=|G|$ | `claim` and `result` | `result` proves the complete quantified proposition. |","title":"Lemos–Moura–Ribas–Silva Conjecture 3.4","url":"https://the-omega-institute.github.io/trureturing-pages/research/lemos-moura-ribas-silva-2024-consecutive-davenport-conjecture-3-4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09c7297265a45b638b1f647dfbb876c53e63dd085977bd7a0af2b1429636d3e7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:len-2022-noisy-metrology-cat-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. L. Len, T. Gefen, A. Retzker and J. Kołodyński, *Quantum metrology with\nimperfect measurements*, Nature Communications 13, 6971 (2022),\narXiv:2109.01160v2. An imperfect measurement multiplies the quantum Fisher\ninformation by\n\n> $\\gamma_\\cM=\\max_{\\ket{\\xi},\\ket{\\xi_{\\perp}}} \\sum_x \\frac{\\mathrm{Re}\\!\\left\\{\\langle\\xi_{\\perp}|M_{x}|\\xi\\rangle\\right\\}^{2}}{\\langle\\xi|M_{x}|\\xi\\rangle}$\n\nand for $N$ probes measured independently $M_{\\xvec}=M_{x_1}\\otimes\\cdots\\otimes\nM_{x_N}$. The supplement conjectures:\n\n> for any classical noise channel, $M_{x}=\\sum_{i}p\\!\\left(x|i\\right)\\Pi_{i}$ that is applied independently on each of the $N$ probes, the optimal $|\\zeta^{N} \\rangle,|\\zeta_{\\perp}^{N}\\rangle$ take the form of ``cat states\": $|\\zeta^{N} \\rangle =\\cos\\left(\\theta\\right)|j\\rangle^{\\otimes N}+\\sin\\left(\\theta\\right)|k\\rangle^{\\otimes N}$, $|\\zeta_{\\perp}^{N} \\rangle =-\\sin\\left(\\theta\\right)|j\\rangle^{\\otimes N}+\\cos\\left(\\theta\\right)|k\\rangle^{\\otimes N}$\n\nHere $\\zeta,\\zeta_\\perp$ are the images of $(\\psi\\pm\\psi_\\perp)/\\sqrt2$ under the\nencoding unitary, so the optimized pair is\n$\\xi=(\\zeta+\\zeta_\\perp)/\\sqrt2$, $\\xi_\\perp=(\\zeta-\\zeta_\\perp)/\\sqrt2$.","title":"Optimality of cat pairs for metrology with noisy measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/len-2022-noisy-metrology-cat-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/986a9f52e6ae5fd34012016ff51cbac8eeeb76e81591596ae8ea9c35e39636a9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lesaulnier-vijay-2011-beta3-lower-density","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"LeSaulnier and Vijay, arXiv:1004.1740, *Discrete Mathematics* 311 (2011), p. 207,\ndefine a subset of the positive integers to be 3-free when it admits a permutation containing\nno three-term arithmetic progression as a subsequence. Its lower density is\n\n\\[\n\\underline d(S)=\\liminf_{n\\to\\infty}\\frac{|S\\cap[1,n]|}{n},\n\\qquad\n\\beta(3)=\\sup\\{\\underline d(S):S\\text{ is 3-free}\\}.\n\\]\n\nTheir Theorem 3 establishes \\(\\beta(3)\\ge1/4\\), and they conjecture \\(\\beta(3)=1/4\\).\nThe formal claim is the universal upper bound: for every \\(S\\subseteq\\mathbb N\\) with\n\\(0\\notin S\\), if there is an injective \\(\\pi:\\mathbb N\\to\\mathbb N\\) whose range is\nexactly \\(S\\) and such that\n\n\\[\n\\forall i We checked by a direct calculation that for all configurations of up to $N=10$ the established reverse level order labeling leads to an adjacency matrix, where each vector $A_{i+1:N,i}$ contains only\n> one non-zero component and therefore the respective quadrature combinations (\\ref{uxpi}) contain at most two non-zero terms. This leads us to a conjecture\n> that this statement holds generally for an arbitrary $N$-vertex tree.\n\nThe source construction, lines 828–835, roots the tree at its center vertex or at an endpoint of its central edge, assigns label $N$ to the root, and assigns decreasing labels level by level, from left to right within each level. The paper assumes $N\\ge3$ for its entanglement application (lines 429–431). The source quotations and bibliographic entry are also in [the literature note](../Library/GraphInvariants/leskovjanovamista2025gaussiangme.md). The checked TeX member has MD5 `b1e8a86c3fefec6858c4ac36ca46f9fd`.\n\nIssue [#21778](https://github.com/the-omega-institute/trureturing/issues/21778) fixes the graph statement with an arbitrary root and arbitrary order within each level. In the module, for every $n\\in\\mathbb N$, every simple graph $g$ on $\\operatorname{Fin} n$ with decidable adjacency and $g.\\mathrm{IsTree}$, every root $r\\in\\operatorname{Fin} n$, and every bijection $\\ell:\\operatorname{Fin} n\\simeq\\operatorname{Fin} n$ satisfying\n\n$$\n\\operatorname{dist}_g(r,u)<\\operatorname{dist}_g(r,v)\n\\ \\Longrightarrow\\ \\ell(v)<\\ell(u),\n$$\n\ndefine $A^\\ell_{ji}$ to be the natural-valued adjacency entry of $g$ at $\\ell^{-1}(j),\\ell^{-1}(i)$: it is $1$ for adjacent vertices and $0$ otherwise. Then `claim` states\n\n$$\n\\forall i\\in\\operatorname{Fin} n,\\quad\ni.\\mathrm{val}+1 No claim is made that either set $S$ or $\\hat S$ is globally \\textbf{minimal} for the class of events with which it is concerned. That is to say: it has not been proved that there does not exist some other set $S_?$ containing only one or two event variables but which retains all the stated properties.\n> Nonetheless, we \\textit{conjecture} that there is indeed no smaller set $S_?$ and that $S$ and $\\hat S$ are therefore examples of \\textbf{minimal} sets.\n\nThe quotation is from the downloaded v6 TeX member `introduction.tex` (md5 `2b21a1456fcb7bb5df295f300f8b61ae`). The definitions and source locators are collected in [the literature note](../Library/Geometry/lesterhaddadingripaios2022parity.md).\n\nAn event is an ordered quadruple $(p,q,a,b)$ of real four-vectors. Write $u\\cdot v=u_0v_0-u_1v_1-u_2v_2-u_3v_3$ and $u^2=u\\cdot u$. The forward cone consists of vectors with $u_0\\ge0$ and $u_1^2+u_2^2+u_3^2\\le u_0^2$, including zero. The two domains are:\n\n- $\\Omega_{(pq)(ab)X}$: all quadruples of forward-cone vectors (`allEvents`);\n- $\\Omega_{(pq)\\to(ab)X}$: those quadruples additionally satisfying $p^2q^2<(p\\cdot q)^2$ (`collisionEvents`).\n\nThe symmetry uses a common matrix $\\Lambda$ satisfying $\\Lambda^{\\mathsf T}\\eta\\Lambda=\\eta$, $\\det\\Lambda=1$ and $\\Lambda_{00}>0$, where $\\eta=\\operatorname{diag}(1,-1,-1,-1)$, together with independent permutations of $(p,q)$ and $(a,b)$. Parity $\\mathscr P$ fixes each energy and negates each spatial momentum. An event is chiral when no such symmetry carries it to its parity image. These match `prelims.tex`, lines 25–27, 190–193, 228–243, 265–268, 433–443 and 656–666; neither domain adds momentum conservation or fixed masses.\n\nIssue [#25338](https://github.com/the-omega-institute/trureturing/issues/25338) fixes the assertion: for either domain and every pair $f_1,f_2:\\mathrm{event}\\to\\mathbb R$ continuous on that domain, invariant under these symmetries there, and parity odd there, some chiral event in the domain satisfies $f_1=f_2=0$. This assertion is proved.","title":"Three continuous invariant parity-odd variables are necessary for two interchangeable pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/lester-haddadin-gripaios-2022-chiral-event-parity-minimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f5caae42c938aa572479e7d32e6b388b2c8a7c8eaacd4b4dc5ac54dd9f74798.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:levens-2025-pascalian-galois-square-discriminant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Owen Levens, *Polynomials Arising from Sorted Binomial Coefficients*,\narXiv:2511.03082v1, Section 6.2, Conjecture 6.4 states:\n\n> P_n(z) has full Galois group for even n. For n = 2m + 1, the Galois group of P_n(z) is the mth hyperoctahedral group.\n\nThe source defines the Pascalian numbers by\n$\\langle n,k\\rangle=\\binom n{\\lfloor(n-k)/2\\rfloor}$ for $0\\le k\\le n$\nand $P_n(z)=\\sum_{k=0}^n\\langle n,k\\rangle z^{n-k}$.\nThe target is the second sentence, with $m\\ge1$. The counterexample is\n$m=24$, $n=49$. Preregistration: [#19007](https://github.com/the-omega-institute/trureturing/issues/19007).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Pascalian coefficient and full polynomial | `P` over rational polynomials, summing every index from zero through n | Natural division is the stated floor; the exponent uses natural subtraction in the source range. |\n| n = 2m + 1 | `claim` substitutes `2 * m + 1` into `P` | Every natural m with `1 ≤ m` is included; 24 is included. |\n| mth hyperoctahedral group | `hyperoctahedral m`, the centralizer of `signFlip m` on `Fin m × Bool` | The two Boolean values represent the signs. These are precisely signed permutations. |\n| The Galois group is that group | Existence of an abstract multiplicative equivalence from `Polynomial.Gal` | This weakest isomorphism reading is refuted, hence so is the permutation-group reading. |\n| Refutation of the universal odd sentence | `result : Not claim` | No extra mathematical hypothesis. The even sentence is outside `claim`. |","title":"A square discriminant refutes the odd clause of Levens Conjecture 6.4","url":"https://the-omega-institute.github.io/trureturing-pages/research/levens-2025-pascalian-galois-square-discriminant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b172cfd0c1f3105861f1d2463ca84d09596a54360b9820a2eca09a76594683b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lewis-won-crossword-rook-counts","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joel Brewster Lewis and Robert Won, *Non-attacking rook placements on crossword grids*, arXiv:2609.03081v1, Question 4.2:\n\n> For which positive integers r does there exist a crossword grid which admits exactly r rook placements? Can the crossword grids which\n> admit exactly r rook placements be characterized?\n\nThe paper observes that block-diagonal juxtaposition multiplies counts, so that it suffices to treat prime `r`, and it conjectures that\npermutation grids miss infinitely many values.","title":"Every Natural Number Counts the Rook Placements of a Crossword Grid","url":"https://the-omega-institute.github.io/trureturing-pages/research/lewis-won-crossword-rook-counts/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81f44d5dc5e61d7e76d462eb4ec009abd1bac3726da5394ef17144d4969740bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lewis-won-permutation-grid-counts-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joel Brewster Lewis and Robert Won, *Non-attacking rook placements on crossword grids*, arXiv:2609.03081v1, Section 3.3:\n\n> Conjecture 3.10. A positive integer r occurs as the rook placement count of a permutation grid if and only if r ≠ 4, r ≠ 12, and\n> r ≢ 3 (mod 4).\n\nThe paper supports the conjecture by exhaustive computation over permutations with `n ≤ 8`, recorded for the counts `r ≤ 108`.","title":"A Permutation Grid with 155 Rook Placements","url":"https://the-omega-institute.github.io/trureturing-pages/research/lewis-won-permutation-grid-counts-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/098468a5b0af4294f30980ec0acf93fb8a52e33f489b5298f73de6a5f1c3eaf1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lewis-won-skew-merged-rook-placements","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joel Brewster Lewis and Robert Won, *Non-attacking rook placements on crossword grids*, arXiv:2609.03081v1, Section 3.3:\n\n> Conjecture 3.9. A permutation w ∈ S_n is skew-merged if and only if Grid(w) admits exactly one or two rook placements.\n\nGrid(w) is the n × n board whose black squares are (i, w(i)); a rook placement is a set of white squares meeting every across word and every\ndown word exactly once. A permutation is skew-merged when its entries split into an increasing and a decreasing subsequence. The paper proves\nthe forward direction (Proposition 3.8) and characterizes exactly one placement (Theorem 3.7).","title":"Skew-Merged Permutations Are Exactly Those With One or Two Rook Placements","url":"https://the-omega-institute.github.io/trureturing-pages/research/lewis-won-skew-merged-rook-placements/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c55ba02a1d1df82d63535170c4b7d70dd03c3151d9eacb018cc4a580b6ddcc7c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-2025-coefficient-matrix-forms-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Li, *A necessary and sufficient condition for genuinely entangled n-qubit states with six non-zero\ncoefficients*, arXiv:2510.16561v1, Section 6, closes with the question: \"What to do next is how to find\nthe forms of the corresponding coefficient matrices. When $m=4$, there are two forms [Li-24]. When $m=6$,\nthere are four forms. How many forms of the corresponding coefficient matrices are there when $m=2p$?\"\nHere an $n$-qubit state with $m=2p$ non-zero coefficients $b_i$ on basis states $B_1<\\cdots The proposed conjecture is that **the w-optimization problem has arbitrary\n> even-order convexity**, more specifically, for $k \\in \\{1,2,3,\\cdots\\}$ we always have\n\n$$\\frac{d^{2k}}{dw^{2k}}\\ln\\det(P(w))\\geq 0,\n\\qquad \\frac{d^{2k}}{dw^{2k}}\\operatorname{tr}(P(w))\\geq 0.$$\n\nIssue [#11592](https://github.com/the-omega-institute/trureturing/issues/11592)\npreregisters the source quotations, quantified statement, Tier 1 classification\nand literature-search boundary. The reading retains both inequalities for\nevery positive even order and every interior weight, with the semidefinite\nhypotheses of Section 2.1.","title":"Arbitrary even-order convexity of split covariance intersection","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-2026-split-cif-even-order-convexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a61060ae2217c88d93f26cfe31fb31b4b66cbaadd2c696fc47de03ee305606e3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-guo-signless-laplacian","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wen-Jun Li and Ji-Ming Guo, *A new conjecture on Laplacian eigenvalues and degree\nsequence of a graph*, Filomat 39:13 (2025), 4551–4564, Conjecture 2.22, asserts\n\n$$\n\\sum_{q_i\\geq2}(q_i-2)^2\\leq\n\\left(1-\\frac1{d_1}\\right)\\sum_{i=1}^n d_i(d_i-1).\n$$\n\nThe graph is finite, simple, undirected and loopless; $Q=D+A$ is its signless\nLaplacian, and all eigenvalues occur with algebraic multiplicity. Degrees are\nordered decreasingly in the source. The denominator restricts the assertion to\n$d_1=\\Delta(G)>0$. There is no connectedness or regularity hypothesis.\nPreregistration: [#23307](https://github.com/the-omega-institute/trureturing/issues/23307).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A graph with $n$ vertices | `n : ℕ`, `G : SimpleGraph (Fin n)` in `claim` | All finite labelled simple graphs; no additional structural restriction. |\n| $Q=D+A$ and the eigenvalues $q_i$ | Existing `sourceMatrix G .signless` and `graphSpectrum G .signless` | Actual real eigenvalues, retaining multiplicity. |\n| Sum over $q_i\\geq2$ | `TQ` | Sum of $\\max(q-2,0)^2$ over the entire spectral multiset; the summand at $q=2$ is zero. |\n| $d_1$ | `SimpleGraph.maxDegree` | The maximum vertex degree. |\n| Degree-sequence sum | `R` | Sum over vertices; permutation of the degree sequence preserves it. |\n| $1/d_1$ is defined | `0 < G.maxDegree` | Positive denominator domain; an edgeless graph is excluded. |\n| The displayed upper bound | `TQ G ≤ R G` | Every degree is cast to $\\mathbb R$ before subtraction and division. |\n| Universal assertion is false | `result : ¬ claim` | A triangle contradicts the unchanged assertion. |\n\n`TQ`, `R` and `claim` retain the literal binding definitions in #23307. Graph\nmatrices, degrees and multiplicity-preserving spectra reuse their existing owners.","title":"Li–Guo Conjecture 2.22","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-guo-signless-laplacian/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e04a044ad0b56958defe2f2651ace4fa5474491f0f39fd9dce40e46c52b77136.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-jiang-2026-fixed-noise-polar-pair-optimality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Li and L. Jiang, *High-Rank Encoding Can Improve Approximate Quantum Error Correction*, arXiv:2609.00778v1, compare encoders of a logical qudit into two qudits under a family of noise channels N_p (Eqs. (18)–(23)).\n- **The pair.** For a unit vector χ, the encoder C_{V_p(χ)} uses the polar isometry V_p(χ) = C_p(χ)[C_p(χ)†C_p(χ)]^{−1/2} with C_p(χ) = Q_p(I ⊗ |χ⟩) (Eq. (S60)), and the decoder is the partial trace tr_S. The supplement shows that this rank-one pair attains the optimal quadratic coefficient of the rank-one gap as p → 0 (Eqs. (S63)–(S64)).\n- **The question.** \"Exact optimality of this pair at fixed p > 0 remains open\" (after Eq. (S64)).\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/lijiang2026highrank.md).\n\nIssue [#13324](https://github.com/the-omega-institute/trureturing/issues/13324) fixes the reading. For every d ≥ 2, 0 < p < 1 and unit χ, the pair is claimed to be optimal against every pair of completely positive trace-nonincreasing encoder and decoder whose encoder has input-first Choi rank at most one (the source's rank-one class, Eq. (13)), for the entanglement fidelity at τ = I/d. The question is answered negatively.","title":"The polar rank-one encoder of Li–Jiang is not exactly optimal at fixed noise","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-jiang-2026-fixed-noise-polar-pair-optimality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c99b73c8cbca4b83a7786b6dc3042c27cf74b996b71d4a20434fb24f650c7b9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-luo-zhang-2025-fisher-discord-partial-trace-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Li, S. Luo and Y. Zhang, *Fisher discord as a quantifier of quantum\ncomplexity*, arXiv:2510.15375v1; published as *Fisher discord as a quantifier\nof quantum metrological complexity*, Phys. Rev. A 114, 032449 (2026).\n\n§VI (Summary), verbatim:\n\n> Both the quantum Fisher information based on the symmetric logarithmic derivative and the skew information does not increase under the partial traces, that is, I_F(ρ₁₂, H ⊗ 1) ≥ I_F(ρ₁, H), I_W(ρ₁₂, H ⊗ 1) ≥ I_W(ρ₁, H) … It is natural to consider such an open question: Whether quantum Fisher discord C(ρ₁₂, H ⊗ 1) = I_F(ρ₁₂, H ⊗ 1) − I_W(ρ₁₂, H ⊗ 1) for bipartite states does not increase under the partial trace.\n\n| Source object | Formal reading |\n|---|---|\n| Density matrix | A positive semidefinite finite complex matrix of trace one, `SLDSpectralEnergy.IsDensity`. |\n| Second-factor partial trace | `Defs.partialTrace₂ ρ i j = ∑ b, ρ (i,b) (j,b)`. |\n| Local observable | Hermitian `H`, lifted as `Defs.liftLeft H = H ⊗ 1`. |\n| Equation (6) | `Defs.IsSLD ρ H L`: Hermitian `L` with $i[\\rho,H]=(L\\rho+\\rho L)/2$; `Defs.fisherSLD ρ L` is $\\tfrac14\\operatorname{Re}\\operatorname{tr}(\\rho L^2)$. |\n| Equation (7) | `Defs.skewWY ρ H` is $-\\tfrac12\\operatorname{Re}\\operatorname{tr}[\\sqrt\\rho,H]^2$, with the positive matrix square root. |\n| Equation (10) | $C=I_F-I_W$, expressed as the difference of those two real quantities. |\n| Universal comparison | `Defs.claim` quantifies over both finite dimensions, every joint density matrix, every Hermitian local observable and every Hermitian SLD solution on both spaces, and requires $C(\\operatorname{tr}_2\\rho,H)\\le C(\\rho,H\\otimes1)$. |","title":"Partial trace can increase Fisher discord","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-luo-zhang-2025-fisher-discord-partial-trace-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d29c5d04c79897b020b01f7e13a9f72e3bca54a138daf8b92705768094d3c5a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-morgan-siegrist-2026-inflection-points-conjecture-6-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Li, Morgan and Siegrist, *Concavity structure of fractional differential equations and zeros of Mittag–Leffler functions*, Conjecture 6.1, asks whether every solution under Theorem 5.2 has exactly one sign-changing zero of its second derivative in $(0,1)$. The source explicitly states that uniqueness remains open. The verbatim source and literature screen are in [#18772](https://github.com/the-omega-institute/trureturing/issues/18772), and the source locators are in [the Library note](../Library/Certificates/li2026concavity.md).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Interpretation |\n| --- | --- | --- |\n| Definition 1.1, the left-sided Riemann–Liouville fractional integral | `rlIntegral` | Lower endpoint zero, `Real.Gamma`, real powers and the interval integral; positive order and positive argument give the source interpretation. |\n| Equation (7), the solution and its boundary constant | `solution` | The source expression with the boundary constant substituted. Equivalence with the fractional Sobolev weak solution is supplied by the paper. |\n| $1<\\alpha<2$ | Two order hypotheses of `claim` | All real orders in this interval. |\n| $f\\in C([0,1])\\cap C^1((0,1])$ | First two conjuncts of `Hypotheses` | `ContinuousOn` on the closed interval and `ContDiffOn` on the half-open interval. |\n| $f(1)=0$, $f\\geq0$, $f\\not\\equiv0$ | Next three conjuncts of `Hypotheses` | Endpoint equality, pointwise nonnegativity and one nonzero point. |\n| Weighted right derivative limit zero | Last conjunct of `Hypotheses` | `deriv f`, right-sided neighborhood filter and exponent $(3-\\alpha)/(2-\\alpha)$. |\n| Exactly one zero in $(0,1)$ | Existence and uniqueness conjuncts of `claim` | The derivative is `iteratedDeriv 2`; every interior zero equals the selected zero. |\n| Changes sign at that zero | Last conjunct of `claim` | Negative product at every sufficiently close pair on opposite sides. |\n\nThere are no additional hypotheses on the public conclusion. Lean's integral and real powers are totalized; the displayed source interpretation uses their stated domain conditions.","title":"li morgan siegrist 2026 inflection points conjecture 6 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-morgan-siegrist-2026-inflection-points-conjecture-6-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c9a74b6ab156b903b493c5f0511ea193642347f49d608641a6d8d7dc6ef33d03.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-tan-circulant-positivity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mao-Sheng Li and Yi-Xi Tan, *On the Bargmann invariants for quantum imaginarity*, arXiv:2412.08022v1, Section III, page 5, state:\n\n> For the case when n = 3, we deduced that the positive semidefiniteness of C_H follows from the positive semidefiniteness of H. This leads us to the conjecture that the positive semidefiniteness of H implies the positive semidefiniteness of C_H (which may depend on n) for general n.\n\nThe universal implication is **Refuted**, even on real positive-entry correlation matrices at $n=4$. [Preregistration #19081](https://github.com/the-omega-institute/trureturing/issues/19081) specifies the source, conventions and literature screen. The [Library note](../Library/QuantumContext/litan2024bargmann.md) quotes the definitions of $\\mathbf H_n$, $C_n$ and $z_k$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every positive dimension $n$ | `claim`: every natural `n` with `[NeZero n]` | The source matrices have positive order |\n| Every complex matrix $H\\in\\mathbf H_n$ | Complex matrices indexed by `Fin n`, `H.PosSemidef`, and every diagonal entry equal to one | Hermitianity is included in `Matrix.PosSemidef`; no extra restriction |\n| $l\\oplus k$ | `l + Fin.ofNat n k` in `z` | Modular addition with zero-based labels |\n| $r_{ij}\\ge0$ and principal $\\theta_{ij}\\in(-\\pi,\\pi]$ | Complex norm and `Complex.arg` | The source's modulus and phase conventions |\n| Geometric modulus and arithmetic phase means | `z`, including the real $1/n$ power | All indices occur exactly once |\n| $C_H=\\sum_{k=0}^{n-1}z_k C_n^k$ | `CH`, with the source shift written entrywise | The shift has a one exactly when $j=i+1$ |\n| Positive semidefiniteness of $C_H$ | `(CH n H).PosSemidef` | Full positivity conclusion |\n| Refutation | `result : ¬ claim` | One admissible order-four instance disproves the universal assertion |\n\nAll binders and conjuncts implement these clauses. The unit-diagonal restriction makes the claimed implication weaker than an unrestricted matrix implication, so its refutation also refutes that stronger assertion.","title":"Li–Tan circulant positivity is false at order four","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-tan-circulant-positivity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e951ff241902f2b8318fd4ed0bfdda6e6dd1c04559b9fbc0c6111bdf1b6d6cee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-theil-harrow-chuang-2026-purify-and-clone-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z. Li, E. Theil, A. W. Harrow and I. Chuang, *An Exponential Sample-Complexity Advantage for Coherent Quantum Inference*, [arXiv:2605.21457v2](https://arxiv.org/abs/2605.21457v2), Section S3 A, p. 9, states:\n\n> This channel is optimal for m ≤ n (see Ref. [S2]), and we conjecture that it is also optimal for m > n.\n\n[Preregistration #19410](https://github.com/the-omega-institute/trureturing/issues/19410) fixes $n=1$, $m=2$ and the one-site worst-case squared-infidelity reading. The universal optimality assertion is refuted by $d=r=2$. A strict-improvement theorem for every $d\\ge r\\ge2$ is not part of this settlement. The [Library note](../Library/QuantumContext/litheilharrowchuang2026coherent.md) identifies the source and conventions.\n\n### Statement echo\n\n| Source clause or binding convention | Lean counterpart |\n| --- | --- |\n| Squared fidelity and infidelity | `sqFidelity` is the squared real trace of the positive square root; the loss uses one minus that value. |\n| All purifications, equation (S34) | `purifications` contains every positive trace-one rank-one matrix with the specified environment partial trace. |\n| One joint system-environment site | `oneSiteMarginal` retains both registers of either output pair. |\n| One common target before the maximum, equation (S13) | `oneSiteLoss` takes the infimum over the convex hull of purifications after forming the maximum of the two infidelities. |\n| Worst-case rank-at-most-$r$ domain, equation (7) | `worstCaseOneSiteRisk` takes the supremum over every canonical density state with matrix rank at most $r$. Infima and suprema use `EReal`. |\n| Totally symmetric output restriction | `TotallySymmetricSupport` asserts $S\\Xi S=\\Xi$, with $S=(I+F)/2$, on both sides. |\n| Werner cloner and the one-input RP map | `wernerOneToTwo`, `randomPurifyOne` and `purifyAndClone` retain the source matrix expressions, with cloning dimension $dr$. |\n| Genuine channel competitors | `AdmissibleChannel` uses `QuantumChannel` and the support restriction on every admissible input. |\n| Positive dimensions $1\\le r\\le d$ | `claim` retains these quantifiers; the counterexample instantiates them at two. |\n| Claimed optimality | `claim` compares the two worst-case risks for every admissible channel; `result` proves its negation. |","title":"Purify-and-clone fails one-site worst-case optimality","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-theil-harrow-chuang-2026-purify-and-clone-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b84476a6097094e2a59f9a71504dc8906739a2f371d6e5b6dfbf24f6fb40703a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:li-uncu-finite-andrews-gordon-companion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Runqiao Li and Ali K. Uncu, *A MacMahon Analysis View of Cylindric Partitions*, arXiv:2501.19272v1, Section 1,\nConjecture 1.3, equation (1.5): for integers n ≥ 0, k ≥ 5 and k > i ≥ 1,\nΣ_{n₁≥⋯≥n_{k−1}≥n_k=0} q^{n₁²+⋯+n_{k−1}²+n_i+⋯+n_{k−1}} ∏_{j=1}^{k−1} [2n − 2Σ_{ℓ Thus, we conjecture that this property holds for arbitrary even number N,\n> i.e., the Bloch state |ξ1,1,...,1(−π)⟩ (there are N/2 − 1 1’s) should have\n> the largest weight in |ψHKNN(N)⟩.\n\nEq. (6), p. 4, defines the unnormalized HKNN vector by summing the products\nof singlets over every pair partition, with each pair's smaller endpoint\nfirst and no permutation sign. The singlet is\n[i,j] = |↑⟩i|↓⟩j − |↓⟩i|↑⟩j. The normalized Bloch states are the nonzero\ntranslation sums of computational-basis configurations, divided by their\nHilbert norms; Eq. (7) gives the N=6 examples.","title":"The largest Bloch weight in the HKNN state","url":"https://the-omega-institute.github.io/trureturing-pages/research/li-wu-2026-hknn-block-bloch-largest-weight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d62a66d8066881a4310d013263927b313e607829077b8d6daf1f6227fe06b493.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liabotro-2017-signed-pauli-sum-norm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"O. Liabøtrø (arXiv:1607.02667, quant-ph; Phys. Rev. A 95, 052315 (2017))\nstudies maximal `(4^m − 1, m, p)` quantum random access codes. Their\nworst-case success probability is governed by the signed sums\n`Σ(β) = Σ_{k=1}^{4^m−1} (−1)^{β(k)} σ_{c_1(k)} ⊗ ⋯ ⊗ σ_{c_m(k)}` of all\nnon-identity Pauli words, for arbitrary signs `β`. The paper states:\n\n> We have checked numerically that no $\\Sigma(\\beta)$ has an eigenvalue less\n> than $1-(1+\\sqrt(3))^m$ for $m=1,2$. … We conjecture that this is the case for\n> all $m$, i.e.:\n> $||\\sum_{k=1}^{4^m-1}(-1)^{\\beta(k)}\\bigotimes_{i=1}^m\\sigma_{c_i(k)}||\\leq (\\sqrt{3}+1)^m-1$\n\nIssue #11468 fixes the reading. `k ↦ (c_1(k), …, c_m(k))` is a bijection onto\nthe non-identity words, `‖·‖` is the operator norm, and the conjecture is the\ndisplayed inequality for every `m` and every `β`. The formal `claim` is that\ninequality; the result refutes it.","title":"A signed sum of three-qubit Pauli words of norm 21","url":"https://the-omega-institute.github.io/trureturing-pages/research/liabotro-2017-signed-pauli-sum-norm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77c8152bf837422b9e72762bf3875d9611411712d11e8fb1c6e41de65b89056b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liang-sagan-2026-pinnacle-middle-order","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Liang and B. E. Sagan, *Log-concavity and log-convexity via distributive\nlattices*, Order 43 (2026), Article 21, DOI 10.1007/s11083-026-09735-2,\narXiv:2408.02782v2, Section 8, printed pages 34–35, Question 8.6\nand its preceding definition:\n\n> Let σ be a permutation of distinct positive integers. Let Pin_n(σ) = {π ∈ S_n | the pinnacles of π are the elements of σ in that order}. … Computation of examples raise the following questions.\n> Question 8.6. For all n and σ:\n> (a) Is the restriction of the middle order to Pin_n(σ) a distributive lattice?\n> (b) Is the sequence (pin_n(σ)) log-concave?\n\nSection 6, “Permutations”, printed pages 17–18, defines inversions and inversion tops:\n\n> Given π = π1 π2 . . . πn ∈ Sn in one-line notation, its set of inversion (values) is Inv π = {(πi, πj) | i < j and πi > πj}.\n\n> We also say that πi is an inversion top if (πi, πj) ∈ Inv π for some πj.\n\nOn printed page 18 the inversion coordinate and middle order are specified separately:\n\n> where ιi = # of inversions in which i is an inversion top.\n\n> We can now use the bijection just given to define the middle order (Sn, ⪯) by π ⪯ σ if and only if ι(π) ≤ ι(σ) where the order on inversion tables is component-wise.\n\nPinnacles are the values π_j with π_{j−1}<π_j>π_{j+1}.\n\n### Formal reading\n\nLean names have prefix `D5.S0.Certificates.Combinatorics.LiangSaganPinnacle.Result`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Permutation word | `Equiv.Perm (Fin n)` | Increasing positions; Fin value k represents k+1. |\n| Ordered pinnacle sequence | `pinnacleWord` | Values at internal strict local maxima, from left to right. |\n| Inversion-top table | `iota` | Number of smaller values to the right of a value. |\n| Middle order | `middleLe` | Componentwise comparison of `iota`. |\n| Ordered pinnacle class | `Pin` | Fibre of `pinnacleWord`. |\n| Universal distributive-lattice assertion | `claim` | A distributive-lattice structure with exactly this order on every fibre. |\n| Refutation | `result` | Exactly ¬ claim. |\n\nThe source uses distinct positive integers. Repeated values cannot occur\nin a pinnacle word. A source sequence containing a value outside [n]\nhas empty fibre; the counterexample uses only distinct in-range values.","title":"A general obstruction to the pinnacle middle-order lattice","url":"https://the-omega-institute.github.io/trureturing-pages/research/liang-sagan-2026-pinnacle-middle-order/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92b9826df5bc0e8acd23131747f5d1e52701ee405251277f558470e43640641b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lichtenfelz-modin-preston-2026-zeitlin-ricci-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Lichtenfelz, K. Modin and S. C. Preston, *Ricci curvature for hydrodynamics on the sphere*,\narXiv:2508.09833v1; Commun. Math. Phys. 407, 37 (2026), Section 2.2, Conjecture 2: \"For each fixed\n$\\ell > 1$, the averaged Ricci curvature $\\tilde r_\\ell(N)$ of the Zeitlin metric on $SU(N)$ becomes\nnegative for sufficiently large $N$, and in the limit $\\tilde r_\\ell(N)\\to-(H_\\ell-1)/2$, as $N\\to\\infty$.\"\nHere $\\tilde r_\\ell(N)=(r_\\ell(N)^+-r_\\ell(N)^-)/(N^2-1)$ with $r_\\ell(N)^\\pm$ the finite sums of squared\nWigner 6j symbols $\\{\\ell\\,k\\,k';\\,s\\,s\\,s\\}$, $s=(N-1)/2$, over $1\\le k,k' A sublattice dismantling is a subdivision of a lattice into two non-empty sublattices. A lattice is sublattice dismantlable if it can be decomposed into 1-element lattices by consecutive sublattice dismantlings. We have checked with the help of SageMath that $\\Pi(n)$ is sublattice dismantlable for $n=$ 1 to 5. Is this always the case?\n\nThe question quantifies over every positive integer $n$. The source's priority\nforests are increasing rooted forests on $0,\\ldots,n$ with interval components.\nTheir order is edge containment, with an additional greatest element.\nThe existing `PrincipalFilters.Pi` is this literal carrier, and `coreOrderIso`\nidentifies it with `WithTop (IntervalForest n)`.\n\nA subdivision is read as a disjoint partition into nonempty sublattices, closed\nunder the ambient binary meet and join, without requiring the ambient bounds.\nThe identical definition and the interval/complement split in\n[Sage's immutable implementation](https://github.com/sagemath/sage/blob/133eb21d995e6aa5b6de2cd8db798a442e5cc033/src/sage/combinat/posets/lattices.py#L4032)\nsupport this standard meaning. The source does not separately say “disjoint”.\nThe authors' historical Sage version and exact call are unknown. Evidence of an\nintended overlapping-cover meaning would invalidate this semantic bridge.","title":"Sublattice dismantling of the priority lattice","url":"https://the-omega-institute.github.io/trureturing-pages/research/lillo-rosas-2026-priority-lattice-sublattice-dismantling/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd2d65c3ea08a0f5a4a1803157ca863cd808f8df63dcefc72f939370c2d15a57.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lin-2013-operator-wielandt-norm-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Minghua Lin, *On an operator Kantorovich inequality for positive linear maps*, J. Math. Anal. Appl. 402 (2013), 127–132; arXiv:1212.5690v1, §3, pp. 6–7.\n\n§3, p. 6, Theorem 3.1 states verbatim:\n\n> Theorem 3.1. [3] Let 0 < m ≤ A ≤ M and let X, Y be two partial isometries on H whose final spaces are orthogonal to each other. Then for every 2-positive linear map Φ, Φ(X*AY)Φ(Y*AY)⁻¹Φ(Y*AX) ≤ ((M−m)/(M+m))² Φ(X*AX).\n\n§3, pp. 6–7, immediately before Conjecture 3.4:\n\n> Despite the invalidity of (3.6), there is some evidence that the following assertion could be true. We have been unable to prove (or disprove) it.\n\n§3, p. 7, Conjecture 3.4 states verbatim:\n\n> Conjecture 3.4. Under the same condition as in Theorem 3.1. Then for every 2-positive linear map Φ, ‖Φ(X*AY)Φ(Y*AY)⁻¹Φ(Y*AX)Φ(X*AX)⁻¹‖ ≤ ((M−m)/(M+m))².\n\n§3, p. 6, defines 2-positivity verbatim:\n\n> Let Φ be a unital positive linear map between C*-algebras. We say that Φ is 2-positive if whenever the 2 × 2 operator matrix [[A, B],[B*, C]] is positive, then so is [[Φ(A), Φ(B)],[Φ(B*), Φ(C)]].\n\nP. Zhang, *Remarks on an operator Wielandt inequality*, arXiv:1506.00737, describes Conjecture 3.4 as still open.","title":"Lin's operator Wielandt norm conjecture fails","url":"https://the-omega-institute.github.io/trureturing-pages/research/lin-2013-operator-wielandt-norm-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d8ac0eaf3ddb2876f420b90c2a3fc9040d0c8bbda7271728feab0c27f649759.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lin-2016-psi-map-properties","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The fixed source is M. Lin, *New properties for certain positive\nsemidefinite matrices*, arXiv:1609.08231v1, §3, Example 3.6; the journal\nversion is *Linear Algebra and its Applications* 520 (2017), 32–43.\nThe setup states:\n\n> we assume that [A X; X∗ B], where A, X, B are n × n, is positive semidefinite.\n\nDefinition 1.1(iii) states:\n\n> M is said to have a-property if 2sj (M12 ) ≤ λj (M11 + M22 ), j = 1, . . . , n.\n\nDefinition 1.1(iv) states:\n\n> M is said to have g-property if sj (M12 ) ≤ λj (M11 ♯ M22 ), j = 1, . . . , n.\n\nExample 3.6 states:\n\n> If we consider the map Ψ : X ↦ 2(trX)In − X, then using the approach in [14] we can show that the matrix [Ψ(A) Ψ(X); Ψ(X∗) Ψ(B)] is PPT. Though there are strong numerical evidence suggesting that this block matrix also has g-property, yet we have not even been able to show that it has a-property.\n\nThe concluding §5 states:\n\n> Besides the challenging problems described in Example 3.5 and Example 3.6, …\n\nFor every natural n, complex n-by-n blocks A,X,B with\n[A X; X* B] positive semidefinite, and every decreasing index j, prove\nboth inequalities\n\n$$\n2s_j(\\Psi(X))\\le\\lambda_j(\\Psi(A)+\\Psi(B)),\n\\qquad\ns_j(\\Psi(X))\\le\\lambda_j(\\Psi(A)\\#\\Psi(B)),\n\\quad \\Psi(T)=2(\\operatorname{tr}T)I-T.\n$$","title":"Lin: both spectral properties of the map Ψ","url":"https://the-omega-institute.github.io/trureturing-pages/research/lin-2016-psi-map-properties/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e9d419bd1f6154ede95e526b59c259a3cfc12c292412f6def626df2dcb5286f5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lin-2016-trace-augmentation-g-property","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The original locator is Conjecture 3.1, printed page 124, of the paper\nidentified by DOI 10.13001/1081-3810.3108. The fixed restatement is\narXiv:1609.08231v1, §3, Example 3.5. Its setup states:\n\n> we assume that [A X; X∗ B], where A, X, B are n × n, is positive semidefinite.\n\nDefinition 1.1(iv) states:\n\n> M is said to have g-property if sj (M12 ) ≤ λj (M11 ♯ M22 ), j = 1, . . . , n.\n\nExample 3.5 states:\n\n> [Φ(A) Φ(X); Φ(X∗) Φ(B)] where Φ : X ↦ X+(trX)In , is PPT … Numerical experiments suggest that the matrix [Φ(A) Φ(X); Φ(X∗) Φ(B)] has g-property, which we haven't been able to prove yet.\n\nThe concluding §5 refers to:\n\n> Besides the challenging problems described in Example 3.5 and Example 3.6, …\n\nFor every natural dimension, all complex matrices A, X, B with positive\nsemidefinite block matrix W=[A X; X* B], and every decreasing index j,\nthe target is\n\n$$\ns_j(\\Phi(X))\\le\\lambda_j(\\Phi(A)\\#\\Phi(B)),\n\\qquad \\Phi(T)=T+(\\operatorname{Tr}T)I.\n$$","title":"Lin: the g-property of trace augmentation","url":"https://the-omega-institute.github.io/trureturing-pages/research/lin-2016-trace-augmentation-g-property/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d05b19e280acb00d2b6bb44c11d2f83124d472e8d3560cfac4e9be9c7264011.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lin-2017-majorization-conjecture-4-3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lin, *On a determinantal inequality arising from diffusion tensor imaging*,\nCommunications in Contemporary Mathematics 19 (2017), Conjecture 4.3, asks\nwhether for positive semidefinite matrices $A,B$ and every real $p>0$,\n\n$$\n\\lambda(A^2+|BA|^p) \\succ \\lambda(A^2+|AB|^p).\n$$\n\nHere $|X|=(X^*X)^{1/2}$ and the eigenvalue lists are in descending order.\nThe Lean claim uses the literal carrier `Matrix (Fin n) (Fin n) ℂ`, the\ncontinuous functional calculus expression for $|X|^p$, and the source's\nprefix-sum definition of majorization.\n\n### Statement echo\n\nThe source clause is “Let A and B be n × n positive semidefinite matrices.\nThen λ(A² + |BA|^p) ≻ λ(A² + |AB|^p) for all p > 0.” The declaration\n`D5/S3/Quantum/Matrix/LinDTIMajorizationRefutation.claim` binds $n,A,B,p$,\nthe two positive-semidefinite hypotheses, and the two ordered eigenvalue lists\nwith exactly those operations.","title":"Lin's matrix majorization conjecture fails at exponent four","url":"https://the-omega-institute.github.io/trureturing-pages/research/lin-2017-majorization-conjecture-4-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd7777bc164d7c11712afd98f3ead5600e359f3c1ccedb7a0e492f975619ad2e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lin-wang-cai-2023-broom-spectral-ratio-conjecture-5-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z. Lin, J. Wang and M. Cai, *The Laplacian spectral ratio of connected graphs*,\narXiv:2302.10491v1, Conjecture 5.4, page 16, state:\n\n> Let T be a tree with n ≥ 8 vertices. Then R_L(K_{1,n−1}) ≤ R_L(T) ≤ R_L(B_n^{(n−3)/2}) if n is odd; R_L(B_n^{(n−4)/2}) if n is even. The left (right) equality holds if and only if T = K_{1,n−1} (T = B_n^{(n−3)/2} or T = B_n^{(n−4)/2}).\n\nThe [source note](../Library/GraphInvariants/linwangcai2023laplacianratio.md)\nrecords the definitions and version; the preregistration is\n[#19460](https://github.com/the-omega-institute/trureturing/issues/19460).\nThe upper inequality is false: $R_L(B_{19}^{8}) 4.\n\nIt is proved.","title":"Every data qubit of the delayed remote-controlled circuit violates the dimension witness S ≤ 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/lin-wang-liu-huang-2026-delayed-rcqc-dimension-witness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f943b4ebc4b72a0f00c4ad0f89b0787ecdd47aea036e79bd582e216a70ce7b6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lindblad-guerrero-2025-anderson-symmetry-converse-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"O. Lindblad and E. Guerrero, *Simple Eigenvalues and Non-vanishing Eigenvectors\nof the Anderson Model*, arXiv:2512.00278v1, ask after Theorem 1.3:\n\n> Conversely, we also ask whether every bad potential shares a nontrivial symmetry with the laplacian.\n\nSection 4 also states:\n\n> In general, we do not know whether the converse of lemma 3.2 is true.\n\nOn the cycle with $L>2$, the source's operator is\n$$(H_t\\psi)(j)=\\sum_{k\\sim j}(\\psi(j)-\\psi(k))+t v_j\\psi(j).$$\nDefinition 1.1 calls $V=\\operatorname{diag}(v)$ bad when for every real $t$\nat least one of simple eigenvalues and non-vanishing eigenvectors fails.\nThe target LG-CONV quantifies over every $L>2$ and every\n$v:\\operatorname{Fin}L\\to\\mathbb R$ taking values in $\\{-1,1\\}$.\nIssue #13591 fixes nontrivial shared symmetry as a real orthogonal matrix\n$O\\ne I$, $O\\ne-I$, commuting with both $\\Delta$ and $V$.\nBoth sufficient branches of Lemma 3.2 exclude $-I$, so refuting this weak\nreading also refutes their converse. A one-dimensional counterexample is a\ncounterexample to the general-grid question. An affine change\n$V\\mapsto\\alpha V+\\beta I$, $\\alpha\\ne0$, reparametrizes the coupling and\nshifts eigenvalues without changing eigenvectors or shared commutants;\nnormalization to $\\{-1,1\\}$ therefore represents any two distinct values.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Simple eigenvalues | `bad`: `∀ μ : ℂ, H.charpoly.IsRoot μ → H.charpoly.rootMultiplicity μ = 1`, with `H = (SimpleGraph.cycleGraph L).lapMatrix ℂ + (t : ℂ) • Matrix.diagonal (fun j => (v j : ℂ))` | Each characteristic-polynomial root has algebraic multiplicity one. |","title":"A bad Anderson potential without nontrivial shared symmetry","url":"https://the-omega-institute.github.io/trureturing-pages/research/lindblad-guerrero-2025-anderson-symmetry-converse-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd022499b68c1c109fcd78847a678593971bfb0101ffc63ff6fbabeab3fbb374.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-2025-gce-subsystem-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Liu, J. Knörzer, Z. J. Wang and J. Tura, *Generalized Concentratable\nEntanglement via Parallelized Permutation Tests*, Phys. Rev. Research 7,\nL032022 (2025), arXiv:2406.18517v1. For an $n$-qubit pure state $\\psi$ and a\nset $s$ of qubits,\n\n> $\\mathcal{C}^{(K)}_{\\ket{\\psi}}(s):=\\frac{1}{K-1}\\left(1-\\frac{1}{2^{|s|}}\\sum_{\\alpha\\in\\mathcal{P}(s)}\\Tr(\\rho_{\\alpha}^K)\\right)$ for any $K>1$ […] We take $\\Tr(\\rho^K_{\\varnothing})=1$ in the sum.\n\nThe conjecture (arXiv Conjecture 6, journal Conjecture 1), clause 1:\n\n> $\\mathcal{C}^{(K)}_{\\ket{\\psi}}(s')\\leqslant \\mathcal{C}^{(K)}_{\\ket{\\psi}}(s)$ if $s'\\subseteq s$.\n\nThe paper states that this clause and the subadditivity clause \"have been\nproven to be true for $K=2$\" and conjectures that \"they also hold for $K>1$\".","title":"Subsystem monotonicity of generalized concentratable entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-2025-gce-subsystem-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4fcfab8fbadcc3b016ac396a71de945fcc7beaf1d506da252c43d7bafd55dabd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-2025-qtd-qjs-lower-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Yupan Liu, *Quantum state testing beyond the polarizing regime and quantum triangular discrimination*, arXiv:2303.01952v5, §1.3, “Improved inequalities on the quantum triangular discrimination”, printed p. 7, states:\n\n> We observe that Theorem 1.4(ii) is not tight. Numerical simulations indicate that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) for any states ρ₀ and ρ₁.\n\nAfter Theorem 3.4:\n\n> Numerical simulations suggest that the tight bound is QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁), while we only managed to prove a slightly weaker bound ½QTD²(ρ₀, ρ₁) ≤ QJS₂(ρ₀, ρ₁) ≤ QTD(ρ₀, ρ₁) in Theorem 3.4.\n\nThe paper was published in *Computational Complexity* 34 (2025), article 11. Definition 3.1 specifies the inverse square root on the support. Definition 2.10 defines QJS by the entropy of the average minus the average entropy; QJS₂ uses base-2 logarithms.","title":"QTD squared can exceed base-2 quantum Jensen–Shannon divergence","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-2025-qtd-qjs-lower-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eab4ca7a449a7dc59d976a3c3089351e2c70fc732297987cf5c90fb4213e0ca9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-2026-powered-binary-renyi-monotonicity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Liu, *Computational hardness of estimating quantum entropies via binary entropy bounds*, arXiv:2601.03734v2, Section 1.4(b):\n\n> Can the inequalities in \\Cref{table:Rank2RenyiQEA-hardness} be tightened? For instance, is it possible to prove that $\\rbra[\\big]{\\frac{\\Halpha(x)}{\\ln(2)}}^{2/\\alpha}$ is monotonically non-decreasing in $\\alpha$ for all fixed $x\\in[0,1]$, as suggested by numerical evidence and as a generalization of \\Cref{thm:Renyi-binary-entropy-upper-bound-0leAle2}?\n\nHere H^R_α(x) = ln(x^α + (1−x)^α)/(1−α). The verbatim passages are in [the literature note](../Library/Entropy/liu2026binaryentropybounds.md).\n\nIssue [#19090](https://github.com/the-omega-institute/trureturing/issues/19090) fixes the reading: for every x ∈ [0, 1] and all orders 0 < a < b with a, b ≠ 1, (H_a(x)/ln 2)^{2/a} ≤ (H_b(x)/ln 2)^{2/b}. It is refuted.","title":"The powered binary Rényi entropy is not monotone in the order","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-2026-powered-binary-renyi-monotonicity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa5c6e44c3b87ee2f513d8e24ac7386ef2e62f4f5bb3d524ef0ed3167365ce2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-debrota-2021-fewer-effects-orthogonality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Liu and J. B. DeBrota, *Relating measurement disturbance, information and orthogonality*, arXiv:2105.02074v2 (Phys. Rev. A 104, 052216, 2021), Appendix C, subsection “Fewer effects”, Conjecture 2 (`apssamp.tex`, lines 795–797):\n\n> The above constructed POVM has\n> minimal orthogonality $O$ among POVMs having $n(n Starting from a trivial state $\\rho=\\frac{1}{2^N}(\\mathbb 1^{\\otimes N}+\\sigma_x^{\\otimes N})$, we numerically verify that: when $N=4,6,8$, the resulting steady state is $\\rho_\\infty^{(N)}=\\frac{1}{2^N}(\\mathbb 1_2^{\\otimes N}+\\frac{1}{2^{N/2-1}}U_{CZX}^{(N)})$. We conjecture that this relation holds for general $N$.\n\nFor $n\\in\\mathbb N$, the module indexes computational-basis configurations by $C_n=\\mathrm{Fin}(n)\\to\\mathrm{Fin}(2)$. Its `tensorX n` matrix flips every bit, `cz n` is the diagonal cyclic controlled-$Z$ phase, and `czxUnitary n = cz n * tensorX n`. The four-site channel is embedded at cyclic windows by `windowChannel`; `parentLindbladian n` is the sum of `windowSuperoperator n i - 1` over zero-based even starts, and `evolution n t a` is the matrix exponential of that superoperator acting on the entries of `a`.\n\nThe formal statement is `claim`: for every `n` with `Even n` and $4\\le n$, the evolution from $((2:\\mathbb C)^n)^{-1}\\,\\cdot\\,(1+\\mathrm{tensorX}\\ n)$ tends at `atTop` to $((2:\\mathbb C)^n)^{-1}\\,\\cdot\\,(1+(2:\\mathbb C)^{1-(n/2:\\mathbb N):\\mathbb Z}\\,\\cdot\\,\\mathrm{czxUnitary}\\ n)$.","title":"The CZX parent Lindbladian reaches its conjectured even-ring steady state","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-et-al-2026-czx-parent-lindbladian-steady-state/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/214773e849b871cf99c537d55f878eb4830caf15d4f70d75451e2152e660cb84.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-qiu-dahlsten-vedral-2025-pure-product-signalling-negativity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Liu, Y. Qiu, O. Dahlsten and V. Vedral, *Quantum causal inference with extremely light touch*, arXiv:2303.10544v4 (npj Quantum Information 11, 54, 2025), define the causal monotone in Eq. (causmono) as\n$f(R) := \\lVert R\\rVert_{tr}-1 = \\Tr \\sqrt{ R R^\\dag}-1$ (arxiv.tex lines 143–145). For a channel with Choi-Jamiolkowski matrix\n$M_{12}:= \\sum_{i,j}\\ket{i}\\bra{j}^T \\otimes \\mathcal{M}_{2|1}(\\ket{i}\\bra{j})$ (Eq. (CHOI), lines 161–164), their two-time PDM is\n$R_{12}=\\frac12(M_{12}\\rho+\\rho M_{12})$ with $\\rho=\\rho_1\\otimes\\mathbbm{1}_2$ (Theorem 2timesPDM, lines 166–173). After the null-negativity theorem for channels that do not signal from $B$ to $A$, the paper states:\n\n> We moreover conjecture that if the channel is signalling from B to A, we can always find a pure {\\em product} input state such that $f(R_{B_1A_2} )> 0$. We prove this conjecture for a quite general case of 2-qubit unitary evolutions in the Supplementary Information.\n\nThis is the conjecture at arxiv.tex line 231. The formal statement uses finite nonempty index types $\\alpha_1,\\beta_1,\\alpha_2,\\beta_2$, a `QuantumChannel` $p$ from $\\alpha_1\\times\\beta_1$ to $\\alpha_2\\times\\beta_2$, and the module's `IsSignallingBA` predicate. It asks for vectors $a:\\alpha_1\\to\\mathbb C$ and $b:\\beta_1\\to\\mathbb C$ with $a^*\\!\\cdot a=1$, $b^*\\!\\cdot b=1$, and $0<\\operatorname{traceNorm}(\\operatorname{pdmBA}(p,a,b)^\\mathsf{H})-1$.","title":"Every finite-dimensional signalling channel has a pure product input with positive PDM negativity","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-qiu-dahlsten-vedral-2025-pure-product-signalling-negativity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/642b2837be6454d554a254b0b5218dc10bdbba2943245366c6169f1c8e348a28.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:liu-takagi-lloyd-2019-mixed-unitary-pinching-commutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z.-W. Liu, R. Takagi and S. Lloyd, *Diagonal quantum discord*, J. Phys. A: Math. Theor. 52, 135301 (2019), arXiv:1708.09076v2, Section 3.2, “$\\pi$-commuting local operations and monotonicity”, conjecture following the probabilistic-Hadamard example:\n\n> We conjecture (which is not important for our current purpose) that ${\\bf ISO} = X_A(\\pi_A)$. That is, qubit mixed-unitary channels that are not isotropic all fail to satisfy the condition.\n\nThis quotation is line 463 of the source archive's `revised_jphysa_clean.tex` (md5 `3ae0def434187ebe1875d87961e23048`). The relevant definitions are at lines 261, 325–329 and 347; line 339 assumes the input marginal nondegenerate. See also [the literature note](../Library/Entropy/liutakagilloyd2019diagonaldiscord.md).\n\nThe formal definitions use the following conventions:\n\n- `IsMixedUnitary` means a finite mixture $\\Phi(a)=\\sum_j p_j U_j a U_j^\\dagger$ on every complex $2\\times2$ matrix, with real $p_j\\ge0$, $\\sum_jp_j=1$, and unitary $U_j$.\n- `IsIsotropic` means that, for some fixed $\\gamma\\in[0,1]$ and unitary $U$, either $\\Phi(a)=(1-\\gamma)UaU^\\dagger+\\gamma\\operatorname{tr}(a)I/2$ for every $a$, or $\\Phi(a)=(1-\\gamma)Ua^{\\mathsf T}U^\\dagger+\\gamma\\operatorname{tr}(a)I/2$ for every $a$. Transpose gives the complex linear extension of the antiunitary action on Hermitian states.\n- `localMap` applies $\\Phi$ to each first-qubit block; for these linear maps it is $\\Phi\\otimes\\mathrm{id}_B$. The marginal $\\rho_A=\\operatorname{tr}_B\\rho$ is `partialTraceRight`.\n- `IsNondegenerate` means Hermitian with two distinct eigenvalues. For a Hermitian marginal, `pinchA` is $\\pi_A(\\rho)=\\sum_i(P_i\\otimes I)\\rho(P_i\\otimes I)$, where $P_i$ are its rank-one spectral projectors; its spectral basis is recomputed on the argument to which pinching is applied.\n\nIssue [#22787](https://github.com/the-omega-institute/trureturing/issues/22787) fixes the reading to the quotation's second sentence: for every mixed-unitary qubit map $\\Phi$ that is not isotropic, there is a positive semidefinite two-qubit matrix $\\rho$ of trace one such that $\\rho_A$ and $\\Phi(\\rho_A)$ are nondegenerate and $\\pi_A((\\Phi\\otimes\\mathrm{id}_B)(\\rho))\\ne(\\Phi\\otimes\\mathrm{id}_B)(\\pi_A(\\rho))$. This is proved.","title":"Non-isotropic mixed-unitary qubit channels do not commute with marginal pinching","url":"https://the-omega-institute.github.io/trureturing-pages/research/liu-takagi-lloyd-2019-mixed-unitary-pinching-commutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a5d46c3347b65d55d638bc44fd6a2b8394a13c6c24b0fd2c846aa9a70f6ee96.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lmys511-kyfan","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhen Lin, Lianying Miao, Guanglong Yu and Han Sheng, *On the Ky Fan k-norm of the\nLI-matrix of graphs*, Electronic Journal of Linear Algebra 38 (2022), 680–696,\nConjecture 5.11 (page 694), states:\n\n> Among all connected graphs with n and m edges n ≤ m ≤ 2n−3, the Gₘ,ₙ is the unique graph with maximal value of ‖LI(G)‖F₂ and ‖LI(G)‖F₃.\n\nHere $n$ is the number of vertices, $m$ the number of unordered edges,\n$L(G)=D(G)-A(G)$, $LI(G)=L(G)-(2m/n)I_n$, and the Ky Fan $k$-norm is the sum\nof the $k$ largest singular values, with multiplicity. The assertion is read as a\nseparate maximum and equality characterization for each of $k=2$ and $k=3$.\nThe interpretation involving a unique simultaneous maximizer is outside the claim.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite simple undirected graphs of order $n$ | `SimpleGraph (Fin n)` | Every finite labelled graph of that order is represented. |\n| Connected, with $m$ edges | `G.Connected`, `Nat.card G.edgeSet = m` | Mathlib connectivity and unordered edges. |\n| $n\\le m\\le2n-3$ | The two range hypotheses in `claim` | `m + 3 ≤ 2 * n` avoids truncated subtraction; together the hypotheses force $n\\ge3$. |\n| $LI(G)=L(G)-(2m/n)I_n$ | `LI` | Literal source-specific shift of Mathlib's Laplacian. |\n| $G_{m,n}$ in Figure 2.1 | `Gmn n m` | Vertex $0$ is universal; vertex $1$ meets $2,\\ldots,m-n+2$. Argument order is $n,m$. |\n| Separate assertions for $F_2$ and $F_3$ | `claim` quantifies over $k=2$ or $k=3$ | Both the upper bound and the equality iff isomorphism are retained. |\n| Refutation | `result` | The public theorem proves the negation of that whole claim using its $k=2$ equality clause. |\n\nSection 2 prints $2n-m+3$ pendant edges. That count conflicts with the stated\norder, size and figure; the figure's construction has $2n-m-3$ pendant vertices.\nThe definition uses the figure and order/size convention fixed in #21801.\nThere is no additional mathematical hypothesis in the public result.","title":"Lin–Miao–Yu–Sheng Conjecture 5.11: second-norm uniqueness","url":"https://the-omega-institute.github.io/trureturing-pages/research/lmys511-kyfan/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28e99a8ac2185ae89b177c25e2629c12773c5a4495348ce2e738abc60a6c8efc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lo-monaco-2025-stabilizer-povm-maximal-entanglement-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Lo Monaco, S. Lorenzo, A. Ferraro, M. Paternostro, G. M. Palma and\nL. Innocenti, *The non-stabilizerness cost of quantum state estimation*,\narXiv:2510.00157v2, Section IV, Conjecture 1 (PDF page 8), state:\n\n> For any pure |ψ⟩, the maximal value of s_μ can be obtained with maximally\n> entangled S.\n\nFor n data qubits and m ancillas, a stabilizer measurement is an orthonormal\ncommon-eigenvector basis of a maximal Abelian Pauli subgroup excluding −I.\nIts effective effects are μ_b = (I⊗⟨ψ|)|Φ_b⟩⟨Φ_b|(I⊗|ψ⟩), and s_μ is the\ncomplex dimension of their linear span. The source defines\n2p = n+m−dim(S_n)−dim(S_m), where S_n and S_m are the data-supported and\nancilla-supported subgroups. Maximal entanglement means p=min(n,m).\nThe formal claim retains all n,m and all pure normalized ancillas and\ncompares every stabilizer basis for the same ancilla.","title":"A stabilizer measurement exceeds every maximally entangled one","url":"https://the-omega-institute.github.io/trureturing-pages/research/lo-monaco-2025-stabilizer-povm-maximal-entanglement-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f89e7a22a18f720e23d30bbffd956647c8f00e24cc12028324da61da38fdb8a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lobo-balanzo-juando-pironio-2026-eq36-tightness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. P. Lobo, M. Balanzó-Juandó and S. Pironio, *Generalized measurement incompatibility*,\narXiv:2605.16151v1, call measurements $G$-jointly measurable when a common parent reproduces\nthem while deterministically fixing only the outcomes in the subsets $G_y$ (Definition 2). For two\nlossy projective qubit measurements (Eqs. (13), (18)–(19)) with detection efficiency $\\eta$, whose\naxes are separated by an angle $\\theta\\in[0,\\pi/2]$, and $G=(\\{+,-\\},\\emptyset)$ (case (d) of\n§IV.B), the paper proves $G$-joint measurability for $\\eta\\le 2/(2+\\sin\\theta)$ (Eq. (36)) and\nreports that \"up to machine precision, this bound is tight\", from numerical solutions of the\n$G$-JM semidefinite program. The verbatim statements are in\n[the literature note](../Library/QuantumContext/lobo2026generalized.md).\n\nIssue [#15638](https://github.com/the-omega-institute/trureturing/issues/15638) fixes the\nreading: for all unit Bloch vectors $r_1,r_2$ and every $\\eta\\in[0,1]$, $G$-joint measurability\nimplies $\\eta\\,(2+\\lVert r_1\\times r_2\\rVert)\\le2$, where $\\lVert r_1\\times r_2\\rVert=\\sin\\theta$.","title":"Tightness of the lossy-qubit partial joint-measurability bound η ≤ 2/(2 + sin θ)","url":"https://the-omega-institute.github.io/trureturing-pages/research/lobo-balanzo-juando-pironio-2026-eq36-tightness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08c443065c4a2d2432e71e4bb7306aab8700b4d3fccb53ddb16caec5d045f13a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lonigro-chruscinski-2022-dephasing-two-classicality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Lonigro and D. Chruściński, *On the classicality of quantum dephasing processes*, arXiv:2211.02014v2; Frontiers in Quantum Science and Technology 1, 1090022 (2022), Section 3.2, immediately following Proposition `prop:2classicality_mubs`. The v2 TeX source (`dephasing-v2.tex`, md5 `e2904085775dc9d0af3b2dba4d50c065`), line 347, states:\n\n> It may be conjectured that this is the only measurement choice yielding a genuinely time-dependent $2$-classical process---excepting the particular case in which $\\rho_{t_0}$ is maximally mixed, cf.~the discussion in the next subsection.\n\nThe antecedent is the phase-modified Fourier measurement basis of Eq. `eq:mubs`, whose squared overlaps with every dephasing basis vector are $1/d$ (TeX lines 284–289). Proposition `prop:2classicality_mubs` assumes an initial state diagonal in the dephasing basis and gives two-classicality for that measurement family, for arbitrary bath Hamiltonians and bath state (lines 326–347). The definitions and source locators are also in [the literature note](../Library/QuantumContext/lonigrochruscinski2022dephasing.md).\n\nIssue [#21441](https://github.com/the-omega-institute/trureturing/issues/21441) fixes the following necessity statement, encoded by `claim`. For every $d,m\\in\\mathbb N$, every family of Hermitian bath matrices $H_j$ on $\\mathbb C^m$, positive semidefinite unit-trace matrices $\\sigma$ and $\\rho$ on the bath and system respectively, unitary measurement matrix $V$, and initial time $t_0\\in\\mathbb R$, assume:\n\n- $\\rho$ is diagonal in the dephasing basis and $\\rho\\ne I_d/d$;\n- some rank-one measurement projector $P_x=|m_x\\rangle\\langle m_x|$, with $m_x$ the $x$th column of $V$, does not commute with some dephasing projector $E_j=|e_j\\rangle\\langle e_j|$;\n- for every $t_0\\le t_1\\le t_2$, the one-time law is normalized and both marginals of the two-time law equal the corresponding one-time law;\n- for some fixed outcomes $x,y$, two ordered time pairs at or after $t_0$ give different two-time values.\n\nThe conclusion is $|V_{jx}|^2=1/d$ for every $j,x$. The laws here are the literal joint-system trace formula of the paper's Eq. (2), with Hamiltonian $\\mathbf H=\\sum_jE_j\\otimes H_j$, propagator $e^{-i(t-s)\\mathbf H}$, initial product $\\rho\\otimes\\sigma$, and system Lüders operations $X\\mapsto(P_x\\otimes I_m)X(P_x\\otimes I_m)$. There is no bath reset. The definitions `prob1` and `prob2` take values in $\\mathbb C$; `twoClassical` imposes the stated normalization and marginal equalities. No positive lower bound on $d$ or $m$ is added to `claim`.\n\nThis is a consequence of the conjectured necessity: it restricts to finite baths and time-independent Hamiltonians, asks only for unbiasedness rather than membership in the named Fourier family, and additionally requires actual nonconstancy of the two-time law. It is refuted.","title":"A block measurement refutes mutual-unbiasedness necessity for two-classical dephasing","url":"https://the-omega-institute.github.io/trureturing-pages/research/lonigro-chruscinski-2022-dephasing-two-classicality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec4a82f542281666825d09c7b617eb3afc2e246299c816677ec7110f0436254d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:loopy-degree-sequence-with-loops","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 11.4 of `kirillov2026loopy` asks exactly:\n\n> Does L_G determine the degree sequence of an arbitrary graph G, loops included?\n\nIn repository coordinates, the complete target quantifies independently over\nfinite vertex sets `V W`, edge-occurrence lists `E F`, and accumulated-loop\nfunctions `ell eta`. From `Valid V E ell`, `Valid W F eta`, and\n`loopy V E ell = loopy W F eta`, it concludes\n`degreeMultiset V E ell = degreeMultiset W F eta`. It assumes no equality of\norders. Lists preserve parallel occurrences; both endpoints of a pending loop\nare counted, accumulated loops contribute twice, and mapping all vertices in\nthe carrier includes isolates. No connectedness, nonemptiness or simplicity\nhypothesis is present.","title":"Degree recovery from the ordinary Loopy polynomial with loops","url":"https://the-omega-institute.github.io/trureturing-pages/research/loopy-degree-sequence-with-loops/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/104a1d541158c876826f124bc566d99a8a898bb22514d870696f87de04e0295a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:loopy-generalized-degree-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 11.3 of `kirillov2026loopy` asks exactly:\n\n> Does L_G determine the generalized degree polynomial GD_G?\n\nThe source's conventions admit every finite undirected multigraph, loops and\nparallel edges included, and Problem 11.3 adds no restriction; the sentences\nthat follow it introduce the companion Problem 11.4 about degree sequences,\nwhich Theorem 4.3 settles in the loopless case and Theorem 6.15 settles through\nthe refined polynomial. The pair delivered here carries loops and therefore\nlies inside the problem's own domain.\n\nIn repository coordinates the target quantifies independently over finite\nvertex sets `V W`, edge-occurrence lists `E F` and accumulated-loop functions\n`ell eta`. From `Valid V E ell`, `Valid W F eta` and\n`loopy V E ell = loopy W F eta` it concludes equality of the exponent-triple\nmultisets of the generalized degree polynomial. No looplessness, connectedness,\nsimplicity or equal-order hypothesis is present.","title":"The ordinary Loopy polynomial does not determine the generalized degree polynomial","url":"https://the-omega-institute.github.io/trureturing-pages/research/loopy-generalized-degree-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/985952512fdefe6e8f8482c203a7a5f42460b650e309af7f35090258aa8f18d6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:louvet-2025-symmetric-absolutely-ppt","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Louvet, E. Serrano-Ensástiga, T. Bastin and J. Martin, *Nonequivalence between absolute separability and positive partial transposition in the symmetric subspace*, arXiv:2411.16461v2 and Physical Review A 111, 042418 (2025), state Conjecture 1: every symmetric $N$-qudit state with\n\n$$D = \\binom{N+d-1}{d-1}$$\n\nnon-zero eigenvalues of the form\n\n$$1-(D-1)p/D,\\quad p/D,\\ldots,p/D,$$\n\nwhere $p \\in [p_{\\min},1]$ and\n\n$$p_{\\min}=1/(1+2[D\\binom{N}{\\lfloor N/2\\rfloor}]^{-1}),$$\n\nis symmetric absolutely PPT. The formal claim uses `Matrix.PosSemidef`, the symmetric-column predicate, the full-space characteristic polynomial with $d^N-D$ zero roots, and positivity of every site partial transpose.\n\n### Statement echo\n\n| source clause | Lean counterpart | fidelity |\n| --- | --- | --- |\n| symmetric $N$-qudit state | `IsSymmetricOperator ρ` together with `ρ.PosSemidef` | exact quantified predicate |\n| $D=\\binom{N+d-1}{d-1}$ | `D N d` | literal definition |\n| $p_{\\min}=1/(1+2[D\\binom{N}{\\lfloor N/2\\rfloor}]^{-1})$ | `pmin N d` | literal definition with `Nat.div` |\n| prescribed non-zero spectrum and $d^N-D$ zero roots | the `ρ.charpoly` equality in `claim` | exact characteristic-polynomial factors |\n| $p\\in[p_{\\min},1]$ | `pmin N d ≤ p` and `p ≤ 1` | exact interval hypotheses |\n| symmetric absolutely PPT | `IsSAPPT ρ` | exact quantified conclusion |","title":"Louvet symmetric absolutely-PPT spectrum conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/louvet-2025-symmetric-absolutely-ppt/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fcb186ac5a203cd8c509eb088ec534ea4cdeb749291f2e4b4c6d5d4e3dbc50d2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lovas-andai-2016-dobrushin-infimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lovas and Andai represent a qubit channel `Q : M_2 → M_2`, a completely\npositive trace-preserving map, by its Choi block matrix, whose blocks\n`Q_11 = Q(|0⟩⟨0|)` and `Q_22 = Q(|1⟩⟨1|)` have diagonals `(a, 1−a)` and\n`(f, 1−f)` in the parametrization (eq:matQ). The trace-distance contraction\ncoefficient is\n\n> η^Tr(Q) = sup { Tr|Q(ρ) − Q(σ)| / Tr|ρ − σ| : ρ, σ ∈ M_2 },\n\nand `Q_C(a,f)` denotes the set of qubit channels over the classical channel\n`((1−a, a), (1−f, f))` with respect to that parametrization. The subsection\non the distribution of `η^Tr` over classical channels states\n\n> We conjecture that inf{η(Q) : Q ∈ Q_C(a,f)} = |a−f| which is equal to the\n> trace-distance contraction coefficient of the underlying classical channel.\n\nIssue #10033 fixes the readings: `M_2` is the set of qubit states and a qubit\nchannel is a completely positive trace-preserving map; `Q_C(a,f)` consists of\nthe channels for which `Q(|0⟩⟨0|)` and `Q(|1⟩⟨1|)` have diagonals `(a, 1−a)`\nand `(f, 1−f)`; the source writes positivity of the Choi matrix as `Q > 0`, and\ncomplete positivity is read as a positive semidefinite Choi matrix; the\nsupremum defining `η^Tr` runs over all pairs of states, the ratio of trace\nnorms `Tr|·|` being the ratio of trace distances (the factors `½` cancel), and a\npair `ρ = σ` counts as `0/0 = 0`, which does not change the supremum;\n`a, f ∈ [0,1]`.","title":"Lovas and Andai's conjecture on the least trace-distance contraction over a classical channel","url":"https://the-omega-institute.github.io/trureturing-pages/research/lovas-andai-2016-dobrushin-infimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd367fc5eaa3044bc78072e9efbe6f6289d6c4e62662c594a127b18fbc3f5633.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lu-sun-zhang-2024-beta-invertible-s3-obstruction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D.-C. Lu, Z. Sun and Z. Zhang (arXiv:2406.12151, JHEP 11 (2025) 081) study\nG-ality defects of 2d QFTs through the SymTFT of the finite Abelian symmetry\n`A`. The anyons of the SymTFT are the pairs `(a, â) ∈ A ⊕ Â`, and an anyon\npermutation symmetry is a block matrix `U = (α β; γ δ)` with `β : Â → A` that\npreserves the self-statistics and the braiding; `U` is β-invertible when `β`\nis invertible. A G-ality extension needs a subgroup isomorphic to `G` whose\nnon-identity elements are all β-invertible. For `A = ℤ_N × ℤ_N` and `G = S₃`\nthe paper states (§1.1):\n\n> We find that for any N containing a factor of 2 or 3, there are no\n> β-invertible S_3 symmetries in the SymTFT, thus there cannot be S_3-ality\n> defects for these N. We are able to prove this analytically when N is even,\n> and we conjecture this is true generally for N being a multiple of 3 based\n> on our numerical evidence.\n\nIssue #11228 fixes the reading: `Â` is identified with `ℤ_N × ℤ_N` through\nthe dot product, the symmetries are the invertible 4 × 4 matrices over\n`ZMod N` preserving `Q(a, â) = a·â`, `β` is the upper-right 2 × 2 block, and a\nβ-invertible `S₃` is a homomorphism `ρ : S₃ → GL₄(ZMod N)` into these\nsymmetries with `det β(ρ g)` a unit for every `g ≠ 1`. Only this group-theoretic\nstatement is settled; the physical consequence rests on the\nEtingof–Nikshych–Ostrik classification of G-extensions and is not formalized.","title":"No β-invertible S₃ symmetry in the ℤ_N × ℤ_N SymTFT when 3 ∣ N","url":"https://the-omega-institute.github.io/trureturing-pages/research/lu-sun-zhang-2024-beta-invertible-s3-obstruction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c471f6068d6a60750a4651d77d1112165b944d9468107898902802810faac9a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:luiz-vieira-2024-blanusa-independent-roman-closing-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. G. Luiz and F. A. S. Vieira, *Independent [k]-Roman Domination on Graphs*,\narXiv:2406.11688v1, Discuss. Math. Graph Theory 45 (2025), 1249–1272,\nSection 6, conjecture that “this property holds for all generalized Blanuša\nsnarks,” where the property is $\\gamma_{[kR]}(G)=i_{[kR]}(G)$ for $k\\ge 2$.\nThe Lean claim is the literal quantification over $t\\in\\{1,2\\}$, $i\\ge 1$,\nand $k\\ge 2$ in `D5/S3/Combinatorics/Graph/Domination/LuizVieiraBlanusaIndependentRoman.claim`.\n\n### Statement echo\n\nThe source active-neighbour filter appears literally in `IsRomanDom`; the other\nsource definitions are mirrored by\n`IsIndepRomanDom`, `weight`, `gammaRoman`, `indepRoman`, the block and link\nedge sets, the recursive `verts` and `edges`, and `snark`. The settling\ndeclaration is `D5/S3/Combinatorics/Graph/Domination/LuizVieiraBlanusaIndependentRoman.result : ¬ claim`; it instantiates $t=1$, $i=1$, $k=21$.","title":"The Luiz–Vieira generalized Blanuša closing conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/luiz-vieira-2024-blanusa-independent-roman-closing-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac107280b438b8dc9d65e42f02c9754fa57c2c6e9dc403d48b18d71ed2c461c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:luo-zhen-liu-ebler-dahlsten-2023-neighbourhood-passivity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Luo, Y.-Z. Zhen, X. Liu, D. Ebler and O. Dahlsten, *Bound on annealing performance from stochastic thermodynamics, with application to simulated annealing*, arXiv:2311.10424v1, Phys. Rev. E 108, 054119 (2023), bound the activity A^D of simulated annealing with Glauber acceptance (Eqs. (30)–(36)).\n- **Neighbourhood passivity** (§IV.B, Eq. (41)): P_i = Σ_{j∈N(i)} (γ′_i − γ′_j)(p_i − p_j)/(γ′_i + γ′_j), where primes denote the next step.\n- **Its role.** If P_i ≥ 0 for all i, then A^D ≤ 1/2 (Eq. (40)), which would support the conjecture A^D(k) ≤ 1/2 of Eq. (39).\n- **The expectation:** \"as the system initially staying in a thermal state evolves towards a new thermal state in SA, we expect the passivity at least in the neighbourhoods to be preserved.\"\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/luozhenliueblerdahlsten2023annealing.md).\n\nIssue [#17931](https://github.com/the-omega-institute/trureturing/issues/17931) fixes the reading: for every run from a Gibbs state on a connected regular neighbourhood with distinct energies and a strictly cooling schedule, P_i(k) ≥ 0 for every step k and state i. This expectation is refuted.","title":"Glauber annealing from a thermal state need not preserve neighbourhood passivity","url":"https://the-omega-institute.github.io/trureturing-pages/research/luo-zhen-liu-ebler-dahlsten-2023-neighbourhood-passivity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/debda4b4051dc70506377c8ae7f89566d39b1d6e5789bc37202bdd6e53ed15ca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:luo-zhen-liu-ebler-dahlsten-2023-relaxation-order-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Luo, Y.-Z. Zhen, X. Liu, D. Ebler and O. Dahlsten, *Bound on annealing performance from stochastic thermodynamics, with application to simulated annealing*, Phys. Rev. E 108, 054119 (2023), arXiv:2311.10424v1, bound the performance of simulated annealing (SA) by comparing it with a partial swap (PS) model.\n- **The dynamics.** SA with Glauber acceptance moves p(k + 1) = P(k + 1)p(k), with P_ij = γ_i/(|N(j)|(γ_i + γ_j)) on neighbours (Eqs. (30)–(32)). The PS model moves p′ = p + µ^PS(γ′ − p) (Eq. (43)), and its second largest eigenvalue is 1 − µ^PS (Proposition 2).\n- **The rate.** The energy relaxation rate of SA is µ = (E′_p − E_p)/(E′_γ − E_p) (Eq. (46)), under E_p ≥ E′_p and E_p ≥ E_γ ≥ E′_γ.\n- **The conjecture.** With τ_rel(P) = 1/(1 − λ_2(P)): \"we conjecture that the order of relaxation times of the two processes corresponding to µ and µ^PS is the inverse order of µ and µ^PS, i. e. τ_rel ≤ τ^PS_rel ⇒ µ ≥ µ^PS. (50)\"\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/luozhenliueblerdahlsten2023annealing.md).\n\nIssue [#17238](https://github.com/the-omega-institute/trureturing/issues/17238) fixes the reading: for every regular neighbourhood, energies, strictly cooling schedule, step k and rate 0 < m ≤ 1, m ≤ 1 − λ_2(P(β(k + 1))) implies µ(k) ≥ m. The conjecture is refuted.","title":"A shorter relaxation time does not imply a faster energy relaxation in Glauber annealing","url":"https://the-omega-institute.github.io/trureturing-pages/research/luo-zhen-liu-ebler-dahlsten-2023-relaxation-order-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f85706f3fa061d5b90e68e8857d59cf4d4ac60b18f1261756a55f1d9b3a1ea0f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:lv-zhang-mesh-123-321-joint-symmetry","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Shuzhen Lv and Philip B. Zhang, *Joint Equidistributions of Mesh Patterns 123 and 321 with Symmetric and\nMinus-Antipodal Shadings*, arXiv:2501.00357v3, Section 2.4, Conjecture 2 (printed page 9), pair S21: the mesh\npatterns (123, R) and (321, R) are jointly equidistributed on Sₙ, where R = {0,1,2}² ∪ {(3,3)} as rendered in\nthe source's diagram.\n\nAn occurrence of (123, R) in π is a triple of positions with increasing values such that no other point of π lies\nin a shaded cell; (321, R) is the same with decreasing values. Joint equidistribution means that, for all n, k\nand l, the number of π ∈ Sₙ with k occurrences of (123, R) and l occurrences of (321, R) equals the number with l\nand k.","title":"Joint Symmetry of the Mesh Patterns 123 and 321 with Shading {0,1,2}² ∪ {(3,3)}","url":"https://the-omega-institute.github.io/trureturing-pages/research/lv-zhang-mesh-123-321-joint-symmetry/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de2672441f72714298b07f277458b60e92d1c25d07f5c5977223d8d867143a4f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ma-weigert-2025-log-function-monotone-chain","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rourou Ma and Julian Weigert, *Complete monotonicity of log-functions*, arXiv:2505.04225v2 (Analysis and Mathematical Physics 15, 141, 2025), §4, Conjecture `conj:shrinking`, `infiniteOrderDeriv.tex`, lines 154–160 (TeX macros expanded and display mathematics set inline):\n\n> For any $n\\in \\mathbb N$ the semialgebraic sets $D_k=\\left\\{f\\in \\mathcal{F}_{1,n}\\middle|\\forall x\\in \\mathbb R:\\left(\\frac{-\\mathrm d}{\\mathrm d x}\\right)^kf(x)\\geq 0\\right\\}$ form a descending chain under inclusion, i.e. $D_k\\supseteq D_{k+1}$, which converges to the region $\\mathrm{CM}_{1,n}=\\bigcap_{k=1}^\\infty D_k$.\n\nThe source defines $\\mathcal F_{1,n}$ as the smooth functions $f(x)=p(\\log x)/x$ on $x>0$, where $p$ is a real polynomial of degree at most $n$ (`Intro.tex`, lines 20–23). Although the conjecture's display writes $x\\in\\mathbb R$, the functions' domain and the preceding example's sign condition are $x>0$ (`infiniteOrderDeriv.tex`, line 150). Issue [#21992](https://github.com/the-omega-institute/trureturing/issues/21992) fixes this positive-domain reading.\n\nWrite $f_p(x)=p(\\log x)/x$ and $D_{n,k}=\\{p\\in\\mathbb R[y]:\\operatorname{natDegree}(p)\\le n,\\ \\forall x>0,\\ (-1)^k f_p^{(k)}(x)\\ge0\\}$. The module represents regions by their polynomial parameters. Its `claim` states, for every $n\\in\\mathbb N_0$:\n\n- $D_{n,k+1}\\subseteq D_{n,k}$ for every natural $k\\ge1$;\n- for each real polynomial $p$ with $\\operatorname{natDegree}(p)\\le n$, membership in every $D_{n,k}$ with $k\\ge1$ is equivalent to $(-1)^j f_p^{(j)}(x)\\ge0$ for every natural $j\\ge0$ and every $x>0$.\n\nHere $f_p^{(j)}$ denotes Lean's `iteratedDeriv j (logFunction p)`. The zero polynomial has natural degree zero. The claim is proved.","title":"The Ma–Weigert derivative-sign regions form a descending chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/ma-weigert-2025-log-function-monotone-chain/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb18ad231abde3be581d24472704d1a6074bc20d8ecd25a607083927322b3e9b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ma2013-dsspectral","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bassam Mourad and Hassan Abbas, *On the symmetric doubly stochastic matrices\nthat are determined by their spectra*, arXiv:1310.1273v1, printed page 10,\nConjecture 5.1:\n\n> For $0 $\\Delta_n^s$ are points on the line segments $[I_n,C_n]$, $[I_n,P]$ and\n> $[C_n,P]$ where $P$ is a vertex of $\\Delta_n^s$.\n\nHere $\\Delta_n^s$ is the full polytope of real symmetric nonnegative matrices\nwith every row and column sum equal to one, and $\\Delta_n^s(a)$ is its\ntrace-$a$ slice. A matrix is DS if every similar member of that full polytope\nis permutationally similar to it. Similarity means conjugation by an arbitrary\ninvertible real matrix. All segments include their endpoints.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Real matrices of order $n\\geq2$ | `Matrix (Fin n) (Fin n) ℝ`, order binder in `claim` | Unrestricted real entries; order bound makes $C_n$ meaningful. |\n| Symmetric doubly stochastic matrices | `DeltaSymm`, using `doublyStochastic` and `Matrix.IsSymm` | Full polytope, with nonnegativity and both stochastic sums. |\n| Trace $a$, $0 The semigroup (T(t))t≥0 is called L∞-bounded, if there exists M > 0 such that ‖T(t)f‖∞ ≤ M‖f‖∞, for all f ∈ L²(Ω) ∩ L∞(Ω). (1)\n\nEquation (3) states verbatim:\n\n> (3) ‖gT(t)f‖∞ ≤ ‖gf‖∞ for all f ∈ L∞(Ω) ∩ L²(Ω)\n\nProblem 22 and its comment state verbatim:\n\n> Problem 22 (L∞-contraction for a weight problem). In the setting of Problem 21 assume that the semigroup (T(t))t≥0 is L∞-bounded, i.e., satisfies (1) for some M ≥ 1. Does there exist a function g ∈ L∞(Ω) satisfying g > 0 µ-a.e., 1/g ∈ L∞(Ω), and (3)? Comments. In general, this might be more than one can hope for.\n\nThe setting of Problem 21 is a σ-finite measure space $(\\Omega,\\mu)$ and a closed sectorial sesquilinear form $a$ on a dense $V\\subseteq L^2(\\Omega)$, with associated operator $A$ and bounded analytic semigroup $T(t)$ generated by $-A$. The locator binds to the edited problem list, without assigning individual authorship to this question.","title":"L∞-contraction for a weight","url":"https://the-omega-institute.github.io/trureturing-pages/research/madou-remizov-vafadar-2024-linf-weight-contraction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef56ca312c09d9259b278a0459093cc069e6fb2f3dc444291024f4fa75708c62.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mahalingam-pandoh-2021-scattered-palindrome-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mahalingam and P. Pandoh, *Counting scattered palindromes in a finite word*, arXiv:2108.02729v1, §5, Conjecture 5.6, states:\n\n> The maximum number of scattered palindromic subwords in a word of length $n$ is $3(2^{\\lceil n/2\\rceil-1})-2$ if $n$ is odd, $2^{n/2+1}-2$ if $n$ is even, and this bound is achieved only when $q=\\lceil n/2\\rceil$ if $n$ is odd, $\\frac n2$ and $\\frac n2+1$ if $n$ is even.\n\n$SP(w)$ counts distinct nonempty palindromic scattered subwords, rather than their embeddings. A scattered subword is a subsequence; $q$ counts the distinct letters in $w$. Write $M(n)$ for the displayed maximum.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Scattered containment and palindromes (§2) | `spal` uses `List.sublists.toFinset` and excludes the empty word while requiring reversal equality. | Every word over a decidable alphabet; duplicate embeddings count once. |\n| $SP(w)=|SPAL(w)|$ and $q=|Alph(w)|$ (§2) | `sp` and `alphabetSize`. | Cardinalities of the literal source objects. |\n| The two parity cases of $M(n)$ | `maxSP`. | Natural division $(n+1)/2$ represents the ceiling; subtraction is truncated natural subtraction. |\n| Maximum over all length-$n$ words | First conjunct of `claim`; `result`. | Universal bound over every decidable alphabet, including the empty word. |\n| Attainment of the maximum | Second conjunct of `claim`; `attainment`; `result`. | Every $n\\ge1$ and every permitted $q$; witnesses are natural-number words. |\n| Only the specified alphabet sizes | `permittedQ`, third conjunct of `claim`; `result`. | Every extremal nonempty word; the even alternatives form a disjunction. |\n\nThe positive-length assumption applies to attainment and necessity. The universal bound includes $n=0$ as an additional proved boundary case. Decidable equality enables finite-set cardinalities; it does not restrict a finite word's letter pattern. Natural-number letters provide existence witnesses and impose no restriction on the universal clauses.","title":"Mahalingam–Pandoh maximum number of scattered palindromes","url":"https://the-omega-institute.github.io/trureturing-pages/research/mahalingam-pandoh-2021-scattered-palindrome-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/902004ab5d965f4fd41cf124732d92b4d55f69531103a27c5b1d1ddf9b40e97d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mahmoud-2026-quaternary-extension-minimal-length-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. A. Mahmoud, *Minimal Counterexamples of the MacWilliams Extension Theorem for Stabilizer Codes*, arXiv:2607.26214v1 (2026), §6.2, open problem 2, states:\n\n> \\textbf{Minimal length for general $q$.} Determine the minimal counterexample length for general $q$. For $q=2$ it is $3=q+1$ (Theorem~\\ref{thm:census}); we expect $q+1$ to be minimal in general, in the light of the threshold results for additive isometries in \\cite{Dyshko16,DyshkoGeom}, but we claim it only for $q=2$.\n\nThis is a verbatim excerpt of `MzWi_Ext_Th.tex`, line 429, retrieved from the versioned arXiv source; its md5 is `ee3d0a4f9ba273634eb55aad8ed7b850`. The rest of the item asks separately about full-support minimal lengths. The definitions and source locators are also in [the literature note](../Library/Quantum/mahmoud2026macwilliams.md).\n\nThe source's Definition 2.3 (TeX lines 237–239) is:\n\n> Fix $q$ and $n$. We say the quantum extension property $\\mathrm{QEP}(q,n)$ holds if for every pair $C,D\\le\\A_q^n$ of trace-symplectically self-orthogonal additive codes and every additive bijection $f\\colon C\\to D$ with $\\wt(f(u))=\\wt(u)$ for all $u\\in C$, the map $f$ extends to a monomial transformation of $\\A_q^n$.\n\nHere $A_q=\\mathbb F_q^2$, and additive means $\\mathbb F_p$-linear, where $p$ is the characteristic (TeX lines 211–217). Definition 2.1 (lines 221–223) specifies a coordinate permutation followed by maps in $\\operatorname{Sp}(2,q)$, and explicitly mentions the larger group of all coordinatewise $\\mathbb F_p$-linear automorphisms.\n\nIssue [#20851](https://github.com/the-omega-institute/trureturing/issues/20851) fixes both readings at $q=4$. Write $\\mathrm{QEP}(4,n)$ for the specified symplectic coordinate group and $\\mathrm{QEP}^{\\mathrm{add}}(4,n)$ for the larger additive coordinate group. The formal `claim` is\n\n$\\operatorname{IsLeast}\\{n\\in\\mathbb N:1\\le n\\ \\land\\ \\neg\\mathrm{QEP}(4,n)\\}\\,5\\ \\lor\\ \\operatorname{IsLeast}\\{n\\in\\mathbb N:1\\le n\\ \\land\\ \\neg\\mathrm{QEP}^{\\mathrm{add}}(4,n)\\}\\,5.$\n\n`IsLeast` requires both membership in the failure set and being no larger than any member. This disjunction is refuted: neither reading has least positive failure length $5$.","title":"Five is not the minimal quaternary extension-counterexample length","url":"https://the-omega-institute.github.io/trureturing-pages/research/mahmoud-2026-quaternary-extension-minimal-length-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d20581819c02324e4d9cc50839acd521633e6ebce89d14dacf1d1272fe001581.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mahmoud-2026-stabilizer-pair-local-unitary-inequivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. A. Mahmoud (arXiv:2607.26214v1, 2026) constructs weight-isometric pairs of\nstabilizer codes that are not monomially equivalent. Theorem 5.3 gives the\nfull-support pairs\n\n- (ii) `S_A = ⟨ZZZZI, XXIIX, IIXXX⟩` and `S_B = ⟨ZZIZX, XIZXI, IXXYI⟩`, two\n `[[5,2]]` codes;\n- (iii) `S+ = ⟨XXIIXX, IIXXXX, ZZZZIX⟩` and `S− = ⟨XXZXII, ZZXYII, IZZZXX⟩`,\n two `[[6,3]]` codes.\n\nThe paper then asks:\n\n> Question 5.4 (Extension analogue of LU–LC). Are the codespaces of SA and SB\n> of Theorem 5.3(ii)—or those of S+ and S− of Theorem 5.3(iii)—mapped to one\n> another by some product unitary U1⊗···⊗Un composed with a qubit\n> permutation?\n\nand §6.2 lists it as open problem 1, with the `[[5,2]]` pair as \"the minimal\nopen instance\". Issue #11210 fixes the reading: qubits are numbered from the\nleft; the Pauli matrices are the frozen `qubitX`, `qubitZ` with `Y = i X Z`; a\nword and a product unitary act by the tensor product; the codespace is the\njoint `+1` eigenspace of the three generators; the permutation acts by\n`(P_σ ψ)(x) = ψ(x ∘ σ)`; and \"mapped to one another\" means that the image of\nthe first codespace under `U·P_σ` is the second. The formal `claim` is the\npositive answer for either pair, and the formal `result` refutes it. The\n\"More generally\" part of the question is not addressed.","title":"Local-unitary inequivalence of the Question 5.4 stabilizer pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/mahmoud-2026-stabilizer-pair-local-unitary-inequivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e85a6e4def0b64e4166b518a5dc24336fc4e509475d774ad34bdbf4d8a0a9c2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mahmoud-mortici-2026-wallis-remainder-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 4, Open Question of the source identified in\n`D5/L/Asymptotics/mahmoudmortici2026wallisratio` asks:\n\n> Do the functions μ(x)=−x²−{(3/x)[ln(√x Γ(x+1/2)/Γ(x+1))]^{−1}+24}^{−1}, θ(x)=(23040x⁵/31){3x/(24x²+1)+ln(√x Γ(x+1/2)/Γ(x+1))}, χ(x)=(358400x⁷/257){(120x²+31)/(96x(10x²+3))+ln(√x Γ(x+1/2)/Γ(x+1))} satisfy the property that μ(x) and χ(x) are strictly increasing, while θ(x) is strictly decreasing for all x>0?\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.WallisRemainder`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $B(x)=\\sqrt{x}\\,\\Gamma(x+1/2)/\\Gamma(x+1)$ | `Defs.wallisB` | The exact real Gamma quotient. |\n| $\\mu(x)$ | `Defs.mu` | The displayed nested reciprocal, with exactly the constants 3 and 24. |\n| $\\theta(x)$ | `Defs.theta` | The displayed difference multiplied by $23040x^5/31$. |\n| $\\chi(x)$ | `Defs.chi` | The displayed difference multiplied by $358400x^7/257$. |\n| Implicit reciprocal conditions | First conjunct of `Defs.claim` | For every $x>0$, $\\ln B(x)\\ne0$ and $(3/x)[\\ln B(x)]^{-1}+24\\ne0$. |\n| Strict monotonicity | Remaining conjuncts of `Defs.claim` | For every real $0 For all r ∈ ℤ,\n> Ŝ_{n,k}(−1, 2; r) = Σ_{j=⌊(2−r)/2⌋}^{⌊(n+2−r)/2⌋} C(n − j, n − k) n! C(n + 1, 2j + r − 1).\n\nThe source remarks that the upper argument `n − j` may be negative. Issue\n#10170 fixes the readings: `n, k ∈ ℕ` with `k ≤ n`, `r ∈ ℤ`; `Ŝ` is the\nfinite sum of Theorem 4.1 with the rising factorial; `C(n − j, n − k)` is the\ngeneralized binomial coefficient `Ring.choose`; the sum runs over integers `j`\nbetween the stated floors.","title":"A closed form for the generalized Stirling numbers of boson ordering","url":"https://the-omega-institute.github.io/trureturing-pages/research/maier-2024-boson-ordering-stirling-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/537b6d82f4cc4ee13b8da5c044d445e7373f6b32d6bdbca427fbc480b8f6f6d8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:maity-furuichi-dutta-2026-tsallis-ordering-case-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Srikrishna Maity, Shigeru Furuichi and Supriyo Dutta, *r-deformed α-z-Rényi relative entropy*, arXiv:2607.01805v1, equations (8)–(9), printed page 2:\n\n> For 0 ≤ α < 1 and q > (1 − α), we compare two upper bounds B₁(ρ||σ) = D^{(r)}_{α,z}(ρ||σ) and B₂(ρ||σ) = −Tr[ρ ln_{1−α}((ρ^{−q/2} σ^q ρ^{−q/2})^{1/q})] of the Tsallis relative entropy, numerically. We observe that for non-commutative density operators ρ and σ, D_α(ρ||σ) ≤ B₁(ρ||σ) ≤ B₂(ρ||σ).\n\nSection 4, Case 5, printed page 12:\n\n> Let Tr(ρ) = 1 and Tr(σ) = 1, that is both ρ and σ are density operators. Here, we numerically observe that for any two operators ρ and σ, B₁(ρ||σ) is a tighter bound of Tsallis relative entropy than B₂(ρ||σ).\n\nThe target is the second inequality, $B_1\\le B_2$. [Preregistration #18947](https://github.com/the-omega-institute/trureturing/issues/18947) supplies the source quotations, parameter conventions and literature screen.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Definition 1: $\\ln_r(x)=(x^r-1)/r$ | `lnr` | The literal real-power expression; $r\\ne0$ is imposed by `claim`. |\n| Definition 2: $D^{(r)}_{\\alpha,z}$ is the deformed logarithm of the trace of the powered sandwich, divided by $\\alpha-1$ | `renyiRZ` | The source's $B_1$, using `cfc` and the real trace; no separate forwarding definition. |\n| Equation (6): $D_\\alpha=\\operatorname{Tr}[\\rho-\\rho^\\alpha\\sigma^{1-\\alpha}]/(1-\\alpha)$ | `tsallis` | Literal expression, with real trace. |\n| Equations (8), (47): the inverse-power sandwich and negative weighted deformed-log trace | `B2` | Literal $B_2$, including the order of factors. |\n| Noncommuting density operators | `claim` quantifies $n$, two complex $n\\times n$ matrices, positive definiteness, trace one and noncommutation | Positive definiteness is a sufficient subclass of density operators; it satisfies support and negative-power conditions. Refutation in this subclass refutes the stated assertion. |\n| $0\\le\\alpha<1$, $q>1-\\alpha$; Definitions 1–2 give $r\\ne0$, $z>0$ | All four real parameters and all five hypotheses in `claim` | The counterexample uses plotted parameters $\\alpha=r=1/2$, $z=2$, $q=1$. |\n| $D_\\alpha\\le B_1\\le B_2$ | Both conjuncts of `claim` | `result` negates the full conjunction by contradicting its second conjunct. |","title":"The Maity–Furuichi–Dutta noncommuting Tsallis bound ordering is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/maity-furuichi-dutta-2026-tsallis-ordering-case-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d8702687c5f5baaed907afc2314bc7a469d26d0bd0b9bac189a249150eaf1e0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:maity-krishna-2025-mutual-abelian-border-limits","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Maity and Krishna, “Mutually Abelian-Bordered Binary Words”, arXiv:2509.20773v1,\nSection 5 (Conclusion), question 1, p. 29, ask:\n\n> “Do the limits $\\lim_{n\\to\\infty} \\mathcal{M}(n)/2^{2n}$ and $\\lim_{n\\to\\infty} \\overline{\\mathcal{M}}(n)/2^{2n}$ exist?”\n\nThe counts concern ordered pairs of binary words of the same length. An internal\nabelian border compares a nonempty proper suffix of the first word with a prefix\nof the second; an external border compares a prefix of the first with a suffix\nof the second. Abelian equivalence means equal counts of every letter.\nThe two border lengths may differ and the borders may overlap. The first count\nrequires both borders; the second requires neither.","title":"Maity–Krishna's mutual abelian-border limit question","url":"https://the-omega-institute.github.io/trureturing-pages/research/maity-krishna-2025-mutual-abelian-border-limits/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/25c0e2b4f59c7e05166cc4df1e43b5cd4eb1365b4c302299288bd53ca75d245c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:makuta-kuzaka-augusiak-2023-stabilizer-commutation-rank","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"O. Makuta, B. Kuzaka and R. Augusiak, *Fully non-positive-partial-transpose genuinely entangled subspaces*, arXiv:2203.16902v2; Quantum 7, 915 (2023), Section 3, Conjecture 1 (`final_5.tex`, lines 458–463):\n\n> Given a stabilizer $\\mathbb{S}=\\langle G_{1},\\dots,G_{k}\\rangle$, if the local dimension $d$ is a prime number and the corresponding stabilizer subspace $V$ is genuinely entangled, then\n>\n> $\\dim K(\\mathbb{S}) \\geqslant \\left\\lceil \\frac{N-1}{d-1} \\right\\rceil.$\n\nThe source defines $K(\\mathbb{S})$ using the local commutation exponents (lines 375–391). In the Lean coordinates, write the prime local dimension as $p$. A Pauli word is a phase residue together with $x,z:\\mathrm{Fin}\\,N\\to\\mathbb F_p$, acting as $\\omega^{\\mathrm{phase}}\\bigotimes_n X^{x(n)}Z^{z(n)}$, with $\\omega=\\exp(2\\pi i/p)$. For words $a,b$, the commutation vector has coordinates $z_a(n)x_b(n)-x_a(n)z_b(n)$; `commutationSpace G` is the $\\mathbb F_p$-span of all generator-pair vectors.\n\n`IsStabilizer G` requires pairwise commuting generator matrices and excludes every scalar matrix other than the identity from their generated matrix submonoid. `stabilizerSubspace G` is the intersection of the generators' fixed subspaces. `genuinelyEntangled G` says that every nonzero vector in this intersection fails to factor across every bipartition with two nonempty sides. Factorization means an amplitude of the form $u(x|_Q)v(x|_{Q^c})$; normalization is not required.\n\nIssue [#20105](https://github.com/the-omega-institute/trureturing/issues/20105) fixes the statement for every $N\\ge1$, prime $p$, and finite generator family. `claim` uses natural-number division:\n\n$(N-1+(p-2))/(p-1)\\le\\operatorname{finrank}_{\\mathbb F_p}(\\operatorname{commutationSpace}G)$.\n\nFor $N\\ge1$ and prime $p$, the left side is $\\lceil(N-1)/(p-1)\\rceil$. The conjecture is proved.","title":"The prime-dimensional stabilizer commutation-rank bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/makuta-kuzaka-augusiak-2023-stabilizer-commutation-rank/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7ef6093d57d68263a9917da5d38393ec2bdd32304ee307862e583002e91c360.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:malik-2020-falling-factorial-moment-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. A. Malik and R. Lopez-Mobilia (arXiv:2004.07168, cond-mat.stat-mech) define\nan entropy through the number of samples needed to see a repeated microstate.\nAfter eq. (003.21) they state:\n\n> Conjecture. (n²/k') (1/k')^n k'!/(k'−n)! ≤ 1 for integer n ∈ [0, k'].\n\nwith \"We have not found a proof for this conjecture. However, we have analyzed\nit numerically and it seems to hold.\" Issue #11361 fixes the reading: for every\ninteger `k ≥ 1` and every integer `0 ≤ n ≤ k`, equivalently\n`n² · k(k − 1)⋯(k − n + 1) ≤ k^{n+1}`.","title":"A bound on n² times a falling factorial","url":"https://the-omega-institute.github.io/trureturing-pages/research/malik-2020-falling-factorial-moment-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1fc85b1130edd4db0f154ec83696676c62e818bd8dd32ec34d9813f2817660ac.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:malkoun-2023-subgroup-permanent-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Malkoun, *Do these cousins of permanents satisfy the following inequality?*,\nMathOverflow question 456248, revision 8,\n`382F6B28-9E4F-456B-9254-E0A041A674B5`.\nThe [source note](../Library/QuadraticForms/malkoun2023cousins.md) gives the identity,\nattribution, license and bounded literature comparison.\n\n> Let $H$ denote an $n$ by $n$ hermitian positive semidefinite matrix. Let $G$\n> and $K$ be two subgroups of the symmetric group $\\Sigma_n$. Define\n> $f_{G,K}(H)=\\sum_{(\\sigma,\\tau)\\in G\\times K}\\prod_{i=1}^n\n> h_{\\sigma(i),\\tau(i)}$. Does there exist a positive constant $c(n,G,K)$\n> depending on $n$, $G$ and $K$ such that\n> $\\Re(f_{G,K}(H))\\geq c(n,G,K)\\prod_{i=1}^n h_{ii}$?\n\nThe complete quantifier order is\n\n$$\n\\forall n\\in\\mathbb N\\ \\forall G,K\\leq\\operatorname{Sym}(n)\\quad\n\\exists c\\in\\mathbb R\\quad c>0\\ \\land\\\n\\forall H\\in M_n(\\mathbb C),\\quad\n (H=H^*\\land H\\succeq0)\\Longrightarrow\n c\\prod_i\\Re(h_{ii})\\leq\\Re(f_{G,K}(H)).\n$$\n\n`claim` has this proposition as its definition and `result` has type `¬ claim`.\nNative index $i:\\mathrm{Fin}(n)$ corresponds to source label $i+1$; every\npermutation acts on that same index set. `crossSum` sums over the actual\nsubgroup types, without normalization. No Young-subgroup, rank, entry-sign\nor strictly positive diagonal condition is added to the universal assertion.\n`diagonal_product_eq_real` supplies the Hermitian real-diagonal bridge.","title":"Malkoun: the unrestricted subgroup permanent lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/malkoun-2023-subgroup-permanent-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c5b70a24a8859cd17ac5189692490a2181cca5f194b6e006a2ea5634e3387af.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:malkoun-2025-young-subgroup-conjecture-d","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joseph Malkoun, *Finite graphs and configurations of points* (2025 preprint; arXiv:2508.13472v3; Journal of Geometry and Physics, DOI 10.1016/j.geomphys.2026.105993), Section \"A conjecture in matrix analysis\", defines the setting as follows (TeX line 405):\n\n> Given an equivalence relation, say $\\sim$, on $[n]$, we let $P_\\sim$ be the group of all permutations of $[n]$ which preserve the equivalence classes of $[n]$ under $\\sim$. Let $\\sim_i$, for $i = 1, 2$, be two equivalence relations on $[n] = \\{1, \\dots, n\\}$, with associated permutation groups $P_1$, $P_2$, respectively (we write $P_i$ rather than $P_{\\sim_i}$ to alleviate the notation).\n\nAt TeX lines 407–408:\n\n> Let $H$ be a Hermitian positive semidefinite $n \\times n$ matrix. We define\n> $f_{\\sim_1, \\sim_2}(H) = \\sum_{\\sigma_1 \\in P_1} \\sum_{\\sigma_2 \\in P_2} \\prod_{i = 1}^n H_{\\sigma_1(i), \\sigma_2(i)}.$\n\nConjecture D (TeX lines 410–412) reads, with the source's macro `\\Real` expanded as $\\operatorname{Re}$:\n\n> Under the above hypotheses, we have\n> $\\operatorname{Re}( f_{\\sim_1, \\sim_2}(H) ) \\geq |P_1 \\cap P_2| \\, \\prod_{i = 1}^n H_{i i}.$\n\nThese quotations were checked in `finite_graphs_and_configurations_of_points_v5.tex` from the [v3 source archive](https://arxiv.org/src/2508.13472v3), md5 `3c520a352e066a90cdcffb75a2ee568c`. See also [the literature note](../Library/QuadraticForms/malkoun2025finitegraphs.md).\n\nIssue [#22163](https://github.com/the-omega-institute/trureturing/issues/22163) fixes the reading: for every natural number $n$, every pair of equivalence relations $r_1,r_2$ on $\\mathrm{Fin}\\,n$, and every complex Hermitian positive semidefinite matrix $H$, the proposed inequality holds, with $P_j=\\{\\sigma:\\forall i,\\ r_j(\\sigma(i),i)\\}$. The real-valued formulation uses $\\prod_i\\operatorname{Re}H_{ii}$. The universal assertion is refuted.","title":"Malkoun's Conjecture D fails for Young subgroups","url":"https://the-omega-institute.github.io/trureturing-pages/research/malkoun-2025-young-subgroup-conjecture-d/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/98d9e418336e7c13d5d5cbf7f759d9fdbf02b3d799ba397b8300abdab4f04be1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:malkoun-2026-mo512579-jordan-elementary","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier anchors only the explicit first question of Malkoun's\n*Promoting polynomial equalities for nonnegative variables to elementary symmetric\npolynomial inequalities*, [MathOverflow 512579](https://mathoverflow.net/questions/512579),\nrevision 1, GUID DFD20136-D37A-462C-8A59-FD828AEEFF9A. The source first promotes\nreal nonnegative variables to Hermitian positive semidefinite matrices and sets\n$L=XY+YX$ and $R=X^2+Y^2$. Its question reads (HTML formatting adapted):\n\n> Given an $n$ by $n$ matrix $A$, denote by $\\lambda(A)$ its \"set\" of eigenvalues,\n> taking multiplicity into account. Moreover, denote by $e_k$ the $k$-th elementary\n> symmetric polynomial. Note that both $L$ and $R$ are hermitian, so that their\n> eigenvalues are all real. Is it true that\n> $$e_k(\\lambda(L)) \\leq e_k(\\lambda(R)),$$\n> for any $n$ by $n$ hermitian positive semidefinite matrices $X$ and $Y$ and for\n> any $k \\in \\{1, \\dots, n\\}$?\n\nThe precise assertion is: for every positive integer $n$, every pair of complex\nHermitian positive semidefinite $n\\times n$ matrices $X,Y$, and every integer\n$1\\leq k\\leq n$, the displayed inequality holds. Repeated eigenvalues retain\ntheir algebraic multiplicity. No positivity of $XY+YX$, commutation,\ninvertibility, rank, simple-spectrum or dimension restriction is imposed.\nThe source's later unspecified question about promoting arbitrary symmetric\npolynomial inequalities is not this dossier's assertion.\n\nThe source text is by [Malkoun (user 81645)](https://mathoverflow.net/users/81645)\nand is licensed under [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/).\nSee the [literature note](../Library/Eigenstructure/malkoun2026mo512579.md).","title":"Malkoun's two-matrix elementary symmetric inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/malkoun-2026-mo512579-jordan-elementary/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e3de4feef6b57426541d2975d104e34f28e60bc7a5032436b2dfeb4236ba7bf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mallick-ganguly-majumdar-2026-pea-tensor-nonclosure","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Mallick, N. Ganguly and A. S. Majumdar, *On the characterization of partially entanglement breaking and annihilating channels*, arXiv:2411.19315v2; J. Math. Phys. 67, 082201 (2026), ask in Section VI, Conclusions, PDF page 16:\n\n> We have also shown that $r$-partially entanglement breaking channels are not closed under tensor products, motivating the investigation of whether a similar property holds for $r$-partially entanglement annihilating channels.\n\nThis is the verbatim sentence in `main.tex`, line 1167, from the v2 source archive (member MD5 `02f5d47ee4475c5ba30a52b1e5931b9b`). The definitions are in Section IV.A, PDF pages 9–10, `main.tex` lines 630–632; see also [the literature note](../Library/QuantumContext/mallickgangulymajumdar2026partialannihilating.md).\n\nWrite $\\operatorname{PEA}_{d,r}(E)$ for the module's `IsPEA d r E`: $E$ is a completely positive, trace-preserving complex-linear map on the matrices of $\\mathbb C^d\\otimes\\mathbb C^d$, cannot be written as a tensor product of two individual CPTP maps, satisfies $r For n ∈ Z≥0, we have the following (i) and (ii). (i) If the Grundy number of a position (x, y, z) is 2n, then x ⊕ y ⊕ z = 2n or x ⊕ y ⊕ z = 2n + 1. (ii) If the Grundy number of a position (x, y, z) is 2n + 1, then x ⊕ y ⊕ z = 2n or x ⊕ y ⊕ z = 2n + 1.\n\nThe game is Definitions 4–5: strictly reduce any one of three natural\npile sizes, or merge two piles each at least two, allowing both placements\nof the sum and zero. Preregistration:\n[#19861](https://github.com/the-omega-institute/trureturing/issues/19861).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Three nonnegative pile sizes | Natural arguments of `move`, `grundy` and `claim` | Zero is included |\n| Definition 5 sets (1)–(3) | First three alternatives of `move` | All strict reductions |\n| Definition 5 sets (4)–(6) | Last three alternatives of `move` | Both piles at least two; both placements |\n| Grundy value is mex of option values | `grundy`, with the existing `AbukuEnforceNimberDomination.mex` | Private `move_decreases` and `grundy_eq` remove the well-founded guard on options |\n| Both even and odd implications, for every natural n | Two conjuncts of `claim`, proved by `result` | Neither implication is dropped |\n| Termination measure in the binding conventions | `size`: stone total plus number of nonempty piles | An implementation measure, not an extra game hypothesis |\n\nThe four public definitions preserve the binding expressions in #19861.\nThe measure and recursion guard add no hypothesis to the universal result.","title":"manabe 2024 restricted amalgamation nim conjecture 1","url":"https://the-omega-institute.github.io/trureturing-pages/research/manabe-2024-restricted-amalgamation-nim-conjecture-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bea0e24c2115cb64c6c58582de81cd4d8e2dd4778de487742d9a1b8df13b1015.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:manattu-lakshmanan-2025-cubic-ar-graphs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Manattu and Lakshmanan S., arXiv:2502.11760, Section 7, state: “whether all cubic graphs are AR-graphs is still an open question.” The exact assertion is preregistered in [#14910](https://github.com/the-omega-institute/trureturing/issues/14910).\n\nAn injective positive-integer edge labeling makes a vertex additively rigid when all subset sums of its incident edge labels are distinct. An AR-labeling of a graph with $m$ edges is a bijection from the edges onto $\\{1,\\ldots,m\\}$ making every vertex additively rigid. The target includes every finite simple graph of degree three at every vertex, including the empty graph.","title":"manattu lakshmanan 2025 cubic ar graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/manattu-lakshmanan-2025-cubic-ar-graphs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2cb6735c3051f8f3ddf7d2d8a119cd00bc4fe57d940d481d2c204fbde2e462d8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mandel-robins-2019-minimal-rational-foic-faces","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Mandel and S. Robins, *Dragging the roots of a polynomial to the unit circle*,\narXiv:1908.03208v3, Section 10, Problem 3:\n\n> What are the minimal faces of the FOIC that contain rational points? Most rays clearly do not (except in darga 6). On the other hand, Section 9.2 gives several examples of faces that do. Are there interlace rational polynomials in those minimal faces?\n\nFor $n\\ge2$, write $J=\\{0,\\ldots,\\lfloor n/2\\rfloor\\}$ and\n$O_d=\\{j\\in J:n/\\gcd(n,j)=d\\}$ for $d\\mid n$. For nonempty $S\\subseteq J$,\nlet $F_S$ be the closed face where every index in $S$ attains the maximum\nof the interlace functionals of equation (12). The classification concerns\nnonzero rational points; zero belongs to every closed face.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Darga and sigma coordinates | `n ≥ 2`, `x : Fin (n / 2) → ℝ` | Coordinate k represents sigma index k+1. |\n| Interlace certs and fan faces | `I`, `certIdx`, `interlaceNumber`, `face` | Equations (12)–(15), Proposition 5.5, and the Interlace Formula. |\n| Rational points | `IsRationalPt`, `HasRatPt` | Nonzero rational coordinates. |\n| Minimal rational faces | Second conjunct of `claim` | Maximal cert sets are exactly the complements of individual order classes. |\n| Interlace rational points in those faces | Third conjunct of `claim` | Every nonzero rational point has strict inequality on the omitted class and rational maximum. |","title":"Minimal rational faces of the fan of interlace certs","url":"https://the-omega-institute.github.io/trureturing-pages/research/mandel-robins-2019-minimal-rational-foic-faces/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e49ad38d5a36c39ed28250199bfd82f4dfb3bbf5d8b29195fe5f6b8b77e01f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mane-2026-negative-index-fibonacci-factorization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. R. Mane, *Identically vanishing k-generalized Fibonacci polynomials*,\narXiv:2507.11596v4, p. 19, Section 6, Conjecture 6.4, states:\n\n> For k ≥ 2 and n = −sk, where s ∈ [2, k + 1] (the pattern fails for s ≥ k + 2),\n> 𝓕_{−sk,k}(x) = −x(x^k + 1)^{s−2}(x^k + s). (6.4)\n> Hence 𝓕_{−sk,k}(x) has roots x^k = −s.\n\nThe recurrence (1.2)–(1.3) is\n\n$$\n\\mathcal F_{n,k}(x)=\\sum_{j=1}^k x^{k-j}\\mathcal F_{n-j,k}(x),\\qquad\n\\mathcal F_{1,k}=1,\\qquad \\mathcal F_{n,k}=0\\quad (-(k-2)\\le n\\le0).\n$$\n\nThe frozen definitions `a` and `F` in\n`D5/S3/Zeros/NegativeIndexTribonacciAttainmentRefutation.lean` encode\n$a_k(m)=\\mathcal F_{1-m,k}$ and `F k n = a k (1-n).toNat`.\nThe claim is an identity in $\\mathbb Z[X]$ with natural $k,s$,\n$k\\ge2$ and $2\\le s\\le k+1$; the index is the integer $-sk$.","title":"Mane's negative-index factorization","url":"https://the-omega-institute.github.io/trureturing-pages/research/mane-2026-negative-index-fibonacci-factorization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6575cec2fc4654288beabcb1255a1d88e16bb7fb0c277934a0384db930a99272.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mane-2026-negative-index-fibonacci-root-amplitude","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. R. Mane, *Identically vanishing k-generalized Fibonacci polynomials*,\narXiv:2507.11596v4, Section 6, Conjecture 6.6, states:\n\n> For fixed k ≥ 3 and n < 0, the upper bound is ζ_{n,k} ≤ ⌊|n|/k⌋.\n> The bound is attained whenever r_{n,k} = 1 (equivalently n = −sk,\n> where s ≥ 1), i.e. it is a tight bound.\n\nThe recurrence (1.2)–(1.3) and initial values are\n\n$$\n\\mathcal F_{n,k}(x)=\\sum_{j=1}^{k}x^{k-j}\\mathcal F_{n-j,k}(x),\\qquad\n\\mathcal F_{1,k}=1,\\qquad\n\\mathcal F_{n,k}=0\\quad (-(k-2)\\le n\\le0).\n$$\n\nThe source statistic is the maximum of $|x|^k$ over nonzero complex roots;\nit is zero when there are none, including vanishing polynomials. The Lean\nencoding uses `a k m` for $\\mathcal F_{1-m,k}$, `F k n := a k (1-n).toNat`,\nand the real supremum of the nonzero-root amplitude image. Its empty and\nunbounded totalizations give zero. The guards in `claim` are $k\\ge3$ and\n$s\\ge1$, with integer index $-sk$ and real-valued right side $s$.","title":"The negative-index root-amplitude attainment clause is refuted","url":"https://the-omega-institute.github.io/trureturing-pages/research/mane-2026-negative-index-fibonacci-root-amplitude/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa25f00064be47f231e60c54c791ee9a1fb12b005cae7bb6638f28fdd8713224.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:manetti-ricciardi-2015-higher-antibracket-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Manetti and G. Ricciardi, *Universal Lie Formulas for Higher\nAntibrackets*, SIGMA 12 (2016), 053, arXiv:1509.09032v3, Conjecture 2.4:\n\n> For every $n\\ge 2$ the coefficients $c^n_i$ of Theorem~{\\rm \\ref{thm.standardform}} are given by the formula\n\n$c_i^n=\\frac{\\displaystyle (-1)^{n} \\prod_{j=2}^{i}\\dfrac{n(n-1)-(j-1)(j-2)}{2}}{\\displaystyle \\sum_{h=2}^n h\\left(\\prod_{j=2}^{h}\\dfrac{n(n-1)-(j-1)(j-2)}{2}\\right)\\left(\\prod_{j=h}^{n-1}\\dfrac{(1-j)(j+2)}{2}\\right)}$,\n\n> where every empty product is intended to be equal to $1$. Moreover $(-1)^{n}c^n_i>0$ for every $n\\ge i\\ge 1$.\n\nThe coefficients of Theorem 2.3 are the unique rationals with\n$\\Phi^{n+1}_f=(c_1^n\\rho_1^n+c_2^n\\rho_1^{n-2}\\rho_2+\\dots+c_n^n\\rho_n)f$ for\nevery linear operator $f$ of every graded commutative algebra. The paper\nproves Theorem 2.3 from Theorem 6.4, the same identity\n$\\Phi^{n+1}=(c_1^n\\rho_1^n+\\dots+c_n^n\\rho_n)\\Phi^1$ for the operators\n$\\Phi^{m,i}$ and $\\rho_k$ on $\\mathbb{Q}[x]$, through the $\\mathfrak g$-module\nmorphism of Lemma 6.3; uniqueness in Theorem 2.3 identifies the two\nsequences. The authors verified the formula for $n\\le12$.","title":"The closed form of the higher-antibracket coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/manetti-ricciardi-2015-higher-antibracket-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fadefa51d90399752ed3d39efa05bb0d0af16a30ff32b3420141ae29f432daa1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:manna-mandal-paul-sain-2025-polyhedral-extreme-tests","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Manna, Mandal, Paul and Sain, *An improvement of the\nBlanco-Koldobsky-Turnšek characterization of isometries*, arXiv:2511.20345v1,\nfinal Open Question, S2.p45.1:\n\n> Open Question: In an n-dimensional polyhedral Banach space, does the preservation of Birkhoff-James orthogonality by a bounded linear operator at any n linearly independent extreme points necessarily imply that the operator is a scalar multiple of an isometry?\n\nThis dossier anchors only this question. The standing field is real or complex;\nlocal preservation means every direction and every base-field scalar.\nThe test tuple is chosen independently, rather than requiring preservation\nsimultaneously at every independent tuple. The conclusion permits any one\ncommon scalar, including zero, and a linear isometry of the same carrier.","title":"Independent extreme-point tests in polyhedral Banach spaces","url":"https://the-omega-institute.github.io/trureturing-pages/research/manna-mandal-paul-sain-2025-polyhedral-extreme-tests/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0ab4faf2643306e966dd328e164dd78b38c59c17f332557b650d9704b800f28.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mansour-level-sequences-101-102-catalan","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Toufik Mansour, *Wilf Classes for Level Sequences Avoiding Patterns of Length Three*, Mathematics 14(11),\n1983 (2026), Section 4:\n\n> Problem 1. We have that F_{101,102}(x) = (1 − √(1 − 4x))/(2x) − 1.\n\nA level sequence of length n is a sequence w_1 ⋯ w_n of nonnegative integers with w_1 = 0 and\nw_i ≤ 1 + lev(w_1 ⋯ w_{i−1}), where lev counts adjacent equal entries; the problem asks for the number of\nlevel sequences avoiding the patterns 101 and 102.","title":"Level Sequences Avoiding 101 and 102","url":"https://the-omega-institute.github.io/trureturing-pages/research/mansour-level-sequences-101-102-catalan/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/295aafc89148b8ada092c56ee0911228d69da862a77d65f09e05d56932224e5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mao-wang-2026-delannoy-square-negative-roots","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 4.1 of Mao and Wang, *The Narayana transformation*, asks whether\nthe row polynomials of `A²` and `D²` have only real nonpositive roots.\nThe target here is only the original path-count `D²` clause.\n\nFor every natural `n` and complex `z`, the exact target is\n\n\\[\nG_n(z)=0\\quad\\Longrightarrow\\quad \\operatorname{Im}z=0\n\\quad\\hbox{and}\\quad\\operatorname{Re}z<0.\n\\]\n\nHere `D(n,k)` counts actual E/N/NE paths to `(n-k,k)` and\n`G(n,k)=sum_(j=k)^n D(n,j)D(j,k)`. Row zero is the constant polynomial one.\nThe target has no assumptions about root counts, interlacing, recurrence\nstability or real-rootedness of a surrogate family.","title":"Negative roots of the original Delannoy matrix square","url":"https://the-omega-institute.github.io/trureturing-pages/research/mao-wang-2026-delannoy-square-negative-roots/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6409f7465420693443f490e19cc849c4109c2df4480688b2be0ee123e56bf9d9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marinescu-niculescu-2024-multiplicative-hornich-hlawka","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D.-Ş. Marinescu, C. P. Niculescu, *A survey of the Hornich-Hlawka inequality*, arXiv:2407.03278v1, §6, the paragraph applying Theorem 8 to the logarithm, states:\n\n> Applying Theorem 8 to the function f (x) = ln(1 + x) one obtains the multiplicative analog of the inequality (2.1), (1 + |x|)(1 + |y|)(1 + |z|)(1 + |x + y + z|) ≥ (1 + |x + y|)(1 + |y + z|)(1 + |z + x|), for all x, y, z ∈ R. No extension of this remark to the case of inner product spaces is known.\n\nThe claim replaces absolute values by norms in every real or complex inner product space, with no finite-dimensionality or completeness assumption.","title":"The multiplicative Hornich–Hlawka inequality in inner product spaces","url":"https://the-omega-institute.github.io/trureturing-pages/research/marinescu-niculescu-2024-multiplicative-hornich-hlawka/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5a993e7b7292c18f469efd506f605767a2cbae3c0a7c1acdbda62d889da261c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marinescu-niculescu-2024-variable-exponent-hornich-hlawka","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D.-Ş. Marinescu and C. P. Niculescu, *A survey of the Hornich-Hlawka inequality*, arXiv:2407.03278v1, §3, the paragraph introducing Problem 3 and Problem 3 itself, state verbatim:\n\n> A natural generalization of Lp spaces is provided by the class of Lp(·) spaces with variable exponent. They differ from the classical Lp spaces in that the exponent p is not constant but a function from the measure space under attention to [1, ∞]. See the monograph of L. Diening, P. Harjulehto, P. Hästö and M. Ruzicka [6] for a thorough presentation. Problem 3. Is every Lp(·)(0, 1) space with a variable exponent p(·) : (0, 1) → [1, 2] a Hornich-Hlawka space?\n\nA Hornich–Hlawka space satisfies\n\n$$\n\\|f+g\\|+\\|g+h\\|+\\|h+f\\|\n\\le \\|f\\|+\\|g\\|+\\|h\\|+\\|f+g+h\\|\n$$\n\nfor every triple. The norm considered here is the Luxemburg norm. The journal version is *A New Look at the Hornich-Hlawka Inequality*, Results Math. (2025), doi:10.1007/s00025-025-02464-9; the quotation and locator refer to the arXiv version.","title":"Variable-exponent Lebesgue spaces and the Hornich–Hlawka inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/marinescu-niculescu-2024-variable-exponent-hornich-hlawka/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/beebea5f1930aff77a9e1ff86177802c61ecaf7202bb67ab0fc664313df0ed48.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mariot-manzoni-2022-moca-orthogonal-array-strength-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Mariot and L. Manzoni, *Building Correlation Immune Functions from Sets of Mutually Orthogonal Cellular Automata*, arXiv:2207.08280v1, Section 5, asks to “prove that the binary OA obtained from a set of k-MOCA has always at least strength 3”. The precise target and source quotations are preregistered in [#19624](https://github.com/the-omega-institute/trureturing/issues/19624). The source is described in [the Library note](../Library/StatisticalMechanics/mariotmanzoni2022moca.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Definition 1: sliding windows of length $d$ | `ca` | $n=2b$, $d=b+1$, zero-based finite indices |\n| Bipermutive: $f(y)=y_0+\\phi(y_1,\\ldots,y_{b-1})+y_b$ over $\\mathbf F_2$ | `IsBipermutive` | existential arbitrary central Boolean function |\n| Eq. (6): unequal superposed pairs at distinct cells | `AreOrthogonal` | input halves joined by `Fin.append` |\n| A family of $k$ pairwise orthogonal bipermutive CA | `IsMOCA` | every rule bipermutive, distinct rule indices orthogonal |\n| Eq. (7): juxtaposition of all $k$ output vectors | `mocaArray` | all $2^{2b}$ input rows, columns in `Fin k × Fin b` |\n| Each pattern occurs $N/2^t$ times in any $t$ distinct columns | `IsOrthogonalArray` | injective column selections, subtype cardinality, natural division |\n| Always strength at least three | `claim`, `result` | all $b,k$ and all such families; strength $t=3$ |\n\nFor $b=0$ the output is empty; this degenerate extension exceeds Definition 1's $d\\le n$ parameter range and introduces no three-column selection. The same vacuity applies whenever $kb<3$. The theorem retains the complete source range $b\\ge1$ and does not assert that three columns exist.","title":"Strength three of every Mariot–Manzoni MOCA array","url":"https://the-omega-institute.github.io/trureturing-pages/research/mariot-manzoni-2022-moca-orthogonal-array-strength-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8af8078ee33fad73f7d4ed5589dcc6fc3b96a30e569d3db47852ad77a6f1d32.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marotti-needleman-2026-consecutive-cycle-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Marotti and Needleman, *Consecutive Cycle Sums*, arXiv:2610.08889, Section 5, final paragraph on printed page 6, ask whether the cycle with labels 1, …, n can eventually be completed using at most n/a added edges for every positive integer a. A graph is complete when every integer from 1 through T_n = n(n + 1)/2 is the label sum of a nonempty vertex set inducing a connected subgraph. The label order remains fixed, and mEC(n) is the minimum number of edges added to this cycle to make it complete.\n\nThe precise target is: for every integer a ≥ 1, there exists N such that, for every integer n ≥ N, a complete graph containing the original n-cycle has an added-edge count e satisfying ea ≤ n.","title":"Sublinear edge completion for consecutive cycle sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/marotti-needleman-2026-consecutive-cycle-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a42d5dd853d4acd836f787b7bd99b902c89dce2e335ad157d0a9ce9ffde08a13.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marquette-vanderkamp-quispel-2026-lotka-volterra-birationality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"I. Marquette, P. H. van der Kamp and G. R. W. Quispel, *Subalgebras of integrals, commutants, and superintegrable deformations of Lotka–Volterra systems*, arXiv:2602.17945v1, Section 2:\n\n> In arbitrary dimensions, we have not been able to prove the birationality of the change of variables, but we conjecture this to be the case, and we provide explicit formulas in Appendix \\ref{secA}.\n\nAppendix A restates it: \"We conjecture that the transformation from the $n$ variables $x_i$ to the $n-1$ integrals and distinguished variable $x_n$ is a birational map.\" The integrals F_k and their coordinate reversals G_k are quoted in [the literature note](../Library/FluidDynamics/marquette2026lotkavolterra.md).\n\nIssue [#19118](https://github.com/the-omega-institute/trureturing/issues/19118) fixes the reading: for every n ≥ 2 and every field of characteristic zero, the following n elements generate the rational function field F(x_1, …, x_n):\n- odd n = 2m − 1: F_1, …, F_m, G_2, …, G_{m−1}, x_n;\n- even n = 2m: F_1, …, F_m, G_1, …, G_{m−1}, x_n.\n\nThis is proved.","title":"The Lotka–Volterra integrals are birational coordinates in every dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/marquette-vanderkamp-quispel-2026-lotka-volterra-birationality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f8e58814dc252972c216a9a02589f6d444e7cabdca45d8485808e8b70f19bba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marquez-gonzalez-2026-cp-filter-transpose-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Samuel A. Márquez González, *Feasibility Ordering of Entanglement-Source\nPlacement for Qubit Channels*, arXiv:2609.18803v1, printed PDF p. 5,\nSection VI, Eq. (47), asks:\n\n> This separates the higher-dimensional problem into two levels. The strong\n> question is whether every unital qudit channel $\\Upsilon$ is CP-filter\n> equivalent to its transpose, i.e., whether there exist invertible CP maps\n> $\\mathcal L$ and $\\mathcal R$, with CP inverses, such that\n> $\\Upsilon^T=\\mathcal L\\circ\\Upsilon\\circ\\mathcal R$.\n\nIssue #13180 preregisters the universal question for every dimension\n$d\\geq2$ and the refutation at $d=3$. A channel is completely positive\nand trace preserving, with unitality $F(I)=I$. Both filters have two-sided\ncompletely positive inverses; neither filter must preserve trace.\nThe channel transpose is the source's Eqs. (15)–(16): a finite Kraus\nfamily $K_i$ gives $F^T(X)=\\sum_i K_i^T X\\overline{K_i}$ in the fixed basis.","title":"A qutrit obstruction to strong CP-filter equivalence","url":"https://the-omega-institute.github.io/trureturing-pages/research/marquez-gonzalez-2026-cp-filter-transpose-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4c7e13e431874d84761a09db0f29e7d7706fd3e7c883205eb4ed7c9e228e984.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marquez-gonzalez-2026-positive-filter-transpose-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Samuel A. Márquez González, *Feasibility Ordering of Entanglement-Source\nPlacement for Qubit Channels*, arXiv:2609.18803v1, Section VI, Eq. (48), states\nthe weak question:\n\n> The weaker question, which is sufficient for the source-placement theorem,\n> asks only whether there exist a positive map $\\mathcal{F}$ and a completely\n> positive map $\\mathcal{E}$ such that\n> $\\Upsilon^T=\\mathcal{F}\\circ\\Upsilon\\circ\\mathcal{E}$.\n\nand adds that \"failure of the weak condition would identify the precise\nobstruction to this proof strategy\". The channel transpose is Eq. (16),\n$\\Lambda^T(X)=\\sum_iK_i^TX\\overline{K_i}$. The verbatim statements are in\n[the literature note](../Library/QuantumStates/marquezgonzalez2026feasibility.md).\n\nIssue [#13926](https://github.com/the-omega-institute/trureturing/issues/13926)\nreads Eq. (48) as the assertion that every unital qutrit channel admits the\nweak factorization: for every completely positive, trace-preserving, unital\n$\\Upsilon$ on $M_3$ and every finite Kraus family of $\\Upsilon$, there exist a\npositive $\\mathcal F$ and a completely positive $\\mathcal E$ on $M_3$ with\n$\\Upsilon^T=\\mathcal F\\circ\\Upsilon\\circ\\mathcal E$. Neither filter must\npreserve trace, be unital or be invertible.","title":"A qutrit obstruction to positive/CP factorization of the channel transpose","url":"https://the-omega-institute.github.io/trureturing-pages/research/marquez-gonzalez-2026-positive-filter-transpose-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aaecb5c47ef43d2a7f9df1e017407a59006c6bcdf32d5e671883dd6ec53b209e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:martinez-mirror-gamblers-ruin","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Martinez, *A symbolic computational approach to the generalized gambler's ruin\nproblem in one and two dimensions*, arXiv:2412.07667v1, Conjecture 5.1, page 16:\n\n> Consider the generalization of the gambler’s ruin problem when we add a mirror step. If the particle starts at $x=1$, then\n> $$\\lim_{N\\to\\infty} f_N^{(p)}(1)=\\frac{\\sqrt{(p+1)(1-3p+4p^2)}-(1-2p)(p+1)}{2p(p+1)}$$\n> whenever we restrict the particle moves by either moving from $x$ to $x-1$ with probability $q_1$, or from $x$ to $x+1$ with probability $p$, or from $x$ to $N-x$ with probability $q_2$ where $q_1=q_2=\\frac{1-p}{2}$.\n\nHere $0 f N 1) Filter.atTop …` | Natural-number limit |\n| The displayed radical | `limitValue p` | Literal numerator, denominator and positive square root |\n| Winning-probability family | `∀ f`, satisfying the recurrence for every $N≥2$ | Uniqueness identifies every such family |\n\nThe universal family quantifier introduces no extra restriction on the source's\nsolution: existence follows from the finite formula, and `harmonic_unique`\nproves uniqueness on the interval. The lower bound $N≥2$ ensures the starting\npoint is interior and removes only finitely many indices from the limit.\nNatural subtraction is truncated, and at interior points agrees with integer\nsubtraction. Conjecture 5.1 assigns $p$ to the right step; the symmetric example\nin Section 4 assigns $p$ to the mirror step. The claim uses Conjecture 5.1's\nassignment.","title":"Martinez's mirror-step gambler's ruin limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/martinez-mirror-gamblers-ruin/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7402bf90cd97f0ffe48f9757f429c0b7a818727880e7c26355d91210027d1001.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:marton-2023-double-cglmp-one-bit-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"I. Márton, E. Bene, P. Diviánszky and T. Vértesi (arXiv:2308.10771, quant-ph;\nnpj Quantum Information 10, 79 (2024)) look for Bell-type inequalities that\nlocal hidden variables cannot saturate even with one bit of communication.\nWith `CGLMP_d = P(A₀ ≥ B₀) + P(A₀ ≤ B₁) + P(A₁ < B₀) + P(A₁ ≥ B₁)` and two copies\nplayed in parallel, they state:\n\n> The one-bit bound L1bit(CGLMP_d^{⊗2}) = 12 in the last column is verified by\n> the branch-and-bound algorithm up to d = 10. We conjecture that this is the\n> exact bound for any d ≥ 2.\n\nFor the truncations to the inputs `{00, 01, 11} × {00, 01, 11}` and\n`{00, 01, 11} × {00, 11}` they give \"the conjectured local bound\" 7 and 4,\n\"which we verified up to d = 20\".\n\nIssue #11276 fixes the reading. Local behaviours follow eq. (P_LHV): the hidden\nvariable has an arbitrary probability distribution on `ℝ`, and the responses are\nmeasurable conditional distributions. One-bit behaviours follow eq.\n(P_LHV1bit), with a measurable bit `l(X, λ)`; both directions of communication\nare included, as the paper lists the game as bidirectional. Each bound is the\ngreatest value of the Bell expression over its class.","title":"The one-bit bound of the double CGLMP game","url":"https://the-omega-institute.github.io/trureturing-pages/research/marton-2023-double-cglmp-one-bit-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b24bef7a2d817f9c12566aa425c158b413f2232c3fa170d876d6f56189cdfbae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:masini-theurer-plenio-2021-postprocessed-improvement-unfaithfulness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Masini, T. Theurer and M. B. Plenio, *Coherence of operations and interferometry*, Phys. Rev. A 103, 042426 (2021), arXiv:2102.04863v2, Section IV (Conclusions), `PaperMasiniTheurerPlenio.tex`, line 300:\n\n> Whilst we provided a method to compute the pre-processed improvements and showed that they are not faithful for $\\lambda\\ne\\mu$, it is an open question whether there exist analogous results for the post-processed cases.\n\nThe source defines faithfulness in Section II A, line 153:\n\n> $M(\\Theta)=0$ exactly if $\\Theta$ is a free channel.\n\nHere $\\mu=1-\\lambda$, $\\Delta$ removes off-diagonal entries in the chosen incoherent basis, and the free channels for coherence creation are the maximally incoherent operations, characterized by $\\Psi\\Delta=\\Delta\\Psi\\Delta$ (Section II A, lines 132–134). The phase channel is conjugation by $U=\\operatorname{diag}(e^{i\\phi_k})$ (Section II B, lines 171–175). The post-processed improvement is defined in Section III B, Eq. (14), line 273:\n\n> $N_{\\lambda,\\vec{\\phi}}(\\Theta):=\\max_{\\Psi\\in\\mathcal{MIO}}\\left|\\left|\\left(\\lambda-\\mu \\Lambda_{\\vec{\\phi}}\\right) \\Psi \\Theta\\Delta\\right|\\right|_1-|\\lambda-\\mu |$.\n\nThe source calls the operation norm the induced trace norm (Section III A, line 202). The formal reading uses $\\|\\Phi\\|_{1\\to1}=\\sup_{\\|A\\|_1\\le1}\\|\\Phi(A)\\|_1$ over all complex matrices, with no positivity or Hermiticity restriction on $A$.\n\nIssue [#22607](https://github.com/the-omega-institute/trureturing/issues/22607) fixes the faithfulness half of the question: for every $\\lambda\\in[0,1]$ with $\\lambda\\ne\\tfrac12$, every natural $M\\ge1$, and every $\\vec\\phi\\in\\mathbb R^M$, there exists a qubit-to-qubit channel $\\Theta$ that is not maximally incoherent and has $N_{\\lambda,\\vec\\phi}(\\Theta)=0$, optimizing over all MIO channels from the qubit to dimension $M$. This is proved.\n\nThe definitions and source locators are collected in [the literature note](../Library/Divergence/masinitheurerplenio2021interferometry.md).","title":"Biased post-processed interferometric improvements are not faithful","url":"https://the-omega-institute.github.io/trureturing-pages/research/masini-theurer-plenio-2021-postprocessed-improvement-unfaithfulness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b667de9129c14d5e176c94e7d3dabc7620ef53cfb7ddde31ddc691fc72359f7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-2012-a026023-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A026023 counts the sequences `s(0), …, s(n)` of nonnegative integers with\n`|s(i) − s(i − 1)| = 1` and `s(0) = 3`; by a comment of R. M. Ziff, `a(n)/2^n`\nis the probability that a random walker started at `x = 4` has not been\nadsorbed at `x = 0` by time `n`. Its formula field records:\n\n> Conjecture: (n+4)*(n-1)*a(n) +(n-1)*(n+1)*a(n-1)\n> -2*(n+1)*(2*n+1)*a(n-2) -4*(n-1)*(n+1)*a(n-3)=0. - _R. J. Mathar_, Sep 29\n> 2012\n\nIssue #10761 fixes the reading: `a` is the entry's name read literally, the\nrecurrence is asserted for every `n ≥ 3` and read in `ℤ`.","title":"Mathar's recurrence for the surviving walks A026023","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a026023-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8cabfe87ae3670dd9dddceea44be5ddf24982b41cf76b56c6d9a3b4214284261.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-2012-a068551-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A068551 is `a(n) = 4^n − C(2n, n)`; by a comment of G. Critzer it counts\nthe returns to the axis in all ±1 lattice paths (random-walk bridges) from the\norigin to `(2n, 0)`. Its formula field records:\n\n> Conjecture: n*a(n) + 2*(3-4*n)*a(n-1) + 8*(2*n-3)*a(n-2) = 0. - _R. J.\n> Mathar_, Apr 01 2012\n\nIssue #10754 fixes the reading: `a` is the entry's name, the recurrence is\nasserted for every `n ≥ 2` and read in `ℤ`.","title":"Mathar's recurrence for the lattice-path returns A068551","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a068551-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0dbe26d9e72a97467cb4b54f6cb119c0499e1a2dd927c7b6f4a811d72b721812.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-2012-a108958-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A108958 counts the unordered pairs of distinct binary words of length\n`n` with the same number of 1's, `a(n) = Σ_{k=0}^{n} C(C(n,k), 2)`; for `n`\nspins 1/2 these are the nontrivial zero-quantum transitions. Its formula\nfield records:\n\n> Conjecture: n*(n-2)*a(n) +2*(-3*n^2+7*n-3)*a(n-1) +4*(n-1)*(2*n-3)\n> *a(n-2)=0. - _R. J. Mathar_, Apr 04 2012\n\nIssue #10598 fixes the reading: `a` is defined by the entry's first formula\nline (also at `n = 0`, value `0`), and the recurrence is asserted for every\n`n ≥ 2`.","title":"Mathar's recurrence for the zero-quantum transition pairs A108958","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a108958-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d91174448cc8d6487a04ad36a0ee2425c066d49757340c640333e3bb07edee6d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-2012-a113409-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A113409 is named \"A transform of the central binomial coefficients A001405.\"\nIts FORMULA field defines the sequence literally:\n\n> a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*C(k, floor(k/2)).\n\nThe conjecture in the same field is:\n\n> Conjecture: (n+2)*a(n)-2*(n+1)*a(n-1) +(n-4)*a(n-2) +2*a(n-3) +4*(2-n)*a(n-4)=0. - _R. J. Mathar_, Nov 07 2012\n\nPreregistration issue #13334 fixes the nonnegative-index domain: every natural\n`n >= 4`. The binomial sum and its indices are natural numbers; the displayed\nrecurrence is evaluated in the integers. Its statement is `claim`, and\n`D5/S1/Recurrence/MatharCentralBinomialTransformRecurrence.result` proves it.","title":"Mathar's recurrence for OEIS A113409","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a113409-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0397437a5fc3b790d83a111beb6271adb7b4aa739cb2ea92d9b32906f0af8492.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-2024-nonbonding-domino-pairs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. J. Mathar (arXiv:2404.18806, 2024) counts placements of dominoes on an\n`r × c` board that are non-bonding: every square of one is at L1 distance at\nleast 2 from every square of another. They are hard dimers with\nnearest-neighbour exclusion. His Conjecture 1 (equation (21)) reads:\n\n> D(r,c,2) = 2c²r² − 2(cr² + c²r) + ½(r² + c²) − 22cr + (59/2)(c + r) − 30;\n> r, c ≥ 3.\n\nIssue #11068 fixes the reading. A domino is a set of two board squares at L1\ndistance 1, and `D(r, c, 2)` counts the sets of two pairwise non-bonding\ndominoes. The identity holds in `ℚ` for all `r, c ≥ 3`.","title":"Mathar's Conjecture 1 on two non-bonding dominoes","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2024-nonbonding-domino-pairs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/221d9111ccebed9cd2870fa279b0e1bb3a3c478fd78dd28bd8584b8d92dc45bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-a027182","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A027182 revision 8 states Mathar's FORMULA conjecture for the\nself-convolution of the rows of Kimberling's A027170 revision 32.\nThe verbatim source is in\n[the source note](../Library/RecurrenceComposition/matharkimberling2026a027182.md),\nand the quantified statement is preregistered in\n[#23023](https://github.com/the-omega-institute/trureturing/issues/23023).\n\nFor $n,k\\in\\mathbb N$, define\n$t(n,k)=\\binom{n+2}{k+1}+\\binom{n+1}{k}+\\binom{n+1}{k+1}+\\binom nk$,\n$T(n,k)=t(n,k)-t(0,0)+1$, with natural subtraction, and\n$g_n(X)=\\sum_{k=0}^n T(n,k)X^k\\in\\mathbb N[X]$.\nPut $a_n=[X^n]g_n(X)^2$.\nThe settling theorem proves, for every $n\\ge6$,\n\n$$\n2(n+2)a_n+(-17n-23)a_{n-1}+(47n+36)a_{n-2}\n+2(-22n-17)a_{n-3}+4(-n+20)a_{n-4}\n+8(2n-9)a_{n-5}+24(12n-47)=0.\n$$\n\nThe coefficient arithmetic is in $\\mathbb Z$ and sequence indices use\nnatural subtraction. The proposition asserts this recurrence; it does\nnot assert a separate formal D-finiteness predicate.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Four binomial terms defining $t$ in A027170 | `t` | Same terms, order and natural carrier. |\n| $T(n,k)=t(n,k)-t(0,0)+1$ | `T` | Literal natural subtraction, with the source row domain $0\\le k\\le n$. |\n| Row generating polynomial over $k=0,\\ldots,n$ | `g` | Literal finite sum, including both endpoints. |\n| Central coefficient of the squared row polynomial | `a` | Coefficient $n$ of $g_n^2$. |\n| Six sequence terms and affine forcing term in the FORMULA field | `claim`, `result` | Every coefficient and lag is preserved, over all natural $n\\ge6$. |\n| Published offset $1$ | Total `a` with $a_0=1$ | The extension is explicit; no extra hypothesis restricts the published range. |\n| No additional source clause | Integer casts and total definitions | Coefficient arithmetic is integer arithmetic; totality outside the source row domain does not weaken the quantified claim. |","title":"Mathar and Kimberling's self-convolution recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6aa9306de344e1d9a035d36908e2a5f35346b6dff8d22427987929e3e6532c3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-a174193-order-four-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. J. Mathar's unnumbered FORMULA entry for OEIS A174193 gives the algebraic\nseries\n\\[\n\\frac{1-8x+13x^2+24x^3-48x^4-(1-6x+x^2+34x^3-26x^4-4x^5)\\sqrt{1-4x}}{2x^2(1-x)(1-4x)^2}\n\\]\nand conjectures the displayed order-four recurrence for its coefficients. The\nformal statement uses rational power series, a branch $s$ with constant\ncoefficient one and $s^2=1-4X$, and the cleared equation $DA=P-Bs$. It asserts\nexistence of such a source series and the recurrence for every such series and\nevery natural $n\\ge4$.\n\n### Statement echo\n\nThe Lean `claim` contains the existence conjunct, the universal source-series\npredicate, the guard $4\\le n$, natural lagged indices $n-1$ through $n-4$, and\nthe five rational polynomial coefficient terms exactly. `IsSourceSeries` contains\n$\\exists s$, $s(0)=1$, $s^2=p$, and $DA=P-Bs$.","title":"Mathar's OEIS A174193 order-four recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a174193-order-four-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ddf66d031875fdc896ea680ab018d0afd86f262eb52da8da66f1fba0fd329a2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathar-oeis-a073759-least-prime-factor","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. J. Mathar's conjecture in [OEIS A073759](https://oeis.org/A073759), FORMULA (unpaginated), states: \"Conjecture: a(n) = 2*A046667(n) if n>=5. - _R. J. Mathar_, May 15 2026\". The NAME is \"Largest number < n that is neither a divisor of nor relatively prime to n, or 0 if no such number exists.\" The companion entries define A046666 by \"a(n) = n - (smallest prime dividing n).\" and A046667 by \"a(n) = A046666(n)/2.\" The offsets are respectively 1, 2 and 2. Preregistration: [#18937](https://github.com/the-omega-institute/trureturing/issues/18937).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive number below $n$ | `a`: the finite interval from 1 inclusive to $n$ exclusive | Literal preregistered convention |\n| Neither divisor nor relatively prime | `a`: the conjunction of nondivisibility and failure of `Nat.Coprime` | Both exclusions retained |\n| Largest, or zero if empty | `a`: `Finset.sup id` of the filtered interval | Natural supremum defaults to zero |\n| Smallest prime dividing $n$ | `A046666`: natural subtraction of `Nat.minFac n` | Exact for the source domain $n\\ge2$ |\n| Divide A046666 by two | `A046667`: `Nat.div` by 2 | Integer division; parity supplies exactness |\n| Every $n\\ge5$ | `claim`, proved by `result` | No extra hypothesis or conjunct |\n\nAll names belong to `D5.S3.Arith.Primes.MatharA073759`. The definitions extend to natural indices outside the OEIS offsets; the settlement quantifies only over $n\\ge5$.","title":"mathar oeis a073759 least prime factor","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c678585d15aeb8a15ddb5537813eed17e0882f1e86a3ce38312e9f13402df7b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathoverflow-469781-weighted-orthogonal-minors","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source is meler (MathOverflow user 526848; real name unverified), [MO469781](https://mathoverflow.net/questions/469781), revision 1, GUID `F53D06CE-3805-43B9-9FEB-43A79F67F7AB`, CC BY-SA 4.0, with identical [MSE4902019](https://math.stackexchange.com/questions/4902019) crosspost. The [Library note](../Library/Geometry/meler2024weightedminors.md) owns the source citation. The complete external question is:\n\n> Fix $n \\geq 3$ and take any orthonormal vectors $x,y,z \\in \\mathbb{R}^n$. Let also $A \\in M_n(\\mathbb{R})$ be a symmetric matrix with positive entries ($A_{ij} = A_{ji} > 0$). Is the following inequality true? $$\\sum_{i For every odd $n$ and every nonzero $b$, does the affine hull of\n> $\\mathrm{Supp}(b)$ have dimension at least $n-1$? Equivalently: is\n> $\\mathrm{Supp}(b)$ contained in no affine subspace of codimension $\\ge 2$?\n\nThe [original revision](https://api.stackexchange.com/2.3/revisions/E06A6743-2C15-41CD-9390-04D1862ABA02?site=mathoverflow&filter=withbody)\nis attributed under CC BY-SA 4.0. The title's hyperplane wording is a locator;\nthe target is the dimension lower bound in the body.\n\nPrecisely, let $n$ be any positive odd integer, $b\\in\\mathbb F_2^n$ nonzero,\nand $a\\in\\mathbb F_2^n$. All indices are modulo $n$. Put\n$g_k=a_{k-1}$ and $h_k=a_k+a_{k-2}+b_k$ in $\\mathbb F_2$, and define the\ninteger matrix\n\n$$\nA_k(a)=\\begin{pmatrix}1&1\\\\(-1)^{h_k}&(-1)^{h_k+g_k}\\end{pmatrix}.\n$$\n\nEquivalently, its entry indexed by $s,t\\in\\{0,1\\}$ is\n$(-1)^{h_ks+g_kst}$. With the literal ordered integer product, set\n\n$$\nW_b(a)=\\operatorname{Tr}(A_0(a)A_1(a)\\cdots A_{n-1}(a)),\\qquad\n\\mathrm{Supp}(b)=\\{a\\in\\mathbb F_2^n:W_b(a)\\ne0\\}.\n$$\n\nThe assertion is\n$\\dim_{\\mathbb F_2}\\operatorname{aff}(\\mathrm{Supp}(b))\\ge n-1$.\nLength one, coincident cyclic indices, every nonzero mask, and every tested\nword including zero and all-one are included. There is no length or weight\ncutoff, additional nonempty-support premise, or equality assertion.","title":"Affine support of the odd cyclic sign-matrix trace","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathoverflow-512655-cyclic-sign-trace-affine-support/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d23da9b0266995432c5f443fcd237280bdf484a410531ebfe967e3de5b0d7efe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathoverflow-513954-singular-diagonal-downdate-inertia","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source is Lucid (MathOverflow user 596445), revision 2 of [MathOverflow question 513954](https://mathoverflow.net/revisions/513954/2), revision GUID `1084D752-1BBD-405E-B335-28FBDD1F4D6D`. The [Library note](../Library/QuadraticForms/lucid2026structureddowndate.md) is the sole source acknowledgement and records the CC BY-SA 4.0 license.\n\nThe source asks:\n\n> I ran into this while working through a rank-one perturbation problem and got stuck on the case where $D$ is singular.\n> Let $D = \\operatorname{diag}(d_1, \\dots, d_n)$ be real diagonal, possibly singular, and let $x \\in \\mathbb{R}^n$ satisfy\n> $$\\Delta := x^\\top D x \\neq 0.$$\n> Set $w = Dx$ and\n> $$M = D - \\frac{ww^\\top}{\\Delta}.$$\n> Writing $\\operatorname{inertia}(A) = (n_+(A), n_-(A), n_0(A))$, is the following formula always valid?\n> $$\\operatorname{inertia}(M) = \\begin{cases} (n_+(D) - 1,\\ n_-(D),\\ n_0(D) + 1), & \\Delta > 0, \\\\ (n_+(D),\\ n_-(D) - 1,\\ n_0(D) + 1), & \\Delta < 0. \\end{cases}$$\n> I specifically want a proof that doesn't assume $D^{-1}$ exists.\n> Note: this isn't the general rank-one update $D + \\rho vv^\\top$ for arbitrary $v$ (the classical secular-equation setting) , here $w = Dx$ is tied to $D$ itself, with coefficient exactly $1/\\Delta$, chosen so $x$ becomes an exact null vector of $M$. I'm after the inertia count specifically, including the singular-$D$ case and the index-tracking behavior in the corollary below.\n> Here's what I've tried. Define $H = \\{z : w^\\top z = 0\\}$. Since $w^\\top x = \\Delta \\neq 0$,\n> $$\\mathbb{R}^n = H \\oplus \\operatorname{span}\\{x\\}.$$\n> If $Z$ contains a basis of $H$ and $P = [Z \\ \\ x]$, then\n> $$P^\\top D P = \\begin{pmatrix} Z^\\top D Z & 0 \\\\ 0 & \\Delta \\end{pmatrix}, \\qquad P^\\top M P = \\begin{pmatrix} Z^\\top D Z & 0 \\\\ 0 & 0 \\end{pmatrix}.$$\n> Does Sylvester's law of inertia immediately establish the displayed formula, including all additional null directions when $D$ is singular?\n> A related corollary: suppose $D(y)$, $x(y)$, and hence $M(y)$, vary continuously on $X \\setminus A$, where $X$ is homeomorphic to an open $(n-1)$-simplex, $n \\geq 3$, and $A$ is finite. If $\\Delta(y) \\neq 0$ and $M(y) \\succeq 0$, does path-connectedness imply that $\\Delta$ has constant sign and, when negative, exactly one fixed diagonal index of $D(y)$ remains negative throughout?\n> I've checked the inertia formula numerically for random $D$, $x$ up to $n = 9$, including singular $D$, and haven't found a counterexample.\n\nThis is one full unit: the complete two-sign inertia triple and the continuous-family sign/fixed-negative-index corollary. The matrix clause ranges over every finite dimension and retains singular diagonal matrices; the family clause keeps the open-simplex-minus-finite-points domain, continuous families, nonzero pivot and pointwise positive semidefiniteness. No invertibility, connected-family replacement, dimension cutoff, tolerance or spectral-gap premise is added.","title":"Inertia of a singular diagonal matrix after a structured rank-one downdate","url":"https://the-omega-institute.github.io/trureturing-pages/research/mathoverflow-513954-singular-diagonal-downdate-inertia/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28cb122e0fbc56e2f4f8e88c5ca54abc514d836b1751f5ada0f0545ce93a1901.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mathoverflow-516091-prime-square-local-moment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source is MathOverflow question 516091, revision 1, by **unkown**, GUID `5D544532-802E-47EE-9240-92FCF941D961`. The [attributed source note](../Library/AnalyticClosure/unkown2026primesquaremoment.md) gives the original definitions, question and bounded literature comparison. [Issue #18820](https://github.com/the-omega-institute/trureturing/issues/18820) specifies the quantified target.\n\nAll local Lean names below belong to `D5.S3.Analytic.SeriesInequalities.PrimeSquareMomentApproximation`.\n\n| Source clause | Lean counterpart | Domain and endpoints |\n| --- | --- | --- |\n| $e_p(0)=1$ | `factor p 0` | Index zero is present |\n| Rational $e_p(a)$ for integer $a\\ge1$ | Nonzero branch of `factor` | Every positive natural exponent |\n| Real $s>0$ | `Real.rpow` in `localMoment` | The final real threshold is chosen with $s\\ge1$ |\n| Full infinite sum over $a\\ge0$ | `∑' a : ℕ, (factor p a)^s / p^a` | No truncation or finite proxy |\n| $c=\\log(7/6)/2$, $P=e^{cs}$ | `c`, `Real.exp (c*s)` | $P^2=\\exp(2cs)$ |\n| Sufficiently small fixed $\\delta>0$ | `0 < δ`, `δ < 1 - c3/c` | `c3 = log(5/4)/3`; positive nonempty interval |\n| Every fixed $B>2$ | `∀ B : ℝ, 2 < B →` | Threshold may depend on $δ,B,ε$ |\n| Uniform $o(1)$ as real $s\\to\\infty$ | `∀ ε>0, ∃ S : ℝ, ∀ s≥S, ∀ p, ...` | One threshold precedes both $s$ and $p$ |\n| Prime $p\\equiv1\\pmod4$ | `p : ℕ`, `p.Prime`, `p % 4 = 1` | No change of prime or congruence domain |\n| $P^{1-δ} Supposedly, if e does not vanish in the equation above, S(x) is not globally bounded. However, this question remains open for now.\n\nSection 7 states the parameter condition explicitly:\n\n> The case of $b_2 \\neq \\frac{2a_2b_1-a_1a_2b_0}{a_1}$ is still poorly understood, although we believe it leads to not globally bounded solutions.\n\nThe literal assertion quantifies over every rational tuple\n$(a_1,a_2,b_0,b_1,b_2)$ with $a_1a_2\\neq0$ and\n$b_0\\notin\\{-2,-3,\\ldots\\}$, and every sequence $s:\\mathbb N\\to\\mathbb Q$\nwith arbitrary rational initial values satisfying\n\n$$\n(n+b_0)s_n+(a_1n+b_1)s_{n-1}+(a_2n+b_2)s_{n-2}=0\n\\qquad\\text{for every }n\\geq2.\n$$\n\nIt asserts that\n\n$$\nE=a_1a_2b_0+a_1b_2-2a_2b_1\\neq0\n\\quad\\Longrightarrow\\quad s\\text{ is not globally bounded}.\n$$\n\nDefinitions 1.4 and 1.5 require both of the following for global\nboundedness: a single nonzero integer $C$ for which $C^ns_n$ is an\ninteger for every $n\\geq1$, and a real $R>0$ for which the full complex\nseries $\\sum_{n=0}^{\\infty}s_nz^n$ is summable for every complex $z$\nwith $|z| This leads us to conjecture that the lower bound in Th.~1 is tight for all $n$.\n\n> A description of the worst-case process is an open problem.\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/mayer2021zerofidelity.md).\nIssue [#14119](https://github.com/the-omega-institute/trureturing/issues/14119) reads the two\nstatements for every $n\\ge1$, every choice of single-qubit SIC-POVMs and every finite Kraus family:\nthe lower bound holds with equality exactly when every Kraus operator lies in the span of the\nidentity and the single-qubit Pauli operators, and every value $f\\in[\\frac13,1]$ of $F_0$ is attained\nwith equality.","title":"Tightness of the 0-fidelity lower bound and the worst-case processes","url":"https://the-omega-institute.github.io/trureturing-pages/research/mayer-2021-zero-fidelity-tightness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1663b532d54c36bd30d1e838ae312a61ca13b7a83dd7d841571e6766cdf685ff.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mayumi-kimura-ohno-chruscinski-2024-weighted-bottcher-wenzel-ii","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Mayumi, G. Kimura, H. Ohno and D. Chruściński, “Böttcher-Wenzel inequality for weighted Frobenius norms and its application to quantum physics”, arXiv:2403.04199v2, Conjecture 1, equation (15), ask whether\n\n$$\\|[A,B]\\|_\\omega\\leq\\sqrt{\\frac{\\lambda_m+\\lambda_M}{\\lambda_m}}\\,\\|A\\|_\\omega\\|B\\|$$\n\nholds for all $A,B\\in M_n(\\mathbb C)$ and every positive definite weight $\\omega$. Here $[A,B]=AB-BA$, $\\|A\\|_\\omega^2=\\operatorname{tr}(A^*A\\omega)$, $\\|B\\|^2=\\operatorname{tr}(B^*B)$, and $\\lambda_m,\\lambda_M$ are the attained minimum and maximum eigenvalues of $\\omega$. Issue #13715 preregisters this case as a tier-3 research line.","title":"Weighted Böttcher–Wenzel inequality, case (ii)","url":"https://the-omega-institute.github.io/trureturing-pages/research/mayumi-kimura-ohno-chruscinski-2024-weighted-bottcher-wenzel-ii/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2379a85421b16b5ce0a12448fcce6cd728b108cbff56feaab03a6af5fd84057c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mazorchuk-xu-2026-first-column-deficit-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Volodymyr Mazorchuk and Shuoyang Xu, *On the number of cokernel-closed additive\nsubcategories for uniformly oriented A_n quivers*, arXiv:2610.07999v2,\nSection 5.7, conjecture:\n\n> It seems reasonable to conjecture that $r(n,n+1-i)$ is a polynomial in $n$ of\n> degree $2i$ and leading coefficient $\\frac{1}{i!}$.\n\nThe preceding sentence specifies “any fixed number of missing points.” This dossier\nanchors the whole conjecture: for every natural $i$, one rational polynomial $P_i$\nhas exact degree $2i$, leading coefficient $1/i!$, and satisfies\n$P_i(n)=r(n,n+1-i)$ for every natural $n$ with $i\\le n+1$.\n\nSection 3, before Section 3.1, defines $T(n)=\\{(a,b):0\\le a\\le b\\le n\\}$ and its\nsubsets satisfying Condition (star). Three selected points $(a,b),(c,d),(e,f)$,\nallowing repetitions, with $a\\le c,e\\le b\\le d,f$ force\n$(\\max(c,e),\\max(d,f))$ and, if $b<\\min(d,f)$, also $(b+1,\\min(d,f))$.\nSection 5.1 defines $r(n,k)$ as the number of these subsets with exactly $k$\nselected first-column points. Empty subsets and rows are permitted.\n\n### Source correspondence\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Literal triangular subsets | `triangle`, `Star`, `family`, `count` in `MazorchukXuClosure` |\n| Repeated selected points and both forced conclusions | All three membership premises in `Star`; `star_iff_rules` proves equivalence with the ordered-edge rules |\n| Exact first-column statistic | `firstColumnCard`; `count_eq_edgeFamily_card` preserves its value |\n| Every fixed deficit | Outer `∀ i : ℕ`, followed by `∃ p : Polynomial ℚ` before `∀ n` |\n| Every defined rank | `∀ n : ℕ, i ≤ n+1 → p.eval (n : ℚ) = (count n (n+1-i) : ℚ)` |\n| Exact degree and leading coefficient | `p.natDegree = 2*i` and `p.leadingCoeff = (i.factorial : ℚ)⁻¹` |\n| Boundary cases | No exclusions of `i=0`, `n=0`, `i=n+1`, zero or tied coordinates, or empty rows |\n\nThere is no eventual threshold, fixed-deficit bound, nonempty-row or representation\nhypothesis. The source does not require integer coefficients. The canonical embedding\nof the rational polynomial into the reals preserves evaluation, degree and leading\ncoefficient. The separate recurrence convention at rank $-1$ is outside this domain.","title":"Mazorchuk–Xu: every first-column deficit has a counting polynomial","url":"https://the-omega-institute.github.io/trureturing-pages/research/mazorchuk-xu-2026-first-column-deficit-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f4c40ca0f6181a2c48606b2f791cf3d38a9e157cb5a7f8172aa43464244e9a1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mazurek-horodecki-2018-thermal-process-self-dual-factorization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Mazurek and M. Horodecki, *Decomposability and convex structure of thermal processes*, New J. Phys. 20, 053040 (2018), arXiv:1707.06869v2, Section IV.B, “Detailed balance symmetry”, Conjecture 1:\n\n> If an extremal TP $C$ for $d$ dimensional space is decomposable into a sequence of extremal TPs $A$, $B$, each acting non-trivially on at most $d-1$ dimensional space: $C=AB$, and $C$ is not a direct sum of extremal TPs from lower dimensional subspaces, then $C$ is not self-dual with respect to the conjugation associated with the operator scalar product.\n\nThis quotation is from `manuscript.tex`, lines 425–427, read from the [arXiv v2 source](https://arxiv.org/src/1707.06869v2), md5 `69e7001dd049721e95968ea93ad550c2`. The source's matrix characterization of thermal processes is at line 159, its discussion of proper-subspace factors at lines 208–211, and its conjugation at lines 415–420. These passages are also located in [the literature note](../Library/StatisticalMechanics/mazurekhorodecki2018decomposability.md).\n\nIssue [#20157](https://github.com/the-omega-institute/trureturing/issues/20157) fixes the following precise reading. For every $d\\ge2$ and every positive, pairwise distinct weight vector $g\\in\\mathbb R^d$ with $g_i\\le g_0$ for every $i$, let\n\n$\\mathrm{TP}(g)=\\{T\\in\\mathbb R^{d\\times d}:T_{ij}\\ge0\\text{ for all }i,j,\\ \\sum_iT_{ij}=1\\text{ for all }j,\\ Tg=g\\}$.\n\nFor all extreme points $A,B,C$ of this set, suppose $C=AB$, each of $A$ and $B$ has a diagonal entry equal to one, and there is no nonempty proper set of levels whose two cross-blocks in $C$ are zero. The claim concludes that $C$ is not self-dual: $C_{ij}g_j\\ne C_{ji}g_i$ for some $i,j$. The module calls these predicates `thermalProcesses`, `actsOnFewerLevels`, `irreducible` and `selfDual`, and calls the universal implication `claim`.\n\nThe universal claim is refuted. Its additional restriction to distinct weights makes it weaker than the source's unrestricted conjecture, so refuting this restriction also refutes Conjecture 1.","title":"A self-dual irreducible extreme thermal process can factor on proper subspaces","url":"https://the-omega-institute.github.io/trureturing-pages/research/mazurek-horodecki-2018-thermal-process-self-dual-factorization-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ec6e5b339424d694c952311db1ea172dbe3c5235c9b1c62c0fad2b0c2c3b2e4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mbirika-schrader-spilker-associated-pell-gcd-entry-point-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mbirika, Schrader, and Spilker define the Pell and associated Pell sequences\nin Definition 3 on printed page 4:\n\n> **Definition 3.** The Pell sequence (P_n)_{n≥0} and the associated Pell sequence (Q_n)_{n≥0} are defined by the recurrence relations P_n = 2P_{n−1} + P_{n−2} and Q_n = 2Q_{n−1} + Q_{n−2}, respectively, with initial conditions P_0 = 0, P_1 = 1, Q_0 = 1, and Q_1 = 1.\n\nSection 6.2 on printed page 22 gives the entry-point convention:\n\n> When (S_n)_{n≥0} is the Pell or associated Pell sequence, we have partial results towards closed forms for gcd(S_k, k) that involve the entry point (or rank of apparition), e_S(p), which is the smallest index r > 0 such that p divides S_r where p is a prime.\n\nThe same printed page states:\n\n> **Conjecture 32.** We claim that gcd(Q_k, k) > 1 if and only if there exists a prime p such that p divides k and the rank of apparition (or entry point), e_Q(p) divides k. For example, gcd(Q_21, 21) = 7 and for the prime p = 7, we have p divides 21 and e_Q(p) = 3 divides 21.\n\nThe formal `claim` follows the printed universal biconditional and makes the\nentry point explicit as the least positive index at which the prime divides\nthe associated Pell sequence.","title":"Mbirika-Schrader-Spilker Associated Pell GCD Entry-Point Refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/mbirika-schrader-spilker-associated-pell-gcd-entry-point-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c83caf8e6e926f64fb9af36c1f1d38261e0c37ccf475c4186a5d6a3b434df9e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mbirika-spilker-gibonacci-power-gcd-normalization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 6.1, Question 6.2, PDF p. 14 of arXiv:2204.12626v2 asks:\n\n> Fix $G_0, G_1 \\in \\mathbb{Z}$ and set $d := \\gcd(G_0,G_1)$. Consider the two Gibonacci sequences $(G_n)_{n\\geq 0}$ and $(G_n^\\prime)_{n\\geq 0}$, where $(G_n^\\prime)_{n=0}^\\infty$ is generated by the relatively prime initial conditions $G_0^\\prime = \\frac{G_0}{d}$ and $G_1^\\prime = \\frac{G_1}{d}$. Then the following identities holds:\n>\n> $\\mathcal G_{G_0,G_1}(k)=d\\cdot\\mathcal G_{G'_0,G'_1}(k)$; $\\mathcal G^2_{G_0,G_1}(k)=d^2\\cdot\\mathcal G^2_{G'_0,G'_1}(k)$.\n>\n> The first formula holds from Guyer-Mbirika [4, Theorem 19], while the second formula holds from Theorem 3.6 in this current paper. Will it be the case that $\\mathcal{G}_{G_0, G_1}^n\\!(k) = d^n \\cdot \\mathcal{G}_{G_0^\\prime, G_1^\\prime}^n\\!(k)$ for all $n \\geq 3$?\n\nFor every signed integer seed pair $(a,b)≠(0,0)$, every natural\n$k\\geq1$ and every natural $e\\geq3$, let $G(0)=a$, $G(1)=b$ and\n$G(n+2)=G(n+1)+G(n)$. Define the nonnegative gcd of the entire shift family by\n\n$$C_e(a,b;k)=\\gcd_{t\\in\\mathbb N}\\left(\\sum_{i=1}^{k}G(t+i)^e\\right).$$\n\nWith $d=\\gcd(a,b)>0$, the target is\n\n$$C_e(a,b;k)=d^e C_e(a/d,b/d;k).$$\n\nThe quotients are exact integers. Both gcds satisfy their divisibility iff\nfor every integer divisor, including zero and negative divisors. Individual\nzero seeds, zero windows and odd exponents remain in scope.","title":"Gibonacci power-window gcd normalization","url":"https://the-omega-institute.github.io/trureturing-pages/research/mbirika-spilker-gibonacci-power-gcd-normalization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f6ad4e8ab11a0fb95f772b611fe784ab51794d173b62b8a005c22d4a2b27982.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mckean-2025-toda-primes-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stephen McKean, *Toda primes*, arXiv:2511.19744v1, §5.1,\nConjecture 5.6, printed p. 8:\n\n> Conjecture 5.6. Let d be a Bernoulli denominator.\n> (i) If F (d) = 4a for some integer a, then T (am) = T (a) whenever D₄ₐₘ = D₄ₐ.\n> (ii) If D₄ₐₘ ≠ D₄ₐ = d, then t(am) ≥ t(a).\n\nDefinition 1.1, §1, printed p. 1:\n\n> Definition 1.1. Let n be an integer. A Toda prime of n is an odd prime p such that\n> p − 1 | 4n and gcd(p, 4n/(p−1)) = 1. Denote the set of Toda primes of n by T (n) (see Table 1),\n> and let t(n) := |T (n)| (see Figure 1).\n\nNotation 5.5, §5.1, printed p. 8:\n\n> Notation 5.5. Given an even integer 2m, let P (2m) denote the set of primes such that\n> p − 1 | 2m, so that D₂ₘ = ∏ₚ∈P(2m) p. Let F (d) = min{2m > 0 : D₂ₘ = d} [PW23].\n\n### Formal reading\n\nLean names have prefix `D5.S3.Arith.McKeanTodaPrimes.Result`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $T(n)$ | `todaPrimes` | Odd primes with predecessor dividing $4n$ and coprime quotient. |\n| $t(n)$ | `t` | Cardinality of the finite Toda-prime set. |\n| $D_k$ | `bernDen` | Denominator of Mathlib's rational `bernoulli k`. |\n| $P(k)$ | `denominatorPrimes` | Primes whose predecessor divides $k$. |\n| Conjecture 5.6(i) | `clauseI` | Positive $a,m$, first even index $4a$, denominator unchanged. |\n| Conjecture 5.6(ii) | `clauseII` | Positive $a,m$, denominator changed. |\n| The full conjecture | `claim` | Conjunction of the two universal assertions. |\n| Both negations | `refutedClauses` | Each component assertion is false. |\n| Refutation | `result` | Exactly `¬ claim`. |\n\nThe condition $F(D_{4a})=4a$ is expressed equivalently as the absence of a\nsmaller positive even $k$ with $D_k=D_{4a}$. The equality at $4a$ is automatic.\nFor positive $n$, the range $p<4n+2$ is exhaustive because\n$p-1\\mid4n$ implies $p-1\\le4n$. At $n=0$, the quotient is zero and\nno prime is coprime to zero, so $T(0)$ is empty. All counterexamples have positive indices.\nThe natural-number restriction suffices to refute the integer conjecture.","title":"Refutation of both Toda-prime denominator assertions","url":"https://the-omega-institute.github.io/trureturing-pages/research/mckean-2025-toda-primes-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb7f0243457a160a533944469f90801db7a21ea3c57d417ec6a1c62ccd0ea575.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mcnulty-2025-path-robustness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Daniel McNulty, “A Graph-Theoretic Approach to Quantum Measurement Incompatibility”,\narXiv:2511.15954v1, Conjecture 1, p. 9:\n\n> For all $n\\geq 1$, paths satisfy\n> $\\eta(P_{2n})=\\eta(P_{2n+1})=\\eta(C_{2n+2})=\n> \\frac{2}{2n+2}\\csc\\bigl(\\frac{\\pi}{2n+2}\\bigr)$.\n\nThe source's Definition 1 and Eqs. (8)–(10) fix the observables and the\nvisibility convention. MCN-1 in #13934 fixes the strong statement for every\nfinite-dimensional realization: Hermitian involutions on $\\mathbb C^d$,\n$d\\geq1$, anticommutation on distinct adjacent vertices, and commutation\non distinct nonadjacent vertices. No irreducibility is assumed.","title":"McNulty's path-robustness conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/mcnulty-2025-path-robustness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa469d1c86d1798860365a8319e87cc82740a13c22eb87d7850ca1b216d21337.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meddaugh-stephens-2025-mixing-small-periodic-sets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jonathan Meddaugh and Elyssa Stephens, *Shadowing with small diameter sets\nof bounded cardinality*, arXiv:2509.04312v1, Question 5.6:\n\n> Is there a weakly mixing system which has the neighborhood N-shadowing property (N>1) that does not have dense small periodic sets?\n\nUse the intended **nonempty** meaning of a small periodic set: a nonempty\ncompact set B with f^p(B)=B for a positive natural p. The printed definition\nomits nonemptiness. The phase space is compact metric and the map continuous;\nthe source does not require a homeomorphism.","title":"Weak mixing and neighborhood shadowing without small periodic sets","url":"https://the-omega-institute.github.io/trureturing-pages/research/meddaugh-stephens-2025-mixing-small-periodic-sets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca48015927faa61ef79b8d1ffe013b6733a49fb00277b1a9b4e31897d1554540.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meddaugh-stephens-2025-neighborhood-two-shadowing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jonathan Meddaugh and Elyssa Stephens, *Shadowing with small diameter sets\nof bounded cardinality*, arXiv:2509.04312v1, Question 5.3:\n\n> Are there subshifts with neighborhood 2-shadowing which are not a subshift of quasi-finite type?\n\nUse neighborhood shadowing from Definition 2.1 and exact-length\nquasi-finite type from Definition 3.5, with their full quantifiers as\nretained in the [source note](../Library/Dynamics/meddaugh2025neighborhoodshadowing.md).","title":"Neighborhood two-shadowing without quasi-finite type","url":"https://the-omega-institute.github.io/trureturing-pages/research/meddaugh-stephens-2025-neighborhood-two-shadowing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c764e0f9859c9d86531009f7dba92040167de7a4d418f10e70b049d3d671121.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meer-weeks-2024-bounded-fifth-moment-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. J. Meer and E. R. Weeks, *Bounded distributions place limits on skewness and larger moments*, PLOS ONE 19, e0297862 (2024), arXiv:2308.05006v2, subsection “Higher order generalized moments”, `plos_latex_template.tex`, lines 506–510:\n\n> One can immediately put in a value for $a$ of interest and get a potential limit of $D_n$. For example, for distributions bounded from below by $x_{\\rm min}$ we conjecture\n>\n> $D_5 \\geq D_{5,{\\rm min}} = z^3 - z^1 + z^{-1} - z^{-3}$\n>\n> with $z = \\delta / (1 - x_{\\rm min}/\\mu)$ as above. As with $D_3$, our conjectured $D_{5,{\\rm max}}$ is a similar equation using $z = \\delta / (x_{\\rm max}/\\mu - 1)$.\n\nThe quotation preserves the source's text and formula, with its displayed equation rendered as a separate mathematical paragraph. The v2 TeX was read directly; its MD5 is `4452c28a26c7a76c87642af46fe0c8a9`. The source defines central moments and standardized moments at lines 228–236, the original coefficient of variation at lines 290–295, and the shifted one-sided ratios at lines 397–409. See [the literature note](../Library/StatisticalMechanics/meerweeks2024boundedmoments.md).\n\nWrite $\\nu$ for the probability law, distinguishing it from the paper's mean $\\mu$, take $X(x)=x$, and put\n\n$m=\\int x\\,d\\nu(x)$, $v=\\int(x-m)^2\\,d\\nu(x)$, $\\sigma=\\sqrt v$, $D_5=\\int(x-m)^5\\,d\\nu(x)/\\sigma^5$, and $F(z)=z^3-z+z^{-1}-z^{-3}$.\n\nIn Lean, these are `Defs.mean`, `centralMoment id 2`, `Real.sqrt`, the ratio involving `centralMoment id 5`, and `fifthBound`, respectively. Issue [#21300](https://github.com/the-omega-institute/trureturing/issues/21300) fixes the statement: for every Borel probability law on $\\mathbb R$ with $\\int|x|^5\\,d\\nu(x)<\\infty$ and $v>0$, both of the following implications hold:\n\n- for every real $L$, if $X\\ge L$ almost surely, then $F(\\sigma/(m-L))\\le D_5$;\n- for every real $U$, if $X\\le U$ almost surely, then $D_5\\le-F(\\sigma/(U-m))$.\n\nThis statement is proved by `result : claim`. Attainment by two-point laws, also discussed in #21300, is not part of `claim` or `result`.","title":"The Meer–Weeks fifth-moment bounds for distributions bounded on one side","url":"https://the-omega-institute.github.io/trureturing-pages/research/meer-weeks-2024-bounded-fifth-moment-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6e8f781d8205ccc41ffffe5a1921d28551b6b3ed919f6b6e0aa6124ed7d4971.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meer-weeks-2024-bounded-kurtosis-limit-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"David J. Meer and Eric R. Weeks, *Bounded distributions place limits on skewness and larger moments*, PLOS ONE 19, e0297862 (2024), DOI 10.1371/journal.pone.0297862; arXiv:2308.05006v2, subsection “Kurtosis $D_4$”, `plos_latex_template.tex`, line 467:\n\n> While we have not proven that the bidisperse distribution sets the limits for $D_4$ for all other distributions, this is suggestive that Inequalities \\ref{D4mostgeneral} are indeed limits for the kurtosis for any distribution.\n\nThe quotation is checked against the v2 TeX source, md5 `4452c28a26c7a76c87642af46fe0c8a9`. The referenced inequalities are at lines 461–464; their definitions are at lines 437–460. The moment conventions are at lines 228–236 and 290–293. See [the literature note](../Library/StatisticalMechanics/meerweeks2024boundedmoments.md).\n\nIssue [#21539](https://github.com/the-omega-institute/trureturing/issues/21539) fixes the following universal reading. For every real interval $L0$ and variance $v=\\mathbb E[(X-\\mu)^2]>0$, put\n\n$\\sigma=\\sqrt v$, $\\delta=\\sigma/\\mu$, $\\delta_{\\min}=|1-L/\\mu|$, $\\delta_{\\max}=|1-U/\\mu|$, $\\delta_1=\\min(\\delta_{\\min},\\delta_{\\max})$, $\\delta_2=\\max(\\delta_{\\min},\\delta_{\\max})$, $\\delta'=\\sqrt{\\delta_1\\delta_2}$,\n\nand $D_4=\\mathbb E[(X-\\mu)^4]/v^2$, $D_{4,m}(\\delta)=(\\delta/\\delta_m-\\delta_m/\\delta)^2+1$ for $m=1,2$. The assertion requires both:\n\n- if $\\delta<\\delta_1$, then $1\\le D_4\\le D_{4,2}(\\delta)$;\n- if $\\delta_1\\le\\delta<\\delta'$, then $D_{4,1}(\\delta)\\le D_4\\le D_{4,2}(\\delta)$.\n\nThis reading is refuted. It makes no assertion for $\\delta\\ge\\delta'$; the positive-mean restriction suffices for the refutation.","title":"A three-point law exceeds the Meer–Weeks two-sided kurtosis upper limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/meer-weeks-2024-bounded-kurtosis-limit-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23b8159e79ffbcc41a41a52d108d86ca27d62ce836331fc8d09867a5ef140c13.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meeussen-2004-solid-partition-first-column-tau4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A098052 (Wouter Meeussen, 2004) counts the solid partitions of `n` by\ntheir number of extensions:\n\n> T(n,k) counts the solid partitions of n that can be extended to a solid\n> partition of n+1 in exactly (k+3) ways. Equivalently, the number of solid\n> partitions of n that have exactly k+3 partitions of n+1 majoring them.\n\nand its COMMENTS state\n\n> First column is conjectured to be A007426 = tau_4(n).\n\nThe twin entry A098530 (Meeussen, 2004) counts the solid partitions of `n + 1`\nby the number of solid partitions of `n` they just cover and states\n\n> First column conjectured to be the (beheaded) A007426.\n\nA007426 is `tau_4(n)`, the number of ordered factorizations `n = rstu`.\nIssue #10330 fixes the readings, and issue #10404 registers the statement in\nevery dimension `d`: a `d`-dimensional partition of `n` is read through its\nFerrers diagram, a lower set of `n` cells of `ℕ^d` (for `d = 4` the solid\npartitions of A000293); for the entries, a solid partition of `n` is its\nfour-dimensional Ferrers diagram, a lower set of `n` cells of `ℕ⁴` in the\ncoordinatewise order; an extension of `I` is a solid partition of `n + 1`\ncontaining `I`, and a shrinking of `J` is a solid partition of `n` contained in\n`J`. The statement proved is that for every `d ≥ 1` and `n ≥ 1` exactly `τ_d(n)`\npartitions of `n` have `d` extensions and exactly `τ_d(n + 1)` partitions of\n`n + 1` have one shrinking, `τ_d` being the number of ordered factorizations\ninto `d` factors; `d = 4` settles both conjectures. The same statement at\n`d = 3` is the first column of A098529 (stated there without proof), and A007426\nrepeats the `d = 4` observation (\"Appears to equal the number of solid\npartitions of n that can be extended in exactly 4 ways\").","title":"The first columns of A098052 and A098530","url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b9f4aa469afbad97ca3d5a38a93beb1e0c6e233690ce52c792befb216489961.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meeussen-2025-a381265-two-layer-solid","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381265 counts the solid partitions with two layers whose second layer is\na plane partition of 3. Its COMMENTS line records:\n\n> Conjecture: equal to 3*(2*A000219 -A000990 -2*A000041 +1) tested up to\n> n=20.\n\nIssue #10460 fixes the reading. The index `n` is the size of the first layer,\nas the data and the example `a(3) = 6` show. Plane partitions are lower sets\nof `n` cells of `ℕ³`, and a two-layer solid partition is a pair `P₂ ⊆ P₁` of\nthem. A000990 counts the plane partitions all of whose cells have row index at\nmost 1, and A000041 counts `Nat.Partition n`.","title":"Meeussen's formula for two-layer solid partitions A381265","url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c3786712fbfdb90a2fdfc911e3e7e377e0d5a1c75abe6a721c989d6051e8386.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mehboudi-rezaeinia-rahimi-keshari-2025-loss-incompatibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Mehboudi, F. Rezaeinia and S. Rahimi-Keshari, *Measurement incompatibility under loss*, arXiv:2411.05920v3 (Phys. Rev. Lett. 135, 100202 (2025)), model pure loss by a beam splitter of transmissivity τ with vacuum in its second input, E_τ(ρ) = tr₂(U_BS(ρ ⊗ |0⟩⟨0|)U_BS†), and define the dual by tr(M E_τ(ρ)) = tr(ρ E*_τ(M)). Result 2 shows that for τ ≤ 1/n the images under E*_τ of any n measurements are jointly measurable. Result 3 asserts that some n + 1 measurements stay incompatible at τ = 1/n: \"We prove this result numerically for 2 ≤ n ≤ 13, and conjuncture it to hold for arbitrary n.\" The measurements are the displaced on-off detections of Eq. (4), M^k_1 = |μ_k⟩⟨μ_k| and M^k_2 = I − M^k_1, with coherent states of equal modulus separated by the phase 2π/(n + 1). The verbatim statements are in [the literature note](../Library/QuantumBounds/mehboudi2025lossincompatibility.md).\n\nIssue [#15985](https://github.com/the-omega-institute/trureturing/issues/15985) fixes the reading. For every n ≥ 1 there are n + 1 binary POVMs on the single-mode Fock space ℓ²(ℕ) whose images under E*_{1/n} have no parent POVM. A parent ranges over all 2^{n+1} outcome strings and has exactly the given effects as marginals.\n\nThe Lean statement takes E*_τ in the source's own form: N is the image of M when tr(M E_τ(|ψ⟩⟨ψ|)) = ⟨ψ, N ψ⟩ for every ψ. Here E_τ(|ψ⟩⟨ψ|) = tr₂(U_BS(|ψ⟩⟨ψ| ⊗ |0⟩⟨0|)U_BS†) is built from the frozen beam splitter and partial trace, and the trace is the Fock-basis sum. Pure states determine an operator, so this relation determines E*_τ(M). The statement asserts both that such images exist and that no family of them has a parent.","title":"n + 1 displaced on-off measurements stay incompatible under pure loss of transmissivity 1/n","url":"https://the-omega-institute.github.io/trureturing-pages/research/mehboudi-rezaeinia-rahimi-keshari-2025-loss-incompatibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9a2f84133cdf19f0fdc6c0b91956a31fbe85884bd688c9427ce68eaa7f58fae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:memarzadeh-mancini-2016-truncated-loss-dephasing-optimizer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Laleh Memarzadeh and Stefano Mancini, *Minimum output entropy of a\nnon-Gaussian quantum channel*, arXiv:1605.04525v1, Phys. Rev. A 94,\n022341 (2016), section IV, Conjecture 1, state:\n\n> In a truncated Hilbert space of dimension K+1, the minimal output entropy\n> of the quantum channel (3) is achieved either by binomial states of\n> Eq.(14) or by states |κ_α⟩ of Eq. (11), depending on the values of ε and t.\n\nIssue #11680 specifies the fixed-energy reading: for every K ≥ 1,\nN ∈ [0,K], ε ∈ [0,1] and t ≥ 0, some binomial or κ candidate minimizes\noutput entropy among positive semidefinite trace-one inputs whose number\nexpectation is N. Binomial candidates satisfy 1 ≤ M ≤ K, μ ∈ [0,1] and\nMμ=N; κ candidates allow every real phase α.","title":"Binomial and kappa states do not exhaust truncated entropy minimizers","url":"https://the-omega-institute.github.io/trureturing-pages/research/memarzadeh-mancini-2016-truncated-loss-dephasing-optimizer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/13442ecadc3b66b105b854655ae94dcd9e0f05821c62c502345bec2d9ca08794.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mendonca-oeis-a400584-strict-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. R. G. Mendonça's [OEIS A400584, revision 28](https://oeis.org/A400584) states: “Conjecture: T(n,k) > T(n,k+1) for n >= 3 and 1 <= k < 2^n.” The associated [technical note](https://oeis.org/A400584/a400584.pdf) gives the counting recurrence as Proposition 2. The complete source quotations and conventions are in [preregistration #19612](https://github.com/the-omega-institute/trureturing/issues/19612) and [the Library note](../Library/RecurrenceComposition/mendonca2026a400584.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Complete plane binary tree of depth $n$ | `LabeledTree n`: one leaf at zero, one labeled root with two ordered equal-depth subtrees at successor depth | Left and right positions remain distinct; no quotient by swaps. |\n| Each internal vertex is labeled by addition or minimum | The two constructors of `Op` and the operator parameter of `LabeledTree.node` | Exactly the source's two labels. |\n| Every leaf equals one; each vertex applies its operator | All three recursion cases of `value` | Literal source evaluation. |\n| Number of labelings with root value $k$ | `T n k`, the `Nat.card` of the root-value subtype | Counts labelings rather than possible values; finiteness is constructed privately. |\n| $n\\ge3$ and $1\\le k<2^n$ | All natural-number binders and all three guards of `claim` | The same range, including every adjacent pair through the last column. |\n| $T(n,k)>T(n,k+1)$ | `result : claim` | The strict comparison has only its sides reversed syntactically. |\n| Depth-zero convention $T(0,1)=1$ | The leaf constructor and the private `T_zero` helper | The source recurrence's initial condition; no additional hypothesis on the conjecture. |","title":"Strict decrease in Mendonça's min-plus configuration triangle","url":"https://the-omega-institute.github.io/trureturing-pages/research/mendonca-oeis-a400584-strict-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1d4fefea2c802c6d68c961a1a2542ded057199a35fbc87f6dbeb17fc446d149d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:merca-2011-lifted-residue-sum-prime-order","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Merca defines the remainder operation on printed page 2:\n\n> When m is integer and n is a positive integer the quotient of m divided by n is ⌊m/n⌋ and the value m mod n = m − n⌊m/n⌋ is the remainder (or residue) of the division.\n\nThe multiplicative-order convention appears on printed page 17:\n\n> For every positive integer m and every integer a relatively prime to m, we denote by ord_m(a) the multiplicative order of a modulo m, i.e., the smallest positive integer n such that a^n ≡ 1 (mod m), namely ord_m(a) = min {n ∈ N∗ | a^n ≡ 1 (mod m)}\n\nThe observation leading to Conjecture 2 appears on printed page 23:\n\n> Using Maple to determine the value of some sums as Σ_{i=1}^{ord_m(a)} ((2a^i + m) mod 2m), necessary to determine the arithmetic mean (31) for round function, we notice another interesting identity.\n\nThe conjecture is stated on the same printed page:\n\n> Conjecture 2. Let a and m be relatively prime positive integers. If m is prime and ord_m(a) is even then Σ_{i=1}^{ord_m(a)} ((2a^i + m) mod 2m) = m · ord_m(a).\n\nThe printed display brackets the summand as `((2a^i + m) mod 2m)`: the\nremainder is taken of the whole quantity `2a^i + m`. Formula (39) on the same\npage encloses each residue summand as `(a^i mod m)` in the same typography.\n\nWith the printed bracketing, the proved statement is: for every positive natural\nnumber `a` and every prime natural number `m` coprime to `a`, if the\nmultiplicative order of `a` modulo `m` is even, then\n\n```text\nSum_{i=1}^{ord_m(a)} ((2*a^i + m) mod (2*m)) = m*ord_m(a).\n```","title":"Merca's lifted residue sum at prime order","url":"https://the-omega-institute.github.io/trureturing-pages/research/merca-2011-lifted-residue-sum-prime-order/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73400d15940ad61a053341ff21f53a507de1345aa2fa6d39c3f13c2686896f32.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:merca-2011-residue-sum-even-order-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Merca, Journal of Integer Sequences 14 (2011), Article 11.9.1, defines the\nremainder operation on printed page 2:\n\n> When m is integer and n is a positive integer the quotient of m divided by n is\n> ⌊m/n⌋ and the value m mod n = m − n⌊m/n⌋ is the remainder (or residue) of\n> the division.\n\nSection 3.2 on printed page 17 defines the multiplicative order:\n\n> For every positive integer m and every integer a relatively prime to m, we denote\n> by ord_m(a) the multiplicative order of a modulo m, i.e., the smallest positive\n> integer n such that a^n ≡ 1 (mod m), namely\n> ord_m(a) = min {n ∈ N∗ | a^n ≡ 1 (mod m)}\n\nSection 4 on printed page 23 states the conjecture:\n\n> **Conjecture 1.** Let a and m be relatively prime positive integers. If a−1 and m\n> are relatively prime and ord_m(a) is even then\n> Σ_{i=1}^{ord_m(a)} (a^i mod m) = m · ord_m(a) / 2.\n\nThe formal `claim` retains both positivity conditions, both coprimality\nconditions, even multiplicative order, the inclusive index range from 1 through\nthe order, and the least non-negative remainder operation. Its equality doubles\nboth printed sides to express the exact integer statement in natural numbers.\n\n| Printed expression | Formal expression |\n| --- | --- |\n| positive integers `a`, `m` | `0 < a`, `0 < m` with `a m : ℕ` |\n| `a` and `m` relatively prime | `Nat.Coprime a m` |\n| `a−1` and `m` relatively prime | `Nat.Coprime (a - 1) m` |\n| `ord_m(a)` | Mathlib's `orderOf (a : ZMod m)` |\n| `ord_m(a)` is even | `Even (orderOf (a : ZMod m))` |\n| `Σ_{i=1}^{ord_m(a)} (a^i mod m)` | `residueSum m a` over inclusive `Finset.Icc 1 (orderOf (a : ZMod m))` |\n| `sum = m · ord_m(a) / 2` | `2 * residueSum m a = m * orderOf (a : ZMod m)` |","title":"Merca's even-order residue-sum conjecture refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/merca-2011-residue-sum-even-order-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6d94384463d84e88993f8dc3742c3fbeb7fb57157f6e1c4e5c48acc53150bf4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:metya-mukherjee-independent-convolution-l1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nilava Metya and Satyaki Mukherjee, *Approximate Uniformity in Finite\nConvolution Models*, arXiv:2609.38243v1, §4.3,\n[Conjecture 4.7](https://arxiv.org/html/2609.38243v1#S4.Thmlemma7), states:\n\n> The optimal value of (21) is (2n−1)⁻¹ for n≥3.\n\nThe source declares [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/).\nThe quotation and mathematical source correspondence here are attributed to\nthat version under the same license. The proof below is a repository\nformalization of the sharp bound.\n\nEquation (21) minimizes the full L1 norm of the deviation from uniform over\nordinary convolutions of two independent probability vectors. For every\nnatural n≥3 and real closed-simplex vectors p,q on Fin n, put\n\n\\[\nr_k=\\sum_{i,j Let rho and sigma be noncommuting density matrices, with regularized states\n> rho_epsilon and sigma_epsilon. If N is a commutativity-preserving CPTP\n> channel and the support of N(rho_epsilon) is contained in that of\n> N(sigma_epsilon), then Q(rho_epsilon || sigma_epsilon) is at least\n> Q(N(rho_epsilon) || N(sigma_epsilon)), which is at least zero.\n\nThe literal source formulas and sentences are in\n[the literature note](../Library/QuantumStates/meunson2026cumulant.md).\nThe source's malformed pair `\\rho,\\sigma)` is transcribed literally there;\nthe Blueprint display uses the normalized pair `(rho,sigma)`.\n\nEquation `state_regularized` defines\n\n$$A_\\varepsilon=(1-\\varepsilon)A+\\frac{\\varepsilon}{d}I,\n\\qquad 0<\\varepsilon<1.$$\n\nEquation `regularized_Cu-Q` and the definition of relative quantumness give\n\n$$Q(A\\Vert B)=-S_0^Q(A\\Vert B)\n=\\ln\\operatorname{Tr}\\left[A\\exp(\\ln B-\\ln A)\\right].$$\n\nDensity means positive semidefinite with trace one. CPTP means complete\npositivity at every finite amplification and trace preservation on every\nmatrix. CoP quantifies over every pair of commuting density matrices.\nSupport is the operator range. The Lean claim includes both conclusion\ninequalities and all source hypotheses. It quantifies over `d : ℕ` and\ncomplex matrices on `Fin d`; one witness with `d = 2` refutes that universal\nclaim. Every matrix logarithm in the witness is of a positive definite matrix,\nso the refutation uses no convention for a singular output logarithm.","title":"Conjecture 15 fails for a commutativity-preserving qubit channel","url":"https://the-omega-institute.github.io/trureturing-pages/research/meunson-deesuwan-2026-cop-relative-quantumness-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8cefd9d6a57aa2d50cd61632ed61e6035e7fb23023e2858c22399d8a49bc9d5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:meunson-deesuwan-2026-cumulant-renyi-qdpi","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Meunson and T. Deesuwan, *Cumulant-based quantum relative Rényi\nfunctional*, arXiv:2606.31205v1, define for $\\alpha>1$ (Definition 3,\nSection IV)\n\n$$S_\\alpha^{Q}(\\rho\\Vert\\sigma)=\\frac{1}{\\alpha-1}\\ln\\operatorname{Tr}\n\\left(\\rho\\,e^{(\\alpha-1)(\\ln\\rho-\\ln\\sigma)}\\right)$$\n\non pairs with $\\operatorname{supp}\\rho\\subseteq\\operatorname{supp}\\sigma$.\nThe abstract and the conclusion state that the quantum data-processing\ninequality (QDPI) for arbitrary CPTP maps remains open. The verbatim\nsentences and Definition 3 are in\n[the literature note](../Library/QuantumStates/meunson2026cumulant.md).\nThis question differs from Conjecture 15 of the same paper ($\\alpha=0$,\ncommutativity-preserving channels), settled in\n[the earlier dossier](meunson-deesuwan-2026-cop-relative-quantumness-refutation.md).\n\nIssue [#13439](https://github.com/the-omega-institute/trureturing/issues/13439)\nfixes the reading. `QDPI α` asserts\n$S_\\alpha(\\mathcal N\\rho\\Vert\\mathcal N\\sigma)\\le S_\\alpha(\\rho\\Vert\\sigma)$ for\nevery dimension $d$, every pair of positive definite density matrices and\nevery CPTP map $\\mathcal N$ with positive definite outputs. Matrix logarithms\nare continuous functional calculus, as in the frozen $\\alpha=0$ module.\n`claim` asserts that some $\\alpha>1$ satisfies `QDPI α`. Faithful inputs meet\nthe support condition, so refuting this restricted claim refutes QDPI for the\npaper's functional at every $\\alpha>1$.","title":"Data processing fails for the cumulant-based Rényi functional at every order above one","url":"https://the-omega-institute.github.io/trureturing-pages/research/meunson-deesuwan-2026-cumulant-renyi-qdpi/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c95df2f4e089d711c6b4dfcf79dd9a9dd98ead18529847cb0682cc52cf5f7e9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mi-xu-yang-2025-partially-restricted-stack-maximal-sort-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mi, Xu and Yang, ECA 5:3 (2025), Article #S2R22, introduce the stack\nmachines used in the conjecture. Printed page 1 states:\n\n> The map sends permutations through a stack that always avoids the permutation\n> 21 when read from top to bottom.\n\n> We further generalize s to s_{(T,k)}. We define the maps s_{(T,k)} that avoid\n> containing more than k distinct permutations from T in the stack at once.\n> More specifically, the map sorts a permutation π via the following stack\n> sorting algorithm: If adding the leftmost element of the input to the stack\n> keeps the stack (T, k)-avoiding, push that element into the stack. Otherwise,\n> pop the top element off the stack and append it to the output.\n\nPrinted page 2 defines containment and avoidance:\n\n> Given two permutations π and σ, we say π contains σ if there exist\n> a_1, a_2, . . . , a_k such that the sub-permutation\n> π_{a_1} π_{a_2} . . . π_{a_k} is order-isomorphic to σ. Otherwise, we say π\n> avoids σ.\n\n> Furthermore, for a set T of permutations and a nonnegative integer k, we say\n> π is (T, k)-avoiding if π contains at most k elements of T.\n\nThe same page fixes the second map:\n\n> Throughout the paper, we use t to denote s_{({12,21},1)}.\n\nPrinted page 6 defines the sorting time:\n\n> Let m_{(s∘t)}(π) be the smallest nonnegative integer j such that\n> (s ∘ t)^j(π) = id_{|π|}.\n\nPrinted page 7 gives the proved sufficient direction:\n\n> Lemma 5.5. For n ≥ 3 and π ∈ S_n, if π = 2σ1n where\n> σ ∈ Av_{n−3}(213) then m_{(s∘t)}(π) = 2n − 5.\n\nIt then states the named open assertion:\n\n> Conjecture 5.2. The permutation π takes 2n − 5 sorts to be sorted to the\n> identity by (s ∘ t) if and only if π = 2σ1n and\n> σ ∈ Av_{n−3}(213).\n\nThe formal reading takes the quantifier to be `n ≥ 3`, reads\n`m_{(s∘t)}(π) = 2n−5` as `TakesSorts π (2n−5)`, and reads\n`π = 2σ1n` with `σ` a permutation of `{3,…,n−1}` avoiding 213. The printed\nconjecture carries no lower-bound clause; the `n ≥ 3` reading is taken from\nLemma 5.5 and Conjecture 5.1 and was preregistered in issue #9318. The reading\nuses `s := s_{({21},0)}`, the page-1 West machine. Figure 1 on printed page 1\nshows a different restricted map and supplies no evidence about `t`.","title":"Mi-Xu-Yang Conjecture 5.2 refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/mi-xu-yang-2025-partially-restricted-stack-maximal-sort-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5de2c59bb204e43828393904cbc2504bf3aac6d57d48e4ee74802a22eb6de6e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:milekhin-adamska-preskill-2025-spacetime-schatten-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Milekhin, Z. Adamska and J. Preskill, *Observable and computable entanglement in time*, arXiv:2502.12240v2, define the spacetime density matrix T_AB of a subsystem A at time 0 and a subsystem B at time t.\n- **The definition.** Tr(T_AB (O_A ⊗ 1_B)(1_A ⊗ O_B)) = ⟨O_A(0) O_B(t)⟩ = Tr(ρ O_A U† O_B U) (Eqs. (4), (9)); A and B \"do not have to be related to each other in any particular way\". The Schatten norm is ||M||_p = (Tr (M†M)^{p/2})^{1/p} (footnote a).\n- **The conjecture.** After Audenaert's inequality for density matrices (Theorem 1, Eq. (24)), §IV item 9 states: \"Conjecture 1 For any T_AB on a finite-dimensional H_A ⊗ H_B and p > 1: 1 + ||T_AB||_p ≥ ||Tr_A T_AB||_p + ||Tr_B T_AB||_p. (25)\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/milekhinadamskapreskill2025observable.md).\n\nIssue [#17385](https://github.com/the-omega-institute/trureturing/issues/17385) fixes the reading: for every finite dimension, every density matrix ρ, every unitary U between the factorizations H_A ⊗ H_Ā and H_B ⊗ H_B̄ of the whole system, every T satisfying Eq. (9) for all O_A and O_B, and every real p > 1, inequality (25) holds.\n\nThe conjecture is refuted.","title":"The spacetime density matrix violates the Audenaert-type Schatten subadditivity","url":"https://the-omega-institute.github.io/trureturing-pages/research/milekhin-adamska-preskill-2025-spacetime-schatten-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08ee948d7f63ce6f19d4ca919595c9006e189fb3d353afda9cf44c0623a54fdb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:milne-jennings-jevtic-rudolph-2014-chsh-centre-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Milne, D. Jennings, S. Jevtic and T. Rudolph, *Quantum correlations of two-qubit states with one maximally mixed marginal*, Phys. Rev. A **90**, 024302 (2014), arXiv:1404.3951v2, define the CHSH nonlocality in §II as β(ρ) = max_B |tr(ρB)|, with B = α·σ ⊗ (β + β′)·σ + α′·σ ⊗ (β − β′)·σ and the maximum over all directions α, α′, β, β′. §V, Conjecture 1:\n\n> Let ρ be a general two-qubit state with E centred at c. The CHSH nonlocality is tightly bounded as β(ρ) ≤ max[2√(2(1 − c)), 2].\n\nHere E is Alice's steering ellipsoid and c = |c⃗| is the length of its centre. The verbatim statements and the centre convention are in [the literature note](../Library/QuantumBounds/milne2014fullyentangledfraction.md). The repository settled Conjecture 2 of the same paper in [the fully-entangled-fraction dossier](milne-jennings-jevtic-rudolph-2014-fully-entangled-fraction.md).\n\nIssue [#16558](https://github.com/the-omega-institute/trureturing/issues/16558) fixes the reading:\n- β is the supremum of |tr(ρB)| over four unit vectors of ℝ³;\n- the centre is the companion module's `centre`: (a − Tb)/(1 − |b|²) for |b| ≠ 1 and a for |b| = 1.\n\nThe conjecture is proved in this form. \"Tightly bounded\" is proved as a supremum at every centre length t ∈ [0, 1]: for every ε > 0 some state with c = t has β > max(2√(2(1 − t)), 2) − ε.","title":"CHSH nonlocality is tightly bounded by the steering-ellipsoid centre","url":"https://the-omega-institute.github.io/trureturing-pages/research/milne-jennings-jevtic-rudolph-2014-chsh-centre-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f823cea5fe5780522abf2c5b8e7979060cac94bbad0f37ab192b0c902eef9658.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:milne-jennings-jevtic-rudolph-2014-fully-entangled-fraction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Milne, D. Jennings, S. Jevtic and T. Rudolph, “Quantum correlations of\ntwo-qubit states with one maximally mixed marginal”, Physical Review A 90,\n024302 (2014), arXiv:1404.3951v2, Section V, Conjecture 2:\n\n> Let $\\rho$ be a general two-qubit state with $\\mathcal{E}$ centred at $\\vec c$. The fully entangled fraction is tightly bounded as $f(\\rho)\\leq 1-c/2$.\n\nHere $f(\\rho)=\\max_{|\\phi\\rangle}\\langle\\phi|\\rho|\\phi\\rangle$ over maximally\nentangled $|\\phi\\rangle$ (Section III), $\\mathcal E$ is Alice's steering\nellipsoid, and $c=|\\vec c|$. The centre is given in the companion paper\nS. Jevtic, M. Pusey, D. Jennings and T. Rudolph, Physical Review Letters 113,\n020402 (2014), arXiv:1303.4724v2:\n$\\vec c=(\\boldsymbol a-T\\boldsymbol b)/(1-b^2)$, and $\\vec c=\\boldsymbol a$\nwhen $b=1$.","title":"The fully entangled fraction is at most one minus half the steering-ellipsoid centre","url":"https://the-omega-institute.github.io/trureturing-pages/research/milne-jennings-jevtic-rudolph-2014-fully-entangled-fraction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b22c1f9ce571fce8b43c7e7c1564b9333e3b408a856f6678a8660f1225a098e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:minac-nguyen-tan-question-3-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The primary source is [Question 3.5](https://arxiv.org/html/2607.00404v1#S3.Thmthm5):\n\n> Suppose that F has the n-sum property with respect to U. Is it true that F[G] also has the n-sum property with respect to U_{ε_G}?\n\nThe complete statement quantifies over a finite field F with more than two\nelements, every finite group G, a subgroup U of F's units containing minus\none, and the original natural number n at least two. If every field element\nis a sum of exactly n elements of U, every element of the actual group\nalgebra must be a sum of exactly n genuine units whose coefficient-sum\naugmentations lie in U. Repetition is allowed. Modular characteristic,\nnoncommutative groups, characteristic two with more than two field elements,\nand the trivial group remain in scope.","title":"Mináč–Nguyen–Tân Question 3.5","url":"https://the-omega-institute.github.io/trureturing-pages/research/minac-nguyen-tan-question-3-5/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bbfd89a4bb291434a07c5883c50efe650cf9279caf0e4542a422784c2ab33f05.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:minadiaz-rubin-2025-chebyshev-faber-equipotential","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Miña-Díaz and O. Rubin, *Chebyshev polynomials on equipotential\ncurves*, arXiv:2505.03967v1, Section 2, Problem 2.2:\n\n> Suppose the compact set $K$ is such that for some $n\\in\\mathbb N$, $T_{n,K}=\\hat F_n$, where $\\hat F_n$ is the $n$th monic Faber polynomial associated to the map $\\phi$ that satisfies (1.2). Is it true then that $T_{n,L_r}=\\hat F_n$ for all $r>1$?\n\nHere $\\Omega$ is the unbounded component of $\\mathbb C\\setminus K$,\n$g_\\Omega(z,\\infty)=\\log|\\phi(z)|$, and\n$\\phi(z)=cz+c_0+c_{-1}/z+\\cdots$, with\n$c=\\operatorname{cap}(K)^{-1}$. When $\\Omega$ is simply connected,\n$\\psi=\\phi^{-1}$. The equipotential curve is\n$L_r=\\{z\\in\\Omega:|\\phi(z)|=r\\}=\\psi(\\{|w|=r\\})$.\nThe polynomial $F_n$ is the polynomial part of $\\phi^n$ at infinity,\n$\\hat F_n=c^{-n}F_n$, and $T_{n,E}$ is the unique monic degree-$n$\nsup-norm minimizer on $E$.\n\nThe Lean `claim` quantifies over exterior maps $\\psi$ differentiable\nand injective on $|w|>1$, a positive capacity $\\kappa=1/c$, compact\ncomplement $K$, and a bounded correction $\\psi(w)-\\kappa w$ for\n$|w|>2$. Its `IsMonicFaber` requires a monic degree-$n$ polynomial\n$F$ with $F(\\psi(w))-(\\kappa w)^n\\to0$ at infinity. Its\n`IsChebyshev` compares every uniform real bound for every monic\ndegree-$n$ competitor. For compact nonempty sets this expresses\nsup-norm minimization. The implication asks whether such an $F$\nChebyshev on $K$ remains Chebyshev on every $L_r$, $r>1$.","title":"Miña-Díaz–Rubin's Chebyshev–Faber equipotential question","url":"https://the-omega-institute.github.io/trureturing-pages/research/minadiaz-rubin-2025-chebyshev-faber-equipotential/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/510084d213a6a0d7eb627e88e6f04543710612351a82838aa49c6dafac353565.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:miro-roig-tran-strict-sign","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Miró-Roig and Tran, immediately after the proof of Proposition 3.12 in\narXiv `2001.06143v1`, conjecture that for every integer `n>=2`,\n\n```text\nsum_{k=0}^n (-1)^k binom(2n+2,k)\n * (2n^2 - 1 - (2n-1)k)^(2n-1) < 0.\n```\n\nThe sum, affine subtraction, and power in the formal owner are literal integer\noperations. Proposition 3.12(c) has a separate `n>=4` premise for a preceding\nweak-Lefschetz implication; the conjectural display following the proof\nexplicitly returns to every `n>=2`. The paper's finite computation through\n`n=400`, empirical monotonicity suggestion, and WLP consequence are not the\nformal target.","title":"The Miró-Roig--Tran Strict-Sign Conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/miro-roig-tran-strict-sign/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d1f094f8bc78d3024bdebb5ddea917154bd4f591440724aede266009cd71ca5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mirzavaziri-yaqubi-2026-gaussian-quotient-monotonicity-problem-7-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mirzavaziri and Yaqubi, *Graph binomial coefficients and a positivity conjecture of Guo and Krattenthaler*, arXiv:2610.19249v1, Problem 7.2, p. 16, ask:\n\n> Characterize the quadruples $(N,k,a,b)$ for which $\\genfrac{[}{]}{0pt}{}{N}{k}_q/([a]_q[b]_q)$ is a polynomial whose only negative coefficients are $-1$ at $q^1$ and at $q^{D-1}$. In particular, is this always the case when $\\genfrac{[}{]}{0pt}{}{N}{k}_q/[a]_q$ is weakly increasing from $q^1$ to its middle coefficient?\n\nThe answer to the final sufficiency question is negative. The full characterization remains open.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Gaussian polynomial and q-integers | Existing `LiUncuDefs.gauss` and `QProductNecessityRefutation.qInteger` | The q-Pascal Gaussian and finite geometric sum are reused; q-integers are mapped from natural to integer coefficients. |\n| The two ordinary quotients are polynomials | `claim` quantifies integer polynomials $U,P$ with the two exact multiplication identities | These identities state exact division, without rational-function ambiguity. |\n| Weak increase from $q^1$ to the middle | `increasingFromOne` | Every adjacent comparison at $2\\le s\\le\\lfloor\\deg(U)/2\\rfloor$; no comparison from degree zero is imposed. |\n| Only negative coefficients are $-1$ at $1,D-1$ | `twoExceptionSignPattern` | Both exceptional coefficients equal $-1$; every other natural index is nonnegative, including the zero tail. Here $D=\\deg(P)$. |\n| Is this always the case? | `claim`, negated by `result` | Universal natural indices $N,k,a,b$, with $k\\le N$ and positive $a,b$, and all exact polynomial quotients. |\n| Domain conventions | Natural subtraction and division; $k\\le N$, $1\\le a,b$ | Subtraction is truncated. These are the preregistered conventions; the numerical screen's narrower conditions are not added as hypotheses. |\n\nThe private `adjacent_iff_monotoneOn` correspondence connects the literal adjacent condition to `MonotoneOn` on the closed interval. It changes no public definition.","title":"Gaussian quotient monotonicity does not force two exceptional signs","url":"https://the-omega-institute.github.io/trureturing-pages/research/mirzavaziri-yaqubi-2026-gaussian-quotient-monotonicity-problem-7-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3434ff71eb3cfb6d9cd0231f0a6764aa95244a590e45cb3cf58f8ee0d7cbbb5d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mirzavaziri-yaqubi-cyclic-full-symmetry-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Madjid Mirzavaziri and Daniel Yaqubi, *Cyclic Latin Eulerian Numbers*, arXiv:2609.28808v1, Remark 3.9:\n\n> Cyclically shifting π naturally explains an order-n cyclic symmetry for ⟨⟨n k⟩⟩_c. However, computations for n ≤ 5 (Table 1) reveal a\n> surprising fact: ⟨⟨n k⟩⟩_c is actually invariant under the full symmetric group action on k … Identifying the hidden mechanism behind\n> this full invariance remains an open problem.\n\nHere `⟨⟨n k⟩⟩_c` counts permutations `π` whose row-reordered cyclic square `L_π` has column-ascent vector `k`.","title":"The Cyclic Latin Eulerian Numbers Are Not Fully Symmetric","url":"https://the-omega-institute.github.io/trureturing-pages/research/mirzavaziri-yaqubi-cyclic-full-symmetry-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57cf1c3f4e5cfabd348f1d91d6ac1f334e7237a0ad936ceeb14326c3b481fc54.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mirzavaziri-yaqubi-latin-eulerian-multiples","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Madjid Mirzavaziri and Daniel Yaqubi, *Latin Eulerian Numbers*, arXiv:2609.25100v1, Remark 4.16:\n\n> Precisely two things remain: (i) that every multiple of n strictly between n and (n − 1)^2 − 1 is attained by some Latin square\n> which Proposition 4.14 reduces to an explicit, checkable question about the function F, verified for n ≤ 11 but open in general;\n> (ii) unimodality of (T_n(m)) …\n\nHere `Σ(L)` is the sum over the columns of an order-`n` Latin square of the number of ascents read from top to bottom. Part (i)\nasks, for every `n ≥ 5` and `2 ≤ k ≤ n − 3`, for an order-`n` Latin square with `Σ(L) = kn`.","title":"Every Interior Multiple of n Is a Latin-Square Ascent Total","url":"https://the-omega-institute.github.io/trureturing-pages/research/mirzavaziri-yaqubi-latin-eulerian-multiples/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/338d46111478dbeee7fba2a7e46ede9c6613c18b43efd6310a609fc309d251f3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:misawa-nishimura-harmonic-strength-pair-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Misawa and Nishimura, *Spherical Designs on S¹ of Finite Harmonic Strength*,\narXiv:2505.06893v2, Conjecture 3.3, printed page 6:\n\n> Let p ≠ q be integers with p, q > 1. Then N({p, q}, 2) = 5.\n\nThe quantity N(T, 2) is defined on the same printed page 6:\n\n> For a nonempty finite set T ⊂ N, define\n> N(T, 2) := min{|X| | X ⊂ S¹, Hst(X) = T}.\n\nand the harmonic strength is taken in the paper's working form (printed\npages 2--3):\n\n> Finally, we identify S¹ with {z ∈ C | |z| = 1} and write P_k(X) := Σ_{x∈X} x^k\n> for the k-th complex moment of a finite set X ⊂ S¹. By [2, Lemma 1], for\n> X ⊂ S¹ one has Hst(X) = {k ∈ N | P_k(X) = 0}.","title":"Misawa--Nishimura harmonic-strength pair refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/misawa-nishimura-harmonic-strength-pair-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38d7b05e6fb764d052e02085918b8aeeea22561c3ab82affe38911a56cfbc270.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:miska-ulas-2022-sigma-three-conjecture-4-8","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Miska and Ulas, *On the Diophantine equation σ₂(X̄ₙ) = σₙ(X̄ₙ)*, arXiv:2203.03942v1, page 16, Conjecture 4.8 states:\n\n> Let n ∈ N≥4 and suppose that m ∈ N satisfies m = σ_3(X_n) = σ_n(X_n). Then m ≤ (1/12)(3021n⁶ − 77575n⁵ + 920361n⁴ − 6235705n³ + 24764202n² − 53664304n + 48986640) and x_n ≤ (1/12)(27n⁴ − 190n³ + 471n² − 500n + 216).\n\nWrite $Q(n)$ and $D(n)$ for the two bounds, respectively. The source defines $\\sigma_j$ as the elementary symmetric sum of products over $j$ distinct increasing indices, and a solution as a nondecreasing tuple of positive integers. The target is the conjunction as printed in version 1. Preregistration: [#19678](https://github.com/the-omega-institute/trureturing/issues/19678).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $n\\ge4$ | `claim` quantifies `n : ℕ` with `hn : 4 ≤ n` | Exact |\n| Positive integer tuple $X_n$ | `X : Fin n → ℕ` and positive entries | Exact; zero-based indices |\n| $x_i\\le x_{i+1}$ | `Monotone X` | Equivalent on the finite linear order |\n| $m=\\sigma_3(X_n)=\\sigma_n(X_n)$ | Both equalities to `MvPolynomial.eval X (MvPolynomial.esymm (Fin n) ℕ j)` | Literal subset-product sum; no new symmetric-sum alias |\n| $m\\le Q(n)$ and $x_n\\le D(n)$ | The two rational inequalities in `claim`; last entry indexed by `n - 1` | Both conjuncts retained |\n| Refutation | `result : ¬ claim` | The first conjunct fails at $n=11$ |\n\nThe rational casts preserve the integer inequalities and interpret the source's factor $1/12$ exactly. The index proof follows from $n\\ge4$ and adds no mathematical hypothesis.","title":"Refutation of Miska–Ulas Conjecture 4.8","url":"https://the-omega-institute.github.io/trureturing-pages/research/miska-ulas-2022-sigma-three-conjecture-4-8/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe6d9f839ea57906d5545141897158864ef8f057d2a82e91b8e1ca5b615cf728.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mista-fiurasek-2006-qudit-partial-estimation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Mišta Jr. and J. Fiurášek, *Optimal partial estimation of quantum states from several copies*, arXiv:quant-ph/0606112v1, Phys. Rev. A 74, 022316, Introduction, Eq. (2), states:\n\n> Moreover, we also conjecture that the optimal fidelity trade-off for an ensemble of N identical pure states of a d-level system has the form\n\n$$\n\\sqrt{F-\\frac1{N+d}}=\n\\sqrt{(d-1)\\left(\\frac{N+1}{N+d}-G\\right)}+\n\\sqrt{N\\left(G-\\frac N{N+d}\\right)}.\n$$\n\n[Preregistration #16777](https://github.com/the-omega-institute/trureturing/issues/16777) fixes the literal covariance-reduced PSD-seed model. The source is documented in [the Library note](../Library/QuantumChannels/mista2006partialestimation.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $N$ copies of a $d$-level pure state, $N\\ge1$, $d\\ge2$ | `claim` quantifies natural `N`, `d` with precisely these two inequalities | Every allowed dimension and copy count is retained. |\n| Symmetric input space tensored with one output qudit | The carrier is the product of the literal occupation subtype $\\{n:\\mathrm{Fin}(d)\\to\\mathbb N\\mid\\sum_i n_i=N\\}$ with `Fin d` | Occupation carriers occur inline; no alternate certificate space replaces them. |\n| Eq. (12), the partial transpose of the symmetric projector | `OccupationPartialTransposeSpectral.RF`, `symmetricProjector`, `occupationEmbedding` | The normalized word-sector projector, partial transpose and compression are the definitions. Entry formulas are proved afterward. |\n| Eq. (10), reference projection and output partial trace | `OccupationPartialTransposeSpectral.RG`, `outputReferenceProjector`, reused `PartialTraceMutualInformation.partialTraceRight` | The matrix product, partial trace and tensor with the output identity are retained. |\n| Positive seed and Schur normalization | `chi.PosSemidef` and trace equal to the complex coercion of $\\binom{N+d-1}{d-1}$ | No rank restriction is imposed on competitors. |\n| Output and estimation fidelities | Real parts of `trace (chi * RF N d)` and `trace (chi * RG N d)` | Both scores use the literal source operators. |\n| $N/(N+d-1)\\le G\\le(N+1)/(N+d)$ | The two bounds on real `G` in `claim` | The complete closed interval, including endpoints, is retained. |\n| Optimal curve, Eq. (2) | `IsGreatest` of the set of all feasible output scores at fixed `G` | The displayed squared curve is both attained and an upper bound. Its square-root terms are nonnegative on the stated interval, so it is equivalent to Eq. (2). |\n| Proof of the conjecture | `QuditPartialEstimationTradeoff.result : claim` | There are no additional hypotheses. |","title":"The Mišta–Fiurášek qudit partial-estimation trade-off","url":"https://the-omega-institute.github.io/trureturing-pages/research/mista-fiurasek-2006-qudit-partial-estimation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9a7d2bab557c0313c1bd4e7b43775023fbd77a95bee5793b57643ac15a30e48.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mitra-2021-broadcasting-channel-extrema","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Mitra, *Layers of classicality in the compatibility of measurements*, Phys. Rev. A 104, 022206 (2021), arXiv:2101.05752v4, Section III.A asks whether the broadcasting channels of an observable have a highest element and whether they have a lowest element under precomposition by channels that are nondisturbing for that observable. The source repeats both questions in its Conclusions.\n\nThe formal convention uses an observable with outcomes `Fin m`, input matrices on `Fin d`, output matrices on `Fin d × Fin d`, finite Kraus channels, the broadcasting predicate, the nondisturbing predicate, and the precomposition relation. The settling declaration `D5/S3/Quantum/Measurements/BroadcastingChannelExtremaRefutation.result` has type `¬ claim` and carries `OpenProblemResolutionClaim(Refuted)`.\n\n### Statement echo\n\nThe source's broadcasting condition is the pair of marginal trace equalities in `Broadcasts`; its nondisturbance condition is equality of each observable effect's trace after channel application in `Nondisturbing`; `ConstructedFrom` is precomposition by such a channel. `HasHighest` and `HasLowest` quantify over broadcasting channels and this relation. `claim` combines observability, existence of a broadcasting channel, and the disjunction of the two extrema. The fixed witness uses `d = 3`, `m = 2`, and the observable `A`; the remaining binders and hypotheses are those displayed by the corresponding Scribe nodes.","title":"Mitra broadcasting-channel extrema","url":"https://the-omega-institute.github.io/trureturing-pages/research/mitra-2021-broadcasting-channel-extrema/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/660b4ba7b710d21f337a23de9beaea4fbae77e64a845da7d421e3e5479070cdb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mitra-2022-unsharpness-white-noise-monotonicity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Mitra, *Quantifying unsharpness of observables in an outcome-independent way*, Int. J. Theor. Phys. 61, 236 (2022), arXiv:2201.02578v3, defines instrument-independent unsharpness measures of an observable A = {A_i}.\n- **The measures.** With the X-matrix X^A = Σ_i ‖A_i‖A_i, they are E(A) = ‖I − X^A‖ and E′(A) = Tr[(I/d)(I − X^A)] (Eqs. (44) and (48)).\n- **Monotonicity criteria.** Theorems 3 and 4 show that E and E′ are non-decreasing under the white noise A^λ_i = λA_i + (1 − λ)I/n exactly when Σ^A_1 ≥ Σ^A_2 (Eq. (50)) and Σ′^A_1 ≥ Σ′^A_2 (Eq. (60)) hold. Here Σ^A_1 = x^A_min − 1/n and Σ^A_2 = Σ_i ‖A_i‖/n − x^A_min, with x^A_min the lowest eigenvalue of X^A.\n- **The conjecture.** Propositions 1–2 prove both inequalities for dichotomic qubit observables. Then: \"Conjecture 1. For any qubit observable A, inequality Σ^A_1 ≥ Σ^A_2 and inequality Σ′^A_1 ≥ Σ′^A_2 hold and therefore, E(A^λ) ≥ E(A) and E′(A^λ) ≥ E′(A) for all 1 ≥ λ ≥ 0.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/mitra2022unsharpness.md).\n\nIssue [#17118](https://github.com/the-omega-institute/trureturing/issues/17118) fixes the reading: every qubit POVM with any number n ≥ 1 of outcomes, the operator norm, and all four clauses. The conjecture is refuted.","title":"Mitra's instrument-independent unsharpness measure is not monotone under white noise for qubit observables","url":"https://the-omega-institute.github.io/trureturing-pages/research/mitra-2022-unsharpness-white-noise-monotonicity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc39bb07c6e8ab3ac202e38630f20e14942e65c07a30ecfe31f51f6f40191f3c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mitra-badhani-ghosh-2023-completely-useless-concatenation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Mitra, H. Badhani and S. Ghosh, *Improvement in quantum communication using quantum switch*, Phys. Scr. 98, 045101 (2023), arXiv:2108.14001v6, Section III.B, Conjecture 1:\n\n> Concatenation of two completely useless channels is always completely useless.\n\nThe preceding definition is:\n\n> A useless channel is completely useless if under both C+ and C− it remains useless.\n\nHere uselessness concerns entanglement transfer and means entanglement breaking. Equation (8) gives the two completely positive switch branches with Kraus operators $(K_iK_j\\pm K_jK_i)/2$. These branches need not preserve trace. The conjecture concerns normalized channels, together with entanglement breaking of their original map and both branches.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Qubit Kraus matrices and completeness | `IsKrausChannel`, complex matrices on `Fin 2`, and the sum of $K_i^\\dagger K_i$ equal to the identity | Literal normalization; no extra hypothesis |\n| Kraus sum action | Existing `MatrixMap.of_kraus K K` | The sum of $K_iXK_i^\\dagger$; no new forwarding definition |\n| Equation (8), plus branch | `switchPlus` | Anticommutator Kraus operators divided by two |\n| Equation (8), minus branch | `switchMinus` | Commutator Kraus operators divided by two |\n| Complete uselessness | `CompletelyUseless` | All three entanglement-breaking conjuncts, using the existing `IsEB` |\n| Concatenation, second after first | `composeKraus L K` | The pair $(i,j)$ carries $L_jK_i$ |\n| Every pair of such channels | `claim` | Every pair of finite index types, both Kraus families, both normalization hypotheses and both complete-uselessness hypotheses |\n| Refutation | `result : ¬ claim` | A single pair satisfying all premises and violating the conclusion suffices |\n\nFinite index types are the Kraus-family convention of [#16400](https://github.com/the-omega-institute/trureturing/issues/16400). `Fintype` arguments are typeclass arguments, with no extra decidability assumption in the claim. The six public definitions state those conventions; the counterexample construction and all proof helpers are private.","title":"Two completely useless channels with a useful concatenation switch branch","url":"https://the-omega-institute.github.io/trureturing-pages/research/mitra-badhani-ghosh-2023-completely-useless-concatenation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b5a3c2b6005203d4b462b9785e350cbcaed3e1b5683d924e6b9cef6931812b2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mmd2205-seidelchain","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Santanu Mandal, Ranjit Mehatari and Kinkar Chandra Das, *On the spectrum and energy of Seidel matrix for chain graphs*, arXiv:2205.00310v1, Conjecture 4.14:\n\n> Let $G$ be a chain graph of order $n$ with the binary string $b=0^{s_1}1^{t_1}0^{s_2}\\ldots0^{s_k}1^{t_k}$ such that $\\sum_i(s_i+t_i)=n$, $s_i\\ge1$ and $t_i\\ge1$. Then $SE(G)\\ge SE(H)$, where $H\\in\\Gamma_{n,k}$. Moreover, the equality holds if and only if $G\\in\\Gamma_{n,k}$.\n\nHere $\\Gamma_{n,k}$ has binary word $0^{s_1}(10)^{k-1}1^{t_k}$ with positive endpoints, each at most $n-2k+1$, and $s_1+t_k=n-2k+2$. Seidel matrices are $J-I-2A$ and energy sums the absolute eigenvalues with multiplicity. The theorem quantifies over every comparator $H$.\n\n### Statement echo\n\nThe local names below belong to `D5.S3.Combinatorics.Graph.KyFanGap.MandalMehatariDasSeidelMinimum`; the five graph conventions lie in its `MandalMehatariDasSeidel` namespace.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive alternating runs $0^{s_1}1^{t_1}\\cdots0^{s_k}1^{t_k}$ | `binaryString`, positive hypotheses in `claim` | All run lengths and their order are retained. |\n| A later 1 is adjacent precisely to earlier 0-vertices | `chainGraph` | The symmetric adjacency relation has no loops. |\n| Order $n$ | Sum hypothesis in `claim` | The vertices are the positions of the binary list; its length is the sum of the runs. |\n| $S=J-I-2A$ | `seidel` | Literal all-ones matrix, identity and scalar adjacency matrix. |\n| $SE$ is the sum of absolute eigenvalues | `energy`, `energy_eigenvalues` | Real characteristic roots with multiplicity; every graph matrix used is real symmetric. |\n| Every $H\\in\\Gamma_{n,k}$ | Universal list `h` and `Gamma n k h` in `claim` | No representative comparator is substituted. |\n| Minimum and equality precisely in $\\Gamma_{n,k}$ | Both conclusions of `claim`, proved by `MandalMehatariDasSeidel.result` | Full inequality and equivalence, including $k=1$. |\n| $k\\ge1$ and $2k\\le n$ | Explicit guards in `claim` | These follow from nonempty positive runs and introduce no source restriction. |\n| Finite ordered vertices | `Fin` and the finite type instances | These encode the finite graph and its construction order. |\n\nStandalone `Gamma` outside $1\\le k$ and $2k\\le n$ is outside the source interpretation: natural subtraction truncates. The settling proposition has both guards. The literal binary and Seidel conventions are retained; the proofs also use shared spectral owners.","title":"Minimum Seidel energy of chain graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/mmd2205-seidelchain/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad9d2bcb283162334079443b4b5f4f0eab99c9cb26be8ca9c7842c1d483be3d9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mo516232-zero-sum-count-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every positive integer sequence satisfying a constant integer recurrence,\nlet $P$ be the characteristic polynomial of its minimal recurrence. For\nevery finite $D\\subset\\mathbb Z$, count the ordered original-coordinate\nassignments $S_n(D)=\\#\\{d\\in D^n:\\sum d_i a_i=0\\}$. The assertion is\nthat all these count sequences are C-finite if and only if $P$ has no\nunit-circle roots, all its expanding roots are simple, and all ratios of\ntwo expanding roots are roots of unity. The target is the entire displayed\niff of MathOverflow 516232 revision 2.","title":"The full zero-sum prefix-count characterization","url":"https://the-omega-institute.github.io/trureturing-pages/research/mo516232-zero-sum-count-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/42bfee9dae91c89b38f82f053cdd873e584eaacad37ccc53ac2191b4ea83409f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mohan-neetu-small-doubling-conjecture-61-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 6.1 in Mohan and Neetu, arXiv:2607.11194v1, section 6, asserts\nthat a nonempty finite subset S of a torsion-free group, partitioned into two\ndisjoint abelian sets A and B, generates an abelian subgroup whenever\n`|S^2| <= 3k - 3`. Here `k = |S|`, `S^2 = {xy : x,y in S}`, and an abelian set\nmeans one whose generated subgroup is abelian. The formal `claim61` quantifies\nover the group and all three finite sets with these premises; `result61` proves\nits negation.","title":"Mohan-Neetu Conjecture 6.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/mohan-neetu-small-doubling-conjecture-61-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84d9023ac4cae008ed7ea6fd5ec5f8715d84a69060068a0188ecfbb8cbc31693.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mohan-neetu-small-doubling-conjecture-62-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 6.2 in Mohan and Neetu, arXiv:2607.11194v1, section 6, asserts\nthat a nonempty finite subset S of a torsion-free group, partitioned into\nthree pairwise disjoint abelian sets A, B and C, generates an abelian subgroup\nwhen `|S^2| <= 3k - 4`. Here `k = |S|`, `S^2` is the product set, and an\nabelian set has an abelian generated subgroup. The formal `claim62` retains\nevery premise; `result62` proves its negation.","title":"Mohan-Neetu Conjecture 6.2","url":"https://the-omega-institute.github.io/trureturing-pages/research/mohan-neetu-small-doubling-conjecture-62-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c650840a0709a0c7cab24af5b4a9edba70f112bc11122e971047d89dc51c7664.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mohan-neetu-small-doubling-conjecture-63-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 6.3 in Mohan and Neetu, arXiv:2607.11194v1, section 6, asserts\nthat a finite subset S of a torsion-free group with `|S| >= 3`, identity in\nS, and `|S^2| <= 3|S| - 3` generates an abelian subgroup. The formal\n`claim63` states these quantified premises and conclusion; `result63`\nproves `not claim63`.","title":"Mohan-Neetu Conjecture 6.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/mohan-neetu-small-doubling-conjecture-63-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8791805f274a8ed8f80c5740bc5b59d8c1102670c5239725696520e1d935d9a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mohanty-sharma-sazim-pradhan-pati-2022-luders-coherence-optimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. D. Mohanty, G. Sharma, Sk Sazim, B. Pradhan and A. K. Pati, *Creation of quantum coherence with general measurement processes*, Quantum Information Processing 21, 48 (2022), arXiv:2004.10875v2, Section 3.2, `coh-measure.tex`, line 298:\n\n> We couldn't get the expression for $\\max_{\\{\\vec{a},a_{\\pm}\\}}C_{l_1}(\\rho'_{\\pm})$ analytically, so we have plotted it against the initial state coherence $C_{l_1}(\\rho_{\\pm})$.\n\n> However in this case, we don't know yet for which states there can not be any coherence gain and for which states the coherence gain will be maximum.\n\nThe Conclusion, line 505, asks:\n\n> As a future line of work, it would be interesting to obtain the analytical results for two outcome and n-outcome POVMs.\n\nThe source defines $C_{l_1}(\\rho)=\\sum_{i\\ne j}|\\rho_{ij}|$ at line 109 and the nonselective Lüders action $\\Phi_E(\\rho)=\\sum_i\\sqrt{E_i}\\rho\\sqrt{E_i}$ at lines 127–130. A qubit density matrix is a positive semidefinite complex $2\\times2$ matrix of trace one; an $n$-outcome POVM is a family of positive semidefinite effects indexed by $\\mathrm{Fin}\\,n$ whose sum is $I$.\n\nPut $s=C_{l_1}(\\rho)$, $R=\\sqrt{(\\operatorname{Re}(\\rho_{00}-\\rho_{11}))^2+4|\\rho_{01}|^2}$ and $\\lambda=(s+R)/2$. Issue [#22649](https://github.com/the-omega-institute/trureturing/issues/22649) fixes the statement: for every qubit density matrix and every integer $n\\ge2$, $\\lambda$ is the attained greatest output coherence; the optimal gain $\\lambda-s$ vanishes exactly when $\\rho_{00}=\\rho_{11}$, is at most $1/2$, and equals $1/2$ exactly for $\\operatorname{diag}(1,0)$ and $\\operatorname{diag}(0,1)$. This is proved.\n\nThe definitions and source locators are also in [the literature note](../Library/QuadraticForms/mohantysharmasazimpradhanpati2022coherence.md).","title":"The attained coherence optimum for arbitrary finite qubit Lüders measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/mohanty-sharma-sazim-pradhan-pati-2022-luders-coherence-optimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/05c6bcd2fd536a1f958b3e3e73458dbcfc2579c053c0e2cb3cb9045839d24ca0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:molinari-2022-honeycomb-trapezium-huckel-determinant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Molinari's Conjecture 2 in *Graphene nanocones and Pascal matrices* (arXiv:2206.14428v2, section 5, page 9) asks whether the determinant of the Hückel matrix of a honeycomb trapezium with rows of lengths `2k+1, 2k+3, ..., 2n+1` equals the determinant of the displayed reduced Pascal matrix of size `n+1-k`. The weights are arbitrary boundary parameters.\n\nThe formal statement is\n\n```\nclaim := ∀ (R : Type) [CommRing R] (k n : ℕ), k ≤ n → ∀ (x y : ℕ → R),\n (huckel k n x y).det = (reducedPascal k n x y).det\nresult : claim\n```","title":"Molinari's honeycomb trapezium determinant","url":"https://the-omega-institute.github.io/trureturing-pages/research/molinari-2022-honeycomb-trapezium-huckel-determinant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a591bb651e515166ac8809d594017082a95c0048943ff958e992c1550a38968.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:monbroussou-et-al-2025-hamming-weight-pattern-spectral-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Monbroussou, E. Z. Mamon, J. Landman, A. B. Grilo, R. Kukla and E. Kashefi, *Trainability and Expressivity of Hamming-Weight Preserving Quantum Circuits for Machine Learning*, Quantum 9, 1745 (2025), DOI 10.22331/q-2025-05-15-1745; arXiv:2309.15547v4, Conjecture `conj:spectral-gap`, `Quantum.tex` lines 1980–1985:\n\n> Let $(U_{0,n})_{n\\geq2}$ be a sequence of connected patterns of RBS/FBS gates (\\cref{def:connected-pattern}), where each $U_{0,n}$ is such a pattern over $n$ qubits.\n> If the number $J_n$ of gates in the pattern $U_{0,n}$ satisfies $J_n \\in \\mathcal{O}\\left(\\mathrm{poly}(n)\\right)$, then for any fixed Hamming weight $k \\in \\mathcal{O}(1)$, the associated $d_{k,n} \\times d_{k,n}$ stochastic matrix $T_n:=T^{(U_{0,n}^k)}$ (\\cref{eq:stochastic-matrix-T-def}) satisfies\n>\n> $\\Delta_n \\in \\Omega\\left(1/\\mathrm{poly}(n)\\right)\\,,$\n>\n> where $\\Delta_n:=1-|\\lambda_2(T_n)|$ denotes the spectral gap of the stochastic matrix $T_n$.\n\nThe versioned arXiv source archive contains `Quantum.tex` (2252 lines, md5 `057c1aeb9e7f5c3b57445d20f924f4fc`). Its line 1975 states: “The actual proof of \\cref{conj:spectral-gap} is left for future work.”\n\nHere $X_{n,k}=\\{A\\subseteq\\mathrm{Fin}\\,n:|A|=k\\}$ is the weight-$k$ basis, with $N=\\binom nk$. A gate on distinct qubits $a,b$ exchanges their labels by $\\tau_{ab}$ and has stochastic matrix $P_{ab}=(I+S_{ab})/2$, where $S_{ab}$ is the permutation matrix on $X_{n,k}$ induced by $\\tau_{ab}$. A configuration containing both labels or neither is fixed. The pattern matrix is $T=P_J\\cdots P_1$, with its first listed gate acting first. The source defines these matrices by squaring the RBS/FBS entries at angle $\\pi/4$ (lines 158–192, 1455–1458 and 1486–1489), and its gap by the second-largest eigenvalue modulus (lines 1397–1405).\n\nIssue [#22140](https://github.com/the-omega-institute/trureturing/issues/22140) fixes the nontrivial sectors $1\\le k We note that the configurational entropy Sc = S + (−SN) is a sum of a concave and a convex function. Therefore, one cannot directly apply majorization to Sc based on the already obtained results for S and SN. However, this does not rule our that Sc is also a concave function of the parameters {Ck}. Similar to the original Shepp-Olkin paper [22] we will here just investigate the cases M = 2 and M = 3. Based on these results, we conjecture that Sc({Ck}) is indeed Schur concave and majorization thus applies.\n\nFor independent Bernoulli variables X₁,…,X_M with probabilities c₁,…,c_M, put N=Σk Xk, S(c)=Σk h(ck) and SN(c)=H(N). Natural logarithms and 0 log 0=0 are used. The claim is that F(c)=Sc(c)=S(c)−SN(c) is Schur concave on [0,1]^M for every natural M: if d has the same coordinate sum as c and every descending partial sum of d is no larger than the corresponding one of c, then F(c)≤F(d).","title":"Monkman–Sirker configurational entropy is Schur concave","url":"https://the-omega-institute.github.io/trureturing-pages/research/monkman-sirker-2023-configurational-entropy-schur-concavity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb950de1e7cb8528a3b2ded4ec4c6f06478bf2248431a2c568674a1cd9487bbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:monterde-2022-strong-cospectral-multiplicity-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Monterde, *Strong cospectrality and twin vertices in weighted graphs*, Electronic Journal of Linear Algebra 38 (2022), 494–518, DOI 10.13001/ela.2022.6721; arXiv:2111.01265v2, Section 3, immediately following Theorem 3.11:\n\n> We mention an interesting result of {\\'{A}}rnad{\\'{o}}ttir and Godsil which states that if $X$ is a normal Cayley graph and $n$ is the largest multiplicity of an eigenvalue of $A$, then the number of pairwise adjacency strongly cospectral vertices is bounded above by $|V (X)|/n$ \\cite[Theorem 6.1]{Arn2021}. It would be interesting to check whether this bound holds for other classes of graphs, and in general whenever $M=\\textbf{A}$.\n\nThis quotation is from line 727 of the downloaded v2 TeX, `ELA-sample1_copy.tex` (1444 lines, MD5 `21a1ee53b9e64ae0b3a6773aa4fdbb45`). The definitions and source locators are collected in [the literature note](../Library/GraphInvariants/monterde2022strongcospectralitytwins.md).\n\nIssue [#21638](https://github.com/the-omega-institute/trureturing/issues/21638) fixes the general-extension reading for ordinary adjacency: for every $n\\in\\mathbb N$, every connected simple graph $G$ on $\\operatorname{Fin}n$ with decidable adjacency, and every finite set $S$ of pairwise adjacency strongly cospectral vertices,\n$|S|\\le n/m(G)$.\nHere $V_\\theta=\\{x\\in\\mathbb R^n:A(G)x=\\theta x\\}$ and\n$m(G)=\\operatorname{sSup}\\{\\dim_{\\mathbb R}V_\\theta:\\theta\\in\\mathbb R\\}\\in\\mathbb N$, the module's `maxMultiplicity`. Cardinalities and $m(G)$ are cast to the reals before division. The universal extension is refuted.","title":"Two strongly cospectral vertices of $K_{2,3}$ exceed the multiplicity bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/monterde-2022-strong-cospectral-multiplicity-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7beaa745e87ecc781e6531361ecb334004322ef0fa9d710c48daca70ad59038.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:monterde-2023-path-twin-sharp-sedentariness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Monterde (arXiv:2401.00362, math.CO and quant-ph; Discrete Math. 349(4)\n(2026) 114959) studies the quantum walk `U(t) = e^{itA}` of a graph with\nadjacency matrix `A`. A vertex `u` is sharply `C`-sedentary if\n`inf_{t>0} |U(t)_{u,u}| = C` with `0 < C ≤ 1`. The graph `P_n'` is the path\n`1, …, n` with an added vertex `n + 1` joined to vertex `2`, so that `1` and\n`n + 1` are non-adjacent twins. Example 18 proves `|U(t)_{1,1}| ≥ 1/n` for odd\n`n` and states:\n\n> We conjecture that $u$ is sharply $(\\frac{1}{n})$-sedentary in $P_n'$ for\n> all odd $n\\geq 5$.\n\nIssue #11607 fixes the reading: `U(t) = e^{itA} = hamiltonianPropagator A (−t)`,\n`u = 1`, and the claim is `inf_{t>0} |U(t)_{1,1}| = 1/n` for every odd\n`n ≥ 5`. The result refutes it at `n = 9`.","title":"The twin end vertex of P'_9 is not sharply 1/9-sedentary","url":"https://the-omega-institute.github.io/trureturing-pages/research/monterde-2023-path-twin-sharp-sedentariness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/afbeadd3fb3bba48ee4d4b4a63ea6b7aadcf8d2490b537fcec11cffed90fdcf6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:monterde-pal-kirkland-2025-blowup-matching-adjacency-transfer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Monterde, H. Pal and S. Kirkland, *Laplacian quantum walks on blow-up graphs* (2025), arXiv:2504.11585v1, DOI 10.1016/j.laa.2025.10.001, ask in Section 11, Future work (`main.tex`, line 814):\n\n> Lastly, we ask: does Theorem \\ref{pt1} hold for the adjacency case? That is, if $n\\equiv 0$ (mod 4), will the addition of an appropriate matching in $\\up{n}G$ induce adjacency LPST?\n\nTheir Laplacian Theorem `pt1`, Section 10, Edge perturbation (`main.tex`, lines 666–668), states:\n\n> Let $n\\equiv 0$ (mod 4) and $v$ be a vertex in $G$ with $\\sigma_v(G)\\subset\\Zl.$ Then the addition of a matching in $T_v$ results in LPST in $\\up{n}G$ at $\\frac{\\pi}{2}$ between the end vertices of every edge inserted. Moreover, the resulting graph is periodic at all vertices in $T_v$ that are not incident to the newly added edges with period $\\frac{\\pi}{2}.$\n\nThese quotations retain the source macros: `\\up{n}G` denotes the $n$-fold blow-up and `\\Zl` denotes the integers. In the source, $\\sigma_v(G)$ is Laplacian eigenvalue support. The quotations were checked against the downloaded v1 TeX member `main.tex`, md5 `b21c68f0d57c75bbf8eb7530da0c4d04`; the definitions are discussed in [the literature note](../Library/Eigenstructure/monterdepalkirkland2025blowup.md).\n\nIssue [#22707](https://github.com/the-omega-institute/trureturing/issues/22707) fixes the reading settled here. Let $G$ be a finite simple graph with adjacency matrix $A_G$, let $v$ have integral **adjacency** eigenvalue support, and let $4\\mid n$. Choose a fixed-point-free involution $\\rho$ of the $n$ clone indices and insert its perfect matching in **every** clone set. With vertices indexed by $(v,j)$, the resulting matrix is\n$H=A_G\\otimes J_n+I_m\\otimes P_\\rho$.\nFor every clone index $j$, the settlement proves adjacency perfect state transfer at $\\pi/2$ between $(v,j)$ and $(v,\\rho j)$. This answers the appropriate-matching reading; it does not assert the literal variant with a matching inserted only in $T_v$.","title":"Integral adjacency support gives transfer through uniform clone matchings","url":"https://the-omega-institute.github.io/trureturing-pages/research/monterde-pal-kirkland-2025-blowup-matching-adjacency-transfer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f68c8711eb90fc961ca65ade80d7964b571744c16c0d1783d46cf19e31c100f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:moradi-rampersad-shallit-2026-fibonacci-shift-linear-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Delaram Moradi, Narad Rampersad, and Jeffrey Shallit, *Complexity of Linear\nSubsequences of Fibonacci-Automatic Sequences*, arXiv:2603.21645v1,\nSection 4, [Problem 1](https://arxiv.org/html/2603.21645v1#Thmproblem1):\n\n> Prove the number of states in a minimal automaton generating\n> (t(i+c))ᵢ≥₀ is Θ(c).\n\nThis dossier anchors only that complete statement. Put F₀ = 0, F₁ = 1 and\nFₙ₊₂ = Fₙ₊₁ + Fₙ. For an MSD-first word x = x₀…xₗ₋₁ over {0,1}, set\nval(x) = Σⱼ<ₗ xⱼFₗ₊₁₋ⱼ, with val(ε) = 0. Valid words contain no adjacent\nones; arbitrary leading-zero padding is valid. The value t(n) is the parity\nof the number of occupied digits in the canonical Zeckendorf representation\nof n.\n\nFor c ∈ ℕ, s(c) is the attained minimum number of live states of a finite\ndeterministic partial MSD automaton with Boolean state outputs. Every\ncounted state is reachable. The start state is counted and has a zero\nself-loop; a rejecting sink is omitted. For every binary word x, the output\nof the partial run is some(t(val(x)+c)) when x is valid and none otherwise.\nThus every valid padded word is covered, including ε and all-zero words,\nwhich output t(c). Invalid execution is undefined, rather than an output\nchosen without a constraint. These conventions follow Sections 2.1 and 2.2\nof the source.\n\nThe full eventual uniform proposition is\n\n$$\n\\exists a,b\\in\\mathbb R,\\quad a>0\\land b>0\\land\n\\exists c_0\\in\\mathbb N,\\quad\n\\forall c\\in\\mathbb N,\\quad c\\ge c_0\\Longrightarrow\nac\\le s(c)\\le bc.\n$$\n\nThe exact Lean definition is\n`D5/S1/Digit/ZeckendorfProblem1Refutation.Problem1`.\nRequiring c₀ ≥ 1 gives the same eventual statement, since the threshold can\nbe replaced by max(c₀,1). The settling theorem\n`D5/S1/Digit/ZeckendorfProblem1Refutation.problem1_refuted : ¬ Problem1`\nnegates the entire proposition with this original machine class.","title":"Moradi–Rampersad–Shallit Problem 1: shifted Fibonacci–Thue–Morse state complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/moradi-rampersad-shallit-2026-fibonacci-shift-linear-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43278d1105bf0089ea48b56f1b516482f4f80b62f6310659bae55c301d9677a2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:morelli-yamasaki-huber-tavakoli-2022-imprecise-chsh-optimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Morelli, H. Yamasaki, M. Huber and A. Tavakoli, *Entanglement detection with imprecise measurements*, Phys. Rev. Lett. 128, 250501 (2022), arXiv:2202.13131v2, main text, `Manuscript_epsilonEW.tex`, line 111:\n\n> However, as we show in Supplementary Material through an explicit separable model that we conjecture to be optimal, this fact does not qualitatively improve the robustness of idealised ($\\varepsilon=0$) entanglement detection to small inaccuracies.\n\nThe witness and candidate value are given at lines 111–115; the supplement, Section “Entanglement detection based on the CHSH quantity” (`AppCHSH`, lines 232–256), specifies two qubits, targets $X=\\sigma_X$ and $Z=\\sigma_Z$ on both sites, and a common inaccuracy bound $\\varepsilon$. See [the literature note](../Library/QuadraticForms/morelliyamasakihubertavakoli2022imprecise.md).\n\nIn the formal statement, each of $A_1,A_2,B_1,B_2$ is a complex $2\\times2$ matrix satisfying `IsBinaryObservable`: it is Hermitian and $I-A$ and $I+A$ are positive semidefinite. `Admissible` additionally requires\n\n$\\operatorname{Re}\\operatorname{Tr}(A_1X),\\operatorname{Re}\\operatorname{Tr}(A_2Z),\\operatorname{Re}\\operatorname{Tr}(B_1X),\\operatorname{Re}\\operatorname{Tr}(B_2Z)\\ge 2-4\\varepsilon$.\n\n`Separable` means membership in the real convex hull of $\\sigma\\otimes\\tau$, where $\\sigma,\\tau$ are positive semidefinite complex $2\\times2$ matrices with trace one. The witness is\n\n$W(\\rho;A_1,A_2,B_1,B_2)=\\operatorname{Re}\\operatorname{Tr}[\\rho(A_1\\otimes B_1+A_1\\otimes B_2+A_2\\otimes B_1-A_2\\otimes B_2)]$.\n\nPut $\\varepsilon_0=\\frac12-\\frac1{2\\sqrt2}$. The definition `separableValue` is the source's candidate:\n\n- for $\\varepsilon\\le\\varepsilon_0$, $V(\\varepsilon)=4(1-2\\varepsilon)\\sqrt{\\varepsilon(1-\\varepsilon)}+\\sqrt{2-16\\varepsilon(1-\\varepsilon)(1-2\\varepsilon)^2}$;\n- for $\\varepsilon>\\varepsilon_0$, $V(\\varepsilon)=2$.\n\nIssue [#22442](https://github.com/the-omega-institute/trureturing/issues/22442) fixes the reading: for every $\\varepsilon\\in[0,1]$, $V(\\varepsilon)$ is a greatest element of the set of witness values attained by separable states and admissible observables. This is `claim`, proved by `result : claim`. Greatestness includes both attainment and an upper bound on every value in that set.","title":"The separable CHSH optimum under imprecise qubit measurements is attained","url":"https://the-omega-institute.github.io/trureturing-pages/research/morelli-yamasaki-huber-tavakoli-2022-imprecise-chsh-optimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4df369993bb553258c769ec7b6f591061c69e40c2d339533c0a468ea610db5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:moutsinas-2021-democracy-coefficient-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Moutsinas, C. Shuaib, W. Guo and S. Jarvis (arXiv:1908.04358,\nphysics.soc-ph; Sci. Rep. 11, 13943 (2021)) define hierarchical levels of the\nvertices of a weighted simple directed graph. With in-degrees\n`d_j = Σ_i a_ij` and `M = (diag(d) − A)ᵀ`, the forward levels `g` are the\nminimum-norm minimizer of `‖M x − d‖₂`, and the forward democracy coefficient\nis `η_f = 1 − Mean(g_j − g_i)`, the mean over the arcs `i → j` weighted by\n`a_ij`; the backward coefficient `η_b` uses out-degrees. The paper proves\n`η_f, η_b ≥ 0` for weakly connected graphs and `η_f = η_b = 1` for balanced\ngraphs, and states:\n\n> Let $G$ be a weakly connected directed graph. Then the following are true:\n> $\\eta_f(G) \\le 1$ and $\\eta_b(G) \\le 1$; $\\eta_f(G)=\\eta_b(G)=1$ if and only\n> if the graph is balanced.\n\nIssue #11800 fixes the reading: weighted simple directed graphs with\nantiparallel arcs allowed, weak connectivity of the underlying graph, the\nlevels as in the paper's Definition 3.1 without assuming uniqueness, and the\nfirst bullet's forward inequality as the target. The result refutes it.","title":"A directed graph with democracy coefficient above one","url":"https://the-omega-institute.github.io/trureturing-pages/research/moutsinas-2021-democracy-coefficient-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b286486e08d8b0ebb8e5a7f6b386f18ec5a3d59720fd022845d1f8fc913d6861.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:movahedi-2025-diminished-sombor-energy-integer","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Movahedi, *Diminished Sombor matrix, spectral radius, and energy of the\ngraphs*, arXiv:2508.06531v1, Section 5, page 19, states:\n\n> Conjecture 5.1 There does not exist a graph whose diminished Sombor energy is an integer value.\n\nFor vertex degrees $d_i$, the real symmetric matrix has entry\n$\\sqrt{d_i^2+d_j^2}/(d_i+d_j)$ on an edge and zero otherwise.\nIts energy is the sum of the absolute values of all eigenvalues, counted\nwith multiplicity. The conjecture places no edge or connectedness\nrestriction on the graph. Corollary 4.3, page 17, explicitly includes the\nedgeless graph as an equality case. Issue #13380 preregisters the positive\norder, finite simple graph statement and its edgeless counterexample.","title":"Integer diminished Sombor energy of edgeless graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/movahedi-2025-diminished-sombor-energy-integer/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3388929984fcb5968d9e5bb718f95418c624e60705b1180e1c1310b5be007ced.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mub-six-fourth-basis","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Does there exist a family indexed by `Fin 4` of orthonormal bases of\nsix-dimensional complex space such that vectors from distinct bases have\nsquared inner-product magnitude exactly `1 / 6`? Dimension six is known to\nadmit three mutually unbiased bases, while a complete family would contain\nseven. The MUB form of Zauner's conjecture asserts that at most three exist.\n\nThis dossier deliberately anchors Problem 10.2 of arXiv:2410.23997v2: \"Show\nthat no set of four MU bases exists when d = 6.\" It retains Conjecture 2.1 as\nZauner's original formulation, whose affine quantum-design parameters have\n`b = g = 6`, `r = 1`, `lambda = 1 / 6`, and `k = 4`. Problem 10.1 is the\ncomparison statement for `Fin 7`: \"Show that no complete set of seven MU bases\nexists when d = 6.\" Conjecture 1.1 likewise concerns complete MUB families in\ncomposite dimensions that are not prime powers. Nonexistence at `Fin 4` implies\nnonexistence at `Fin 7`, while nonexistence at `Fin 7` does not by itself\nexclude `Fin 4`; the former is therefore the stronger nonexistence statement.\n\nHere \"Zauner's conjecture\" means this MUB upper-bound conjecture. It is not the\nsame conjecture as the identically named SIC-POVM existence conjecture in all\ndimensions.\n\nThe review also reports fourteen mathematically equivalent formulations of the\ncomplete-set existence problem. In dimension six they concern `Fin 7`, not the\n`Fin 4` fourth-basis problem. They are source-side routes, not fourteen\nrepository theorems.","title":"A fourth mutually unbiased basis in dimension six","url":"https://the-omega-institute.github.io/trureturing-pages/research/mub-six-fourth-basis/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ae230a47669e820b194220a43fd184436f2c7fc4decc7ac797564b67b879614.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mukherjee-banerjee-sengupta-sen-2021-bubble-eigenstate-degeneracy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Mukherjee, D. Banerjee, K. Sengupta and A. Sen, *Minimal model for Hilbert space fragmentation with local constraints* (Phys. Rev. B 104, 155117, 2021; arXiv:2106.14897v3), section “Construction of bubble eigenstates”, TeX `fragmentation_prb_v2.tex`, lines 729–731, immediately following Table I:\n\n> While we have not been able to find a closed form expression for the degeneracy of the different bubble eigenstates for an arbitrary $L$, we can still place a lower bound on the degeneracy of such states when $L \\gg 1$.\n\nThe counting prescription is explicit in the same source, lines 670–674:\n\n> The number of bubble eigenstates at an integer energy $E=n_1-n_2$ can be calculated for a given system size from the number of configurations that satisfy the hard core constraint of at least two consecutive $\\downarrow$ spins separating any of the $n_1$ $X$ and $n_2$ $Y$ units on the periodic one-dimensional lattice.\n\nHere $X$ and $Y$ are three-site units with energies $+1$ and $-1$, respectively (the source's construction, lines 634–647). Issue [#26367](https://github.com/the-omega-institute/trureturing/issues/26367) fixes the combinatorial reading: sites are $\\mathrm{Fin}\\,L$, a centre set $B$ is admissible when every ordered pair of distinct centres has clockwise separation at least five, and a subset $x\\subseteq B$ specifies the $X$ labels, with the other centres labelled $Y$. Its energy is $2|x|-|B|$. Translations are counted separately, and the empty placement contributes one configuration at energy zero. Let $D_B(L,E)$ denote this count, defined by `bubbleCount`.\n\nFor $1\\le n\\le\\lfloor L/5\\rfloor$, write\n\n$$C(L,n)=\\left(L\\binom{L-4n-1}{n-1}\\right)\\mathbin{\\mathrm{div}}n,$$\n\nwhere $\\mathrm{div}$ is natural-number division, and put\n\n$$A(n,E)=\\begin{cases}\\displaystyle\\binom{n}{(n+E)/2},&n+E\\text{ is even and }|E|\\le n,\\\\0,&\\text{otherwise}.\\end{cases}$$\n\nThe module proves `claim`: for every natural $L\\ge5$ and every integer $E$,\n\n$$D_B(L,E)=[E=0]+\\sum_{n=1}^{\\lfloor L/5\\rfloor}C(L,n)A(n,E).$$\n\nHere $[E=0]$ is one when $E=0$ and zero otherwise; the equality is evaluated in the integers by casting the natural count and placement coefficients. The guarded index $(n+E)/2$ is a nonnegative integer. This settles the stated closed-form gap for labelled configurations; identifying these configurations with eigenstates of $H_3$ uses the source's construction and is not part of the Lean theorem.","title":"A closed formula for the bubble-eigenstate configuration count","url":"https://the-omega-institute.github.io/trureturing-pages/research/mukherjee-banerjee-sengupta-sen-2021-bubble-eigenstate-degeneracy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/714c627391d0d2316400ae75a11281c1bf8a405ffc37ba04f5d86a7994fb34c6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mukherjee-ganguly-2025-non-unital-network-condition-implication","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Mukherjee and N. Ganguly, *Network nonlocality breaking channels*, arXiv:2510.26417v1 (2025), Section IV.B, `Network_Nonlocality.tex`, lines 733–734:\n\n> Comparing Eq.(\\ref{resnon1}) with that of the conditions provided by Eq.(\\ref{nu3}) over the channel $\\mathcal{N}_{NU},$ it is observed that there exist no region in parameter space for which Eq.(\\ref{resnon1}) is violated whereas Eq.(\\ref{nu3}) is satisfied. However, a formal proof of the same is beyond the scope of present work. We thus provide the following conjecture:\n\nConjecture 1 reads:\n\n> Any non-unital channel from $\\mathcal{N}_{NU}$( Eqs.(\\ref{nu1},\\ref{nu2})) acts as a non $n$-locality breaking channel in $\\mathbf{N}_{lin}.$\n\nThe quotations retain the source's TeX reference labels. The corresponding PDF passage is on page 12; `nu3` is Eq. (6), `resnon1` is Eq. (38), and `nu1`, `nu2` are Eqs. (4), (5). The source TeX has MD5 `6bb7806236c7d6b0bcaa73d3eed0bc28`.\n\nIssue [#20159](https://github.com/the-omega-institute/trureturing/issues/20159) fixes the statement to be settled: for every real $t,\\lambda_1,\\lambda_3$, the three conditions\n\n$|\\lambda_1|\\leq1$, $|\\lambda_3|+|t|\\leq1$, and $\\lambda_1^2+t^2\\leq(1-|\\lambda_3|)^2$\n\nimply both\n\n$(|t|+|\\lambda_3|)^2+\\lambda_1^2\\leq1$\n\nand\n\n$2t^2\\lambda_1^2+\\lambda_1^4+(|t|+|\\lambda_3|)^4\\leq1$.\n\nThis parameter implication is proved. Applying it to channels requires the source's Theorem 3, which is not formalized here. The source passage and parameter correspondence are also recorded in [the literature note](../Library/QuantumContext/mukherjeeganguly2025network.md).","title":"The non-unital channel conditions imply both network bounds","url":"https://the-omega-institute.github.io/trureturing-pages/research/mukherjee-ganguly-2025-non-unital-network-condition-implication/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59dff8d458921206af6b5b9ccb0c805c32bacc2cd16d83d08dda044e82bfa39f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mukhopadhyay-sazim-pati-2018-qutrit-magic-coherence-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Mukhopadhyay, S. Sazim and A. K. Pati, *Coherence makes quantum systems 'magical'*, J. Phys. A 51, 414006 (2018), arXiv:1801.04807v3, Eq. (18), stated from numerical simulation:\n\n> The following condition on quantum coherence, quantified via the $l_1$-norm, and 'magic', quantified by the sum negativity, holds for qutrit pure states $M_{SN}[|\\psi\\rangle] \\geq \\frac{C_{l_{1}}[|\\psi\\rangle]}{2}\\sqrt{1- \\frac{C_{l_{1}}[|\\psi\\rangle]}{2}}$\n\nHere $M_{SN} = \\sum_u |W_u| - 1$ for the discrete Wigner function $W$ of the paper's Eq. (12), and $C_{l_1}$ is the sum of the moduli of the off-diagonal entries. The verbatim statements are in [the literature note](../Library/QuantumChannels/mukhopadhyay2018coherencemagic.md).\n\nIssue [#18729](https://github.com/the-omega-institute/trureturing/issues/18729) fixes the reading: the bound for every normalized x ∈ ℂ³ with ρ = x x†. This is proved.","title":"A coherence lower bound on the magic of pure qutrits","url":"https://the-omega-institute.github.io/trureturing-pages/research/mukhopadhyay-sazim-pati-2018-qutrit-magic-coherence-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/375c626f0f14f32b05d9cbe3b4faddb444e2e3520cf4fb1d2b1cc006309aac42.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:mulas-zhang-zucal-2023-non-backtracking-gap-min-degree","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Mulas, D. Zhang and G. Zucal, *There is no going back: Properties of the\nnon-backtracking Laplacian*, arXiv:2303.00373v2, Section 5, Conjecture 5.3:\n\n> If $G$ is not the cycle graph, then $\\varepsilon\\leq\\frac{1}{\\delta-1}$.\n\nThe source assumes a finite simple graph of minimum degree $\\delta\\ge2$.\nIts transition matrix is $P=\\mathcal D^{-1}\\mathcal A$, and its\nnon-backtracking Laplacian is $\\mathcal L=I-P$. The gap from $1$ is\n$\\varepsilon=\\min_{\\lambda\\in\\sigma(P)}|\\lambda|$.","title":"Non-backtracking Laplacian gap and minimum degree","url":"https://the-omega-institute.github.io/trureturing-pages/research/mulas-zhang-zucal-2023-non-backtracking-gap-min-degree/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cab0e5fea354fadee7b66a843945e97f4a4619e095cd33e0945b051be5c8288e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:muller-saniga-giorgetti-de-boutray-holweck-2022-linear-doily-odd-negative-lines","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Muller, M. Saniga, A. Giorgetti, H. de Boutray and F. Holweck, *Multi-qubit doilies: enumeration for all ranks and classification for ranks four and five*, arXiv:2206.03599v2 (J. Comput. Sci. 64, 101853, 2022), §5.4, “Specific behavior of linear doilies”, state:\n\n> From our results on three-, four- and five-qubit cases it follows that a linear doily (a) always features an odd number of negative lines, and (b) does not share a perp-set with the distinguished quadric.\n\n> We conjecture that Property (a) holds for any number of qubits $N \\geq 2$, but we have not yet found of proof of it; we surmise that it has something to do with the fact that a linear doily is ``squeezed'' into a PG$(3,2)$, compared to a quadratic doily that enjoys more degrees of freedom being stretched out in a PG$(4,2)$.\n\nThese quotations are from `xiv-MSGDH2.tex`, lines 2255–2257 and 2259–2263, downloaded from the [v2 source](https://arxiv.org/src/2206.03599v2), MD5 `ba1ee66980eb180ebdcbe7fcd5d8a411`. Definitions and source locators are in [the literature note](../Library/QuantumContext/mullersanigagiorgettideboutrayholweck2022doilies.md).\n\nIssue [#20146](https://github.com/the-omega-institute/trureturing/issues/20146) fixes the reading: for every $N\\ge2$ and every four-dimensional subspace $U\\le\\mathbb F_2^{2N}$ with nondegenerate restricted symplectic form, an odd number of its isotropic triples $\\{u,v,u+v\\}$ have their three canonical, unsigned Pauli observables multiplying to $-I_{2^N}$. This is proved.","title":"Every linear N-qubit doily has an odd number of negative lines","url":"https://the-omega-institute.github.io/trureturing-pages/research/muller-saniga-giorgetti-de-boutray-holweck-2022-linear-doily-odd-negative-lines/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a23d7487303cbcd4a76851da401c8337b6194222a9c769f11c4cb43951d0128d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:muney-gilbreath-component-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Leila Muney, *Holes in Valid-Extension Sets of Finite Gilbreath Sequences*,\narXiv:2606.23721v2, Section 16(4), asks:\n\n> Is max_{S∈G_n} #comp(K_S)=2^(n-4) for all n≥5? Verified for n≤10.\n\nThe integer triangle is s_a^0=s_a and\ns_a^b=|s_(a+1)^(b-1)-s_a^(b-1)| at every valid index. A prefix is Gilbreath\nwhen every positive-row left entry is one. The family G_n consists of every\nstrictly increasing Gilbreath integer prefix of length n beginning with two and\nthree.\nK_S contains every integer k for which appending k is Gilbreath; no increasing\ncondition is imposed on this appended integer. Components are literal maximal\nruns of common difference two in the finite one-parity extension set.","title":"Finite Gilbreath extension component maximum","url":"https://the-omega-institute.github.io/trureturing-pages/research/muney-gilbreath-component-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbe29577c0bd78d132529e31fe78207c3397e9691fcd1c7c1bc0a3e7adf4bfb8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:murshid-gupta-russo-bandyopadhyay-2026-no-repetition-unambiguous-discrimination","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Murshid, T. Gupta, V. Russo and S. Bandyopadhyay, *Quantum nonlocality without entanglement and state discrimination measures*, arXiv:2506.20560v3; Quantum 10, 2174 (2026), consider the uniform ensemble $\\mathcal E_{N,k}$ of ordered products of $k$ distinct members of $N$ normalized states $\\psi_i\\in\\mathbb C^N$ with $\\langle\\psi_i,\\psi_j\\rangle=s\\in(0,1)$ for $i\\ne j$.\n\nTheir conjecture appears in the TeX source `revised_02.tex`, lines 970–984, subsection “Conjecture for multipartite systems”. The quotations retain the source macro `\\E`, meaning $\\mathcal E$, and its equation reference; the displays are written inline.\n\n> For the family of ensembles of multipartite pure product states $\\E_{N,k}$ given by Equation \\eqref{eq:gensemble} the following holds for all $N \\geq 3$ and $2 \\leq k \\leq N$\n\n> $p_{_{L}}^{me}\\left(\\E_{N,k}\\right) \n> $p_{_{L}}^{ud}\\left(\\E_{N,k}\\right) =p_{_{G}}^{ud}\\left(\\E_{N,k}\\right).$\n\n> Moreover, the optimal unambiguous discrimination probability of $\\E_{N,k}$ is given by\n\n> $p_{_{G}}^{ud}\\left(\\E_{N,k}\\right) = \\begin{cases} (1-s)^k, & \\text{ when } k < N \\\\ (1-s)^{N-1}(1+\\left(N-1\\right)s), & \\text{ when } k=N. \\end{cases}$\n\nT. Gupta, S. Murshid and S. Bandyopadhyay, *Unambiguous discrimination of sequences of quantum states*, arXiv:2402.06365v2; Phys. Rev. A 109, 052222 (2024), pose the earlier no-repetition question in `GMBjournal-revision-02.tex`, lines 883–903:\n\n> In our scenario, as well as in more general ones (which we could not solve), repetitions of states are allowed in a sequence. The third problem considers the situation where it is not. Restrict $k Now assume that the inner products of the states $\\left|\\psi_{i}\\right\\rangle$ are equal and positive, say, $s>0$. Then $S_{N,k}^{\\prime}\\subset S_{N,k}$, where $S_{N,k}$ is the set of sequences considered in this paper. Under these restricted conditions, numerical experiments ($N=3$, $k=3$) still suggest that the optimal probability for distinguishing between the elements of $S_{N,k}^{\\prime}$ unambiguously once again obeys $(1-s)^{k}$. However, for arbitrary values of both $N$ and $k$, whether one could benefit from collective measurements in this scenario is an interesting problem to consider in the future.\n\nThe source definitions and locators are also recorded in [the 2026 literature note](../Library/QuantumContext/murshidguptarussobandyopadhyay2026nonlocality.md) and [the 2024 literature note](../Library/QuantumContext/guptamurshidbandyopadhyay2024sequences.md).\n\nIssue [#21233](https://github.com/the-omega-institute/trureturing/issues/21233) fixes the settlement as the unambiguous clauses of the conjecture, together with the 2024 question for $2\\le k Cooling maps are equivalent to low-temperature thermal operations, with regard to the feasibility of state transitions.\n\nThe next sentence distinguishes state-transition feasibility from equality of channel sets:\n\n> Note that this could be true even if the set of cooling maps is strictly larger than that of thermal operations---there could still be a thermal operation achieving every state transition that is possible through cooling maps.\n\nIssue [#26067](https://github.com/the-omega-institute/trureturing/issues/26067) fixes the following reading of the cooling-to-thermal direction. For every finite dimension $d$, every strictly increasing real energy spectrum $E$ with distinct positive gaps, and every pair of density matrices $\\rho,\\sigma$, a cooling-map transition $\\rho\\mapsto\\sigma$ has a low-temperature thermal realization. Here:\n\n- Density matrices are positive semidefinite complex matrices of trace one (`IsDensity`).\n- A cooling map has a finite Kraus family, each member diagonal or of the form $c|j\\rangle\\langle k|$ with $j For $S_h=\\langle a,ah,ah+d,ah+2d,\\ldots,ah+kd\\rangle$, $\\Delta(S_h)=E(e,d)$.\n\nSection 3 assumes $\\gcd(a,d)=1$, $11$, and puts $e=h-1$. All four parameters are positive integers. Every displayed generator, including the redundant $ah$, is retained as a separate nonnegative factorization coordinate. The delta set is the union over represented natural numbers of positive differences between consecutive distinct lengths. The subtractive Euclidean algorithm starts at $(e,d)$; its final equal pair may be repeated without adding visited positive coordinates. These interpretations resolve the unbound $s$, omitted endpoint order and malformed initialization in Definitions 3.1–3.2, as fixed by [#23185](https://github.com/the-omega-institute/trureturing/issues/23185).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| All positive $a,h,d,k$ | The four natural-number binders and four positivity premises of `claim` | No parameter restriction is omitted. |\n| $\\gcd(a,d)=1$, $11$ | `Nat.Coprime a d` and the three strict inequalities of `claim` | Literal restrictions. |\n| $(a,ah,ah+d,\\ldots,ah+kd)$ | `generators`, indexed by `Fin (k + 2)` | The $ah$ coordinate is retained. |\n| Nonnegative coordinate factorizations with weighted sum $n$ | `Z` | The complete coordinate equation is retained. |\n| Length is the coordinate sum; $L(n)$ contains all such lengths | The image in `L` | No minimal-generator substitution. |\n| Positive successive length differences, over represented $n$ | `delta` | Strict endpoint order and no intermediate member are explicit. |\n| Subtractive Euclidean states and visited positive coordinates | `euclidStep`, `euclidState`, `E` | The initial and terminal coordinates are included. |\n| $\\Delta(S_h)=E(h-1,d)$ | The equality in `claim`; `result` proves its universal assertion false | The witness is $(3,2,2,2)$. |\n\nThe displayed finite sums use Mathlib directly; no separate coordinate-sum alias is introduced. The implementation carrier is a product of a natural number and a finite list. Every public definition displays its complete body in the Blueprint. There are no additional source-independent hypotheses in the public claim.","title":"Narayan's expanded-generator delta-set conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/narayan-delta-set/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46618868bb93722c1214413c987f403dff33962d8288cd1e1dcf1cb003741002.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nathanson-2026-problem-12-affine-invariance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Melvyn B. Nathanson, *Problems and results on intersections of product sets and sumsets in semigroups*, arXiv:2604.04781v2, Problem 12, p. 21, asks:\n\n> Let $(A_q)$ and $(A'_q)$ be asymptotically strictly decreasing sequences of sets of integers such that $A'_q=sA_q+t$ for all $q\\in\\mathbb N$ and some $s,t\\in\\mathbb Z$ with $s\\notin\\{0,\\pm1\\}$. Does this imply $H(A_q)=H(A'_q)$?\n\nThe paper takes $\\mathbb N=\\{1,2,3,\\ldots\\}$, defines $A=\\bigcap_q A_q$ and $H(A_q)=\\{h\\in\\mathbb N:hA=\\bigcap_q hA_q\\}$, and uses $hB$ for the $h$-fold sumset. Asymptotically strict decrease means $A_{q+1}\\subseteq A_q$ at every index, with infinitely many strict changes. The answer to Problem 12 is affirmative.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Two sequences of integer sets indexed by positive integers | `A A' : ℕ+ → Set ℤ` in `claim` | The paper's positive-natural indexing is literal. |\n| Both sequences asymptotically strictly decreasing | `AsymptoticallyStrictlyDecreasing A` and its primed counterpart | The definition retains adjacent inclusion and infinitely many unequal neighbours. |\n| The $h$-fold sumset | Pointwise `h • B` | This is repeated set addition, with independent summands. |\n| Positive sum lengths and the sum-intersection equality | `H` | Its natural-number carrier is restricted by `0 < h`. |\n| $A'_q=sA_q+t$ at every index | The affine `Set.image` hypothesis in `claim` | It is the literal set $\\{sa+t:a\\in A_q\\}$. |\n| Integer $s,t$ and $s\\notin\\{0,\\pm1\\}$ | Integer parameters and the exclusion hypothesis in `claim` | All three excluded values are retained. |\n| $H(A_q)=H(A'_q)$ | Conclusion of `claim`; `result : claim` | No source premise is dropped from the delivered statement. |\n\nEvery explicit binder and conjunct comes from these clauses. The Boolean-free formulas in the Scribe display the three definition bodies and the complete proposition of the settling theorem.","title":"Nathanson's Problem 12: affine invariance of sum intersection sets","url":"https://the-omega-institute.github.io/trureturing-pages/research/nathanson-2026-problem-12-affine-invariance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac7880aa4abceebc54c0b7acace3be08f193ee11d1fe0cd70bc5e88f10994bd0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nathanson-2026-problem9-torsion-nontrivial-spectrum-existence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Melvyn B. Nathanson, *Problems and results on intersections of product sets and sumsets in semigroups*, arXiv:2604.04781v2, Section 4, Problem 9, asks:\n\n> Let G be an infinite torsion group. Describe the product intersection sets H_Q(G). Must there exist X in H_N^*(G) with X != {1} and X != N?\n\nThis dossier anchors exactly the final existence question. The preceding classification\nrequest is outside its selected unit. The source uses N = {1,2,3,...}, ordered h-fold\nproducts with repetitions, and H_N^*(G) for the spectra of asymptotically strictly decreasing\nsequences of subsets of the same group G. Empty common intersections are allowed.\n\n### Source correspondence\n\n| Source clause | Canonical counterpart |\n| --- | --- |\n| Every infinite torsion group | `result` quantifies over `{G : Type u} [Group G] [Infinite G]`; it proves the statement for all infinite groups, so every torsion instance is included. |\n| Arbitrary cardinality and group law | There is no `Countable G`, `CommGroup G`, bounded-exponent or subgroup assumption. |\n| Positive-indexed subsets of G | `A : PNat → Set G`; `q+1` is positive successor. |\n| Adjacent inclusion and infinitely many strict changes | `ASD A` is `(∀ q, A (q+1) ⊆ A q) ∧ {q | A q ≠ A (q+1)}.Infinite`. |\n| Arbitrarily late strict changes | Infinitude of that subset of positive naturals implies `∀ q₀, ∃ q ≥ q₀, A (q+1) ≠ A q`; each constructed family is in fact strictly decreasing at every index. |\n| Ordered h-fold products, repetitions allowed | Mathlib pointwise set multiplication and natural powers; both factor orders and all triple support and sign patterns are covered. |\n| Core A and full positive spectrum | `spectrum A = {h : Nat | 0 < h ∧ (⋂ q, A q)^h = ⋂ q, (A q)^h}`. |\n| X in H_N^*(G) | Take `X = spectrum A`; `ASD A` supplies the defining membership. |\n| X differs from {1} and from N | Both inequalities in `result` concern the complete set `spectrum A`, with exponent zero excluded. |\n\nThe exact torsion-specialized, arbitrarily-late formulation is an application of this\nsingle theorem and receives no additional retained theorem or resolution claim.","title":"Nathanson Problem 9: existence of a nontrivial proper spectrum","url":"https://the-omega-institute.github.io/trureturing-pages/research/nathanson-2026-problem9-torsion-nontrivial-spectrum-existence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2087f285527b79ce507c1784425045cc16a10031704ee3a4f971c0c5df3b1211.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nathanson-additive-h-bases-problem-12","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 12(2) of arXiv:2605.26425v3 asks whether, for integers `h >= 2` and\n`k >= 2`, every `k`-element integer set `A` with a negative element satisfies\n`ell_h(A) < n-flat_h(k)` whenever `ell_h(A)` is defined.","title":"Refutation of Nathanson's additive h-basis strict inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/nathanson-additive-h-bases-problem-12/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/835f142161bd72210b50ab32f643cc9d3786cf2086bfb32dd553a70cb6f6b34b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:negami-2026-jones-temperley-lieb-jordan-obstruction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Negami, *Quantum gates from the middle convolution of twisted Burau\nrepresentations*, arXiv:2610.00293v1, builds braid representations for quantum\ngates by the Katz–Long–Moody construction: a seed representation of\n$F_n\\rtimes B_n\\cong B_{1,n}$ on $V$ gives operators $S_i$ on $V^{\\oplus n}$,\n$S_i=s_i^{\\oplus n}\\,\\mathrm{diag}(I,R_i,I)$ with\n$R_i=\\begin{pmatrix}0&g_i\\\\I&I-g_{i+1}\\end{pmatrix}$, and a quotient by\n$K+L_\\lambda$, $K=\\bigoplus_j\\mathrm{Ker}(g_j-I)$. Remark 6.7(b) tests seeds\nfrom the endpoint sectors of the Jones–Temperley–Lieb path representations\n($\\rho(\\sigma_i)=\\alpha_\\ell I+\\alpha_\\ell^{-1}E_i$, $\\alpha_\\ell=ie^{-\\pi\ni/(2\\ell)}$) for $n=3$, $\\ell\\in\\{4,5,6,7,8,10,12,16\\}$ and $n=4$,\n$\\ell\\in\\{5,7\\}$: in sectors with $K=0$ finite-precision rank tests suggest\nnon-trivial Jordan blocks of the ambient $S_i$, which \"by themselves …\nestablish neither non-semisimplicity nor unboundedness\". Problem 6.8 ends:\n\"Also prove the Jordan obstruction for the Jones--Temperley--Lieb\nendpoint-sector seeds\". The verbatim statements are in\n[the literature note](../Library/QuantumStates/negami2026middle.md).\n\nIssue [#13581](https://github.com/the-omega-institute/trureturing/issues/13581)\nfixes the reading: the obstruction is a Jordan chain of length two of $S_1$ (all\n$S_i$ are conjugate to it in the braid group) in a sector with $K=0$, with the\nstandard path-model operators $E_i$ (which the source names without writing\ntheir entries), and it is proved for every $n\\ge3$, $\\ell\\ge4$ and every\nendpoint sector of dimension at least two.","title":"The Jordan obstruction for the Jones–Temperley–Lieb endpoint-sector seeds","url":"https://the-omega-institute.github.io/trureturing-pages/research/negami-2026-jones-temperley-lieb-jordan-obstruction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/882520bcb9c50a72a61fabcb972b33d5bb2723e0479fa1e2890e3eaa1d307593.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:negari-2610-01860-high-degree-constant-sharpness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A.-R. Negari, F. Salek, Z. Zimborás, A. Harrow, P. Hayden and J. Eisert,\n*Approximation theorems for fermionic Gaussian states*, arXiv:2610.01860v1\n(2026), Section VI, printed page 26, state:\n\n> It remains open, however, whether the constants in the resulting high-degree energy and free-energy estimates are optimal, since their derivation also uses edge averaging, Cauchy–Schwarz, and only the operator-norm normalization of the interactions.\n\nThe source estimates are Theorem III.1, equation (44), and Theorem III.5,\nequation (79), both with prefactor $\\sqrt{2m/D}$. The formal claim asks whether,\nfor every $m\\ge1$, every $c<1$ and every degree cutoff $D_0$, a finite\nregular graph and admissible quadratic fermionic edge interactions attain the\nenergy value $\\sqrt{2m/D}$ and have a finite positive inverse temperature whose\nfree-energy gap is larger than $c\\sqrt{2m/D}$.","title":"Sharpness of the high-degree fermionic Gaussian constants","url":"https://the-omega-institute.github.io/trureturing-pages/research/negari-2610-01860-high-degree-constant-sharpness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fcbdf72ec5705263edb6053b1955f6adaba928a157146a01ed74aa49071e37b6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nepomechie-raveh-2024-qutrit-dicke-pruning-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rafael I. Nepomechie and David Raveh, *Qudit Dicke state preparation*, Quantum Information and Computation 24 (2024), 37–56, arXiv:2301.04989v4, §4.5, “Simplifying the circuit”, propose occupation-dependent ranges for deleting gates from their qutrit circuit. The v4 source `qdit_v3.tex` (md5 `fcb86a69ac9cdb2a7353251021ea3983`), line 1006, introduces the formulas labelled `W3genericsimple`, `W3specialsimple` and `kkdefs`:\n\n> we conjecture that, similarly to the $d=2$ case \\eqref{W2explicit2},\n\nLine 1031 states:\n\n> We have not yet succeeded to prove the result \\eqref{W3genericsimple}-\\eqref{kkdefs}, which we found through experimentation.\n\nThe reading fixed in issue [#21375](https://github.com/the-omega-institute/trureturing/issues/21375) is the following universal preparation identity: for every natural $n\\ge2$ and all natural occupations $k_0,k_1,k_2$ with $k_0+k_1+k_2=n$, the specified pruned circuit maps $|\\mathrm e(\\vec k)\\rangle=|0\\rangle^{\\otimes k_0}|1\\rangle^{\\otimes k_1}|2\\rangle^{\\otimes k_2}$ to $|D^n(\\vec k)\\rangle$.\n\nMore precisely, put $l_1=k_1+k_2$, $l_2=k_2$, and $\\tilde k=k_2$ if $k_0=0$, otherwise $\\tilde k=\\max(k_1,k_2)$. The ranges in that reading are inclusive:\n\n- The single-index product retains $\\mathrm I_{m,l}$ for $\\max(\\tilde k+m-n,1)\\le l\\le\\min(\\tilde k,m-1)$, applied in increasing $l$.\n- The double-index product retains $\\mathrm{II}_{m,a,b}$ for $\\max(l_2+m-n,1)\\le b\\le\\min(l_2,m-2)$ and $\\max(l_1+m-n,b+1)\\le a\\le\\min(l_1,m-1)$, applied with increasing $b$ and increasing $a$ within each $b$.\n- Each $\\mathcal W_m$ applies the single-index product before the double-index product; $\\mathcal U_n$ applies the blocks in time order $n,n-1,\\ldots,2$, each on the leftmost $m$ qutrits. Empty ranges contribute the identity.\n\nThese are `prunedTilde`, `prunedTildeTilde`, `prunedW` and `prunedU`. The universal equality is `claim`; it is refuted. The definitions and source locators are also documented in [the literature note](../Library/QuantumContext/nepomechieraveh2024quditdicke.md).","title":"The proposed pruning ranges do not prepare every qutrit Dicke state","url":"https://the-omega-institute.github.io/trureturing-pages/research/nepomechie-raveh-2024-qutrit-dicke-pruning-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff1aa450f4d16e57a18c6663e8d3584f457390a3944e5ddc4dbadfd3c1fc307a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nersissian-2026-rule84-mod-three-center-column","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Tigran Nersissian, *Diagonal Bases and Diagonal Periods of Elementary\nCellular Automata*, arXiv:2609.25078v1, Remark 6, page 12, states:\n\n> (The outstanding Rule 84 case) The polynomial of Rule 84 is (a + b + ab)(1 + c). Modulo three, its center column begins 1, 1, 2, 2, 1, 2, 2, … and follows the repeating block (1, 2, 2) after the first entry for the tested times 0 ≤ t < 2048. All center values are also units modulo 9 and 27 on that range, as Corollary 2 predicts. An all-time proof of the observed pattern is not supplied here. If Rule 84 is universal modulo three, it is universal at every power of three; three could not be the sole exceptional modulus.\n\nSection 19, page 42, states:\n\n> The classification reduces the remaining single-seed universality question to the eight rules in E at odd primes. The observed Rule 84 pattern modulo three requires an invariant or an all-time recurrence proof. A finite nonzero prefix is insufficient.\n\nThis dossier anchors exactly that all-time pattern. Definition 1, page 4,\nfixes the canonical positive-coefficient polynomial lift and the synchronous\ninteger-lattice orbit. For the modulus 3 and the single seed at zero, let\n`A 0 x = if x = 0 then 1 else 0` and\n`A (t+1) x = (a+b+a*b)*(1+c)`, where\n`a = A t (x-1)`, `b = A t x` and `c = A t (x+1)` in `ZMod 3`.\nThe statement is\n`A 0 0 = 1 ∧ ∀ s : ℕ, A (3*s+1) 0 = 1 ∧ A (3*s+2) 0 = 2 ∧ A (3*s+3) 0 = 2`.\nIt includes every natural time, with no time cutoff or extra hypothesis.","title":"The Rule 84 center column modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/nersissian-2026-rule84-mod-three-center-column/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/575850a5eae0770bb8812e92d01e5d82be3dbf91d7952fecdec0b20abde32462.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nikandish-subspace-orthogonality-clique","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 4 of the cited source asks: \"Determine the exact value of\nomega(O_n*) for general n.\" For every integer `n >= 1`, the vertices\nare all nonzero linear subspaces of `F_2^n`. Distinct vertices are adjacent\nwhen every vector in one is orthogonal to every vector in the other under\nthe standard symmetric, nonalternating dot form.","title":"Nikandish Problem 4.2","url":"https://the-omega-institute.github.io/trureturing-pages/research/nikandish-subspace-orthogonality-clique/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/897b43c027ac8dcbd72239729c780a0875a90a1c6120e1e189b3a5451d727b6f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nikiforov-2015-ky-fan-asymptotic-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source is V. Nikiforov, *Extrema of graph eigenvalues*,\narXiv:1502.00359v1, §4, Conjecture 4.7. Its definition is:\n\n> ξk (n) = max λ∗1 (G) + · · · + λ∗k (G)\n\nThe maximum is over graphs of order n. Starred eigenvalues denote\nsingular values, equivalently the absolute eigenvalues of the real\nsymmetric adjacency matrix, in decreasing order. The text preceding\nthe conjecture states:\n\n> Theorem 4.3 does not shed any light on the case when k is not an exact square, so we suggest the following concrete conjecture.\n\nConjecture 4.7 states:\n\n> There exist infinitely many integers k such that lim_{n→∞} ξk (n)/n < (1+√k)/2.\n\nThe repeated statement is Conjecture 3.20 of *Beyond graph energy: norms\nof graphs and matrices*, arXiv:1510.02850v2, §3.3.","title":"Nikiforov: asymptotic graph Ky Fan gaps","url":"https://the-omega-institute.github.io/trureturing-pages/research/nikiforov-2015-ky-fan-asymptotic-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e5566a9320431eb1b3643f737bdf9d62bc25b220b393a6743486ee88cbfc4124.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nikolov-2023-snake-polynomial-chebyshev-expansion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Kryvonos (ed.), *Open problems on polynomials, their zero distribution and related questions: 2023*, arXiv:2312.07754v1, section “Snake polynomials with positive Chebyshev expansions”, states:\n\n> Consider a majorant µ ∈ C[−1, 1], µ(x) ≥ 0, x ∈ [−1, 1]. If there exists P ∈ Pn, P ≠ 0, such that −µ(x) ≤ P(x) ≤ µ(x), x ∈ [−1, 1], then there exists a unique (up to orientation) polynomial ωµ ∈ Pn called snake polynomial associated with µ, which oscillates most between ±µ.\n\n> The n-th snake polynomial ωµ is uniquely determined by the following properties: a) |ωµ(x)| ≤ µ(x) x ∈ [−1, 1]; b) There exists a set δ* = (τi*)_{i=0}^n, 1 ≥ τ0* > · · · > τn* ≥ −1, such that ωµ(τi*) = (−1)^i µ(τi*), i = 0, . . . , n\n\n> Conjecture (G.N.) If µ ≥ 0 is a continuous even convex function in [−1, 1], then the associated with µ snake polynomials have non-negative expansion in the Chebyshev polynomials of the first kind.\n\nThe assertion is repeated as Conjecture 7.1 (Geno Nikolov) in *Open problems UP24*, arXiv:2504.04845, §7. Here $\\mathcal P_n$ denotes polynomials of degree at most $n$, and $T_k$ denotes the first-kind Chebyshev polynomial.","title":"Nonnegative Chebyshev expansions of snake polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/research/nikolov-2023-snake-polynomial-chebyshev-expansion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b7f7d74057b486eb20a668811fa68a5d3d9fd73c4795e394d7f026b923512840.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nikolov-savov-2023-renewal-minorant-maximality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Nikolov and M. Savov, *Properties and conjectures regarding discrete renewal sequences*, arXiv:2307.00545v2, §3, Conjecture 3.7, states:\n\n> For any k ≥ 3, Q_k is a maximal element in 𝒜_k and it is largest in 𝒜̂_k.\n\nThe formal claim settles the first clause:\n\n```text\nclaim := ∀ k ≥ 3, Q_k ∈ 𝒜_k ∧ ∀ P ∈ 𝒜_k, Q_k ≺ P → P = Q_k.\n```\n\nHere `A_k` is the simplex of nonnegative first `k−1` masses with sum at most one, `u` is the renewal sequence of Eq. (2.2), `Q_k` is the product of partial sums in Eq. (2.5), and `𝒜_k` is the degree-bounded class of polynomial lower bounds in Eq. (2.7). The hatted class is not used.","title":"Refutation of Nikolov–Savov Conjecture 3.7 at k = 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/nikolov-savov-2023-renewal-minorant-maximality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9184839ee911f6d220ea02e1c6f28bb15c1b7ab000002856fdcacc96bda75fb2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nilforoushan-parvaresh-conjecture-8-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 8.1 of Arman Nilforoushan and Farzad Parvaresh, *Improved upper\nbound on the number of distinct k-decks for any k and alphabet size by counting\nthe independent parameters*, arXiv:2609.23106v1, states:\n\n> D_{q,k}(n) = Theta(n^{E_q(k)}) for every fixed q >= 2, k >= 1.\n\nThe paper defines `D_{q,k}(n)` as the number of distinct exact `k`-decks of\nlength-`n` words over a `q`-letter alphabet and defines\n`E_q(k) = sum_{j=1}^k j L_q(j) - 1`, where `L_q(j)` is the number of actual\nlength-`j` Lyndon words over that alphabet. Preregistration issue #9512 records\nthis complete target, not one of the cases already proved in the source.\n\nThe formal target quantifies over an arbitrary finite linearly ordered type\n`A`, assumes exactly `2 <= Fintype.card A`, takes every natural `k` with\n`1 <= k`, and proves `Real.IsTheta` at `Filter.atTop` for the real-valued\ncardinality of `exactKDeckImage A k n`. Its exponent is\n`weightedLyndonExponent A k - 1`, with\n`weightedLyndonExponent A k = sum_{i=0}^k i * actualLyndonCount A i` and\n`actualLyndonCount A i` the cardinality of the subtype of actual Lyndon words\nof length `i`. The zero term vanishes, so this is the source exponent. There is\nno free proxy parameter and no extra hypothesis on `n` or on `k`.","title":"Nilforoushan-Parvaresh Conjecture 8.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/nilforoushan-parvaresh-conjecture-8-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/efbc52bfebdad567181218c20059aac11c40f032d39b7953b8babc27407268f1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nishijima-nakata-2024-gdnn-inclusion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The cited article, arXiv:2204.12119v4, Section 5.1, Conjecture 5.1, asks for\nthe inclusion of the NN-type generalized doubly nonnegative cone in the\nBD-type cone for products of a nonnegative orthant and second-order cones.\nThe statement reads verbatim: “If K is a direct product of a nonnegative\northant and second-order cones, DNN^NN(K) ⊆ DNN^BD(K) holds.” The article\nnotes that the case of a single second-order cone follows from its\nProposition 4.12 and that the general inclusion relationship is unknown.\n\n### Formal reading\n\nLet $p,r\\ge0$ and $k_j\\ge1$ for $1\\le j\\le r$, and put\n\n$$K=\\mathbb R_+^p\\times\\prod_{j=1}^r\n\\{(t,z)\\in\\mathbb R\\times\\mathbb R^{k_j}:\\|z\\|_2\\le t\\}.$$\n\nLet $q$ square each orthant coordinate and send each Lorentz block $(t,z)$\nto $(t^2+\\|z\\|_2^2,2tz)$. Write $\\Sigma_{n,4}$ for the convex hull of\nsquares of homogeneous quadratic polynomials. With the standard coordinate\npairing and $n=p+\\sum_j(k_j+1)$, define\n\n$$\\mathcal K_{\\mathrm{NN},0}(K)\n=\\{A\\in\\mathbb S^n:q(x)^\\mathsf{T}Aq(x)\\in\\Sigma_{n,4}\\}.$$\n\nEvery $X\\in\\mathbb S^n$ satisfying\n$\\operatorname{tr}(XA)\\ge0$ for every $A\\in\\mathcal K_{\\mathrm{NN},0}(K)$\nis positive semidefinite and satisfies $Xs\\in K$ for every extreme ray\ngenerator $s$ of $K^*$. Thus $\\mathrm{DNN}^{\\mathrm{NN}}(K)\\subseteq\n\\mathrm{DNN}^{\\mathrm{BD}}(K)$.\n\n| Source object | Lean counterpart | Correspondence |\n| --- | --- | --- |\n| Orthant and Lorentz product | `Defs.Idx`, `Defs.cone` | Orthant coordinates and a time coordinate plus $k_j$ spatial coordinates per block. |\n| Jordan square | `Defs.jsq` | Homogeneous quadratic polynomials with the displayed block formula. |\n| Quartic SOS cone | `Defs.sos4` | Real convex hull of homogeneous quadratic squares. |\n| Zeroth approximation | `Defs.quartic`, `Defs.knn0` | Symmetric matrices with SOS quartic pullback. |\n| NN-type dual | `Defs.dnnNN` | Symmetric trace dual of exactly this approximation. |\n| BD-type cone | `Defs.dualCone`, `Defs.extremeRays`, `Defs.dnnBD` | Positive semidefiniteness and preservation of every dual extreme ray generator. |\n| All product dimensions | `Defs.claim`, `Result.result` | All $p,r$ and all positive spatial dimensions; no factor is omitted. |","title":"Generalized doubly nonnegative cone inclusion","url":"https://the-omega-institute.github.io/trureturing-pages/research/nishijima-nakata-2024-gdnn-inclusion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2fca32b03d927b589a2aa27b85df68a9829fc362f04fa998d16a1ffb4868d9b7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nishioka-sato-2021-bernoulli-harmonic-zeta-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Nishioka and Y. Sato (arXiv:2101.02399, hep-th; JHEP 05 (2021) 074)\ncompute the free energy of a conformally coupled free scalar on hyperbolic\nspace. In the derivative of the spectral zeta function for even dimension they\nneed, for every `k ≥ 0`,\n\n`−2^{−2k−2}/(k+1) H_{2k+1} − Σ_{m=1}^{k} 2^{−2k−2}(2^{2m}−2)/(k−m+1) · B_{2m}/(2m)\n+ Σ_{j=0}^{2k+1} (−1)^j/2^{2k−j} · C(2k+1, j) H_j ζ(−j) + (1 − 2^{−2k−1}) H_{2k+1} B_{2k+2}/(k+1) = 0`\n\n(eq. (C.12) of the published text), and state: \"We confirmed (C.12) up to\nk = 100 numerically. However, we do not know a proof of (C.12).\" Issue #11322\nfixes the reading: `H_j` the harmonic numbers, `B_n` the Bernoulli numbers\nwith `B_1 = −1/2`, `ζ` the Riemann zeta function.","title":"A Bernoulli–harmonic–zeta identity for the free scalar on hyperbolic space","url":"https://the-omega-institute.github.io/trureturing-pages/research/nishioka-sato-2021-bernoulli-harmonic-zeta-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d0b553e925702e8a92324180d0a1dfd2c7d1e4c95210f5cc076e9bb73f1627d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:noe-2013-a222759-prime-cube-growth","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. D. Noe, OEIS A222759, revision 12, November 2, 2024.\nThe NAME is:\n\n> Conjectured number of primes p for which binomial(n*p,p) (mod p^3) does not equal n.\n\nThe entire first COMMENT is:\n\n> It appears that, for k > 2 and n >= prime(prime(k)^3), then a(n) >= k.\n\nLet $P(j)$ be the one-based $j$-th prime for positive $j$, with $P(1)=2$.\nFor every integer $k\\ge3$ and every integer $n\\ge1$ with\n$n\\ge P(P(k)^3)$, the event is the existence of $k$ pairwise distinct\npositive primes $q_1,\\ldots,q_k$ such that\n$\\binom{nq_i}{q_i}\\bmod q_i^3\\ne n$ for every $i$.\nThe remainder is the least nonnegative integer remainder, compared with\n$n$ itself. The threshold includes equality and cubes $P(k)$ inside the\nouter prime index. The event asserts at least $k$ witnesses without\nassuming finiteness of the entire set satisfying the predicate.\n\nThe formal domain is exactly the positive integer domain represented by\nnaturals with $3\\le k$ and $1\\le n$. An injection $q:\\mathrm{Fin}(k)\\to\n\\mathbb N$ supplies distinct witnesses, each explicitly positive and prime.\nThe total extension $P(0)=2$ is unused on the event's indices; the natural\nremainder convention at modulus zero is unused on prime witnesses.\nThe source identity and operator correspondence are in\n`Library/Factorization/noe2013a222759.md`.","title":"The complete first growth assertion of OEIS A222759","url":"https://the-omega-institute.github.io/trureturing-pages/research/noe-2013-a222759-prime-cube-growth/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/df989cbd4f98cb08e77404821115d58f326761ae944a8dfaca81278c6fb7097b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:novaes-2023-strong-absorption-cumulants","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Novaes, *Time delay statistics for chaotic cavities with absorption*,\narXiv:2305.18235v1, Section 5, p. 7, states “we conjecture that, for n > 1,”\nfollowed by Eq. (58):\n\n$$\n(-1)^n k_n(\\gamma)\n=\\frac{(n-1)!}{M^{2n-2}\\gamma^{2n}}\n-\\frac{(2n-1)n(n-1)!}{M^{2n-2}\\gamma^{2n+1}}\n+O(\\gamma^{-2n-2}).\n$$\n\nHere $M\\geq1$ is an integer channel number and $n\\geq2$ is an integer\ncumulant order. The symmetry class is $\\beta=2$, with perfect coupling and\nuniform absorption. The journal publication is *Journal of Statistical\nPhysics* 190, 182 (2023), DOI 10.1007/s10955-023-03183-7; its Eq. (65)\nis the same conjecture.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Every positive integer channel number | `claim` quantifies every natural `M` with `1 ≤ M`. |\n| Every cumulant order greater than one | Every natural `n` with `2 ≤ n`. |\n| Grabsch's inverse-rate law at $\\beta=2$, $N=M$ | `Foundations.weight` is the literal joint density below; `Foundations.law` divides its measure by total mass. |\n| Wigner time delay in dwell-time units | `Foundations.tauW` is $M^{-1}\\sum_i(\\gamma+x_i)^{-1}$. |\n| The $n$th cumulant | `Foundations.kappa` is the $n$th derivative at zero of the observable's cumulant generating function. |\n| Both coefficients of Eq. (58), including $(-1)^n$ | The literal unsimplified factorial, channel and absorption powers in `claim`. |\n| $O(\\gamma^{-2n-2})$ as $\\gamma\\to\\infty$ | There exist real $C,\\gamma_0$ with $\\gamma_0>0$, and the absolute error is at most $C/\\gamma^{2n+2}$ for every $\\gamma\\geq\\gamma_0$. Constants may depend on $(M,n)$. |\n\nGrabsch, arXiv:1909.01002v2, Eqs. (3)–(4), supplies the model law.\nWriting the inverse delay rates as $\\Gamma_i=\\gamma+x_i$, its unnormalized\nweight on $x_i\\geq0$ and $0\\leq y_j\\leq\\gamma$ is\n\n$$\ne^{-M(\\sum_i x_i+\\sum_j y_j)}\\Delta(x)^2\\Delta(y)^2\n\\prod_{i,j}(\\gamma+x_i-y_j).\n$$\n\nThe five conventions `weight`, `law`, `tauW`, `kappa`, `claim` are literal\nto [#16825](https://github.com/the-omega-institute/trureturing/issues/16825).\nThe probability normalization and bounded observable are proved for this\nlaw. Identification of this law with the physical random-matrix model is\nthe cited mathematical-model premise, rather than a theorem about a\nphysical cavity's dynamics.","title":"Novaes's strong-absorption cumulant conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/novaes-2023-strong-absorption-cumulants/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/750aff99149950989f8ace2db767fdc15a6ddc1285e8a21e95992e234792ce0f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:novaes-2023-strong-absorption-spectral-moments","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Novaes, *Time delay statistics for chaotic cavities with absorption*,\narXiv:2305.18235v1, Section 5, p. 7, states: “We conjecture that\n$(1/M)\\langle\\operatorname{Tr}(Q^n)\\rangle$ is given by” Eq. (54) and that\n$(1/M^n)\\langle[\\operatorname{Tr}(Q)]^n\\rangle$ is given by Eq. (55), for every\npositive integer $n$:\n\n$$\n\\frac1{\\gamma^n}-\\frac n{\\gamma^{n+1}}+\\frac{n^2}{\\gamma^{n+2}}\n-\\frac{(n+1)\\bigl(n(5n-2)M^2+n(n+2)\\bigr)}{6M^2\\gamma^{n+3}}\n+O(\\gamma^{-n-4}),\n$$\n\nand\n\n$$\n\\frac1{\\gamma^n}-\\frac n{\\gamma^{n+1}}\n+\\frac{n\\bigl((n+1)M^2+(n-1)\\bigr)}{2M^2\\gamma^{n+2}}\n+O(\\gamma^{-n-3}).\n$$\n\nHere $M\\geq1$ is the channel number, $n\\geq1$ is the order, and the model is\nthe finite-channel $\\beta=2$ law with uniform absorption. The journal version\nis *Journal of Statistical Physics* 190, 182 (2023), DOI\n10.1007/s10955-023-03183-7.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Every positive channel number | `claim54` and `claim55` quantify $M$ with $1\\leq M$. |\n| Every positive integer order | Both claims quantify $n$ with $1\\leq n$. |\n| Spectral observable $(1/M)\\operatorname{Tr}(Q^n)$ | `traceQPow` is $(M:ℝ)^{-1}\\sum_i(\\gamma+p_1(i))^{-n}$. |\n| Wigner observable $(1/M^n)[\\operatorname{Tr}(Q)]^n$ | `tauW M γ p ^ n` is the normalized Wigner delay power. |\n| Expectation under the stated law | Both integrals use `law M γ`. |\n| The displayed coefficients and remainder orders | The two absolute remainder inequalities in `claim54` and `claim55` use the literal coefficients and powers. |\n| The same external source conjectures both formulas | `claim` is the conjunction `claim54 ∧ claim55`. |","title":"Novaes's strong-absorption spectral and Wigner moment conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/novaes-2023-strong-absorption-spectral-moments/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7fe34877be6402d6a62a060dc47f72ab8c63469d94bdf00de5d6b286ad81195a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:nuradha-mishra-leditzky-wilde-measured-fidelity-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Nuradha, H. K. Mishra, F. Leditzky and M. M. Wilde, *Multivariate fidelities*, arXiv:2404.16101v3, Remark 5.45, page 34, ask:\n\n> Furthermore, it is interesting to determine whether there exists a tuple of states for which F_M(ρ_1, …, ρ_r) > F_U(ρ_1, …, ρ_r)—we leave this as an open question.\n\nThe answer is affirmative. For the three qubit states $R=I/2$, $A=(I+Z/2)/2=\\operatorname{diag}(3/4,1/4)$ and $B=(I+X/2)/2$, the proof gives $F_U(R,A,B)<24/25\\le F_M(R,A,B)$. Here $F(\\rho,\\sigma)=\\|\\sqrt\\rho\\sqrt\\sigma\\|_1$, $F_U$ is the average over unordered pairs, and $F_M$ takes one infimum over all finite POVMs of every outcome size.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Proposition 3.1: root fidelity | `uhlmann` | The existing trace norm applied to the product of positive square roots. |\n| Equation (5.13): average over unordered pairs | `FU` | Real normalization $2/(r(r-1))$ and `Finset.Ioi`. |\n| Definition 5.40: the measurement channel | The two diagonal trace expressions in `FM` | The literal measurement expression is inlined; no second channel definition. |\n| Infimum over finite POVMs of every size | The dependent-sum index in `FM` | A single real infimum over size and POVM together; empty outcome sizes do not supply artificial zero costs. |\n| Existence of a tuple with strict inequality | `claim`, `result` | Natural dimensions and tuple sizes and positive trace-one matrix states; witness dimension two and tuple size three. |\n\nThe public definition bodies are the binding expressions of [#19068](https://github.com/the-omega-institute/trureturing/issues/19068), with the measurement action expanded. No hypothesis is added to the existential question.","title":"A strict gap between measured and Uhlmann average fidelities","url":"https://the-omega-institute.github.io/trureturing-pages/research/nuradha-mishra-leditzky-wilde-measured-fidelity-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a2c7da768c57227be067502d64037e26240d64d10164fdd718c9812a65a5bab.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:obryant-2024-golden-minimizer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kevin O'Bryant, *Visualizing the Sum-Product Conjecture*, arXiv:2411.08139v2,\n§5.7 asks:\n\n> What is the smallest possible value of |A + A|, where A is a set of n\n> positive real numbers with |AA| ≤ 3n − 4? Is it achieved by\n> {φ^i : 1 ≤ i ≤ n}, where φ = (1 + √5)/2?\n\nThe Lean `claim` retains the universal quantifiers over n and finite positive\nreal sets and compares the pointwise sumsets. The theorem `result : ¬ claim`\nrefutes this universal minimizer assertion.","title":"O'Bryant §5.7: the golden minimizer","url":"https://the-omega-institute.github.io/trureturing-pages/research/obryant-2024-golden-minimizer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e998de087d09e965f2c0f58e44ac38ec65c7a9d66664866ebec08ac957c8c2b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:obryant-2024-sum-product-seven-point-cell","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kevin O'Bryant, *Visualizing the Sum-Product Conjecture*,\narXiv:2411.08139v2, §5.2:\n\n> We believe that (23, 13) is not in SPP_{R+}(7).\n\nFor a domain X, the spectrum is\n\n$$\n\\operatorname{SPP}_X(n)\n = \\{(|A+A|,|AA|): A\\subseteq X,\\ |A|=n\\}.\n$$\n\nHere R+ denotes the strictly positive reals, and A+A and AA denote the\npointwise sumset and product set. The exact assertion is that every finite\nset A of positive reals with seven elements satisfies\n\n$$\n\\neg\\bigl(|A+A|=23\\ \\land\\ |AA|=13\\bigr).\n$$\n\nThe Lean `claim` retains these quantifiers and hypotheses. Its theorem\n`result : claim` follows from the stronger statement\n\n$$\n|AA|=13\\quad\\Longrightarrow\\quad |A+A|\\geq24.\n$$","title":"O'Bryant's seven-point positive-real sum-product cell","url":"https://the-omega-institute.github.io/trureturing-pages/research/obryant-2024-sum-product-seven-point-cell/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c145af89b300fda01b50428e9f6deaa768a75bbb0a5affcb82f104464d7f41cb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oconnor-ramgoolam-2025-tensor-high-temperature-leading-classes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. O'Connor and S. Ramgoolam, *Gauged permutation invariant tensor quantum mechanics, least common multiples and the inclusion-exclusion principle*, arXiv:2506.18813v2 (2025), Section 7.2, state:\n\n> Computations with Mathematica and the analysis of special families give strong evidence that the leading contributions in the high temperature limit $ x \\rightarrow 1 $ come from $ p = [ 1^N ] $ followed by $ p = [ 2,1^{ N-2} ] $.\n\nSection 9 asks:\n\n> It would be very interesting to find the proof of the general $s$ conjecture for the leading two terms.\n\nThese quotations are from lines 1068 and 1322 of the v2 e-print member `Perm-Invt-Tensor-Counting-Functions-Journal.tex`, md5 `202b33a8e55dec1592ee90b7e5fbd8d9`.\n\nFor a partition $p$ of $N$, let $A(p)$ be its set of distinct part values and $m_a$ the multiplicity of $a$. The module's rational-valued `degree` is the first sum in the source's equation (7.21), Section 7.2, TeX lines 1079–1087:\n\n$$D_s(p)=\\sum_{t:\\mathrm{Fin}\\,s\\to A(p)}\n\\frac{\\prod_{j\\in\\mathrm{Fin}\\,s}t_jm_{t_j}}\n{\\operatorname{lcm}\\{t_j:j\\in\\mathrm{Fin}\\,s\\}}.$$\n\nTuple entries may repeat. The part values, multiplicities and natural-number least common multiple are cast to the rationals for this expression.\n\nWrite $I_N=[1^N]$ and $T_N=[2,1^{N-2}]$. Issue [#24869](https://github.com/the-omega-institute/trureturing/issues/24869) fixes the reading: for every natural $N\\ge2$ and $s\\ge1$,\n\n$$D_s(I_N)=N^s,\\qquad D_s(T_N)=\\frac{N^s+(N-2)^s}{2} The prime numbers.\n\nGary Detlefs's FORMULA of May 25, 2014 (verbatim):\n\n> Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(n-1) mod n = 1}. - _Gary Detlefs_, May 25 2014\n\nHis FORMULA of May 28, 2014 (verbatim):\n\n> Conjecture: Sequence = {5 and n <> 5| ( Fibonacci(n) mod n = 1 or Fibonacci(n) mod n = n - 1) and 2^(3*n) mod 3*n = 8}. - _Gary Detlefs_, May 28 2014\n\nFor natural numbers, define\n`fibTest(n) := (Fibonacci(n) mod n = 1) or\n(Fibonacci(n) mod n = n-1)`, define\n`fermatTest(n) := 2^(n-1) mod n = 1`, and define\n`inDetlefsSet(n) := (n=5) or\n(n!=5 and fibTest(n) and fermatTest(n))`. The literal refuted statement is\n`claim := for every natural n, 0 Conjecture: for k a positive odd integer, the sequence {a(k^n): n >= 1} is a strong divisibility sequence; that is, for n, m >= 1, gcd(a(k^n), a(k^m)) = a(k^gcd(n,m)).\n\nThe exact formal statement is:\n\n```lean\ntheorem result (k n m : ℕ) (hk : Odd k) (hn : 0 < n) (hm : 0 < m) :\n Nat.gcd (Nat.fib (k ^ n) - 1) (Nat.fib (k ^ m) - 1) =\n Nat.fib (k ^ Nat.gcd n m) - 1\n```\n\nOddness of a natural k supplies its positivity. The source's k=1 and\nevery pair of positive indices are included; there is no cutoff,\nchanged offset, or conditional prime restriction.","title":"Bala's Fibonacci-power strong divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000071-bala-strong-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aae3ffbe2a08357e4c61378b9631e4490714cb5a3d699e80ea15f94734bc0218.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a000364-secant-number-pure-period-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A000364, NAME (verbatim from `Library/Recurrence/bala2023a000364.md`):\n\n> Euler (or secant or \"Zig\") numbers: e.g.f. (even powers only) sec(x) = 1/cos(x).\n\nCOMMENT (verbatim; attributed in that Library note to Peter Bala, May 08 2023):\n\n> Conjecture: taking the sequence [a(n) : n >= 1] modulo an integer k gives a purely periodic sequence with period dividing phi(k). For example, the sequence taken modulo 21 begins [1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, 20, 16, 2, 1, 5, 19, ...] with an apparent period of 6 = phi(21)/2.","title":"Refutation of the A000364 pure-periodicity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000364-secant-number-pure-period-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ef6bbb19ad163162d7a1baa271e21e89b073ce4d29869a137ef2a972abfe8bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a000680-schulte-half-factorial-prime-criterion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A000680, NAME (`%N`, verbatim):\n\n> a(n) = (2n)!/2^n.\n\nThe settled COMMENT (`%C`, verbatim) is:\n\n> Conjecture: For n > 0 (a(n) + 2^n) is a multiple of (2*n + 1) if and only if (2*n + 1) is a prime number. - _Werner Schulte_, Oct 05 2025\n\nThe offset is `%O 0,3`, and the AUTHOR line (`%A`, verbatim) is:\n\n> _N. J. A. Sloane_\n\nThe full-quantifier reading is: for every `n : Nat`, if `0 < n`, then\n`(2 * n + 1) ∣ a n + 2 ^ n` if and only if `(2 * n + 1).Prime`. The scope wall\nis exactly this biconditional for every positive natural `n`; no additional\nsequence values, implications, or historical claims are included.","title":"Schulte's A000680 half-factorial primality criterion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000680-schulte-half-factorial-prime-criterion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02d35d29163fe12793d6d95dcf539ce4876ebdddb9d881997e6d490638dcc523.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A001108, NAME (`%N`, verbatim):\n\n> a(n)-th triangular number is a square: a(n+1) = 6*a(n) - a(n-1) + 2, with a(0) = 0, a(1) = 1.\n\nJaroslav Krizek's COMMENT conjecture (`%C`, verbatim, with its preceding\ncontext sentence):\n\n> The squares of NSW numbers (A008843) interleaved with twice squares from A084703, where A008843(n) = A002315(n)^2 and A084703(n) = A001542(n)^2. Conjecture: Also numbers n such that sigma(n) = A000203(n) and sigma(n-th triangular number) = A074285(n) are both odd numbers. - _Jaroslav Krizek_, Aug 05 2016\n\nThe literal proved statement is\n`forall n : Nat, 0 < n -> (IsSquare (n * (n + 1) / 2) iff\n(Odd (sigma 1 n) and Odd (sigma 1 (n * (n + 1) / 2))))`.\nThe recurrence for A001108, Pell-type completeness, the NSW/A084703\ninterleaving structure, and all other A001108 comments are NOT claimed. The\nguard excludes `n=0`: although `T_0=0` is a square, Mathlib defines\n`sigma(0)=0`, which is even, so the unguarded parity equivalence is false.","title":"The A001108 triangular-square indices have two odd divisor sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ea174126238846ddf8dc09dc3ef51a6e707dbccac3b558722a79ed3f558a86f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a001222-schulte-omega-binomial-convolution","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A001222, NAME (`%N`, verbatim):\n\n> Number of prime divisors of n counted with multiplicity (also called big omega of n, bigomega(n) or Omega(n)).\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: Let f(n) = (x+y)^a(n), and g(n) = x^a(n), and h(n) = (x+y)^A046660(n) * y^A001221(n) with x, y complex numbers and 0^0 = 1. Then f(n) = Sum_{d|n} g(d)*h(n/d). This is proved for x = 1-y (see Dressler and van de Lune link). - _Werner Schulte_, Feb 10 2018\n\nThe A001222 offset (`%O`) is `1,3`. A046660, NAME (`%N`, verbatim):\n\n> Number of prime factors of n counted with multiplicity minus number of distinct prime factors. Omega(n) - omega(n).\n\nA001221, NAME (`%N`, verbatim):\n\n> Number of distinct primes dividing n (also called omega(n)).\n\nWrite Ω for `ArithmeticFunction.cardFactors` and ω for\n`ArithmeticFunction.cardDistinctFactors`. The full-quantifier reading is\n\n```lean\ntheorem result (x y : ℂ) (n : ℕ) (hn : 0 < n) :\n (x + y) ^ Ω n = ∑ d ∈ n.divisors,\n x ^ Ω d * ((x + y) ^ (Ω (n / d) - ω (n / d)) * y ^ ω (n / d))\n```\n\nEvery complex pair x, y and every positive natural n are quantified.\nThe sum uses all positive divisors, including 1 and n; `n / d` is natural\ndivision and is exact for these divisors. Exponent subtraction is natural\nsubtraction; ω(m) ≤ Ω(m) follows from deduplication of the prime-factor\nlist, so it equals the ordinary nonnegative difference. Natural powers\nimplement `0^0 = 1`. The statement includes n = 1 and allows x, y, and x+y\nto vanish, without any nonzero-base assumption.\n\nOnly the general x, y assertion in the quoted Conjecture sentence is\nsettled. The historical x = 1 − y case is Dressler and van de Lune,\nProc. AMS 41 (1973), 403–406, doi 10.1090/S0002-9939-1973-0340191-8.\nOther OEIS assertions and historical priority are outside the formal claim.","title":"Schulte's Ω/ω binomial convolution","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a5d8315816fc6f0a2f71419cee06404ec2fd8c7ff06b5e99b83181fc2c8a6f9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a002000-bala-cubic-three-adic-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A002000, NAME (`%N`, verbatim):\n\n> a(n+1) = a(n)*(a(n)^2 - 3) with a(0) = 7.\n\nThe settled FORMULA line (`%F`, verbatim), inside Peter Bala's Nov 15 2022\n`(Start)` ... `(End)` block:\n\n> Conjecture: a(n+1) == a(n) (mod 3^(n+r+2)) for n >= r.\n\nAUTHOR (`%A`, verbatim):\n\n> _N. J. A. Sloane_\n\nThe offset is `%O 0,1`. Both `n` and `r` range over the natural numbers,\nincluding zero, and the only hypothesis is `r <= n`. The sequence has\nvalues in the integers and is defined by the NAME recurrence. Congruence\nmeans that `3^(n+r+2)` divides the integer difference `a(n+1)-a(n)`.\n\nThe exact Lean statement is:\n\n```lean\ntheorem result (n r : ℕ) (hr : r ≤ n) :\n (3 : ℤ) ^ (n + r + 2) ∣ a (n + 1) - a n\n```\n\nOnly the Conjecture line is settled, for every `n` and every `r <= n`.\nThe block's Lucas representation, 3-adic limit, and product formula are\nnot claimed. In particular, `a` is not defined through Lucas numbers.","title":"Bala's cubic recurrence and its growing three-adic congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002000-bala-cubic-three-adic-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4266ebf9272cd48cda2df4eaf383212d0ae38e864fb5114b2b81ff8900e07bf4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a002487-adamson-stern-farey-inverse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A002487, NAME (`%N`, verbatim):\n\n> Stern's diatomic series (or Stern-Brocot sequence): a(0) = 0, a(1) = 1; for n > 0: a(2*n) = a(n), a(2*n+1) = a(n) + a(n+1).\n\nGary W. Adamson's December 18, 2023 COMMENT (`%C`, verbatim):\n\n> It appears that a(n) is equal to the multiplicative inverse of A007305(n+1) mod A007306(n+1). For example, a(12) is 2, the multiplicative inverse of A007305(13) mod A007306(13), where A007305(13) is 4 and A007306(13) is 7.\n\nThe A002487 AUTHOR line (`%A`, verbatim):\n\n> _N. J. A. Sloane_\n\nOEIS A007305, NAME (`%N`, verbatim):\n\n> Numerators of Farey (or Stern-Brocot) tree fractions.\n\nOEIS A007306, NAME (`%N`, verbatim):\n\n> Denominators of Farey tree fractions (i.e., the Stern-Brocot subtree in the range [0,1]).\n\nThe listing has 0/1 and 1/1 at indices 0 and 1. Each later level lists its\nnew mediants in increasing order in [0,1]. For\n`r = 2^m + j`, `1 <= j <= 2^m`, entry `r` is\n`fareyRow (m+1) (2*(j-1)+1)`. Equivalently, for `r >= 2`, take\n`m = Nat.log 2 (r-1)` and `k = r-1-2^m`, then select row `m+1`,\nposition `2*k+1`. The first and second coordinates are `fareyNum` and\n`fareyDen`.\n\nThe literal Lean statement is:\n\n```lean\ntheorem result (n : ℕ) (hn : 0 < n) :\n stern n * fareyNum (n + 1) % fareyDen (n + 1) = 1 % fareyDen (n + 1)\n```\n\nOnly Adamson's sentence is settled, for positive natural indices. The\nYurramendi frequency conjecture, Torres's conjectures and other Farey-tree\nproperties are not claimed.","title":"Stern values invert the next Farey-tree numerator","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/14cddd9223213bcd9f285b270683ac088d66959d13eb1380c14814171a763802.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a002620-forberg-exponential-floor","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A002620, revision 880, contains the formula attributed to Richard R. Forberg:\n\n> Empirical: a(n-1) = floor(n/(e^(4/n)-1)). - _Richard R. Forberg_, Jul 24 2013\n\nThe entry defines the quarter-squares by\n$a(m)=\\lfloor m/2\\rfloor\\lceil m/2\\rceil=\\lfloor m^2/4\\rfloor$, with offset zero.\nThe complete assertion is\n\n$$\n\\forall n\\in\\mathbb N,\\quad 1\\le n\\ \\Longrightarrow\n\\left\\lfloor\\frac{n}{\\exp(4/n)-1}\\right\\rfloor\n=\\left\\lfloor\\frac{(n-1)^2}{4}\\right\\rfloor.\n$$\n\nAll operations inside the floors are in $\\mathbb R$, with $n$ coerced before\ndivision and subtraction. The exponential is `Real.exp`; both floors are\n`Int.floor` and the equality is in $\\mathbb Z$. The index shift is $m=n-1$.\nThe assertion includes $n=1$, both parities, and arbitrarily large indices,\nwith no additional hypothesis.","title":"Forberg's exponential formula for the quarter-squares","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002620-forberg-exponential-floor/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d62c2d7c6a66f49362ae9871db1673d9881408d3cf19a95254273e3625f1b35.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a003319-kurkov-double-array","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The source audit for this resolution read the canonical OEIS A003319 entry and\nthese immutable OEIS-data revisions:\n\n- A003319 at commit `dacce77176c40534c4309db03464e7f4c08d83dd`;\n- A370380 at commit `a32b213583dbdd91a31cc0975e203471b3afe0fb`;\n- A370381 at commit `ed2582877dea2190aa163abe737afe7d42179fae`.\n\nKurkov's April 26, 2024 conjecture is quoted there verbatim as:\n\n> Conjecture: a(n) = A370380(n-2, 0) = A370381(n-2, 0) for n > 1 with a(0) = a(1) = 1. - _Mikhail Kurkov_, Apr 26 2024\n\nThe sequence `a(n)` counts permutations of `{1,...,n}` that preserve no\nproper nonempty initial interval. The formal subtype uses permutations of\n`Fin n`; translating labels by one gives the same condition. Its literal\nformal count is the finite cardinality\n`c(n) = Fintype.card (IndecomposablePerm n)`, rather than a sequence defined\nby either recurrence.\n\nFor natural row and column indices, the two source arrays are formalized\nliterally as\n\n```text\nU(0,k) = 1\nU(m+1,k) = (k+2) U(m,k+1) + Sum_{j=0..k} U(m,j)\n\nV(0,k) = 1\nV(m+1,k) = Sum_{j=0..k+1} binomial(k+2,j+1) V(m,j).\n```\n\nThe exact frozen theorem is\n\n```lean\nc 0 = 1 ∧ c 1 = 1 ∧\n ∀ n : ℕ, 2 ≤ n → c n = U (n - 2) 0 ∧ c n = V (n - 2) 0\n```","title":"Kurkov's A003319 double-array conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c59d47cb543f4b4bd13566d716798136cd5bde8d72278457a14cb9f70e179767.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a003558-marcus-tent-map-suborder","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A003558, NAME (`%N`, verbatim):\n\n> Least number m > 0 such that 2^m == +-1 (mod 2n + 1).\n\nThe settled COMMENT sentence (`%C`, verbatim):\n\n> It appears that a(n) is the length of the cycle starting at 2/(2*n+1) for the map x->1 - abs(2*x-1). - _Michel Marcus_, Jul 16 2025\n\nAUTHOR (`%A`, verbatim):\n\n> _N. J. A. Sloane_\n\nThe offset is `%O 0,3`. The claim is read for every natural `n>0`, with\n`N=2n+1`: the least positive return time of `2/N` under the tent map is\nthe least positive `m` such that `2^m` is congruent to `1` or `-1` modulo\n`N`. The residue `-1` is written `2n` in natural congruence notation.\n\nThe exact Lean statement is:\n\n```lean\ntheorem result (n : ℕ) (hn : 0 < n) :\n IsLeast {m : ℕ | 0 < m ∧ (2 ^ m ≡ 1 [MOD 2 * n + 1] ∨\n 2 ^ m ≡ 2 * n [MOD 2 * n + 1])}\n (Function.minimalPeriod (tent (2 * n + 1)) 2)\n```\n\nHere `tent N k := if 2*k <= N then 2*k else 2*N - 2*k` is a natural\nnumerator map. For `N>0` and `k<=N`, its branches divided by `N` are\n`2k/N` and `(2N-2k)/N`, exactly the two branches of the rational tent map.\nSince `k<=N`, the second natural subtraction is not truncated. The orbit\ninvariant supplies this bound for every iterate from 2. Division by the\nfixed positive `N` is injective, so numerator and rational return times\ncoincide. This rational interpretation is an algebraic explanation; the\nkernel theorem is the displayed numerator statement.\n\nOnly the quoted Marcus sentence is settled. A003558's other comments\nabout Kaprekar cycles, `x^2-2` iteration, and the Kappraff-Adamson base\nconjecture are not claimed. The hypothesis `0 Number of generalized weak orders on n points.\n\nOEIS A004123 COMMENT (Peter Bala, Jul 08 2022):\n\n> Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).\n\nThe finite formula for this entry is `a(n) = T (fun k => (2 : ℤ)^k) (n - 1)`; its proved onset is `n ≥ m + 1`\nfor every positive modulus `m`. The entry has offset 1, so the transform index is n - 1.\n\nThe companion entry is recorded separately with its own Library note and dossier:\n\nA064618 NAME (offset 0):\n\n> Stirling transform of (n!)^2.\n\nA064618 COMMENT (Peter Bala, Jan 15 2018):\n\n> Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010.","title":"A004123 totient-period conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a004123-generalized-weak-orders-totient-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09f961ca3a756769eaa36ef34d059cc20c065a25cf335d49b8490dbf25aa34bd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a005590-zero-set-fibbinary-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A005590, NAME (`%N`, verbatim):\n\n> a(0) = 0, a(1) = 1, a(2n) = a(n), a(2n+1) = a(n+1) - a(n).\n\nRalf Stephan's FORMULA conjecture (`%F`, verbatim):\n\n> Conjecture: a(3n)=0 iff n in A003714. - _Ralf Stephan_, May 02 2003\n\nOEIS A003714, NAME (`%N`, verbatim):\n\n> Fibbinary numbers: if n = F(i1) + F(i2) + ... + F(ik) is the Zeckendorf representation of n (i.e., write n in Fibonacci number system) then a(n) = 2^(i1 - 2) + 2^(i2 - 2) + ... + 2^(ik - 2). Also numbers whose binary representation contains no two adjacent 1's.\n\nThe literal proved statement is `∀ n, r(3n)=0 ↔ No11 n`, rendered in Lean as\n`∀ n : ℕ, r (3 * n) = 0 ↔ No11 n`, where `r : ℕ → ℤ` satisfies `r(0)=0`,\n`r(1)=1`, `r(2n)=r(n)`, and `r(2n+1)=r(n+1)-r(n)`, and where\n`No11(n) := ∀ j : ℕ, (n >>> j) % 4 ≠ 3`. The growth rate attributed to\nReznick and the partial recurrences recorded by Chai Wah Wu are NOT claimed.","title":"The A005590 zero set at multiples of three is characterized","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd5d28cb20d0da39d67f23006de37972b801faabcbcbd75c904c3eee56552093.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A006068, NAME (`%N`, verbatim):\n\n> a(n) is Gray-coded into n.\n\nMikhail Kurkov's conjecture (`%F`, verbatim):\n\n> Conjecture: a(n) = a(A053645(A063946(n))) + A053644(n) for n > 0 with a(0) = 0. - _Mikhail Kurkov_, Sep 09 2023\n\nThe AUTHOR line (`%A`, verbatim) is:\n\n> _N. J. A. Sloane_\n\nHanna's unconjectured XOR-prefix formula (`%F`, June 4, 2002) is:\n\n> a(n) = n XOR [n/2] XOR [n/4] XOR [n/8] ... XOR [n/2^m] where m = [log(n)/log(2)] (for n>0) and [x] is integer floor of x.\n\nHis inverse formula (`%F`, January 18, 2012) is:\n\n> a(n) XOR [a(n)/2] = n.\n\nThe exact formal result is:\n\n```lean\n(∀ n : ℕ, gray (a n) = n) ∧ a 0 = 0 ∧\n∀ n : ℕ, 0 < n →\n a n = a (complementSecondBit n - msb (complementSecondBit n)) + msb n\n```\n\nHere `gray` is imported from the frozen\n`D5/S1/Digit/Admissibility/GrayCodeBinaryRecurrenceClosedForm`.\nThe definition of `a` is Hanna's XOR-prefix formula written as the\nrecursion `n XOR a(n/2)`, terminating at zero. For positive inputs,\n`msb n := 2 ^ Nat.log 2 n` is A053644, and subtraction of this value is\nA053645. The totalized formula gives `msb 0 = 1`, while A053644(0) is 0;\nevery `msb` argument in the positive recurrence is positive.\n\nThe definition of `complementSecondBit` is A063946, fixing zero and one.\nIts two arithmetic `%F` cases, proved and used inside `result`, are:\n\n> If 2*2^k <= n < 3*2^k then a(n) = n + 2^k; if 3*2^k <= n < 4*2^k then a(n) = n - 2^k","title":"Kurkov's highest-bit recurrence for inverse Gray code","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29e987f16bbcbf9f6dbe04ddc673a447a642b21cb2b65386ec1b38fce6cb8048.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a006472-schulte-factorial-square-prime-criterion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A006472, NAME (`%N`, verbatim):\n\n> a(n) = n!*(n-1)!/2^(n-1).\n\nThe offset is `%O 1,3`. The settled COMMENT line (`%C`, verbatim) is:\n\n> Conjecture: For n > 1, n divides 2*a(n-1) + 4 if and only if n is prime. - _Werner Schulte_, Oct 04 2020\n\nThe full-quantifier reading is: define, for every natural `m`,\n`a(m) = m!*(m-1)!/2^(m-1)` using natural-number division; then for every\nnatural `n` with `2 <= n`, `n` divides `2*a(n-1)+4` if and only if `n` is\nprime. The denominator divides the factorial product for every `m >= 1`, so\nall sequence values used by the criterion are exact quotients.\n\nOnly the quoted Conjecture sentence is settled, as a biconditional for every\nnatural `n >= 2`. No other OEIS comment, formula, or stronger composite\ndivisibility statement is claimed.","title":"Schulte's factorial-square primality criterion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006472-schulte-factorial-square-prime-criterion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a4515d2ddef352f7ce681fc5c7a60e0dfcfb45d98e29d3bdb49d1ee83a2569f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a006972-rotondo-lucas-carmichael-sufficient-condition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A006972, NAME (`%N`, verbatim):\n\n> Lucas-Carmichael numbers: squarefree composite numbers k such that p | k => p+1 | k+1.\n\nDavide Rotondo's COMMENT (`%C`, verbatim) is:\n\n> Conjecture: if k = p*q*r, p = a*d - 1, q = b*d - 1, r = c*d - 1 are distinct odd primes, with d = gcd(p + 1, q + 1, r + 1) and a*b*c*d divides k + 1, then k is a Lucas-Carmichael number. - _Davide Rotondo_, Dec 23 2020\n\nThe full-quantifier reading takes `a,b,c,d,p,q,r,k` in the natural numbers,\nkeeps the stated odd-prime, distinctness, product, gcd, and divisibility\nhypotheses, adds the derivable positivity hypotheses, and concludes that `k` is squarefree, composite,\ngreater than one, and satisfies `s+1 | k+1` for every prime divisor `s`.\nThe equations `p+1=a*d`, `q+1=b*d`, and `r+1=c*d` are equivalent to the\nsource's subtraction equations under the corresponding prime hypotheses:\nprimality gives positivity, which removes the natural-subtraction truncation.","title":"Rotondo's A006972 Lucas-Carmichael sufficient condition","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006972-rotondo-lucas-carmichael-sufficient-condition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/164ab50291c3a2e52dd0ccb62aa1f56dc1d754494919ac9bec29442f691712d5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a008287-schulte-quadrinomial-alternating-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A008287, NAME (`%N`, verbatim):\n\n> Triangle of quadrinomial coefficients, row n is the sequence of coefficients of (1 + x + x^2 + x^3)^n.\n\nThe settled FORMULA (`%F`, verbatim) is:\n\n> T(n,k) = Sum_{j=0..k} (-2)^j*binomial(n,j)*binomial(3*n-2*j,k-j) for n >= 0 and 0 <= k <= 3*n (conjectured). - _Werner Schulte_, Sep 09 2015\n\nThe full-quantifier reading is: for all `n k : ℕ`, if `k ≤ 3 * n`,\nthe integer coefficient `T n k` of `X^k` in\n`(1 + X + X^2 + X^3 : ℤ[X]) ^ n` equals\n`∑ j ∈ Finset.range (k + 1), (-2 : ℤ)^j * (n.choose j : ℤ) *\n((3 * n - 2 * j).choose (k - j) : ℤ)`. Subtraction in binomial indices\nis truncated natural subtraction, and `n.choose j = 0` for `j > n`.\nThe scope wall is only this Schulte 2015 formula. The Shevelev 2010 and\nBala 2013 formulas in the same entry are different and are not settled here.","title":"Schulte's A008287 alternating-binomial expansion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008287-schulte-quadrinomial-alternating-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1ad6e51b9bb060850a249b6a6dba482d4b06618454c21147be0045820db84a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a008365-detlefs-rough-residue-characterization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A008365, NAME (verbatim):\n\n> 13-rough numbers: positive integers that have no prime factors less than 13.\n\nGary Detlefs's COMMENT of December 30, 2011 (verbatim):\n\n> Conjecture: Numbers n such that n^24 is congruent to {1,421,631,841} mod 2310. - _Gary Detlefs_, Dec 30 2011\n\nFor natural numbers, define\n`isRough13(n) := for every natural p, Prime(p) implies p divides n implies\n13 <= p`, and define\n`inResidueSet(n) := n^24 mod 2310 is one of {1,421,631,841}`. The literal\nrefuted statement is `claim := for every natural n, 0 If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j).\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: for m > 4, by iterating the map m -> A008474(m) one always reaches 5 [tested up to m = 320000]. - _Ivan N. Ianakiev_, Nov 10 2014\n\nFORMULA (`%F`, verbatim):\n\n> Additive with a(p^e) = p + e.\n\nAUTHOR (`%A`, verbatim):\n\n> _Olivier Gérard_\n\nThe first terms (`%S`, verbatim excerpt) are\n`0,3,4,4,6,7,8,5,5,9,12,8,14,...`.\n\nFor every natural n, define\n`F(n) = Sum (p + factorization(n)(p))` over `p` in `primeFactors(n)`. The\nliteral proved statement is\n`forall m : Nat, 4 < m -> exists t : Nat, F^[t](m) = 5`. Thus \"reaches 5\"\nmeans that some finite iterate equals 5. The values follow the cycle\n`5 -> 6 -> 7 -> 8 -> 5`; four is a fixed point, so `m > 4` is sharp; and\n`2 -> 3 -> 4` never reaches five. The generating function and the identities\nwith A001222 and A008472 are NOT claimed.","title":"The A008474 prime-exponent sum iteration reaches five","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1bd051229f589e6ba899a1386ec8aea38e6acd8815f5506e609565f05b50e0e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a008590-gcd-sum-parity-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A008590, NAME (`%N`, verbatim):\n\n> Multiples of 8.\n\nLechoslaw Ratajczak's COMMENT conjecture (`%C`, verbatim):\n\n> From _Lechoslaw Ratajczak_, Sep 03 2017: (Start)\n>\n> Conjecture: let gcd_2(b,c) be the second greatest common divisor and lcd_2(b,c) be the second least common divisor of not coprime integers b and c. Consecutive elements of this sequence (for a(n) > 0) are consecutive integers m for which both Sum_{k=1..m, gcd(k,m)<>1} gcd_2(k,m) and Sum_{k=1..m, gcd(k,m) <>1} lcd_2(k,m) are even numbers.\n>\n> a(1) = 8 because 1+2+1+4 = 8 (8 is even) and 2+2+2+2 = 8 (8 is even).\n>\n> a(2) = 16 because 1+2+1+4+1+2+1+8 = 20 (20 is even) and 2+2+2+2+2+2+2+2 = 16 (16 is even).\n>\n> a(3) = 24 because 1+1+2+3+4+1+1+6+1+1+4+3+2+1+1+12 = 44 (44 is even) and 2+3+2+2+2+3+2+2+2+3+2+2+2+3+2+2 = 36 (36 is even).\n>\n> The conjecture was checked for 5*10^4 consecutive integers. (End)\n\nFor `d = gcd(k,m)`, the example at `m=24` fixes\n`gcd2(k,m) = d / minFac(d)` and `lcd2(k,m) = minFac(d)`. Thus `gcd2`\nis the greatest proper divisor of a nontrivial gcd and `lcd2` is its least\ndivisor greater than one. Define\n`G(m) = Sum_{1<=k<=m, gcd(k,m) != 1} gcd2(k,m)` and\n`L(m) = Sum_{1<=k<=m, gcd(k,m) != 1} lcd2(k,m)`.\n\nThe literal proved statement is\n`forall m : Nat, 1 < m -> ((Even (G m) and Even (L m)) iff 8 divides m)`.\nAt `m=1` both filtered sums are empty and equal zero, hence both are even,\nwhile one is not a positive multiple of eight; this is the unique mismatch in\nthe independent scan on `[1,6000]`, and the theorem explicitly assumes\n`1 < m`. The knight-move, odd-square-difference, lattice-pair, square-frame,\nand Catalan/Berdellé eight-odd-squares comments in A008590 are NOT claimed.\nValues of `gcd2` and `lcd2` on coprime pairs are also NOT claimed.","title":"The A008590 gcd-filtered sums characterize the positive multiples of eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008590-gcd-sum-parity-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59d11e033e1a46a1aa2520f0c73684e0cfd8646e4ed290570b43b0112280670f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a018800-least-prime-prefix-suffix-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A018800, NAME (verbatim):\n\n> Smallest prime that begins with n.\n\nCOMMENTS conjecture line (verbatim; Amarnath Murthy, May 01 2002):\n\n> Conjecture: If a(n) = (n concatenated with k) then k < n. - _Amarnath Murthy_, May 01 2002\n\nThe AUTHOR line is:\n\n> _David W. Wilson_\n\nThe sequence begins\n`11, 2, 3, 41, 5, 61, 7, 83, 97, 101, 11, 127, 13, 149, 151, 163, 17, 181, 19, 2003, 211, ...`.\nFor a natural number `n`, the formal sequence value is\n`a(n) = sInf {p : Nat | Prime(p) and exists m : Nat, n * 10^m <= p and p < (n + 1) * 10^m}`.\nThe interval is the decimal-prefix reading of \"begins with n\". The literal\nrefuted statement is\n`for all n >= 1, m >= 1, k < 10^m, a(n) = n * 10^m + k -> k < n`.\nThe condition `m >= 1` discloses the nonempty-suffix reading of\n\"n concatenated with k\".\n\nThe existence of `a(n)` in general, the FORMULA line\n`a(n) = prime(A085608(n))`, and an equivalence between the interval encoding\nand a `Nat.digits` formulation are not claimed.","title":"Refutation of the A018800 least-prime prefix suffix-bound conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a018800-least-prime-prefix-suffix-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d3a5a331b0268c9f2582eb02666463b76d51f9d318944317ee5bd653298bae2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a023887-yanev-sigma-radical-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A023887, NAME (`%N`, verbatim):\n\n> a(n) = sigma_n(n): sum of n-th powers of divisors of n.\n\nVelin Yanev's FORMULA conjecture (`%F`, verbatim):\n\n> Conjecture: sigma_m(n) = sigma(n^m * rad(n)^(m-1))/sigma(rad(n)^(m-1)) for n > 0 and m > 0, where sigma = A000203 and rad = A007947. - _Velin Yanev_, Aug 24 2017\n\nAUTHOR (`%A`, verbatim):\n\n> _Olivier Gérard_\n\nThe exact formal statement, with all variables natural, is\n`forall n m, 0 < n -> 0 < m ->\nsigma m n * sigma 1 (primeRadical n ^ (m - 1)) =\nsigma 1 (n ^ m * primeRadical n ^ (m - 1))`, where\n`D5/S1/Deficit/AlmostAdditivity.primeRadical n = ∏ p ∈ n.primeFactors, p`\nis the frozen natural radical (A007947).\nBoth sides are positive under these hypotheses. The denominator is also\npositive, so this multiplied-out form is equivalent to the quoted identity,\nincluding its exact natural-number quotient. Subtraction in `m - 1` is\nnatural subtraction; the positive-`m` hypothesis makes it the usual predecessor.","title":"Yanev's general sigma-radical identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc57b3924c1bb82e8dd6e9b77a30749d493e5ec887e168c800beea8457d02f2f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a030101-yanev-binary-reversal-position-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A030101, OFFSET (`%O`, verbatim):\n\n> 0,4\n\nOEIS A030101, initial sequence line (`%S`, verbatim):\n\n> 0,1,1,3,1,5,3,7,1,9,5,13,3,11,7,15,1,17,9,25,5,21,13,29,3,19,11,27,7,\n\nOEIS A030101, NAME (`%N`, verbatim):\n\n> a(n) is the number produced when n is converted to binary digits, the binary digits are reversed and then converted back into a decimal number.\n\nRalf Stephan's A030101 recurrence (`%F`, verbatim):\n\n> a(n) = 0, a(2n) = a(n), a(2n+1) = a(n) + 2^(floor(log_2(n)) + 1). For n > 0, a(n) = 2*A030109(n) - 1. - _Ralf Stephan_, Sep 15 2003\n\nVelin Yanev's A030101 conjecture (`%F`, verbatim):\n\n> Conjecture: a(n) = 2*w(n) - 2*w(A053645(n)) - 1 for n > 0, where w = A264596. - _Velin Yanev_, Sep 12 2017\n\nOEIS A030101, AUTHOR (`%A`, verbatim):\n\n> _David W. Wilson_\n\nOEIS A264596, NAME (`%N`, verbatim):\n\n> Let S_n be the list of the first n nonnegative numbers written in binary, with least significant bits on the left, and sorted into lexicographic order; a(n) = position of n in S_n, starting indexing at 0.\n\nHeinz's A264596 recurrence (`%F`, verbatim):\n\n> a(2n) = a(n), a(2n+1) = a(n) + n+1, a(0) = 0. - _Alois P. Heinz_, Nov 19 2015\n\nOEIS A053645, NAME (`%N`, verbatim):\n\n> Distance to largest power of 2 less than or equal to n; write n in binary, change the first digit to zero, and convert back to decimal.\n\nOEIS A053645, defining formula (`%F`, verbatim):\n\n> a(n) = n - 2^A000523(n).\n\nThe formal definitions are:\n\n```lean\ndef w : ℕ → ℕ :=\n Nat.binaryRec 0 (fun bit n previous => if bit then previous + n + 1 else previous)\n\ndef stripTop (n : ℕ) : ℕ := n - 2 ^ Nat.log 2 n\n\ndef rev : ℕ → ℕ :=\n Nat.binaryRec 0 fun bit n previous =>\n if bit then if n = 0 then 1 else previous + 2 ^ (Nat.log 2 n + 1) else previous\n```\n\nThe proved statement is:\n\n```lean\ntheorem result (n : ℕ) (hn : 1 ≤ n) :\n rev n + 2 * w (stripTop n) + 1 = 2 * w n\n```\n\nThe source writes the result with subtraction on the right. Lean natural\nsubtraction truncates, so the theorem moves the two subtracted quantities to\nthe additive side. It proves the source equality together with the\nnon-negativity needed for the source subtraction. The printed recurrence\nbegins `a(n) = 0`, rather than `a(0) = 0`; its initial sequence value and the\nrest of the recurrence identify the intended zero boundary used by `rev`.\n\nNot claimed: A264596 contains a separate Yanev conjecture dated Sep 12 2017.\nThat conjecture is outside this dossier and is not asserted or resolved here.","title":"Yanev's position identity for binary reversal","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8baa1a0ddbffc297ee4a5f295aaf9f1bb402ccbba4ef994e73c1e491cb837cab.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a035513-wythoff-column-residue-pairs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Clark Kimberling's 4 June 2025 comment on OEIS A035513 states:\n\n> Conjecture: If m >= 2, then {(T(n,1), T(n,2)) mod m} has cardinality m^2.\n\nHere `n>=1` ranges over positive row indices and `phi=(1+sqrt(5))/2`.\nThe first two columns are\n`T(n,1)=floor(floor(n*phi)*phi)` and\n`T(n,2)=floor(floor(n*phi)*phi^2)`.\nFor every integer modulus `m>=2`, their residue image is the full product\n`(Z/mZ)^2`. Equivalently, for every `a,b` modulo `m` there is a positive\nrow with those two residues. The formal statement also includes `m=1`.","title":"Wythoff column residue pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a035513-wythoff-column-residue-pairs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d1d6ef268af29af03bd8d471b10fd520fda2716f697be6f0b27b84478735a53c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a038867-cube-gap-five-no-square","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Take the difference between two cubes whose bases are five apart. Klaus\nPurath conjectured on May 15, 2026 that no such difference is a perfect\nsquare, noting verification through one hundred thousand indices.","title":"The A038867 no-square conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a038867-cube-gap-five-no-square/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/243990cd56f75e053409d4d5878163c4f7d492345627d80bfc56105982721636.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A046528, NAME (`%N`, verbatim):\n\n> Numbers that are a product of distinct Mersenne primes (elements of A000668).\n\nThe known COMMENT (`%C`, verbatim) states:\n\n> n is a product of distinct Mersenne primes iff sigma(n) is a power of 2: see exercise in Sivaramakrishnan, or Shallit.\n\nJaroslav Krizek's COMMENT (`%C`, verbatim) states:\n\n> Supersequence of A051281 (numbers n such that sigma(n) is a power of tau(n)). Conjecture: numbers n such that sigma(n) = tau(n)^(a/b), where a, b are integers >= 1. Example: sigma(93) = 128 = tau(93)^(7/2) = 4^(7/2). - _Jaroslav Krizek_, May 04 2013\n\nThe literal proved statement is\n`∀ n : ℕ, 0 < n → (isMersenneProduct n ↔ ratPow n)`, where\n`isMersenneProduct n` means\n`∃ S : Finset ℕ, n = ∏ p ∈ S, p ∧ ∀ p ∈ S, p.Prime ∧ ∃ k : ℕ, 0 < k ∧ p + 1 = 2 ^ k`,\nand `ratPow n` means\n`∃ a b : ℕ, 0 < a ∧ 0 < b ∧ (σ 1 n) ^ b = (σ 0 n) ^ a`.\nHere `σ 1` is the divisor-sum function and `σ 0` is the divisor-count\nfunction tau. Since both bases are positive integers for positive `n`, this\ninteger-power equality is equivalent to Krizek's rational-exponent form.\n\nNo statement about A051281 beyond the displayed equivalence is claimed. The\ninfinitude or distribution of Mersenne primes and novelty of the known\nsigma-equals-a-power-of-two classification are also NOT claimed.","title":"Products of distinct Mersenne primes satisfy exactly the sigma-tau rational-power relation","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/353f2de995adb6dfa46735ac19d3a4c5d5b045c07aafa34ec91db8c9e483be80.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A049473, NAME (`%N`, verbatim):\n\n> Nearest integer to n/sqrt(2).\n\nClark Kimberling's COMMENT (`%C`, verbatim, including the complete sentence\nand attribution):\n\n> Let s(n) = zeta(3) - Sum_{k=1..n} 1/k^3. Conjecture: for n >=1, s(a(n)) < 1/n^2 < s(a(n)-1), and the difference sequence of A049473 consists solely of 0's and 1, in positions given by the nonhomogeneous Beatty sequences A001954 and A001953, respectively. - _Clark Kimberling_, Oct 05 2014\n\nOEIS A001953, NAME (`%N`, verbatim):\n\n> a(n) = floor((n + 1/2) * sqrt(2)).\n\nOEIS A001954, NAME (`%N`, verbatim):\n\n> a(n) = floor((n+1/2)*(2+sqrt(2))); winning positions in the 2-Wythoff game.\n\nWriting `a(n) = round(n/sqrt(2))`, `lower(k) = floor((k+1/2)*sqrt(2))`,\nand `upper(k) = floor((k+1/2)*(2+sqrt(2)))`, the literal proved clause says\nthat every `a(n+1)-a(n)` is zero or one, that it is one exactly when\n`n = lower(k)` for some natural `k`, and that it is zero exactly when\n`n = upper(k)` for some natural `k`.\n\nThe first clause of the same comment, the zeta(3) tail squeeze\n`s(a(n)) < 1/n^2 < s(a(n)-1)`, is NOT claimed. No other property of A001953\nor A001954 is claimed. The general complementarity of nonhomogeneous Beatty\nsequences is classical 2-Wythoff theory and is cited as a literature\nprerequisite, not claimed as a new result.","title":"The differences of A049473 occupy two shifted Beatty sequences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb0d103b541a4455c50e13e42bd1c2d8e0dd4f165bd93299eaefebac19cc4dea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a049591-prime-gap-divisor-characterization-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A049591, NAME (verbatim):\n\n> Odd primes p such that p+2 is composite.\n\nBenoit Cloitre's COMMENT of April 13, 2002 (verbatim):\n\n> Sequence appears also to give all n > 1 such that there is no prime p satisfying the inequality n < p < n+tau(n)^2 where tau(n)=A000005(n).\n\nFor natural numbers, define\n`Term(n) := Odd(n) and Prime(n) and not Prime(n+2)`, and define\n`NoPrimeInGap(n) := for every natural p, Prime(p) implies not\n(n < p and p < n + card(divisors(n))^2)`. The literal refuted statement is\n`for every natural n, 1 < n implies (Term(n) iff NoPrimeInGap(n))`.\n\nThe NAME definition is not refuted. The Bernoulli, `f(2p)=p`, Ianakiev 2019\nquotient, and Ordowski 2020 congruence comments are not claimed or assessed.","title":"Refutation of the A049591 divisor-count prime-gap characterization","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049591-prime-gap-divisor-characterization-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e872cc00cfb72f9ac17f5d8027993456cbe6b5ab297d5969a674fba104c6c046.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A051903, NAME (`%N`, verbatim):\n\n> Maximum exponent in the prime factorization of n.\n\nVelin Yanev's FORMULA conjecture (`%F`, verbatim):\n\n> Conjecture: a(n) = a(A003557(n)) + 1. This relation together with a(1) = 0 defines the sequence. - _Velin Yanev_, Sep 02 2017\n\nAUTHOR (`%A`, verbatim):\n\n> _Labos Elemer_, Dec 16 1999\n\nWith `a n := n.primeFactors.sup n.factorization` and the frozen\n`D5/S1/Deficit/AlmostAdditivity.primeRadical`, the literal theorem is\n\n```lean\na 1 = 0 ∧ (∀ n : ℕ, 1 < n → a n = a (n / primeRadical n) + 1) ∧\n ∀ b : ℕ → ℕ, b 1 = 0 → (∀ n : ℕ, 1 < n → b n = b (n / primeRadical n) + 1) →\n ∀ n : ℕ, 0 < n → b n = a n\n```\n\nHere primeRadical is A007947, and A003557(n) = n / primeRadical(n) on the positive sequence\ndomain. The quotient uses natural-number division. The empty supremum gives\n`a 0 = 0`, an out-of-range extension; no recurrence at zero or one is claimed.\n\nThe third conjunct formalizes the %F line's second sentence: the relation\ntogether with `a(1) = 0` determines the sequence on positive indices. Every\nnatural-valued function `b` with the same base value and recurrence agrees with\n`a` at every positive index; its value at zero is unrestricted.","title":"Yanev's maximum prime exponent recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0d0430533cabe0a4f236b631d856a8b57f5ccb2cbd8e61ebe54e029dcfc89b40.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a054096-stephan-triangle-second-column","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A054096, NAME (`%N`, verbatim):\n\n> T(n,2), array T as in A054090.\n\nRalf Stephan's target COMMENT (`%C`, verbatim):\n\n> Conjecture: sequence is A006183 shifted right. - _Ralf Stephan_, Jan 15 2004\n\nOEIS A054090, NAME (`%N`, verbatim):\n\n> Triangular array generated by its row sums: T(n,0) = 1 for n >= 0, T(n,1) = r(n-1), T(n,k) = T(n,k-1) - (-1)^k * r(n-k) for k = 2, 3, ..., n, n >= 2, r(h) = sum of the numbers in row h of T.\n\nOEIS A054090, FORMULA (`%F`, verbatim):\n\n> T(n, k) = T(n, k-1) - (-1)^k * Sum_{j=0..n-k} T(n-k, j), with T(n, 0) = 1, and T(n, 1) = Sum_{j=0..n-1} T(n-1, j).\n\nOEIS A006183, NAME (`%N`, verbatim):\n\n> a(n) = (n+1)*a(n-1) + (2-n)*a(n-2).\n\nThe literal proved statement is\n\n`forall T B, IsA054090Triangle T -> IsShiftedA006183 B -> forall n, 2 <= n -> T n 2 = B (n - 1)`.\n\nA054096 has offset 2 and A006183 has offset 1, and their `%S` and `%T` data\nlines are term-for-term identical. Thus \"shifted right\" means\n`A054096(n) = A006183(n-1)` for `n >= 2`. Writing `B(j) = a(j-1)` turns\nA006183's recurrence into\n`B(j) = j*B(j-1) + (3-j)*B(j-2)`, exactly the recurrence encoded by\n`IsShiftedA006183`.\n\nNot claimed here are other columns of A054090; the row-sum sequence A054091;\nthe generating-function form of A006183, including Gladkovskii's 2013\ncontinued fraction `%F`, or its equivalence to the recurrence; or any\nparticular term of A054096.","title":"Stephan's A054096 second-column conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/094080ce8012040f85d5971f18b634e8704f8982297453dd87a3449f9d4e4eb5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a059894-stephan-complement-reverse-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A059894, NAME (`%N`, verbatim):\n\n> Complement and reverse the order of all but the most significant bit in binary expansion of n. n = 1ab..yz -> 1ZY..BA = a(n), where A = 1-a, B = 1-b, ... .\n\nRalf Stephan's recurrence conjecture (`%F`, verbatim):\n\n> a(1) = 1, a(2n) = a(n) + 2^(floor(log_2(n))+1), a(2n+1) = a(n) + 2^floor(log_2(n)) (conjectured). - _Ralf Stephan_, Aug 21 2003\n\nRalf Stephan's digit-count conjecture (`%F`, verbatim):\n\n> A000120(a(n)) = A000120(A054429(n)) = A023416(n) + 1 (conjectured). - _Ralf Stephan_, Oct 05 2003\n\nAUTHOR (`%A`, verbatim):\n\n> _Marc LeBrun_, Feb 06 2001\n\nThe definition `a` applies the quoted transformation to the little-endian\nlist `Nat.digits 2 n`: drop its last entry, reverse the remaining list, map\nnatural subtraction `1 - d`, append `[1]`, and evaluate with `Nat.ofDigits 2`.\nThe definition `complementRest` omits only the reversal and represents A054429\non positive inputs. A000120 counts one bits and A023416 counts zero bits in\nthe positive canonical binary expansion.\n\nThe literal theorem statement is:\n\n```lean\na 1 = 1 ∧\n(∀ n : ℕ, 0 < n → a (2 * n) = a n + 2 ^ (Nat.log 2 n + 1)) ∧\n(∀ n : ℕ, 0 < n → a (2 * n + 1) = a n + 2 ^ (Nat.log 2 n)) ∧\n(∀ n : ℕ, 0 < n →\n (Nat.digits 2 (a n)).count 1 = (Nat.digits 2 (complementRest n)).count 1 ∧\n (Nat.digits 2 (a n)).count 1 = (Nat.digits 2 n).count 0 + 1)\n```\n\nHere `Nat.log 2 n` agrees with the floor of the base-two logarithm on the\npositive domain. The counting and recurrence clauses have the explicit\nguard `0 < n`; no such clauses at zero are asserted.","title":"Stephan's A059894 complement-reversal conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a059894-stephan-complement-reverse-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd45c94424bdecf1e8a5a2d333bee3f821a3d9f235a8c74274419bc2dd49fae2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a060693-schroeder-peak-alternating-quadratic-moment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A060693, NAME:\n\n> Triangle (0 <= k <= n) read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks.\n\nCOMMENT:\n\n> T(n,k) is the number of Schroeder paths (i.e., consisting of steps U=(1,1), D=(1,-1), H=(2,0) and never going below the x-axis) from (0,0) to (2n,0), having k peaks. Example: T(2,1)=3 because we have UU*DD, U*DH and HU*D, the peaks being shown by *.\n\nFORMULA:\n\n> Conjecture: Sum_{k=0..n} (-1)^k*T(n,k)*(n+1-k)^2 = 1+n+n^2. - _Werner Schulte_, Jan 11 2017","title":"Schulte's alternating quadratic peak moment for Schröder paths","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a060693-schroeder-peak-alternating-quadratic-moment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a01514366f949caebffa6fdf7acfbbe1030ce0de1f277a6d1f00bb520c4c7a3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a064618-stirling-transform-totient-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A064618 NAME:\n\n> Stirling transform of (n!)^2.\n\nOEIS A064618 COMMENT (Peter Bala, Jan 15 2018):\n\n> Conjecture: for fixed k = 1,2,..., the sequence a(n) (mod k) is eventually periodic with the exact period dividing phi(k), where phi(k) is the Euler totient function A000010.\n\nThe finite formula for this entry is `a(n) = T (fun k => (k.factorial : ℤ)) n`; its proved onset is `n ≥ m`\nfor every positive modulus `m`. The entry has offset 0.\n\nThe companion entry is recorded separately with its own Library note and dossier:\n\nA004123 NAME (offset 1):\n\n> Number of generalized weak orders on n points.\n\nA004123 COMMENT (Peter Bala, Jul 08 2022):\n\n> Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).","title":"A064618 totient-period conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a064618-stirling-transform-totient-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac778d0815a89b9d7424829727bdf08f3b0f2870657bc60fd5f0c705de3cdb71.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a068012-correction-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let a(n) count subsets of {1,...,n} whose element sum is zero modulo six.\nDavid A. Corneth's September 13, 2025 FORMULA contribution states:\n\n> a(3*k+1) = 2*a(3*k) - 2^(k-1), k >= 1, 3*k + 1 = n.","title":"Corneth's second recurrence for OEIS A068012","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a068012-correction-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/42bc42b21dbfd07f584388377b7150146c2f4b238b30b1f67b7ffafbad326f29.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a068012-subset-sum-doubling","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let a(n) count subsets of {1,...,n} whose element sum is zero modulo six.\nDavid A. Corneth conjectured on September 13, 2025 that a(n)=2a(n-1)\nwhenever n>2 and three does not divide n-1.","title":"The A068012 subset sum doubling conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a068012-subset-sum-doubling/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2c5ecc999921dcbd59c7f521b07dd9d7bbb5fc1381f0b523fa61bd7565bef11e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a072872-power-minus-index-prime-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A072872, NAME (verbatim):\n\n> a(n) is the smallest positive number k such that n divides 2^k - k\n\nCOMMENTS (verbatim; Benoit Cloitre, Jul 28 2002):\n\n> If n is a power of 2, a(n) = n. Conjecture : if n > 47, a(n) < prime(n). - _Benoit Cloitre_, Jul 28 2002\n\nThe literal claim is `∀ n > 47, a(n) < prime(n)`, where `prime(n)` is the\nn-th prime. The formal definition uses\n`a(n) = sInf {k ∈ ℕ | 0 < k ∧ n ∣ 2^k - k}` and discloses the convention\n`sInf ∅ = 0`; no general existence claim for this set is made.","title":"Refutation of the A072872 prime-index upper-bound conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a072872-power-minus-index-prime-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a890dd4bff28e6bf823d471acab6ea7c3f8a9add9efdbccd1610e4eb43269c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A074724, NAME (`%N`, verbatim):\n\n> Highest power of 3 dividing F(4n) where F(k) is the k-th Fibonacci number.\n\nMarcus's FORMULA (`%F`, verbatim):\n\n> a(n) = 3^A051064(n) (conjectured). - _Michel Marcus_, May 17 2022\n\nBala's FORMULA (`%F`, verbatim):\n\n> Conjecture: a(n) = (sigma(3*n) - sigma(n))/(sigma(3*n) - 3*sigma(n)), where sigma(n) = A000203(n). Equivalently, a(n) = A088838(n) - A074724(n). - _Peter Bala_, Jun 10 2022\n\nThe AUTHOR line (`%A`, verbatim) is:\n\n> _Benoit Cloitre_, Sep 04 2002\n\nWith `a n = 3 ^ padicValNat 3 (Nat.fib (4 * n))`, the exact claim is\n`forall n : Nat, 0 < n ->\n(a n = 3 ^ (padicValNat 3 n + 1) and\na n * (sigma 1 (3 * n) - 3 * sigma 1 n) = sigma 1 (3 * n) - sigma 1 n)`.\nHere A051064(n) is `v_3(3n) = v_3(n)+1`. Bala's denominator is multiplied out\nin natural numbers; its positivity is proved within the proof. The separate\n\"Equivalently\" remark involving A088838 is not part of this claim. The\npositive-index guard excludes the totalized Lean value at zero.","title":"The Marcus and Bala formulas for the power of three in F(4n)","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/96a17bc1914f0442d0685fe488ccdc6d5e3a4415a9e6fce87a4b03164e89b71b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a076502-nested-recurrence-floor-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A076502 %N (verbatim):\n\n> a(1)=1, a(n)=n-a(n-a(n-a(n-1)))\n\n%C (verbatim; Benoit Cloitre, Nov 08 2002):\n\n> Conjecture : a(n) - c*n is bounded and a(n)=floor(c*n) + 0 + 1 or +2 - _Benoit Cloitre_, Nov 08 2002\n\n%F (verbatim; Benoit Cloitre, Nov 08 2002):\n\n> a(n) is asymptotic to c*n with c=0.5698..is the positive root of x^3-x^2+2*x-1 - _Benoit Cloitre_, Nov 08 2002\n\nThe literal objects are the nested recurrence, `c` as the unique positive root\nof `x^3-x^2+2*x-1`, and the conjunction that `a(n)-c*n` is bounded and\n`a(n) = floor(c*n) + 0, 1 or 2` for every positive `n`. The formal theorem\nrefutes only the second conjunct, which refutes the conjunction.","title":"Refutation of the A076502 nested-recurrence floor-offset conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a076502-nested-recurrence-floor-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f29cc2d73940835ed722f228544040be0bfced2dd2f20fc3683156176110fef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3).\n\n> Triangle of trinomial coefficients T(n,k) (n >= 0, 0 <= k <= 2*n), read by rows: n-th row is obtained by expanding (1 + x + x^2)^n.\n\n> a(0)=1, a(1)=2, a(2)=5, a(3)=11, a(n)=2*a(n-1)+a(n-2)-a(n-3)-a(n-4) for n>3. - _Philippe Deléham_, Oct 25 2006\n\n> Conjecture: a(n) = Sum_{j=0..n/2} A027907(n+1-j,2*j+1), n >= 0. - _Werner Schulte_, Sep 29 2015","title":"The trinomial odd diagonal of OEIS A077864","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4df312a3b1650320b7dbda7516d5bb66896f17e471ce9a4120839ba2efa4ca6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a077866-least-two-subset-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kimberling's 2022 OEIS A077866 comment asks whether the number `b(N)` of\nsubsets of `{1,...,N}` with more than one element, whose least two elements\nsum to the maximum, satisfies `b(0)=b(1)=b(2)=0` and\n`b(n+3)=A077866(n)` for every `n>=0`.","title":"Least-two-element subset count","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077866-least-two-subset-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8675182429fb33c1ed808e4e179c500ccbdbaadd4d76d0191d62d621acb41e75.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a079063-sqrt-prime-gap-lower-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A079063, NAME (verbatim):\n\n> Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1\n\nCOMMENT (verbatim):\n\n> Inspired by Andrica's conjecture. If it is true, a(n)>1 for all n.\n\nFORMULA (verbatim; Benoit Cloitre, Feb 02 2003):\n\n> Conjecture: there is a constant c>0 such that for n large enough, a(n)>c*sqrt(n) and we can take c=0.4. More precisely, there are 2 constants A and B such that A=lim sup n ->infinity a(n)/sqrt(n) exists = 0.75....; B=lim inf n ->infinity a(n)/sqrt(n) exists =0.46.... - _Benoit Cloitre_, Feb 02 2003\n\nThe literal object is `a(n)`, the least `k >= 1` such that\n`sqrt(p_(n+k)) - sqrt(p_n) > 1`, where `prime(n) = p_n` is the n-th\nprime. The refuted claim is its weakest quantified eventual lower-bound form:\nthere exist `c > 0` and `N` such that every `n >= N` satisfies\n`a(n) > c*sqrt(n)`. This refutes the proposed `c = 0.4` and the claimed\npositive `liminf = 0.46...` a fortiori. The existence or value of either\nlimsup or liminf is not formalized.","title":"Refutation of the A079063 square-root prime-gap lower-bound conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079063-sqrt-prime-gap-lower-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61efa8fdce20c908ebfefef3a38a9f4dc4f75d03cdfa05b132a92b24b06adba7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A079278 states (verbatim):\n\n> %N A079278 Define a rational sequence {b(n)} as b(1) = 1, b(n) = b(n-1) + 1/(1 + 1/b(n-1)) for n > 1; a(n) is the denominator of b(n).\n> %F A079278 Conjecture (Quet): a(m+1) = a(m)^2 + a(m)^3 / a(m-1)^2 - a(m)*a(m-1)^2 for m >= 2.\n> %D A079278 Suggested by _Leroy Quet_, Feb 14 2003.\n> %A A079278 _N. J. A. Sloane_, Feb 16 2003\n> %O A079278 1,2\n\nThe literal proved statement is\n\n```lean\ntheorem result (m : ℕ) (hm : 2 ≤ m) :\n (b (m - 1)).den ^ 2 ∣ (b m).den ^ 3 ∧\n (b (m + 1)).den = (b m).den ^ 2 + (b m).den ^ 3 / (b (m - 1)).den ^ 2 -\n (b m).den * (b (m - 1)).den ^ 2\n```\n\nThe public definition `b` is the rational iteration in `%N`, with only the\nharmless total extension `b 0 = 0`. The public theorem `result` states both\nconjuncts directly with Lean's reduced-rational denominator projection\n`(b k).den`. Consequently the public declarations alone identify `a(k)` with\nthe denominator of `b(k)` and entail Quet's `%F`; no private bridge theorem is\nneeded by an external reader to connect a separately named integer sequence.\n\nHere `/` and subtraction are natural-number operations. The divisibility\nconjunct proves that the quotient in Quet's formula is exact, rather than\nsilently interpreting an inexact quotient by floor division. The proof also\nestablishes the inequality required for the natural-number subtraction.\n\nNot claimed here are a recurrence for the numerator sequence A079269; any\nresult about A355615, A080581, or A080582; exhaustive literature coverage;\npriority for the proof; or that finite computations replace the kernel proof.","title":"Quet's A079278 reduced-denominator recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0901e3daa33ca0d843392932c09cce70ff361f3269ad176af62145a23ce12572.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a084068-pell-companion-gcd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Joseph A. Stocke's July 28, 2025 conjecture in OEIS A084068 states\n`A084068(n) = gcd(A001108(n), A001109(n))` for every n at least one.\nThe entry still labels this formula a conjecture when inspected on\nSeptember 8, 2026.","title":"The A084068 Pell gcd conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a084068-pell-companion-gcd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/352453640dbb232b7a202183824e04b7753f0f14d2825759466a313ec879eaa1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a089610-square-interval-prime-count-eventual-increase-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A089610 %N (verbatim):\n\n> Number of primes between n^2 and (n+1/2)^2.\n\nOEIS A089610 %C (verbatim sub-conjecture):\n\n> There exists an n_1 such that a(n) is < a(n+1) for all n >= n_1.\n\nOEIS A089610 %A (verbatim):\n\n> _Cino Hilliard_, Dec 30 2003\n\nThe literal refuted statement is\n`∃ N : ℕ, ∀ n : ℕ, N ≤ n → a(n) < a(n+1)`, where\n`a(n) = ((Finset.Ioc (n*n) (n*n+n)).filter Nat.Prime).card`. Since\n`(n+1/2)^2 = n^2+n+1/4`, this natural interval selects exactly the integer\nprimes between the endpoints printed in the NAME, as T. D. Noe's comment\nnotes. The separate conjecture that `a(n) > 1` after `n = 17` and Oppermann's\npositivity conjecture `a(n) > 0` are not claimed or resolved here.","title":"Refutation of the A089610 eventual strict-increase conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a089610-square-interval-prime-count-eventual-increase-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8cf836f47b9c29fa0d8ad3e3b7e55345c815b50f24c221226f64c2316e506857.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a090079-yanev-run-compression-closed-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A090079, NAME (`%N`, verbatim):\n\n> In binary expansion of n: reduce contiguous blocks of 0's to 0 and contiguous blocks of 1's to 1.\n\nYanev's FORMULA conjecture (`%F`, verbatim):\n\n> Conjecture: a(n) = (2^(A005811(n)+1) + (1-(-1)^n)/2 - 2)/3. - _Velin Yanev_, Dec 12 2016\n\nThe AUTHOR line (`%A`, verbatim):\n\n> _Reinhard Zumkeller_, Nov 20 2003\n\nThe proved statement is\n`forall n : Nat, 3 * a n = 2 ^ (runs n + 1) + n % 2 - 2`, where\n`runs n = ((Nat.digits 2 n).splitBy (· == ·)).length` and\n`a n = Nat.ofDigits 2 (((Nat.digits 2 n).splitBy (· == ·)).map List.head!)`.\nThe list is little-endian; reversing a word preserves its run count and\nreverses the order of its compressed digits. Thus these definitions implement\nthe literal operation. The number of binary runs is the quantity A005811(n)\nused in the conjecture. The integer identity `(1-(-1)^n)/2 = n mod 2`\nidentifies the parity term. The statement is multiplied by 3 to stay in ℕ.\nFor positive n the numerator is nonnegative; at zero both sides are zero.\nLean subtraction is truncated natural subtraction.","title":"Yanev's closed form for binary run compression","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090079-yanev-run-compression-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8e6150e89d6566bcc13a11dc7a0c88300fe7fabbf89acecd7766b7709092fae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a090825-bernoulli-integrality-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A090825, NAME (verbatim):\n\n> Nonprimes n such that (3/2)*(1/n)*(2*n+1)*(3^n+1)*B(2*n) is an integer, where B(k) denotes the k-th Bernoulli number.\n\nBenoit Cloitre's COMMENT of February 11, 2004 (verbatim):\n\n> Conjecture: composite numbers with all prime factors in A053176 are in the sequence.\n\nFor natural numbers, define\n`F(n) := (3/2)*(1/n)*(2*n+1)*(3^n+1)*B(2*n)` in the rationals and\n`A053176(p) := Prime(p) and not Prime(2*p+1)`. The literal refuted statement\nis: for every composite natural `n > 1`, if every prime divisor `p` of `n`\nsatisfies `A053176(p)`, then `F(n)` is an integer.\n\nThe separate COMMENT clause asserting `F(p) = 1 (mod p)` for primes, and the\nquestion whether the subsequence beginning 91, 247 is finite, are not claimed\nor assessed.","title":"Refutation of Cloitre's A090825 Bernoulli-integrality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090825-bernoulli-integrality-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e77e4684ca9247222daeeae168c6896f3040ec9260150af956f47a91072e30e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a091190-somos-prime-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> More generally, given A(x) satisfies x*A(x)^p = B(x*A(x^p)) where B(x) = x/(1-p*x), then it appears that A(x) is an integer series only when p is prime.\n\nFor every natural $p\\geq2$, a rational solution with $A(0)=1$ exists for $xA(x)^p=B(xA(x^p))$, where $B(x)=x/(1-px)$. Every such solution has integer coefficients if and only if $p$ is prime. The coefficientwise expression for $B$ is inlined in `IsGF`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| x*A(x)^p = B(x*A(x^p)) | IsGF preserves the outer and inner substitutions | exact registered convention |\n| B(x)=x/(1-p*x) | IsGF inlines coefficient zero at n=0 and p to the power n-1 otherwise | exact registered convention |\n| offset-zero first term one | IsGF requires constant coefficient one | exact registered convention |\n| integer only when prime | claim proves equivalence with primality for every normalized solution | exact registered convention |\n| exponents in the registered domain | claim requires p at least two and asserts rational existence | exact registered convention |","title":"The Somos prime integrality family in OEIS A091190/A091200","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091190-somos-prime-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/103106be0f6ac6749660808d8016e8aa3381c762fa312fd2c3098a226513c0bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a091259-reduced-numerator-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Take the ratio of the divisor sum of cubes to the ordinary divisor sum, reduce\nit to lowest terms, and read the numerator modulo three. Michel Marcus\nconjectured on August 11, 2024 that the result is the indicator of a\nprime-exponent condition: one exactly when every prime congruent to two modulo\nthree occurs to an even exponent, zero otherwise.","title":"The A091259 congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091259-reduced-numerator-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11c248beb657715f9505baa7aa700cb5e455e0d8636830b6965dddbf132f5f24.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a091915-stephan-even-product-partition-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A091915 states (`%N`, `%F`, `%A`, `%O`, and `%Y`, verbatim):\n\n> %N A091915 Maximum of even products of partitions of n.\n> %F A091915 For n>6, a(n+3) = 3a(n) (conjectured). - _Ralf Stephan_, Dec 02 2004\n> %A A091915 _Jon Perry_, Feb 12 2004\n> %O A091915 1,2\n> %Y A091915 Cf. A000792, A091916.\n\nThe literal proved statement is\n\n```lean\ntheorem result (n : ℕ) (hn : 6 < n) :\n ∃ a : ℕ,\n IsGreatest\n {q : ℕ | ∃ p : Nat.Partition n, Even p.parts.prod ∧ p.parts.prod = q} a ∧\n IsGreatest\n {q : ℕ | ∃ p : Nat.Partition (n + 3), Even p.parts.prod ∧ p.parts.prod = q}\n (3 * a)\n```\n\nFor `n = 1`, no partition has an even product, so the feasible set is empty.\nThe existential formulation avoids assigning an arbitrary total-function value there:\nfor every `n > 6`, it supplies a greatest even product `a` at total `n` and proves that\n`3*a` is the greatest even product at `n+3`. It retains attainment and the upper\nbound through `IsGreatest`, rather than asserting only membership.\n\nNot claimed here are any odd-product statement or result about A091916; any\ngenerating-function form; any parity result about the number of partitions; an\nexhaustive literature result or priority claim; or a greatest-even-product value\nfor an empty feasible set.","title":"Stephan's A091915 maximum even partition-product conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d3a98a0727ec3da751632e83c2caa25bd7bfc5d205bf28e59320b1f0687427d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a092028-least-self-divisor-power-minus-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A092028, NAME (`%N`, verbatim):\n\n> a(n) is the smallest m > 1 such that m divides n^m-1\n\nCOMMENTS (`%C`, verbatim; Farideh Firoozbakht, Mar 26 2004):\n\n> Conjecture 1: All terms of this sequence are primes.\n\n> Conjecture 2: a(n) is the smallest prime factor of n-1 or for n>2, A092028(n) = A020639(n-1).\n\nThe literal formal claim is `∀ n > 2, a(n) = minFac(n − 1)`, where\n`a(n) = sInf {m ∈ ℕ | 1 < m ∧ m ∣ n^m − 1}`. In the natural numbers,\n`sInf ∅ = 0`; for every `n > 2`, the defining set is nonempty. Conjecture 1\nis a corollary because `minFac(n − 1)` is prime, and is not formalized as a\nseparate public theorem.","title":"The least self-divisor exponent in OEIS A092028","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a092028-least-self-divisor-power-minus-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2a8c90a34e13090cdc22fa43b4699416030b145267813c38a62a3bdead088567.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The OEIS entry states, verbatim:\n\n> %N A094802 a(n) = smallest k such that all of 1 through n divides k!.\n> %C A094802 It is conjectured that after n=4 the sequence is prime for n prime or the previous prime for n not prime.\n> %A A094802 _Jon Perry_, Jun 11 2004\n> %O A094802 1,2\n\nThe literal proved statement is: for all natural n and p, if `5 <= n`,\n`Nat.Prime p`, `p <= n`, and every prime q at most n satisfies `q <= p`,\nthen `IsLeast {k : Nat | Nat.lcmUpto n | k !} p`. For a prime n, p is n;\notherwise it is the previous prime. Thus p is precisely the largest prime\nat most n. The entry's example `a(6)=5 as 5!=120 which is divisible by\n1,2,3,4,5 and 6.` agrees. The bound is necessary: `a(4)=4` is not prime.\nMathlib defines `Nat.lcmUpto n` as `(Finset.Icc 1 n).lcm id`, so the `%N`\ndivisibility clause is precisely `Nat.lcmUpto n | k !`.\n\nNot claimed: a general formula for n at most four beyond the computed\nvalues, any general theorem about the Kempner function away from this\nprefix-lcm input, or an asymptotic estimate for `lcmUpto`.","title":"Perry's A094802 least factorial index conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecf14b62317c2211fa91500a739c8cf1a1a8f078955066a78e7e27347571e560.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a096127-factorial-square-divisibility-prime-power","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A096127, NAME (verbatim):\n\n> a(n) is the largest k such that (n^2)!/(n!)^k is an integer.\n\nCOMMENT (verbatim):\n\n> Conjecture: a(n)=n+1 only when n is prime or a power of a prime. [Verified for n=2..5000. - _Amiram Eldar_, Apr 06 2021]\n\nOEIS A096126 states the same assertion. The A096127 offset is 2.","title":"Exact factorial-square divisibility in A096127","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a096127-factorial-square-divisibility-prime-power/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9792ba8b50b42f6de30b63487674869e40a8317f15bb4b4e67f3cd2c8048daf9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a097602-square-counting-recurrence-square-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A097602, NAME (`%N`, verbatim):\n\n> a(n+1) = a(n) + number of squares so far; a(1) = 1.\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: a(n) = m^2 iff m mod 3 > 0.\n\nCOMMENT (`%C`, verbatim):\n\n> a(n) is a square iff n is congruent to {1, 4} mod 9. - _Vladeta Jovovic_, Aug 30 2004\n\nFORMULA (`%F`, verbatim):\n\n> a(9*n+1) = (3*n+1)^2; a(9*n+4) = (3*n+2)^2.\n\nThe conjecture is read as a statement about attained values: for every positive\nm, the value m^2 occurs as a(n) at some positive index n if and only if 3 does\nnot divide m. The literal reading with independent free n and m is false: at\nn=2 the sequence has a(2)=2, while for example m=1 makes the right side true\nand a(2)=m^2 false.","title":"Square values in the A097602 square-counting recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a097602-square-counting-recurrence-square-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77a3ae2f89f28f360b9aac4ca72f34e948334a4455015d5fdedc2856db3dc80d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a100952-log-bounded-semiprime-partition-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A100952, NAME (verbatim):\n\n> Numbers that cannot be written as p*q+r with three distinct primes p, q and r.\n\nCOMMENT (verbatim; Jonathan Vos Post, Nov 25 2004):\n\n> A weaker conjecture: every integer greater than 60 (or some larger value based on further search) may be partitioned into a prime p and a semiprime qr, where the prime p is bounded by log(min(q,r)). Chen (1978) showed that all sufficiently large even numbers are the sum of a prime and the product of at most two primes. Zumkeller's conjecture effectively extends this from \"even\" to both even and odd integers. - _Jonathan Vos Post_, Nov 25 2004\n\nAUTHOR (verbatim):\n\n> _Reinhard Zumkeller_, Nov 23 2004\n\nThe literal refuted statement is\n`exists B : Nat, 60 <= B and forall m : Nat, B < m -> Rep m`, where\n`Rep m` means that there exist natural primes `p`, `q`, and `r` satisfying\n`m = p + q*r` and\n`(p : Real) <= Real.log ((min q r : Nat) : Real)`. Here `Real.log` is the\nnatural logarithm, equal factors `q = r` are allowed, and no distinctness\ncondition is imposed. The refutation therefore covers every stricter reading\nthat requires the three primes to be distinct. It does not resolve the\ncompleteness conjecture for A100952 or Seidov's `p*q-r` conjecture.","title":"Refutation of the A100952 log-bounded semiprime-partition conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a100952-log-bounded-semiprime-partition-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db6fa8b6fe64da7a5e0fdaa3c07af1fb6fb23a79f6a49aed59b10c0ebe945b9f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a103585-sloping-binary-period-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A103585, NAME (verbatim):\n\n> Consider numbers k such that (A102370(k)-k)/2 = 1; read them mod 4 to get the sequence.\n\nRalf Stephan's COMMENT of May 18, 2007 (verbatim):\n\n> Sequence appears to have period 43.\n\nFor natural numbers, define\n`s(k) := k + Sum_{m in Icc(1,k+1)} (if (k+m) mod 2^m = 0 then 2^m else 0)`,\nthe finite form of the A102370 formula; define `P(k) := (s(k) = k+2)`; and\ndefine `A(i) := nth(P,i) mod 4`. Thus `A` is zero-indexed and OEIS `a(m)`\nequals `A(m-1)`. The literal refuted statement is\n`for every natural i, A(i+43) = A(i)`.\n\nThe definition of A102370, the `%N` definition of A103585, any true period,\nand eventual aperiodicity are not claimed or assessed.","title":"Refutation of the A103585 period-43 comment","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fc2af22ff35432cea5343c71692226caa32ada6a1f510e25b122301f984b62a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a104863-floor-sqrt-recurrence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A104863, NAME (verbatim):\n\n> a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=10, a(2)=30.\n\nFORMULA conjecture line (verbatim; Ralf Stephan, Nov 15 2010):\n\n> For n>=17, a(n) = a(n-2) + a(n-4) + 1 (conjectured). If true then for m>5, a(2*m+1) = 4*F(m) + 25*F(m+1) + 1 and a(2*m+2) = 8*F(m) + 30*F(m+1) + 1 with F(n) = A000045(n). - _Ralf Stephan_, Nov 15 2010\n\nThe sequence begins with `a(1) = 10` and `a(2) = 30`, and each later\nterm is the natural floor square root of the sum of the squares of the two\npreceding terms. The literal conjecture is\n`∀ n ≥ 17, a(n) = a(n−2) + a(n−4) + 1`.","title":"Refutation of the A104863 floor-square-root recurrence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a104863-floor-sqrt-recurrence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/632eb7ed7d131d152feecaac85577dbf99dbfa1911ef95f2075c691295e0cd1a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a105595-barry-triangle-row-sums-odd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A105595, NAME (`%N`, verbatim):\n\n> Row sums of number triangle A105594.\n\nIts COMMENT conjecture (`%C`, verbatim) is:\n\n> Conjecture : all terms are odd.\n\nIts FORMULA (`%F`, verbatim) is:\n\n> a(n)=sum{k=0..n, mod(sum{j=0..n, abs(mu(binomial(n, j)))*mod(binomial(j, k), 2)}, 2)}\n\nOEIS A105594, FORMULA (`%F`, verbatim):\n\n> T(n, k) = mod(Sum_{j=0..n}(abs(mu(binomial(n,j)))*mod(binomial(j,k),2)), 2).\n\nThe literal proved statement defines\n`rowEntry n k = (sum j in range (n+1), abs(mu(choose n j)) *\n(choose j k mod 2)) mod 2`, defines\n`rowSum n = sum k in range (n+1), rowEntry n k`, and proves\n`forall n : Nat, Odd (rowSum n)`.\n\nThe following are not claimed:\n\n- any other property of A105594;\n- equivalence between the `%N` matrix-product phrasing\n `abs(A103447)*A047999 mod 2` and the displayed `%F` sum;\n- exhaustive literature coverage or a priority claim.\n\nThe delivery formalizes the two displayed `%F` sums and discloses that source\nchoice rather than identifying the matrix-product description with them.","title":"Every row sum of the A105594 triangle is odd","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/47ff4e0c6cfa764f78f178cedb32875e70ae9dbeb49a34c4174cb7e8b2a02711.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a107928-harmonic-mean-numerator-closed-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A107928, NAME (`%N`, verbatim):\n\n> a(n) is the numerator of harmonic mean of a(n-1) and a(n-2).\n\nFORMULA (`%F`, verbatim):\n\n> a(1)=2; a(2)=3; n>=3: a(n) = numerator(2*a(n-1)*a(n-2)/(a(n-2)+a(n-1))).\n\nCONJECTURE (`%F`, verbatim; Ralf Stephan, Dec 01 2010):\n\n> Conjecture: G.f.: x*(2+3*x+12*x^2+8*x^3+8*x^4)/[(1-2*x)*(1+2*x+4*x^2)]. a(3n) = (3/2)*8^n, a(3n+1) = 3*8^n, a(3n+2) = 2*8^n, for n>0. - _Ralf Stephan_, Dec 01 2010\n\nAUTHOR (`%A`, verbatim):\n\n> %A _Zak Seidov_, Jun 10 2005\n\nThe first terms (`%S`) are 2, 3, 12, 24, 16, 96, 192, 128, 768, 1536,\n1024, 6144, … . The formal claim is\n`∀ m ≥ 1, a(3m) = 12·8^(m−1) ∧ a(3m+1) = 3·8^m ∧\na(3m+2) = 2·8^m`. The Lean definition uses `a 0 = 0` as an offset-one\nsentinel and implements the reduced numerator as `Rat.num ... .toNat`.\nOnly the three-residue closed form is formalized by elementary block\ninduction; no priority claim is made. The printed generating function with\n+8x^4 expands to 2, 3, 12, 24, 32, ... and disagrees with the sequence at\nn = 5 (a(5) = 16), whereas changing the sign to -8x^4 yields the rational\ngenerating function x(2+3x+12x^2+8x^3-8x^4)/(1-8x^3) of the proved closed\nform; the generating-function sentence is neither formalized nor claimed,\nand the discrepancy is recorded here, not corrected upstream.","title":"Closed form of the A107928 harmonic-mean-numerator recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a107928-harmonic-mean-numerator-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fdb6a2b24cb9e55fe1fd120979ff2adfa074127640c0b9dc49038d3b0cfdae1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a110454-distinct-partition-concatenation-zero-indices","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A110454, NAME (verbatim):\n\n> Largest composite number obtained by concatenation of parts of a distinct partition of n, or 0 if no such number exist.\n\nCOMMENT (verbatim; Amarnath Murthy, Aug 04 2005):\n\n> Conjecture a(n) = 0 only for n = 1, 2 and 4.\n\nThe Maple program by R. J. Mathar, dated Feb 08 2008, examines every\npermutation of every distinct partition and accepts a concatenation exactly\nwhen `m > 4` and `m` is not prime. Its loop includes single-part partitions.","title":"The zero indices of A110454","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110454-distinct-partition-concatenation-zero-indices/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/53661a3774d4cb2839cee64b4af59303b085eff3f38b084d1dcfe120e8e33299.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a110491-ratajczak-t-transform-determinant","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lechoslaw Ratajczak's August 4, 2021 COMMENT on OEIS A110491 states:\n\n> Conjecture: {a(n-1), n>=1} is the T-transform of A093178, where T maps a sequence {b(n), n>=1} to the sequence {c(n)} defined by c(n) = det(M_n), where M_n is the n X n matrix with elements M_n(i,j) = b(2*j) for i>j and M_n(i,j) = b(i+j-1) for i<=j.\n\nHere A093178 is `b(r)=1` when `r` is even and `b(r)=r` when `r` is odd.\nThe matrix indices `i,j` in the comment are one-based. Independently, define\nthe integer sequence `a` by `a(0)=1`, `a(1)=2`, and\n\n`a(m+2)=2*a(m+1)+4*(m+1)*m*a(m)`.\n\nFor zero-based finite indices set `i1=i+1` and `j1=j+1`, and define the\norder-`n` integer matrix by the literal source branches\n\n`M_n(i,j)=b(2*j1)` if `i1>j1`, and\n`M_n(i,j)=b(i1+j1-1)` if `i1<=j1`.\n\nThe resolved statement is the unbounded assertion\n\n```lean\ntheorem result (m : ℕ) :\n (sourceMatrix (m + 1)).det = sourceA m\n```\n\nfor every natural `m`. This is exactly the source conjecture at every\npositive order, not a finite prefix, a surrogate matrix, a definition of\n`a` by determinants, or a consequence of an assumed generating function.","title":"Ratajczak's A110491 T-transform determinant conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110491-ratajczak-t-transform-determinant/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5af1f3bc17f3a5dbdcf4bd362b75e9d21715aea331040f9962f98aff0955622c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a112041-dale-triple-scaling-prime-multiples-of-ten","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A112041, NAME (`%N`, verbatim):\n\n> Numbers k such that 1*k + 1, 3*k + 1, 9*k + 1, 27*k + 1 are all primes.\n\nHarvey P. Dale's COMMENT conjecture (`%C`, verbatim):\n\n> Conjecture: all terms except the first are multiples of 10. - _Harvey P. Dale_, Mar 26 2015\n\nThe literal proved statement is\n`forall k : Nat, 0 < k -> (Nat.Prime (k + 1) and\nNat.Prime (3 * k + 1) and Nat.Prime (9 * k + 1) and\nNat.Prime (27 * k + 1)) -> (k = 4 or 10 divides k)`.\nInfinitude of A112041, a characterization beyond this implication, and all\nother comments on A112041 are NOT claimed.","title":"The A112041 prime scalings force the index to be four or divisible by ten","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112041-dale-triple-scaling-prime-multiples-of-ten/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9a4153cb255d60e340bc1167a616124a48f8e95db84d1bb45485293fe9a4d46.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a112302-lerch-somos-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Johannes W. Meijer's June 27, 2016 COMMENTS paragraph in OEIS A112302\nstates the exact external problem:\n\n> With Phi(z, p, q) the Lerch transcendent, define LP(n) = (1/n) * sum(Phi(1/2, n-k, 1) * LP(k), k=0..n-1), with LP(0) = 1. Conjecture: Lim_{n -> infinity} LP(n) = A112302.\n\nThe proof retains all three registered definitions and the complete\nfour-conjunct theorem:\n\n```lean\nnoncomputable def LerchKernel (s : ℕ) : ℝ :=\n ∑' j : ℕ, (1 / 2 : ℝ)^j / ((j : ℝ) + 1)^s\n\nnoncomputable def SomosConstant : ℝ :=\n Real.exp (∑' j : ℕ, Real.log ((j : ℝ) + 1) / (2 : ℝ)^(j + 1))\n\ndef LPSource (a : ℕ → ℝ) : Prop :=\n a 0 = 1 ∧ ∀ n : ℕ, 0 < n →\n a n = (1 / (n : ℝ)) *\n ∑ k : Fin n, LerchKernel (n - k.val) * a k.val\n\ntheorem result :\n (∀ s : ℕ, 0 < s →\n Summable (fun j : ℕ => (1 / 2 : ℝ)^j / ((j : ℝ) + 1)^s)) ∧\n Summable (fun j : ℕ => Real.log ((j : ℝ) + 1) / (2 : ℝ)^(j + 1)) ∧\n (∃! a : ℕ → ℝ, LPSource a) ∧\n (∀ a : ℕ → ℝ, LPSource a →\n Filter.Tendsto a Filter.atTop (nhds SomosConstant))\n```\n\nA112302 denotes the real Somos constant through its decimal expansion.\nThe kernel is Phi(1/2,s,1)=2 Li_s(1/2). For k in Fin n, n-k is positive;\nthe recurrence divides by n only under n>0. Both summability clauses and\nunique source existence are part of the result, not assumed premises.","title":"OEIS A112302: Meijer's Lerch recurrence limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112302-lerch-somos-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a4260954fbe1923f99412954bbd7685be96dbb63e7288f470a966ad8cd368c6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a112399-morken-first-divisor-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A112399, revision 41, source timestamp\n2026-09-19T09:22:12-04:00, NAME defines:\n\n> a(n) = Sum_{k=1..n, gcd(k,n)=1} mu(k), where mu(k) = A008683(k) (the Moebius function).\n\nIts FORMULA section attributes the first of two conjectures to Jamie Morken,\nSep 18 2026:\n\n> a(n) = Sum_{d|n, mu(d)!=0} (Sum_{m=1..n/d, gcd(m,d)=1} mu(m)).\n\nThe question is this equality for every positive integer n. Both intervals\ninclude their endpoints. The function mu is integer-valued and signed, with\nmu(1)=1. The outer sum ranges over positive divisors of n for which mu(d) is\nnonzero, and n/d is exact integer division. The right inner coprimality is\nwith d; the left coprimality is with n. There is no squarefree assumption on n\nand no finite upper bound. The conjecture has AI-assisted provenance, which\nsupplies no mathematical proof. The second A112399 conjecture, also indexed\nby A400203, and the A372706 product-parameter identity are excluded.","title":"Morken's first A112399 divisor-sum conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f34e5ec680f1e12d2a7f282f3608e1bfa3de5530e60d9074313b504f5be1b74f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a115039-prime-square-perfect-power-representation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A115039 and A115038 state the same conjecture:\n\n> Conjecture: There will always be an x,y,n such that x^2 + p = y^n for all primes p.\n\nThe quantified reading is: for every prime natural number `p`, there are\nnatural numbers `x`, `y`, and `n` with `2 <= n` and\n`x^2 + p = y^n`. The lower bound is the standard perfect-power condition;\nwithout it, exponent one makes the equation immediate.","title":"Prime square perfect-power representation","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115039-prime-square-perfect-power-representation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/496354006aec300a4d9bf772f0806537f7f3a71ee2294877d161686b2447d859.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a115770-carryless-seven-five-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"[OEIS A115770](https://oeis.org/A115770) defines the nonnegative integers\n$k$ satisfying $7k=\\operatorname{A048720}(11,k)$ and conjectures that they\nare exactly the integers satisfying $5k=\\operatorname{A048720}(13,k)$.\nThe exact assertion is\n\n$$\n\\forall k\\in\\mathbb{N},\\qquad\n7k=C(11,k)\\ \\Longleftrightarrow\\ 5k=C(13,k).\n$$\n\nThe same $k$ occurs on both sides, $\\mathbb{N}$ includes zero, and there is\nno upper bound on $k$. Multiplication outside $C$ is ordinary multiplication.\nThe operation $C$ is binary coefficient convolution over $\\mathrm{GF}(2)$,\nas defined by [OEIS A048720](https://oeis.org/A048720):\n\n$$\n\\operatorname{bit}(C(u,k),t)\n=\\left(\\sum_{i=0}^{t}\n\\operatorname{bit}(u,i)\\operatorname{bit}(k,t-i)\\right)\\bmod 2.\n$$\n\nDigits are indexed from zero at the least significant bit and have value\nzero or one. The row-five/row-seven conjecture in A115872 expresses this\nsame question.","title":"Seven and five versus binary carryless multiplication","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4549b7d407a35e2e5006c5fcf58395afdc7a5ad4d9807243cc21704a1b84720e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a115801-carryless-multiplier-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"[OEIS A115801](https://oeis.org/A115801) defines the nonnegative integers\n$k$ satisfying $9k = \\operatorname{A048720}(25,k)$ and conjectures that the\nsame integers satisfy $15k = \\operatorname{A048720}(19,k)$.\nThe exact assertion is\n\n$$\n\\forall k\\in\\mathbb{N},\\qquad\n9k=C(25,k)\\ \\Longleftrightarrow\\ 15k=C(19,k).\n$$\n\nHere $\\mathbb{N}$ includes zero, and the multiplications $9k$ and $15k$ are\nordinary natural-number multiplication. The binary digits are indexed from\nthe least significant bit. The operation $C$ is genuine coefficient\nconvolution over $\\mathrm{GF}(2)$:\n\n$$\n\\operatorname{bit}(C(u,k),t)\n=\n\\left(\\sum_{i=0}^{t}\n\\operatorname{bit}(u,i)\\operatorname{bit}(k,t-i)\\right)\\bmod 2.\n$$\n\nThis is the binary polynomial multiplication table\n[OEIS A048720](https://oeis.org/A048720), rather than ordinary multiplication.","title":"Equivalent ordinary and binary carryless multiplier conditions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/45f914e825e8e43824ac84b99516f08ea972df605f4cfdb37e8f2e963179f49d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a115803-carryless-thirteen-eleven-containment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A115805 revision 14 defines the natural numbers k satisfying\n13*k = A048720(29,k), and its cross-reference conjectures containment in\nA115803. A115803 revision 13 defines the natural numbers k satisfying\n11*k = A048720(31,k). Both sequences include zero. The exact question is\n\n$$\n\\forall k\\in\\mathbb N,\\qquad 13k=C(29,k)\\ \\Longrightarrow\\ 11k=C(31,k).\n$$\n\nHere C is the binary carryless product of A048720 revision 62. If a_i and\nb_i are the binary digits of a and b, the t-th binary digit of C(a,b) is\nthe sum of a_i*b_(t-i) over 0 <= i <= t, reduced modulo two. Thus C is\ncoefficient convolution over GF(2). Multiplication outside C is ordinary\nnatural-number multiplication. Every occurrence of k denotes the same\nnatural number, with no positivity or size restriction.","title":"The A115805 to A115803 carryless containment question","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d1168a31976b08374c880734946875810bd33d98ce5d4a3af18730d79c3e7f67.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a115809-carryless-seventeen-nine-containment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A115809 asks whether the integers satisfying its defining condition are\na subset of A115801. In the notation of the source, the exact question is\n\n$$\n\\forall k\\in\\mathbb{N},\\qquad\n17k=C(49,k)\\ \\Longrightarrow\\ 9k=C(25,k).\n$$\n\nHere $\\mathbb{N}$ includes zero and the same $k$ occurs in both sides. The\nproducts $17k$ and $9k$ are ordinary natural-number products. The operation\n$C$ is binary coefficient convolution over $\\mathrm{GF}(2)$:\n\n$$\n\\operatorname{bit}(C(u,k),t)\n=\n\\left(\\sum_{i=0}^{t}\n\\operatorname{bit}(u,i)\\operatorname{bit}(k,t-i)\\right)\\bmod 2.\n$$\n\nThis is the carryless polynomial product described by OEIS A048720, not\nordinary multiplication. The source question and operation are recorded at\n.","title":"Containment of the A115809 carryless multiplier condition","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0cf22c984675ea8cc544479b13b327a7619f353ff6182d48c0f24dba28493d31.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a116184-generalized-harmonic-thirty-seven-cube-progression","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A116184, NAME (`%N`, verbatim):\n\n> Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ].\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: All terms of the arithmetic progression 3+36k belong to a(n).\n\nAUTHOR (`%A`, verbatim):\n\n> _Alexander Adamchuk_, Apr 08 2007\n\nThe first terms (`%S`) are:\n\n> 3, 37, 39, 73, 75, 111, 147, 148, 183, 185, …\n\nThe literal claim is `∀ k ∈ ℕ, 37³ ∣ num(H(36, 3+36k))`, where `Rat.num`\nis the reduced numerator.\nThe same OEIS comment also discusses primes of the form `37m-1` and cases of\ndivisibility by `37^4`; neither assertion is claimed here.","title":"Cubic divisibility of generalized harmonic numerators along Adamchuk's progression","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a116184-generalized-harmonic-thirty-seven-cube-progression/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/35e27592ccc5fb0cc6f1a5772d6cd3b31b356761660eda0b09c33b07a73e6bc4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a119623-divisor-pair-product-prime-preimages","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A119623, NAME (Zak Seidov, Jun 08 2006):\n\n> Composite numbers for which the second elementary symmetric function of divisors (s2) is prime.\n\nOEIS A119623, COMMENTS (Zak Seidov, Jun 08 2006):\n\n> Terms in A119616 are always prime if n is prime p and s2(p)=p, hence it is interesting to find composite numbers for which s2 is also prime. Relative values of s2 are: s2=47,97,163,457,733,2203,3733,7993,10723,11317,21313,22147,26557,33403,57283,61417,67153,79393,101467,149323,160453,162727,174337,272683,296827,318793,358273,432907,440383,486583,551767,639007,832687,843043,911917,961183,1152913,1202017,1277593,1322743,1375303,1462897,1567327,1824997,1878883. Otherwise the sequence s2 gives numbers which appear in A119616 at least twice (and conjecture is that exactly twice).\n\nOEIS A119616, NAME:\n\n> Second elementary symmetric function of divisors of n.\n\nAccordingly, `S2(n)` is defined here verbatim as `e_2(divisors(n))`, the\nsecond elementary symmetric function of the finite divisor multiset. Lean's\n`Nat.divisors 0` is empty, so this definition gives `S2(0)=0`. For positive\n`n`, it is equivalently the sum of `d*e` over unordered pairs of distinct\ndivisors, or `(sigma_1(n)^2 - sigma_2(n))/2` as recorded in A119616 %F; these\nare equivalent descriptions, not the adopted definition.\n\nThe adopted reading is: for every composite `n` for which `p = S2(n)` is\nprime, `{m >= 1 : S2(m) = p} = {n, p}`. This is the only reading consistent\nwith both \"at least twice\" and \"exactly twice\": the identity `S2(p) = p`\nprovides the second occurrence, while \"exactly\" excludes every other\npositive preimage.","title":"Prime values of the divisor-pair product sum have exactly two preimages","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bc9fb8010d6a5dc2e154fe7688e35c87a12a85e7dfd30ea78e19b11b95b3150.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a119690-layman-odd-power-factorial-residue","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A119690, NAME (`%N`, verbatim):\n\n> n! mod n*(n+1)/2.\n\nJohn W. Layman's target COMMENT (`%C`, verbatim):\n\n> It appears that f(n)=(n!)^(2k+1) modulo n(n+1)/2 is n if n is one less than an odd prime, else f(n) is 0, for any integer k. See A175567 for related results involving an even power of n!. - _John W. Layman_, Jul 12 2010\n\nThe entry's FORMULA (`%F`, verbatim):\n\n> a(n) = n if n+1 is an odd prime, a(n) = 0 otherwise.\n\nThe literal proved statement is\n\n`forall n k : Nat, 1 <= n -> (n!)^(2*k+1) mod (n*(n+1)/2) = if\nPrime(n+1) and Odd(n+1) then n else 0`.\n\nThe FORMULA is the `k = 0` case and is already presented by the entry as a\nformula, not a conjecture. The new settlement is Layman's generalization to\nall odd powers `2*k+1`; no priority is claimed for the `k = 0` case.\n\nNot claimed here are A175567 or any result about even powers, the COMMENT\n`All terms are even`, or any other property of A119690.","title":"Layman's odd-power factorial residue classification","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72becc52dabf933fee89eefd666f5c03b97bbcf6f77063ebfe007b5ad2aeb368.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a122399-stirling-power-factorial-prime-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A122399, entry by Vladeta Jovovic, Aug 31 2006; NAME attribution\nsupplied by the orchestrator: Paul D. Hanna, May 31 2022. The following\nNAME and COMMENT text is copied verbatim from the claim in\n`Library/Recurrence/bala2022a122399.md`.\n\nNAME:\n\n> a(n) = Sum_{k=0..n} k^n * k! * Stirling2(n,k).\n\nCOMMENT (Peter Bala, May 31 2022):\n\n> Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.\n\nThe theorem proves that p - 1 is a period for every prime p and every n >= 1.\nIt does not assert that this period is minimal.","title":"Prime periods of the A122399 Stirling power-factorial sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a122399-stirling-power-factorial-prime-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a2a1b9efbdf9c6530dba5225aa069c0a00b896b0750aa89c94e28bef07456970.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a124418-mixed-parity-block-partition-product","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A124418, `%N` (verbatim):\n\n> Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries (0<=k<=floor(n/2)).\n\n`%C` (verbatim):\n\n> Row n has 1+floor(n/2) terms. Row sums are the Bell numbers (A000110). T(n,0)=A124419(n).\n\n`%F` (verbatim; Paul D. Hanna, Nov 08 2006):\n\n> Conjecture: T(n,k) = k!*A049020([n/2],k)*A049020([(n+1)/2],k) where A049020(n,k)=Sum_{i=0..n} S2(n,i)*C(i,k) and S2(n,k)=(1/k!)*Sum_{j=0..k} (-1)^(k-j)*C(k,j)*j^n (the Stirling numbers of 2nd kind).\n\nThe Lean definition `stirling2 r i` counts partitions of an `r`-element set\ninto `i` blocks, which is the object the source names as the Stirling numbers\nof the second kind. The alternating-sum closed form for `S2` printed in the\nFORMULA line is a standard identity for this object and is not formalized here.","title":"Hanna's mixed-parity block partition product in A124418","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f48074731d4e9202df7d86b7a3422adbd7b71be2fd9fa1475ce6f8548f5358da.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a126347-kurkov-reversed-qbell-row-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A126347 has offset `%O 0,6`. Its coefficient data begins (`%S`,\nverbatim):\n\n> 1,1,1,1,1,2,1,1,1,3,3,4,2,1,1,1,4,6,10,9,7,7,4,2,1,1,1,5,10,20,25,26,\n\nIts NAME line (`%N`, verbatim) is:\n\n> Triangle, read by rows, where row n lists coefficients of q in B(n,q) that satisfies: B(n,q) = Sum_{k=0..n-1} C(n-1,k)*B(k,q)*q^k for n>0, with B(0,q) = 1; row sums equal the Bell numbers: B(n,1) = A000110(n).\n\nMikhail Kurkov's conjecture (`%F`, verbatim) is:\n\n> Conjecture: R(n,n) is the (n+1)-th reversed row polynomial where R(0,0) = 1, R(n,k) = R(n-1,n-1) + x^n * Sum_{j=0..k-1} R(n-1,j) for 0 <= k <= n. - _Mikhail Kurkov_, Jul 06 2025\n\nThe AUTHOR line (`%A`, verbatim) is:\n\n> _Paul D. Hanna_, Dec 31 2006, May 28 2007\n\nThe referenced Bell-number entry A000110 has NAME line (`%N`, verbatim):\n\n> Bell or exponential numbers: number of ways to partition a set of n labeled elements.\n\nThe exact formal statement is:\n\n```lean\nforall n : Nat, kurkovR n n = (qBell (n + 1)).reverse\n```\n\nHere `qBell` uses Wagner's q-Bell recurrence. The formal reversal is\nreflection about the natural degree. The locally proved facts that `qBell n`\nis monic of degree `Nat.choose n 2` ensure that this is the fixed-row\ncoefficient reversal meant by the source.","title":"Kurkov's reversed q-Bell row polynomial","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126347-kurkov-reversed-qbell-row-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c90c35cfc5162fdce76c187d9a991cee93d91c0fb987e56e6ce95b9f92cdbd38.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a126762-least-witness-exponent-shift-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A126762, NAME (verbatim):\n\n> a(n) is the least k > n such that the remainder when n^k is divided by k is n.\n\nCOMMENT (verbatim; Thomas Ordowski, Aug 03 2018):\n\n> a(n) is the smallest number k > n such that n^k == n (mod k). Conjecture: a(n) is the smallest number k > n such that n^(k-1) == 1 (mod k). Thus a(n) is coprime to n. - _Thomas Ordowski_, Aug 03 2018\n\nThe published b-file line `363 366` already contradicts the coprimality\nconsequence, since `gcd(363, 366) = 3`.","title":"Refutation of the A126762 least-witness exponent-shift conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126762-least-witness-exponent-shift-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8b55d7dd2f06de343d08fb3d86d52cdf92e4a3aefe7c2d0f123cfc1d8cfe0ddd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a129598-gcd-totient-position-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A129598, NAME (verbatim):\n\n> a(n) = n * A111089(n).\n\nAntti Karttunen's COMMENT of May 1, 2007 (verbatim):\n\n> Conjecture: differs from A050399 at the positions given by A089966. E.g., a(15)=75, instead of A050399(15)=225, a(30)=150, instead of A050399(30)=450, a(33)=363, instead of A050399(33)=1089, a(45)=225, instead of A050399(45)=1225.\n\nThe NAME of A050399 states (verbatim):\n\n> Least k such that n = A009195(k) (= gcd(phi(k), k)).\n\nThe NAME of A089966 states (verbatim):\n\n> Numbers k such that omega(k) = gpf(k) mod lpf(k), where omega=A001221, gpf=A006530 and lpf=A020639.\n\nFor natural numbers, define `g(m) := gcd(m, phi(m))`. Let\n`greatestPrimeFactor(n)` be the last element of the prime-factor list, with\ndefault zero, and define `a(1):=2` and\n`a(n):=n*greatestPrimeFactor(n)` otherwise. Define `b(n)` to be the least\npositive `m` satisfying `g(m)=n` when such an `m` exists, and zero otherwise.\nFinally, define `inA089966(n)` to mean `n=1`, or that `n` is positive and the\nnumber of its distinct prime factors equals\n`greatestPrimeFactor(n) mod minFac(n)`. The literal refuted statement is\n`for every natural n, 1 Numbers m such that z(m)=(0,0) with z as defined in A131851.\n\nCOMMENTS (`%C`, verbatim):\n\n> Intersection of A131854 and A131855: A131851(a(n))=0, A131852(a(n))=0;\n\n> conjecture: a(n) mod 5 = 0.\n\nOEIS A131851, NAME (`%N`, verbatim):\n\n> Real part of the function z(n)=Sum(d(k)*i^k: d as in n=Sum(d(k)*2^k), i=sqrt(-1)).\n\nFORMULA (`%F`, verbatim):\n\n> z(n) = if n=0 then (0, 0) else z(floor(n/2))*(0, 1) + (n mod 2, 0), complex multiplication.\n\nThe literal claim is `∀ m : ℕ, z m = 0 → 5 ∣ m`, where\n`z : ℕ → GaussianInt` is defined by\n`Nat.binaryRec 0 (fun b _ w => (if b then 1 else 0) + (⟨0, 1⟩ : GaussianInt) * w)`.\nThe Lean statement quantifies over every natural m, including zero.","title":"Zeros of the A131853 binary Gaussian evaluation","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ec2b2bc2956dd53c8e8deef020e16cbecb502c2a123b4d767ec08c8f1676de0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a132393-kurkov-nested-stirling-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: T(n+m+1,m+1) = (n+m)! * (Product_{i=1..m} (n+i)^i) * Sum_{j_1=0..n} Sum_{j_2=0..j_1} ... Sum_{j_m=0..j_{m-1}} (-1)^(n*m + j_1 + j_2 + ... + j_m) * (Product_{1 <= q < p <= m} (j_q - j_p + p - q)^2) * Product_{t=1..m} (n+t-1)_{j_t} / ((n-j_t+t)^(m+1) * (j_t+m-t)!) where (x)_{n} is the falling factorial defined as (x)_{n} = Product_{k=0..n-1} (x-k). - _Mikhail Kurkov_, May 24 2026\n\nThe target quantifies over every natural n and every m >= 1, with equality\nin the rationals. T is the unsigned Stirling number of the first kind.\nThe chain is n >= j_1 >= ... >= j_m >= 0. Falling factorial means the\ndisplayed descending product. This is the exact source assertion registered\nbefore formal probes in\nhttps://github.com/the-omega-institute/trureturing/issues/9456.","title":"A132393: Kurkov's nested Stirling identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a132393-kurkov-nested-stirling-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a9a9d1bbebf92f6be57e59b8a78624bed3cb25d9eb870c32b3838799db34ee4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a133901-palindrome-square-reversal-infinitude","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A133901, NAME (verbatim):\n\n> Numbers in A128921 whose square is not a palindrome.\n\nOEIS A128921, NAME (verbatim):\n\n> Palindromes m such that reverse of m^2 is also a square.\n\nOEIS A133901, COMMENT (verbatim):\n\n> Conjecture: Sequence is infinite.\n\nThus membership means a decimal palindrome `p` whose square is not a\npalindrome and whose reversed square is a perfect square. In the formal\nmodule, `rev10` is the numeric value of the reversed decimal string; it is\ndistinct from the digit-list reversal `D` in\n`D5/S3/Arith/SumInConcatenation.lean`.","title":"Palindrome-square reversal infinitude in A133901","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/703cc294d16a53f9477fa678303dcf8107a52e9c21d442f41825bd6a41515823.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A133907, NAME (`%N`, verbatim):\n\n> Least prime number p such that binomial(n+p, p) mod p = 1.\n\nOrdowski's COMMENT (`%C`, verbatim):\n\n> Conjecture: a(n) is the smallest prime p such that Sum_{k=1..n} k^(p-1) == n (mod p). Thus a(n) >= A317358(n). - _Thomas Ordowski_, Jul 29 2018\n\nThe offset is `%O 1,1`. The AUTHOR line (`%A`, verbatim) is:\n\n> _Hieronymus Fischer_, Oct 20 2007\n\nThe related unsigned COMMENT (`%C`, verbatim) is:\n\n> Also the least prime number p such that p divides floor(n/p) or p > n.\n\nDefine `a n = sInf {p : Nat | p.Prime ∧ (n+p).choose p ≡ 1 [MOD p]}`.\nFor each natural `n`, let\n`S(n) = {p : Nat | p.Prime ∧ (∑ k ∈ Finset.Icc 1 n, k^(p-1)) ≡ n [MOD p]}`.\nThe full-quantifier reading is: for every `n : Nat`, if `0 < n`, then\n`a n ∈ S(n)` and, for every `q : Nat`, `q ∈ S(n)` implies `a n ≤ q`.\nThus membership includes primality and the sum congruence, and leastness\ncompares against every prime satisfying that congruence. The exact Lean type is:\n\n```lean\ntheorem result (n : ℕ) (_hn : 0 < n) :\n IsLeast {p : ℕ | p.Prime ∧\n (∑ k ∈ Finset.Icc 1 n, k ^ (p - 1)) ≡ n [MOD p]} (a n)\n```\n\nThe sum includes both endpoints, division is natural-number floor division,\nand `p - 1` is natural subtraction. The unused hypothesis `_hn : 0 < n`\nretains the OEIS domain; no strengthened premise is introduced. The defining\nprime set is nonempty, so the natural infimum is an attained least element.\nOnly the `Conjecture:` sentence is settled. The clause\n`Thus a(n) >= A317358(n)` is a consequence, not claimed separately, and the\nunsigned comment receives no separate public theorem or resolution claim.","title":"Ordowski's binomial and power-sum least-prime conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2798be62883147d20d97a778103912ff8f30df2e8225c94818a1bca35f9debc2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a136162-krizek-next-prime-product-quadruplet","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A136162, NAME (`%N`, verbatim):\n\n> List of prime quadruplets {p, p+2, p+6, p+8}.\n\nJaroslav Krizek's COMMENT (`%C`, verbatim, including both sentences and its\nattribution):\n\n> {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {Q-4, Q-2, Q+2, Q+4} where Q is a product of a pair of twin primes {q, q+2} (for prime q = 3) because numbers Q-2 and Q+4 are for q>3 composites of the form 3*(12*k^2-1) and 3*(12*k^2+1) respectively (k is an integer). Conjecture: {11, 13, 17, 19} is the only prime quadruplet {p, p+2, p+6, p+8} of the form {q*(nextprime(q))-4, q*( nextprime(q))-2, q*( nextprime(q))+2, q*( nextprime(q))+4} where q is a prime (for prime q = 3). - _Jaroslav Krizek_, Jul 07 2017\n\nDefine `nextPrime(q)` as the least prime greater than or equal to `q+1`,\nimplemented by `Nat.find (Nat.exists_infinite_primes (q + 1))`, and define\n`Q(q) = q * nextPrime(q)`. The literal proved statement is\n`forall q : Nat, q.Prime -> ((Nat.Prime (Q q - 4) and\nNat.Prime (Q q - 2) and Nat.Prime (Q q + 2) and Nat.Prime (Q q + 4)) iff\nq = 3)`.\n\nThe preceding twin-prime-product sentence, any other property or\nclassification of prime quadruplets, and every other A136162 comment are NOT\nclaimed. Natural subtraction is truncated subtraction.","title":"The A136162 next-prime product has four prime neighbors only at q=3","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a136162-krizek-next-prime-product-quadruplet/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d71d2c7397ed91f22557e8fe42d500c20d7009931411a5cd9c6e85604ebd28d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a140869-sundaram-triangle-complement-prime","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A140869, NAME (`%N`, verbatim):\n\n> T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1.\n\nCOMMENT (`%C`, verbatim; Vincenzo Librandi, Nov 18 2012):\n\n> Conjecture: If h does not belong to the sequence, then 4*h+5 is prime.\n\nThe literal claim is\n`∀ h, ¬(∃ m n, 1 ≤ n ≤ m ∧ T(m,n) = h) → 4h+5 prime`.\nHere `T(m,n)` and the membership predicate use natural-number truncated subtraction\nand division. On the stated domain `m ≥ n ≥ 1`, the numerator\n`2mn+m+n−2` is at least `2`, so these operations agree with the intended expression.","title":"The A140869 Sundaram-type triangle complement","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a140869-sundaram-triangle-complement-prime/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1eb5ef3fad0820c0e416f7d0793858ee32e83191ee6c9609bf4cc5543f797144.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a144837-bala-lucas-five-adic-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A144837, NAME (`%N`, verbatim):\n\n> a(n) = Lucas(5^n).\n\nPeter Bala's Nov 14 2022 recurrence (verbatim):\n\n> a(n) = a(n-1)^5 + 5*a(n-1)^3 + 5*a(n-1) with a(1) = 11.\n\nThe settled FORMULA line (`%F`, verbatim):\n\n> Conjecture: a(n+1) == a(n) (mod 5^(n+r+1)) for n >= r.\n\nAUTHOR (`%A`, verbatim):\n\n> _Artur Jasinski_, Sep 22 2008\n\nThe offset is `%O 1,1`. Both `n` and `r` range over the natural numbers\nincluding zero, with the sole hypothesis `r <= n`. The sequence is\ninteger-valued and defined by Bala's recurrence, extended to zero with\n`a(0)=1=Lucas(5^0)=Lucas(1)`. This gives `a(1)=11=A144837(1)` and agrees\nwith the OEIS domain `n >= 1`. Congruence means that the integer\n`5^(n+r+1)` divides the integer difference `a(n+1)-a(n)`.\n\nThe exact Lean statement is:\n\n```lean\ntheorem result (n r : ℕ) (hr : r ≤ n) :\n (5 : ℤ) ^ (n + r + 1) ∣ a (n + 1) - a n\n```\n\nOnly the Conjecture line is settled for the recurrence interpretation,\ncovering every `n >= 1` and also `n=0`. The Lucas representation, the\n5-adic limit (A269591), and A268922 are not claimed. The formal definition\ndoes not use Lucas numbers.","title":"Bala's Lucas recurrence and its growing five-adic congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6dc1d23aabe7089bcf493962ef0e8d3cfa7b335eb923f9f9ec7ccb1e3169bb06.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a145110-granvik-redheffer-support-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A145110, NAME (verbatim):\n\n> Number of elements in the Redheffer matrix that contribute to the Moebius function.\n\nCOMMENT conjecture line (verbatim):\n\n> a(11) to a(30) are conjectured to be 11, 23, 13, 18, 19, 25, 17, 29, 19, 30, 26, 26, 23, 45, 27, 30, 32, 39, 29, 48. [From _Mats Granvik_, Dec 21 2008]\n\nFor `n≥2`, let `B_n` have rows and columns indexed by `1,…,n`. Its entry\nis 1 exactly when `j=1` or `i` divides `j`, except that `B_n(n,n)=0`.\nA position `(i,j)` contributes when there is a permutation `σ` of\n`{1,…,n}` with `σ(i)=j` and `B_n(k,σ(k))=1` for every `k`. Let `a(n)`\nbe the number of such positions. Thus the conjecture is the conjunction\nof the twenty displayed equalities, starting with `a(11)=11` and ending\nwith `a(30)=48`.\n\nThe entry's array for `n=8` is the transpose of this matrix convention and\ncontains 13 positions. At `n=1` the literal modified-matrix definition gives\n0 against the entry's datum 1; that index is outside the conjectured list.","title":"The A145110 conjectured Redheffer support counts","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b52bd8f8caf15954484acb5755a3aa8f51af6bdcb79056153ad34f81f49f8b34.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a146557-collinear-triple-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala's July 24, 2025 comment conjectures that n squared divides a(n)\nwhen three does not divide n. Here a(n) counts unordered collinear triples\nin (ZMod n) squared whose first coordinates, and second coordinates, are\npairwise distinct. Collinearity means the difference determinant is zero.","title":"The A146557 collinear triple divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a146557-collinear-triple-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cadb669736dc3fc504f36d3567af77ef5458d9258a32e77e91cd4165b710a686.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a152020-rational-denominator-cube-shift","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A152020, NAME:\n\n> Denominator of 8/(9*n^2) divided by 9.\n\nFORMULA (Andrew Howroyd, Jul 25 2018):\n\n> a(n) = n^2/gcd(n^2, 8). - _Andrew Howroyd_, Jul 25 2018\n\nFORMULA (Andres Cicuttin, Sep 19 2017):\n\n> Conjecture: a(n) = denominator((n-2)^3/n^2). - _Andres Cicuttin_, Sep 19 2017","title":"The rational-denominator cube shift in A152020","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152020-rational-denominator-cube-shift/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa83aefd2249bb4fdd4e638157063fb21cb051604217c353d3ec6d492f82c714.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a152661-mathar-first-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A152661, revision 23, defines a(n), for n >= 2, as the number of\npermutations of {1,...,n} whose first two entries have the same parity.\nIts first conjectural FORMULA, attributed to R. J. Mathar, states\n\n$$\n(3-n)a(n)+2(n-2)a(n-1)+(n-1)(n-2)(n-3)a(n-2)=0.\n$$\n\nThe assertion here is this exact equality in the integers for every natural\nn >= 4. Each sequence value is the actual permutation cardinality, cast to\nthe integers. The domain ensures that n, n-1 and n-2 all lie in the source\ndomain. The adjacent conjecture a(n)=2*A077613(n-1) is excluded.","title":"Mathar's first recurrence for OEIS A152661","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152661-mathar-first-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecd29f7a6ea724e915b72b64473236ce6ecc857788b98d2c480da7b6f03d297f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a160627-israel-laguerre-four-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A160627, NAME (`%N`, verbatim):\n\n> Numerator of Laguerre(n, 4).\n\nRobert Israel's COMMENT conjecture (`%C`, verbatim):\n\n> Conjecture: all terms are odd. - _Robert Israel_, Mar 29 2018\n\nOEIS A160628, NAME (`%N`, verbatim):\n\n> Denominator of Laguerre(n, 4).\n\nRobert Israel's COMMENT conjecture (`%C`, verbatim):\n\n> Conjecture: all terms are odd. - _Robert Israel_, Mar 29 2018\n\nThe literal proved statement defines\n`L(n) = sum_{k=0}^n C(n,k)(-4)^k/k!` in the rationals and proves\n`forall n : Nat, Odd (L n).num and Odd (L n).den`. Here `num` and `den`\nare the numerator and denominator of the reduced rational value. The\ndefinition is the explicit binomial sum for the classical Laguerre polynomial\nat `x=4`; it does not use a Mathlib Laguerre polynomial API.\n\nNo value other than `x=4`, property beyond reduced numerator and denominator\nparity, other A160627 or A160628 comment, or priority claim is included.","title":"Laguerre(n, 4) has odd numerator and denominator","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a160627-israel-laguerre-four-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5cfe88b9363d810f688c99c43f1a1cc6e04f0327f9e5282ad8ae44d6ee559cf3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A163617, `%N` (verbatim):\n\n> a(2*n) = 2*a(n), a(2*n + 1) = 2*a(n) + 2 + (-1)^n, for all n in Z.\n\nFORMULA (verbatim; Velin Yanev, Dec 17 2016):\n\n> Conjecture: a(n) = A003188(n) + (6*n + 1 - (-1)^n)/4.\n\nOEIS A003188, `%N` (verbatim), followed by its `%F` XOR formula:\n\n> Decimal equivalent of Gray code for n.\n\n> a(n) = n XOR floor(n/2), where XOR is the binary exclusive OR operator.\n\nOnly the natural-number half, initialized by `a(0) = 0`, is formalized. The\nnegative-index half is OEIS A163618 and is not claimed.","title":"A Gray-code closed form for the A163617 binary recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a163617-gray-code-binary-recurrence-closed-form/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44dc564ec496bf22849639a65123448c7e5e134d2b871cffbae839a3cd554f83.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a165719-eighth-quadratic-integrality-and-sole-prime","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The OEIS entry states, verbatim:\n\n> %N A165719 Integers of the form k*(k+9)/8.\n> %C A165719 Only one term is a prime number (17). Are all others composite?\n> %F A165719 Conjecture: Valid k values are pairs of the form (8*m-1,8*m) for m=1,2,3.... If true, 17 is the only prime term. - _Bill McEachen_, May 06 2026\n> %A A165719 _Vladimir Joseph Stephan Orlovsky_, Sep 24 2009\n> %O A165719 1,1\n\nFor every positive natural k, `8 | k*(k+9)` is equivalent to the existence\nof a positive natural m with `k+1=8*m` or `k=8*m`. A natural a is an\nintegral term exactly when some positive k satisfies `8*a=k*(k+9)`.\nThe value 17 is a term and a prime; every other term is greater than one\nand not prime. The example's positive-parameter restriction is retained.\nThe equations avoid natural-number floor division and truncated subtraction.\n\nNot claimed: a result for nonpositive parameters, a new method of prime\nclassification, any assertion about arbitrary quadratic forms, an\nexhaustive literature search, or priority beyond the settlement of these\nnamed assertions in the searched scope.","title":"A165719 integrality and the sole prime","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a165719-eighth-quadratic-integrality-and-sole-prime/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/509bc66572a98571f936c978bb9b67d8b8d699f12395cf242a999960a5b67d5f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a167138-logarithmic-power-closure","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: For all integers m > 0, Sum_{n>=0} L(n)^m * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.\n\nA167138, A167139 and A167141 repeat this one general Hanna question.\nAn integer series has integer ordinary coefficients. Let K be any\ncharacteristic-zero field and L any function from natural numbers to K.\nPut E_L(t)=sum L(n)t^n/n! and ell(X)=log(1+X). The exact conclusion is\nintegrality of E_(L^m)(ell(X)) for every positive integer m whenever\nE_L(ell(X)) is integral, at every degree including zero. The natural\npositive exponent in Lean is the equivalent representation of a positive\ninteger exponent. L(0) is arbitrary. No extra integral-L assumption,\nanalytic convergence, finite support or degree/power bound is introduced.","title":"Hanna's universal logarithmic integer-series closure","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a42d49386331fbe14b7b46590513678775a351fe496825d56ecfd87078f129b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a167540-logarithmic-affine-closure","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> For integers m>0, b>=0, Sum_{n>=0} L(m*n+b) * log(1+x)^n/n! is an integer series whenever Sum_{n>=0} L(n)*log(1+x)^n/n! is an integer series.\n\nThe source is the CONJECTURE comment in OEIS A167540. An integer series has\ninteger ordinary coefficients. The exact formal statement concerns every\ncharacteristic-zero field K, arbitrary L:N->K, every positive natural m and\narbitrary natural b, at every ordinary degree including zero. The only\nmathematical premise is integrality of the entire input logarithmic transform.\nIntegral values of L are derived, with arbitrary L(0). There is no sign,\nnormalization, support, degree, dilation, shift or analytic restriction.","title":"Hanna's full logarithmic arithmetic-subsequence closure","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167540-logarithmic-affine-closure/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e7b61c34392988ea0993fb5ec4d6cadd3b0d7d42b32d78c153c5f3b076033bdd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a167857-sorted-divisor-polynomial-three-primes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The Online Encyclopedia of Integer Sequences, A167857, T. D. Noe, defines\nnumbers whose divisors are represented by an integer polynomial:\n\n> That is, these numbers n have the property that there is a polynomial f(x) with integer coefficients whose values at x=0..tau(n)-1 are the divisors of n, where tau(n) is the number of divisors of n.\n\nThe published question is:\n\n> Are there terms with three distinct prime divisors?\n\nThe exact target is existence of a positive natural n with\n`n.primeFactors.card = 3` and `f : Polynomial Int` such that, for every\n`j < n.divisors.card`, evaluation at the integer cast of j equals entry j\nof the actual increasingly sorted finset of all positive divisors of n.\nBoth factor count and divisor list belong to this same n. The list has no\nduplicates; the inputs start at zero. No minimum, degree bound, numerical\ncandidate, divisor subset, alternate order or rational-only conclusion is\nimposed. Preregistration: https://github.com/the-omega-institute/trureturing/issues/16601.","title":"OEIS A167857: three distinct prime divisors","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167857-sorted-divisor-polynomial-three-primes/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2fe7bfc24ca7526912372c697e46c620f63e1a00c68bc40a6a4c1c7b879ff2a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a175406-log-two-floor-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A175406 %N (verbatim):\n\n> The greatest integer k such that (1+1/n)^k <= 2\n\nOEIS A175406 %C (verbatim):\n\n> The sequence of first differences consists of zeros and ones, with no two consecutive zeros and no more than three consecutive ones.\n\nOEIS A175406 %F (verbatim):\n\n> a(n) = n log 2 + O(1). Conjecture: a(n) = floor((n + 1/2) log 2). - _Charles R Greathouse IV_, Apr 03 2012\n\nOEIS A175406 %A (verbatim):\n\n> _Zak Seidov_, May 01 2010\n\nThe literal claim formalized here is\n`∀ n : ℕ, 1 ≤ n → a n = ⌊((n : ℝ) + 1 / 2) * Real.log 2⌋₊`.\nThe definition of `a` uses the literal natural supremum\n`sSup {k : ℕ | (1 + 1 / (n : ℝ)) ^ k ≤ 2}`; in the natural conditionally\ncomplete order, an unbounded set has `sSup = 0`. The result refutes this\nuniversal claim at one explicit positive index.","title":"Refutation of the A175406 floor formula conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a175406-log-two-floor-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce21248f8c9e9ccb858383ed1b770a1f556873aabe8a3f1922b99cc928117e6a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a177680-twin-prime-average-multiple-of-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A177680, NAME (`%N`, verbatim):\n\n> Numbers n such that 6n and 12n are both the average of twin prime pairs.\n\nCOMMENT (`%C`, verbatim; Harvey P. Dale, Sep 26 2017):\n\n> Conjecture: all terms but the first term are multiples of 5. - _Harvey P. Dale_, Sep 26 2017\n\nFORMULA (`%F`, verbatim):\n\n> a(n)=A066388(n)/6.\n\nThe literal claim is\n`∀ n : ℕ, IsMember(n) → n = 1 ∨ 5 ∣ n`, where `IsMember(n)` means\n`1 ≤ n ∧ Prime(6n−1) ∧ Prime(6n+1) ∧ Prime(12n−1) ∧ Prime(12n+1)`.\nLean natural-number subtraction truncates in general, but the hypothesis `n ≥ 1`\nkeeps each displayed subtraction equal to ordinary subtraction here.","title":"The A177680 twin-prime averages modulo five","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a177680-twin-prime-average-multiple-of-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a9cf98a3473a85fb7fe64d9973d1d2eb066a55c621e6665d780524cd663f0cd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a178294-four-grid-collinear-triples","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A178294 records:\n\n> %N A178294 Number of collinear point triples in a 4 X 4 X 4 X... n-dimensional cubic grid.\n\n> %A A178294 _R. H. Hardin_, suggested by R. J. Mathar in the Sequence Fans Mailing List, May 24 2010\n\n> %F A178294 Conjecture: a(n) = 8^n/2-3*4^n/2+6^n. a(n) = +18*a(n-1) -104*a(n-2) +192*a(n-3). G.f.: 4*x*(-1+7*x)/((6*x-1)*(8*x-1)*(4*x-1)). - _R. J. Mathar_, May 24 2010\n\nThe recurrence and generating function on the FORMULA line are consequences\nof the displayed closed form, rather than separate targets here.","title":"Collinear triples in the four-point grid","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a178294-four-grid-collinear-triples/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/246d9b8567adde8c6938d84202e2ddb6544cabbb08b759e38095fb01d10908ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A181278 defines the sequence as follows:\n\n> Number of 2 X n binary matrices M with rows in strictly increasing order and rows of M*Mtranspose (mod 2) in strictly decreasing order.\n\nColin Barker's FORMULA line of March 27, 2018 states verbatim:\n\n> a(n) = 4*a(n-1) + 4*a(n-2) - 16*a(n-3) for n>3.\n\nThe offset is 1, so `a(1) = 0` is a sequence term. For natural-valued terms,\nthe subtraction-free formal claim is the equivalent assertion\n`a(n) + 16*a(n-3) = 4*a(n-1) + 4*a(n-2)` for every `n >= 4`.","title":"Refutation of Barker's A181278 recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a181278-barker-binary-gram-recurrence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff2f388d0ac47e79d163e3a3014880474cfd7a9372e3dafcc8dbeec8c9d2cb03.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a189733-curtz-denominator-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A189733, revision 36, states:\n\n> a(n) = denominator(B(0,n)). Conjecture: a(6*n)=1, a(1+6*n)=1+3*n, a(2+6*n)=1+2*n, a(3+6*n)=2+3*n, a(4+6*n)=1, a(5+6*n)=6+6*n. a(n) = 2*a(n-6) - a(n-12).\n\nAll indices are natural numbers. Define the actual rational triangle by\nT(m,0)=0, T(m,1)=(-1)^m/(m+1), and\nT(m,k+2)=T(m,k+1)+T(m+1,k), with boundary cases taking precedence.\nThe public `triangle` implements these three branches, and the public `a`\nis exactly its zeroth-row reduced denominator. In particular, a zero\nrational has denominator one. The theorem `result` is the equality of\nthe six-tuples for every natural k, including zero, with no extra premise.","title":"Curtz's six-residue denominator conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a189733-curtz-denominator-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc357463393216ec246a6432783da7ec5cb5afe7f175794b6ef3d512b8f34e0d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a192023-comb-wiener-determinant-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A192023, NAME (verbatim):\n\n> The Wiener index of the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices in the graph.\n\nFORMULA (verbatim):\n\n> a(n) = n*(2*n^2 + 6*n - 5)/3.\n\nCOMMENT (verbatim; Clark Kimberling, Mar 31 2012):\n\n> Conjecture: for n>2, A192023(n-2) is the number of 2 X 2 matrices with all terms in {1,2,...,n} and determinant 2n. - _Clark Kimberling_, Mar 31 2012\n\nHere `A192023(n-2)` means the Wiener index of the comb on\n`2(n-2)` vertices, as specified by the NAME, rather than an assumed\nuse of the closed formula.","title":"Refutation of the A192023 comb-Wiener determinant-count conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192023-comb-wiener-determinant-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7430ed97a68c647739086acd8fc7e53cdf7d0c0a6f7ad60a2cca5b2c26b20ed7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a192763-granvik-mertens-array","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Preregistration: [#20559](https://github.com/the-omega-institute/trureturing/issues/20559).\nMats Granvik's OEIS A192763 COMMENTS field states:\n\n> The main diagonal is the Mobius function times the natural numbers A055615 (conjecture). For k>1 the first row is the Mertens function + 2 = A002321 + 2 (conjecture).\n\nThe positive-index array is defined by Michael Somos's Mathematica and PARI programs in the entry.\nIt is symmetric, $T(1,1)=1$, $T(2,1)=2$, and\n\\[\nT(n,1)=\\frac{-T(n-1,1)-\\sum_{i=2}^{n-1}T(n,i)}{n+1}+T(n-1,1)\n\\quad(n\\ge3).\n\\]\nFor $n>k\\ge2$ it satisfies\n$T(n,k)=T(k,((n-1)\\bmod k)+1)$; for $n\\ge2$ its diagonal is\n$T(n,n)=-\\sum_{i=1}^{n-1}T(n,i)$.\nThe formal definition takes values in $\\mathbb Q$ and extends zero-index cells\nby zero. It uses well-founded recursion with measure\n$4\\max(n,k)$ plus phases three for the upper triangle, two for the diagonal,\none for the lower border, and zero for the strict lower interior.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Somos's positive rational recurrence | `GranvikMertensArray.T` | The displayed branch order and inclusive sums; no integer rounding. |\n| Corner and first border cell | `T 1 1 = 1`, `T 2 1 = 2` by definition | OEIS program normalization. |\n| Main diagonal for every positive index | First conjunct of `result` | `∀ n : ℕ, 1 ≤ n → T n n = (n : ℚ) * (ArithmeticFunction.moebius n : ℚ)`. |\n| First row for every index greater than one | Second conjunct of `result` | `∀ k : ℕ, 2 ≤ k → T 1 k = 2 + ∑ j ∈ Finset.Icc 1 k, (ArithmeticFunction.moebius j : ℚ)`. |\n| Mats Granvik and Olivier Gérard's first Mathematica program | Private `granvik`, `T_eq_granvik` and `T_published_unique` | Its literal border and reverse-window equations define exactly `T`. This equality is consumed by `result`. |\n| Math.SE 50719 | Source note | Its printed corner is three; this settlement uses the OEIS program's corner one. Its index slips are resolved by the executable Mathematica and PARI programs in OEIS. |","title":"Granvik's Möbius diagonal and Mertens row","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192763-granvik-mertens-array/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1b2409c2dc5d433202e7251b924e64e8e10320e97625cdeb5de4cdb5c635ceb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a193020-minus-four-eventual-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A193020, revision 13, attributed to John W. Layman and the OEIS\ncontributors, states the separate assertion `a(2^k-4)=k*(k-1)/2` with its\nsufficiently-large lower-bound caveat. For positive n, a(n) counts distinct\ncomplete order-preserving self-shuffle output words of the ordinary binary\nexpansion of n. Different schedules producing the same word count once.\n\nThe exact selected assertion is\n\n$$\\exists K\\in\\mathbb N,\\quad 3\\le K\\ \\land\\\n \\forall k\\in\\mathbb N,\\quad K\\le k\\ \\Longrightarrow\\\n a(2^k-4)=\\frac{k(k-1)}2.$$\n\nThe OEIS entry includes a(0). The selected event uses positive inputs\nthroughout its tail; it does not impose a source convention at zero.\nThis dossier anchors only the minus-four question.","title":"The A193020 minus-four eventual identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-minus-four-eventual-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/712c2d544a44535a33cf45f9816f8d3a508157022ff1c6dccd51a53feba12aa0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a193020-plus-one-eventual-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A193020, revision 13, with the 2011 entry attribution to John W.\nLayman, states the separate assertion `a(2^k+1)=k^2` with its\nsufficiently-large lower-bound caveat. Here a(n), for positive n, counts\nDISTINCT output words obtained by shuffling the ordinary binary expansion\nof n with itself. Each shuffle preserves the order within both copies;\ndifferent schedules producing the same word contribute only once.\n\nThe exact selected assertion is\n\n$$\\exists K\\in\\mathbb N,\\quad 1\\le K\\ \\land\\\n \\forall k\\in\\mathbb N,\\quad K\\le k\\ \\Longrightarrow\\ a(2^k+1)=k^2.$$\n\nThis dossier anchors only that plus-one question. The other identities in\nA193020 are separate questions.","title":"The A193020 plus-one eventual identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-plus-one-eventual-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8662cbc5ea3eb022bcf7fbdedbc4eaeb9d517e5d2260314e4902196877dda92e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a193351-alternating-divisor-sum-prime-square","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A193351, NAME (`%N`, verbatim):\n\n> Numbers k such that A071324(k) is prime.\n\nCOMMENT (`%C`, verbatim):\n\n> Numbers k such that the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is prime.\n\nCONJECTURE (`%C`, verbatim):\n\n> For k > 3, if the alternating sum of all divisors of k (divisors nonincreasing, starting with k) is a prime number, then k is either a square or twice a square.\n\nThe entry also records:\n\n> This conjecture is verified up to k <= 10^8. - _Shreyansh Jaiswal_, Apr 25 2025\n\nWith `T : ℕ → ℤ` denoting that literal alternating sum, the formal claim is\n`∀ k : ℕ, 3 < k → Nat.Prime (T k).toNat → (∃ t : ℕ, k = t ^ 2) ∨\n(∃ t : ℕ, k = 2 * t ^ 2)`. The use of `.toNat` makes the primality predicate\nexplicit; the proof establishes the required nonnegativity before using it.","title":"Prime alternating divisor sums in OEIS A193351","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193351-alternating-divisor-sum-prime-square/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50f4aca5982692eb40fc0e5d3db933bd5c8e61d1cc3a46244dfa17426a09aedc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A196226 defines its terms as the natural numbers `m` for which the sum\nof divisors of `m`, reduced modulo `m`, is `3 + m/2`. John W. Layman's\nconjecture comment gives three universal congruences:\n\n> (1) If m>=14 is a term of this sequence, then sigma(2,m) is congruent to\n> 5 + m/2 modulo m.\n\n> (2) If m>=22 is a term of this sequence, then sigma(3,m) is congruent to\n> 9 + m/2 modulo m.\n\n> (3) If m>=38 is a term of this sequence, then sigma(4,m) is congruent to\n> 17 + m/2 modulo m.\n\nHere `sigma(k,m)` is the sum of the `k`-th powers of the positive divisors\nof `m`. The formal membership predicate makes the entry's evenness condition\nexplicit:\n\n```lean\ndef membership (m : Nat) : Prop :=\n 2 | m /\\ ArithmeticFunction.sigma 1 m % m = 3 + m / 2\n```\n\nNatural-number division is used throughout. The closed `claim` is the\ndisjunction of the three displayed universal statements, so `Not claim`\nasserts that every one of the three conjectures is false.","title":"Refutation of Layman's A196226 divisor-power congruences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a196226-layman-divisor-power-congruence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37b7bb4bd977a205b4a30ecf67a507b8e3154cebb2fcdb7ee8b3c2265a41ddc4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a204061-companion-pell-exp-base-five-residue","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A204061, NAME (`%N`, verbatim):\n\n> G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n ) where A001333(n) = A002203(n)/2, one-half the companion Pell numbers.\n\nCOMMENT conjecture (`%C`, verbatim):\n\n> a(n) == 1 (mod 5) iff n has no 2's in its base 5 expansion (A023729), otherwise a(n) == 0 (mod 5); this is a conjecture needing proof.\n\nFORMULA (`%F`, verbatim):\n\n> G.f.: 1 / ( sqrt(1+x) * (1-6*x+x^2)^(1/4) ).\n\nAUTHOR (`%A`, verbatim):\n\n```text\n_Paul D. Hanna_, Jan 10 2012\n```\n\nThe exact formal target is, for every `n : Nat`,\n`exists z : Z_5, (z : Q_5) = a(n) and\nPadicInt.toZMod z = (if 2 in Nat.digits 5 n then 0 else 1)`.\nHere `a(n)` is coefficient `n` of the defining exponential in `Q_5`.\nIts exponent uses the frozen sequence `Q : Nat → Nat` from\n`D5/S1/Recurrence/PellCompanionGcd`: A001333, one-half the companion Pell\nnumbers, canonically embedded into `Q_5`. This proves the residue conjecture\nin the 5-adic reading.","title":"Hanna's A204061 coefficients modulo five","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecf8b2f517c88c198296197f8e502b3c05a781c783a159f665eb59cdf66f50c3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a208342-joint-polynomial-pair-binomial-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A208342, NAME:\n\n> Triangle of coefficients of polynomials u(n,x) jointly generated with A208343; see the Formula section.\n\nThe defining FORMULA lines are:\n\n> u(n,x) = u(n-1,x) + x*v(n-1,x),\n>\n> v(n,x) = x*u(n-1,x) + x*v(n-1,x),\n>\n> where u(1,x) = 1, v(1,x) = 1.\n\nWith `T(n,k) = [x^(k-1)] u(n,x)`, the conjecture is:\n\n> T(n,k) = Sum_{j=0..floor((k-1)/2)} binomial(k-1-j,j)*binomial(n-k+j,j) for k,n>0 and k<=n (conjectured). - _Werner Schulte_, Mar 07 2017","title":"The A208342 coefficient formula for a jointly generated polynomial pair","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208342-joint-polynomial-pair-binomial-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16a195544f2b0dd9f07e321179c18df6f9175b57e7c46130c154c6f8e15f206c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A208509 revision 28, FORMULA, attributes this conjecture to Knud Werner:\n\n> Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - _Knud Werner_, Jan 11 2022\n\nClark Kimberling authored the sequence entry. Its companion A208508 revision 10\nsupplies the jointly generated first polynomial. The complete defining laws are\nu(1,X)=v(1,X)=1, u(n,X)=u(n-1,X)+X*v(n-1,X), and\nv(n,X)=u(n-1,X)+v(n-1,X)+1 for every n>=2, over Z[X].\nFor every natural n>=1 and every natural k>=0, T(n,k) is the coefficient of\nX^k in this actual v(n,X), including all zero coefficients beyond its degree.\nThe formula's two natural binomial coefficients are cast to integers.\nNo assertion at n=0 is included.","title":"Werner's all-column formula for the affine polynomial pair","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fe1641ecab9b616b78181ed7742209ebe7f710e52eff9f04879d30479d47374.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a219431-hanna-overpartition-log-derivative-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A219431 has Name “Logarithmic derivative of the g.f. of the overpartitions of n^2 (A219430).” Its Comment states: “Conjecture: a(2*n+1) == 2 (mod 4), a(2*n) == 0 (mod 4).” The offset is one. OEIS A219430 states: “Number of overpartitions of n^2; a(n) = A015128(n^2).”\n\nLet $\\bar p(m)$ be the coefficient of $q^m$ in $\\prod_{k\\geq1}(1+q^k)/(1-q^k)$, and put $G(x)=\\sum_{n\\geq0}\\bar p(n^2)x^n$. The settlement proves that every positive-degree coefficient $a(m)=[x^m]xG'(x)/G(x)$ is an integer, with $a(m)\\equiv 2$ for odd $m$ and $a(m)\\equiv0$ for even $m$, modulo four. Preregistration: [#18874](https://github.com/the-omega-institute/trureturing/issues/18874).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Overpartition generating product | `pbar` | At degree $m$, factors with $k>m$ have no effect; the finite product uses $1\\leq k\\leq m$. |\n| Square-indexed coefficients, including index zero | `G` | `PowerSeries.mk` with coefficient `pbar` at $n^2$. |\n| Logarithmic derivative, offset one | `logDerivative` | $xG'(x)G(x)^{-1}$ over the rationals. |\n| Every index at least one | `claim` | All natural $m$ with $1\\leq m$. |\n| Integer coefficients and two congruences | `claim`, `result` | One integer witness equals the rational coefficient and satisfies both parity implications. Integrality is an explicit strengthening, with no additional hypothesis. |\n\nThe four public definitions retain the expressions of the preregistration. Their parameters, coefficient rings and modulus are unchanged. The positive geometric tail and the divisibility-indicator geometric series are reused from their existing owners rather than declared again.","title":"Hanna's overpartition logarithmic derivative congruences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a219431-hanna-overpartition-log-derivative-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17b7d6b0258c04a363ab9cb9793d535b7bcae7c51da123d4f2d4bfa82b2a9b8b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a220181-alternating-weight-stirling-prime-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A220181 gives the following NAME, finite FORMULA, and COMMENT, copied\nverbatim from `Library/Recurrence/bala2022a220181.md`:\n\n> E.g.f.: Sum_{n>=0} (1 - exp(-n*x))^n.\n\n> a(n) = Sum_{k=0..n} (-1)^(n-k) * k^n * k! * Stirling2(n,k).\n\n> Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 0, 3, 0, 0, 1, 1, 0, 3, 0, 0, 1 ...], with an apparent period of 6. Cf. A122399.\n\nThe entry is by Paul D. Hanna, Dec 06 2012; the conjecture COMMENT resolved\nhere is by Peter Bala, Jun 01 2022. The two attributions are recorded\nseparately rather than reassigning the comment to the entry author.","title":"Prime periods of the A220181 alternating-weight Stirling sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a86a9f3838e00493fe8edd0854df814837fb83fada2f7ca91adf3bc4847d326c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a221077-tanh-power-prime-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A221077 (Paul D. Hanna's entry) specifies the following NAME. The\nLibrary note records Peter Bala's conjecture dated Jun 01 2022; that date\nand attribution belong to the conjecture, not to an asserted entry-creation date.\nThe following NAME and COMMENT are quoted verbatim from the supplied Library note.\n\nNAME:\n\n> E.g.f.: Sum_{n>=0} tanh(n*x)^n.\n\nCOMMENT (Peter Bala, Jun 01 2022):\n\n> Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4, 1, 1, 6, 1, 0, 4 ...], with an apparent period of 6.\n\nThis is one conjecture quantified over all primes. It is false at p = 2;\nits substantive period assertion survives for every odd prime.","title":"Refutation of the all-prime A221077 period conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a221077-tanh-power-prime-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40d2194427d07345228ab3b7c8a9425fd0a255ee2a828c3d4476b55afda36288.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a222001-descent-lex-arrays","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For each positive integer `n`, count the actual `n` by 3 arrays whose rows are\npermutations of `1,2,3`, whose adjacent-descent counts are nondecreasing along\nthe original row indices, and whose rows are lexicographically nonincreasing.\nRows may repeat. In the source carrier a row is `Equiv.Perm (Fin 3)` and the\nsource symbol in column `j` is `(r j).val + 1`, so the carrier is exactly the\nsix permutations of `1,2,3`. OEIS A222001 gives the formula\n`2+(n+1)*(n+2)*(n+3)/6` with offset `1,1`.","title":"Three-column permutation arrays with compatible row orders","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222001-descent-lex-arrays/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e3e4a2652c53dfda6de97aad797e86034a98592fadfe4aff044f91176150cc53.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a222013-factorial-product-sum-catalan-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna created OEIS A222013 on Feb 04 2013; the parity conjecture\nis dated Dec 06 2024. These quotations are copied verbatim from\n`Library/Recurrence/hanna2024a222013.md`.\n\nNAME:\n\n> G.f. satisfies: A(x) = Sum_{n>=0} n! * x^n * A(x)^(n*(n+1)/2) / Product_{k=1..n} (1 + k*x*A(x)^k).\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0. - _Paul D. Hanna_, Dec 06 2024\n\nThe infinite sum is formalised with the exact finite window at each coefficient,\nas stated in Route and quoted in Lean syntax in Evidence.","title":"Parity of the A222013 factorial product sum coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222013-factorial-product-sum-catalan-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b562527feb63bf2854bce128ec100112ba123f82644b623e84848467187e749.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a222014-factorial-square-exponent-sum-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A222014 defines its generating series by\n\n> A(x) = Sum_{n>=0} n! * x^n * A(x)^(n^2) / Product_{k=1..n} (1 + k*x*A(x)^n).\n\nPaul D. Hanna's comment of Dec 06 2024 states, verbatim:\n\n> Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.\n\nThe infinite sum is formalized coefficientwise by its exact finite window.","title":"Parity of the A222014 factorial square-exponent sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222014-factorial-square-exponent-sum-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/efb82b50350de3f5babb3224646d8280419395327a15821ca4dd90b1ef6a17e8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a224899-sinh-power-prime-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A224899, NAME:\n\n> E.g.f.: Sum_{n>=0} sinh(n*x)^n.\n\nCOMMENT (Peter Bala, May 29 2022):\n\n> Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1. For example, modulo 7 the sequence becomes [1, 1, 2, 0, 3, 0, 1, 1, 2, 0, 3, 0, ...], with an apparent period of 6. Cf. A245322.\n\nThe Library note attributes the entry to Paul D. Hanna, April 20, 2013,\nand the conjecture to Peter Bala, May 29, 2022. The conjecture quantifies over\nall primes. It is false at p = 2; its substantive content survives for every\nodd prime, so the failure is confined exactly to p = 2.","title":"The prime-period conjecture for OEIS A224899","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a224899-sinh-power-prime-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/322bf4f722242a2ae3c73e5aadf1ebcca47a8f4d0ba47e1fee1844073ac0f4ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a226839-triangular-egf-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A226839, returned revision 11, gives the NAME:\n\n> E.g.f.: exp( Sum_{n>=1} x^(n*(n+1)/2) / n! ).\n\nIts FORMULA is:\n\n> a(n) == 1 (mod 3) (conjecture - valid up to n=1024; if true for n>=0, why?).\n\nThe offset is `0,4`. The immutable preregistration is\n[#23823](https://github.com/the-omega-institute/trureturing/issues/23823#issuecomment-6505813924).\nThe source note is `Library/RecurrenceComposition/hanna2013a226839.md`.\nThe source question and source editorial approval do not supply a proof.\n\n### Statement echo\n\nOver $\\mathbb Q[[X]]$, set\n$T_m=\\lfloor m(m+1)/2\\rfloor$,\n$H(X)=\\sum_{m\\ge1}X^{T_m}/m!$ and $A(X)=\\exp(H(X))$.\nFor every $n\\in\\mathbb N$, including zero, there is a $k\\in\\mathbb Z$ with\n$(n!: \\mathbb Q)[X^n]A=1+3(k:\\mathbb Q)$.\n\nThe public definitions `triangular`, `H` and `A` retain the literal\nexponent, coefficientwise positive-index rational sum and formal exponential\nsubstitution. The denominator is $m!$, not $T_m!$.\nThe private `coeff_H` proves local finiteness using $m\\le T_m$.\nThe public `innerIntegral` is the natural finite sum over $0\\le m\\le n$\nof $(m+1)^{\\overline{n-m}}$ when $0 Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers.\n\nFORMULA conjecture line (`%F`, verbatim):\n\n> Conjecture: a(n) = a(n-3)+a(n-6) for n>12. - _Colin Barker_, Nov 09 2014\n\nEmpirical generating function (`%F`, verbatim):\n\n> Empirical g.f.: -x^2*(x^10 +x^9 +x^8 +2*x^7 +3*x^6 +3*x^5 +3*x^4 +3*x^3 +3*x^2 +2*x +1) / (x^6 +x^3 -1). - _Colin Barker_, Nov 09 2014\n\nCOMMENT (`%C`, verbatim):\n\n> All Fibonacci numbers are in the sequence. The only prime numbers in this sequence are prime Fibonacci numbers.\n\nAUTHOR (`%A`, verbatim):\n\n> _Alonso del Arte_, Jun 19 2013\n\nThe first `%S` terms are `0,1,2,3,4,5,6,8,9,10,13,15,16,21,24,26,34,39,42,55,63,68,89,102,110`.\n\nThe literal proved statement is\n`∀ n : ℕ, 12 < n → a n = a (n - 3) + a (n - 6)`, where\n`mem x := (∃ i j : ℕ, x = Nat.fib i + Nat.fib j) ∧ (∃ r s : ℕ, x = Nat.fib r * Nat.fib s)`\nand `a n := Nat.nth mem (n - 1)`. `Nat.nth` enumerates members from zero,\nso the one-based OEIS index is represented by the subtraction `n - 1` in\nnatural numbers. Equal summands and equal factors are allowed. The empirical\ngenerating-function line and the `%C` corollaries are NOT claimed.","title":"The A226857 Fibonacci sum-product recurrence is proved","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a226857-fibonacci-sum-product-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d9789426494302935dd07556ac8cf160ea1d9095d05b89fc2d450863ab6ff8a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a228558-price-seventeen-four-prime-exponent","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A228558, NAME (`%N`, verbatim):\n\n> Numbers k such that (17^k + 4^k)/21 is prime.\n\nRobert Price's COMMENT (`%C`, verbatim):\n\n> Conjecture: All terms are prime.\n\nThe literal proved statement is\n\n`forall k : Nat, 21 | 17^k + 4^k -> Prime((17^k + 4^k)/21) -> Prime(k)`.\n\nThe divisibility hypothesis is explicit because the OEIS expression denotes an\nexact integer quotient, while natural-number division would otherwise truncate.\n\nNot claimed here are the A228558 comment `a(9) > 10^5`, the A231329 comment\n`a(5) > 10^5`, infinitude of either sequence, primality of any particular term,\nor the classical general factorization fact as a new result. That fact is the\nsame background argument as `2^n-1 prime -> n prime`.","title":"Price's A228558 prime-exponent conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c714ca2b38f1a4442a3b8022d403d3d80e56b1217d4725c0a7de27b42ff4002.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A229140 states (verbatim):\n\n> %N A229140 Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares (A001481).\n> %C A229140 Conjecture: the values between two zeros are always distinct from each other.\n> %A A229140 _Ralf Stephan_, Sep 15 2013\n> %O A229140 1,6\n\nThe literal refuted statement is: for every pair `L Numbers n such that (19^n + 4^n)/23 is prime.\n\nRobert Price's COMMENT (`%C`, verbatim):\n\n> Conjecture: All terms are prime.\n\nThe literal proved statement is\n\n`forall k : Nat, 23 | 19^k + 4^k -> Prime((19^k + 4^k)/23) -> Prime(k)`.\n\nThe divisibility hypothesis is explicit because the OEIS expression denotes an\nexact integer quotient, while natural-number division would otherwise truncate.\n\nNot claimed here are the A228558 comment `a(9) > 10^5`, the A231329 comment\n`a(5) > 10^5`, infinitude of either sequence, primality of any particular term,\nor the classical general factorization fact as a new result. That fact is the\nsame background argument as `2^n-1 prime -> n prime`.","title":"Price's A231329 prime-exponent conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a2149b59c0c6fc02fc7bf2da7ebbe2b635cbb9b143312e81d824a0108ea4709.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a231548-antisigma-decrease-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A231548, NAME (verbatim):\n\n> Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2).\n\nCOMMENT defining antisigma (verbatim):\n\n> Also numbers n such that antisigma(n) < antisigma(n-2), where antisigma(n) = A024816(n) = the sum of the non-divisors of n that are between 1 and n.\n\nCOMMENT conjecture line (verbatim; Jaroslav Krizek, Nov 12 2013):\n\n> Conjecture: there are no numbers n such that antisigma(n) < antisigma(n-3). - _Jaroslav Krizek_, Nov 12 2013\n\nOEIS A024816, NAME (verbatim):\n\n> Antisigma(n): Sum of the numbers less than n that do not divide n.\n\nThe literal object is\n`antisigma(n) = Σ of d ∈ [1,n] with d ∤ n`. Since `n`\nalways divides itself, this agrees with the A024816 interval ending before\n`n`. The conjecture is the claim\n`∀ n ≥ 3, antisigma(n-3) ≤ antisigma(n)`.","title":"Refutation of the A231548 antisigma gap-three conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231548-antisigma-decrease-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d12b7692595599ace657d4f25f74559f9a4e3afb2c76506d4c29931d565fc086.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a239293-ordowski-immediate-successor-weak-pseudoprime","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A239293, NAME (`%N`, verbatim):\n\n> Smallest composite c > n such that n^c == n (mod c).\n\nThe neighbouring terminology COMMENT (`%C`, verbatim):\n\n> a(n) is the smallest weak pseudoprime to base n that is > n.\n\nThe neighbouring sufficient-condition COMMENT (`%C`, verbatim):\n\n> If n is even and n+1 is composite, then a(n) = n+1. [Corrected by _Thomas Ordowski_, Aug 03 2018]\n\nThomas Ordowski's target COMMENT (`%C`, verbatim):\n\n> Conjecture: a(n) = n+1 if and only if n+1 is an odd composite number. - _Thomas Ordowski_, Aug 03 2018\n\nThe literal proved statement is\n\n`forall n : Nat, 1 <= n -> ((1 < n+1 and not Prime(n+1) and n^(n+1) mod\n(n+1) = n mod (n+1)) iff (not Prime(n+1) and Odd(n+1) and 1 < n+1))`.\n\nBy the NAME, `a(n)` is the least composite `c>n` satisfying the displayed\ncongruence. Since `n+1` is the immediate successor of `n`, no natural number\nlies strictly between them. Thus `a(n)=n+1` holds exactly when `n+1` itself\nqualifies; minimality at the immediate successor is automatic. This is the\nreduction used by the theorem, and it does not require a total definition of\n`a` at bases for which no qualifying composite has been supplied.\n\nThe neighbouring sufficient-condition COMMENT is not claimed as a new\nresult. The weak-pseudoprime terminology COMMENT is not claimed as a new\nresult. Totality of `a` is not claimed; proving it would require a different\nexistence theorem asserting a qualifying composite above every base.","title":"The immediate successor is the least weak pseudoprime exactly when it is an odd composite","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239293-ordowski-immediate-successor-weak-pseudoprime/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dfc17f99689e88b9c42f1dfdf81b9d250132e256941189a61ae8d01b0b34bd5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a239907-kurkov-run-alternating-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A239907, NAME (`%N`, verbatim):\n\n> Let cn(n,k) denote the number of times 11..1 (k 1's) appears in the binary representation of n; a(n) = n - cn(n,1) + cn(n,2) - cn(n,3) + cn(n,4) - ... .\n\nThe settled FORMULA (`%F`, verbatim):\n\n> Conjecture: a(n) = n - A329320(n) for n >= 0 (noticed by Sequence Machine). - _Mikhail Kurkov_, Oct 13 2021\n\nAUTHOR (`%A`, verbatim):\n\n> _N. J. A. Sloane_, Apr 07 2014\n\nOccurrences overlap. In the binary word `111`, the counts for block lengths\n1, 2, and 3 are 3, 2, and 1; hence `a(7)=7-3+2-1=5`, as required by the\ndata. Numbering bits from the least significant end preserves counts of\nthe palindromic block of ones. The index range has length `n.size`, the\nbinary length, with the empty range at zero.\n\nFor every natural `n`, define `cn(n,k)` to count starts `i Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer.\n\nCOMMENT conjecture line (verbatim; Jaroslav Krizek, Aug 15 2014):\n\n> Conjecture: Subsequence of A216793.\n\nOEIS A216793, NAME (verbatim; Michel Marcus, Sep 16 2012):\n\n> Numbers n such that gcd(sigma(n), n) > gcd(sigma(m), m) for all m < n.\n\nThe literal claim is\n`∀ n ≥ 1, IsMember n → IsRecord n`, where `IsMember` is the A245786\nintegrality condition and `IsRecord` is the A216793 strict gcd-record\ncondition.","title":"Refutation of the A245786 integrality-implies-gcd-record conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4d3e34ada4a42da8a3bfc250dbb46cfb50d97d09cfc1284374f07b26be0cb7b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a249621-square-prepend-one-append-twenty-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A249621 %N:\n\n> Squares that remain squares when prepended with 1 and appended with 25 in base 10.\n\nOEIS A249621 %C:\n\n> Conjecture: all the numbers in the sequence end in either 00 or 56.\n\n— _Chai Wah Wu_, Nov 02 2014.\n\nFor a natural number `x`, membership means that `x` is a square and, if `d` is\nthe number of its base-ten digits, then `10^(d+2) + 100x + 25` is also a square.\nThis is the literal base-ten concatenation `1||x||25`.","title":"OEIS A249621 terminal digits for square concatenations","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249621-square-prepend-one-append-twenty-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32819f853a37450b0db6e0d4e098085990a32bfa91895c670b7fa71a8e2d3c1f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A249759, revision #40 Sep 10 2025 16:53:46, records these lines\nverbatim:\n\n> %N A249759 Primes p such that sigma(p-1) is a prime q.\n>\n> %S A249759 3,5,17,65537\n>\n> %C A249759 Conjectures: 1) sequence is finite; 2) sequence is a subsequence of A019434 (Fermat primes); 3) sequence consists of Fermat primes p such that sigma(p-1) is a Mersenne prime; 4) a(n) = (A249761(n)+3)/2.\n>\n> %A A249759 _Jaroslav Krizek_, Nov 13 2014\n\nItems 2 and 3 have the following formal joint statement, where `σ 1` is\nMathlib's sum-of-divisors function:\n\n```lean\nopen scoped ArithmeticFunction.sigma in\ntheorem result {p : ℕ} (hp : p.Prime) (hq : (σ 1 (p - 1)).Prime) :\n (∃ m : ℕ, p = 2 ^ 2 ^ m + 1) ∧\n (∃ r : ℕ, r.Prime ∧ σ 1 (p - 1) = 2 ^ r - 1)\n```\n\nNatural subtraction does not weaken this statement. `Nat.Prime.two_le`\ngives `2 ≤ p`, and the proof separately rules out `p = 2` from\n`σ 1 (2 - 1) = σ 1 1 = 1`, which is not prime.\n\nNot claimed: item 1 is a finiteness statement about the resulting class of\nFermat primes and remains open. Item 4 concerns the definition of the\nseparate sequence A249761. Neither item appears on the Lean delivery surface.","title":"Krizek's Fermat-Mersenne classification for A249759","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eee1bd6d72215392d856a2d68b51be2663ffaff0f3935193988443dd5c582c3b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a254074-wu-digit-range-square-not-multiple-of-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A254074, NAME (`%N`, verbatim):\n\n> Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value.\n\nChai Wah Wu's COMMENT (`%C`, verbatim):\n\n> The first digit of a term is either 7, 8 or 9 and the last digit is either 5, 6 or 7. Conjecture: no term is a multiple of 5. - _Chai Wah Wu_, Sep 10 2017\n\nThe literal proved sentence is \"Conjecture: no term is a multiple of 5.\"\nFormally, every positive natural `k` whose decimal digits and square's decimal\ndigits both have minimum 5 and maximum 9 is not divisible by 5. The first\nsentence of Wu's comment, concerning the first and last digits, is NOT\nclaimed. The sparsity, finiteness or infinitude of A254074, and every other\nproperty of the sequence are also NOT claimed.","title":"No A254074 term is a multiple of five","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254074-wu-digit-range-square-not-multiple-of-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37ab24df5ed0921a5a27829375553dde2537e17468db90f564fdaebb46faafc3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A254748, NAME (`%N`, verbatim):\n\n> Numbers without superdivisors: numbers n such that n/k + n fails to divide at least one of (n/k)^(n/k) + n, (n/k)^n + n/k or n^(n/k) + n/k for any divisor k of n.\n\nCOMMENT (`%C`, verbatim):\n\n> Zerosuperdivisor numbers. Numbers n such that A247477(n) = 0.\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: Average of twin prime pairs (A014574) are zerosuperdivisor numbers.\n\nAUTHOR (`%A`, verbatim):\n\n> _Juri-Stepan Gerasimov_, Feb 07 2015\n\nThe A247477 superdivisor definition is: a divisor k is a superdivisor of n when\n`n/k + n` divides each of `(n/k)^(n/k) + n`, `(n/k)^n + n/k`, and\n`n^(n/k) + n/k`, with all division interpreted in `ℕ` (exact when `k ∣ n`).\n\nThe literal claim is\n`∀ p : ℕ, Nat.Prime p → Nat.Prime (p + 2) → ∀ k : ℕ, 1 ≤ k → k ∣ p + 1 → ¬ IsSuperdivisor (p + 1) k`,\nwhere `IsSuperdivisor` is the three-clause predicate above and the quotient is\nnatural-number division. The perfect-number sentence in A254748 is NOT claimed.","title":"The A254748 twin-prime mean has no superdivisor","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ed695bae458e0407bb7c0357725cb86478f2bbae66edf8f375c67a9d97183e6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a258409-divisor-difference-gcd-heinz","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A258409, NAME:\n\n> Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n.\n\nCOMMENT (Gus Wiseman, Oct 16 2019):\n\n> Conjecture: a(n) = A289508(A328023(n)) = GCD of the differences between consecutive divisors of n. See A328163 and A328164.\n\nOEIS A328023, NAME:\n\n> Heinz number of the multiset of differences between consecutive divisors of n.\n\nIts Heinz-number convention is:\n\n> The Heinz number of an integer partition or multiset {y_1,...,y_k} is prime(y_1)*...*prime(y_k).\n\nOEIS A289508, NAME:\n\n> a(n) is the GCD of the indices j for which the j-th prime p_j divides n.\n\nThese OEIS prime indices are one-based, with prime(1)=2. Mathlib's\n`Nat.nth Nat.Prime` is zero-based, so the Lean encoding of prime(delta) is\n`Nat.nth Nat.Prime (delta - 1)`.","title":"The divisor-difference gcd and Heinz decoding identity in A258409","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/665e7f10a989d12d33de1598da0ea22b1d729bda19fa22a4c910f4e17eed8eea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A260310, NAME (verbatim):\n\n> Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members.\n\nCOMMENTS definition line (verbatim):\n\n> Consider pairs (x,y) of numbers where sum(p|x) p + sum(q|y) q = x*sum(p|x) 1/p + y*sum(q|y) 1/q where p, q are primes and sum(p|x) p > sum(q|y) q.\n\nCOMMENTS conjecture line (verbatim; Robert G. Wilson v, Jul 22 2015):\n\n> Conjecture: if a(2n-1) is prime then a(2n) is composite and vice versa. And when a(2n-1) is composite, it is congruent to 0 (mod 6). - _Robert G. Wilson v_, Jul 22 2015\n\nAUTHOR line (verbatim):\n\n> _Juri-Stepan Gerasimov_, Jul 22 2015\n\nThe first three pairs in the SEQUENCE lines are `(3, 8)`, `(7, 16)`, and\n`(11, 18)`. The formal definitions use\n`S(n) = ∑ p ∈ n.primeFactors, p` and the frozen repository definition\n`squarefreeDeriv(n) = ∑ p ∈ n.primeFactors, n / p`, where the division is\nnatural-number division and is exact because each indexed prime divides `n`.\nThe predicate `IsPair(x, y)` means\n`x < y ∧ S(x) + S(y) = squarefreeDeriv(x) + squarefreeDeriv(y) ∧ S(x) ≠ S(y) ∧ S(y) ≠ squarefreeDeriv(y)`.\nThe literal refuted clause is\n`∀ x y : ℕ, IsPair x y → 1 < x → ¬ Nat.Prime x → 6 ∣ x`.\n\nThe prime/composite alternation sentence is not claimed. The separate remark\n`a(2n) < a(2n+2)` is also not claimed; the 2015 b-file already refutes it at\n`a(6552) = a(6554) = 35407`.","title":"Refutation of the A260310 balanced-pair mod-six conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b529e9b6149ef851a0cba0eae3f2b0babb9291e6fdcd08f172333de222131b9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a260407-krizek-all-multiples-of-the-order","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A260407, NAME (`%N`, verbatim):\n\n> Numbers n such that (n-1)^2+1 divides 2^(n-1)-1.\n\nJaroslav Krizek's COMMENT (`%C`, verbatim):\n\n> Conjecture: also numbers n such that ((2^k)^(n-1)-1) == 0 mod ((n-1)^2+1) for all k >= 1. - _Jaroslav Krizek_, Jun 02 2016\n\nDefine `modulus(n) = (n-1)^2+1`, using natural-number subtraction. The\nliteral proved statement is `forall n : Nat, 1 <= n -> (inSequence n iff\nallMultiples n)`, where `inSequence n` means\n`modulus(n) divides 2^(n-1)-1`, and `allMultiples n` means that for every\nnatural `k >= 1`, `modulus(n) divides 2^(k*(n-1))-1`. The equality\n`(2^k)^(n-1) = 2^(k*(n-1))` identifies the COMMENT exponent with the\nformal exponent.\n\nThe claim that `a(7) = 1382401` is the first composite term, the claim that\nthe Fermat numbers `2^(2^k)+1` for `k > 1` form a subsequence, Hasler's\nfurther conjecture that those numbers are the intersection with A260406,\nanything about A260406 or A260072, and every other A260407 comment are NOT\nclaimed. The textbook divisibility fact for arbitrary `m` and `d` is also\nnot claimed as a new result.","title":"The A260407 condition holds at every positive multiple of the order","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac1f99e016e0148946877009895c95a18d5fc788a822375c64c7dc5e4d4a2375.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a265714-positive-threshold-common-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> a(n) = the smallest number k such that floor(Sum_{d|k} 1/sigma(d)) = n.\n\n> Conjecture: a(n) is also the smallest number k such that Sum_{d|k} 1/sigma(d) >= n.\n\n— Jaroslav Krizek, OEIS A265714, revision 23.\n\nThis dossier anchors exactly the common-minimum conjecture. For every\nnatural r>0, let g(k) be the exact rational sum of 1/sigma(d) over all\npositive divisors d of the positive natural k, where sigma(d) is the\nsum of all positive divisors of d. The sets\n{k>0 : floor(g(k))=r} and {k>0 : g(k)>=r} must both be nonempty and have\nthe SAME least element. The floor is ordinary floor, r=1 is included,\nand neither r nor k has a finite upper bound. There is no additional\nexistence or integrality premise. The source's separate integrality\nquestion and proposed numerical minima are outside this single assertion.","title":"Common positive threshold minima for reciprocal-sigma sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a265714-positive-threshold-common-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/909c93d99899dcc467437a04955404d4b6de69e33c7d6ea653d323b746e830fa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a266285-rule13-barker-zizka","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A266285 counts the ON cells among the cells `-n, …, n` of row `n` of the\nRule 13 elementary cellular automaton started from a single ON cell. Its\nformula field records\n\n> Conjectures from _Colin Barker_, Dec 28 2015 and Apr 14 2019: (Start)\n> a(n) = ((-1)^n*(3-2*n)+4*n+1)/4.\n> a(n) = 2*a(n-2)-a(n-4) for n>3.\n> G.f.: (1+x+2*x^3) / ((1-x)^2*(1+x)^2). (End)\n> Conjecture from _Ctibor O. Zizka_, Mar 11 2025: (Start)\n> a(2*n) = n + 1.\n> a(2*n + 1) = 3*n + 1.(End)\n\nIssue #13151 fixes the reading: every cell of the integers is updated at every\nstep, the closed form is read in the rationals, the recurrence is asserted for\n`n > 3` in the integers, and the generating function is stated as the product\nof the series with its denominator, which has constant term 1.","title":"Barker's and Zizka's conjectures for the ON cells of Rule 13","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266285-rule13-barker-zizka/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44047fddd888f8a9192e949c6438fb579cf62987e85128c2bface1c30b57fdd6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a266489-diagonal-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quoted directly from the seat-local file `oeis-A266489.src`\n(`seats/opB-diagidx2/oeis-A266489.src` in the supplied workspace):\n\n> NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1.\n> COMMENT: CONJECTURES:\n> COMMENT: (C1) n divides a(n) for n>=1: A268293(n) = a(n)/n.\n> COMMENT: (C2) a(n) == 0 (mod 2) for n>=2.\n> AUTHOR: _Paul D. Hanna_, Feb 07 2016\n\nThe target is only clause (C1) for this entry.\nThe formal object has `a(0)=a(1)=1` and exponent `e(n)=n`.\nThe formal vanishing condition is imposed for `n>1`, as in the quoted NAME.","title":"A266489: clause C1, index divisibility on the frozen object","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-diagonal-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/751ef95ecd6954f7a7243c2b806a25d71338f61be8b95ccea0f743643df9b004.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a266489-negative-power-diagonal-mod-two","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"From [Library/ArithSums/hanna2016a266489.md](../Library/ArithSums/hanna2016a266489.md),\nOEIS A266489, Paul D. Hanna, Feb 07 2016:\n\nNAME:\n\n> G.f. A(x) satisfies: [x^n] A( x/A(x)^n ) = 0 for n>1.\n\nCOMMENT:\n\n> (C2) a(n) == 0 (mod 2) for n>=2.\n\nThe normalization is `a(0) = a(1) = 1`. Only this congruence clause is claimed.","title":"A266489: the mod-two congruence clause","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b113e829aeb03cd2d4cf5be85f204e1f60858a0a9dc2f9c5b5c23bb8fc388ce8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a266614-rule41-barker","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A266614 counts the ON cells among the cells `-n, …, n` of row `n` of the\nRule 41 elementary cellular automaton started from a single ON cell. Its\nformula field records\n\n> Conjectures from _Colin Barker_, Jan 02 2016 and Apr 18 2019: (Start)\n> a(n) = a(n-2)+a(n-4)-a(n-6) for n>5.\n> G.f.: (1+x^2+3*x^3-2*x^4+5*x^5) / ((1-x)^2*(1+x)^2*(1+x^2)).\n> (End)\n\nIssue #13157 fixes the reading: every cell of the integers is updated at every\nstep, the recurrence is asserted for `n > 5` in the integers, and the\ngenerating function is stated as the product of the series with its\ndenominator, which has constant term 1.","title":"Barker's conjectures for the ON cells of Rule 41","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266614-rule41-barker/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/254d81e8c8b2ac71354e8d448f68ed334b42973bcda3b3ef34f5d78abf23af3b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a267681-rule201-barker-hasler","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A267681 and A267680 read row `n` of the Rule 201 elementary cellular\nautomaton, started from a single ON cell, on the cells `-n, …, n`: in base 2\n(A267681) and as a string of decimal digits (A267680). Their formula fields\nrecord\n\n> a(n) = 5*a(n-1)-20*a(n-3)+16*a(n-4) for n>4.\n> G.f.: (1-5*x+21*x^2+14*x^3-40*x^4) / ((1-x)*(1-2*x)*(1+2*x)*(1-4*x)).\n\n> Conjecture: a(n) = 2*4^n - (n%2*2 + [n]*5)*2^(n-1) - 1, where [n] = 1 iff\n> n > 0; n%2 = 1 iff n is odd. - _M. F. Hasler_, Jul 28 2018\n\nfor A267681 (the first two by Colin Barker, Jan 19 2016), and\n\n> a(n) = 101*a(n-1)-10100*a(n-3)+10000*a(n-4) for n>4.\n> G.f.: (1-101*x+10101*x^2+89910*x^3-101000*x^4) / ((1-x)*(1-10*x)*(1+10*x)*(1-100*x)).\n\nfor A267680 (Colin Barker, Jan 19 2016 and Apr 20 2019). Issue #11102 fixes\nthe reading: every cell of the integers is updated at every step, the\nrecurrences are asserted for `n > 4`, and each generating function is stated\nas the product of the series with its denominator, which has constant term 1.","title":"Barker's and Hasler's conjectures for the rows of Rule 201","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a267681-rule201-barker-hasler/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7092c22c4a20921c7d90cb338c0cefd5333ff347a47fdb7411ad45de038c7a1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a270236-rgf-antepenultimate-label-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A270236, NAME (verbatim):\n\n> Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.\n\nCOMMENT defining the restricted growth functions (verbatim):\n\n> The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i Conjecture: T(n+1,n-1) = 2+n*(n+1)*(3*n^2-5*n+26)/24 for n>1.\n\nRevision #22 records that Alois P. Heinz corrected the left index from\n`T(n,n-2)` to the current `T(n+1,n-1)` on Mar 18 2016.","title":"The antepenultimate-label count in OEIS A270236","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-antepenultimate-label-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d498d9b445720ddc6d613246ea6b7490878249b223cc35ca84afe7ec5e52880d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a270236-rgf-penultimate-label-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A270236, NAME (verbatim):\n\n> Triangle T(n,p) read by rows: the number of occurrences of p in the restricted growth functions of length n.\n\nCOMMENT defining the restricted growth functions (verbatim):\n\n> The RG functions used here are defined by f(1)=1, f(j) <= 1+max_{i Conjecture: T(n,n-1) = 2+n*(n-1)/2 for n>1.","title":"The penultimate-label count in OEIS A270236","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-penultimate-label-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cae661cf1ebd6be54e4295455fafd9e369ed36c418f04f50ec9e9b60f24702bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a271931-hanna-prime-substitution-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> More generally, for prime p there exists an integer series G(x) that satisfies: G(x) = G( x^p + p*x*G(x)^p )^(1/p) with G(0)=0, G'(0)=1 (conjecture).\n\nFor every prime $p$, there is $G\\in\\mathbb Z[[x]]$ with $G(0)=0$, $[x]G=1$, and $G^p=G(x^p+pxG^p)$. The entries A271931, A271932 and A271933 specialize to $p=5,7,11$. The powered equation expresses the normalized root convention.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| every prime exponent | claim quantifies over natural p with Nat.Prime | exact registered convention |\n| integer series | the existential series has coefficient ring integers | exact registered convention |\n| G(0)=0 and G'(0)=1 | constant coefficient zero and coefficient one equal to one | exact registered convention |\n| root equation | the pth-power substitution equation in claim | exact registered convention |","title":"Hanna's prime substitution integrality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/14a50a6533facea50145e6eae2cc12b4532d400f4a14b698aaab91e7cfd133e2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a274010-stechkin-divisor-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A274010, NAME (`%N`, verbatim):\n\n> Boris Stechkin function: a(n) is the number of m with 2 <= m <= n and floor(n(m-1)/m) divisible by m-1.\n\nRidouane Oudra's FORMULA conjecture (`%F`, verbatim):\n\n> Conjecture: a(n) = tau(n) + tau(n-1) - 2, for n>=2. - _Ridouane Oudra_, Feb 28 2020\n\nAUTHOR (`%A`, verbatim):\n\n> _Robert Israel_, Jun 06 2016\n\nThe formal definition counts the natural numbers `m` in the closed interval\nfrom two through `n` for which `m-1` divides the natural quotient\n`n*(m-1)/m`. Natural division is the floor of the corresponding nonnegative\nrational quotient. For every `n >= 2`, the formal theorem states\n`stechkinFunction n + 2 = n.divisors.card + (n-1).divisors.card`.\nBoth divisor finsets contain one, so their cardinalities sum to at least two;\nthe subtraction-free equality is therefore equivalent to the quoted formula.","title":"Oudra's divisor-count formula for the Stechkin function","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a274010-stechkin-divisor-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d288b0d662c17f7c3d53b572c47360b801aafedcc3e14e2ed423576a11900928.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a277829-a278770-catalan-square-hankel-limits","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let\n\n```text\nC_m = binomial(2*m,m)/(m+1),\nD_r(n) = det[(C_(i+j+r))^2] over i,j in Fin n.\n```\n\nThe empty determinant is one. Prove both conjuncts\n\n```text\nlim_(n->infinity) log(D_1(n))/n^2 = 2*log(2),\nlim_(n->infinity) log(D_2(n))/n^2 = 2*log(2).\n```\n\nThese are exactly Kotesovec's November 29, 2016 A277829 conjecture and\nNovember 28, 2016 A278770 conjecture. Their one-based programs use\n`C_(i+j-1)^2` and `C_(i+j)^2`, respectively, so the zero-based shifts are\nexactly one and two. The source assertions were preregistered in\nhttps://github.com/the-omega-institute/trureturing/issues/9530 before the\nproof probes.","title":"A277829 and A278770: squared-Catalan Hankel limits","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a277829-a278770-catalan-square-hankel-limits/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a359bdfce50f239b4bb8e74fcc2f17ec38c6062809389105971f40f271795c2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A280258, NAME (`%N`, verbatim):\n\n> a(n) = Sum_{d|n} pxi(d), where pxi(m) is the product of totatives of m (A001783).\n\nJaroslav Krizek's COMMENT (`%C`, verbatim):\n\n> Conjecture: a(n) is odd for numbers in A183300; a(n) is even for numbers in A001105 (2*n^2).\n\nThe dependent definitions are A001783's product of the integers from one\nthrough `m` that are relatively prime to `m`, A183300's positive integers not\nof the form `2*k^2`, and A001105's integers of the form `2*k^2`. The complete\nquantified reading is\n\n`forall n : Nat, 0 < n -> (Odd (a n) <-> not exists k : Nat, n = 2*k^2)`,\n\nwhere `a(n)` sums the phitorials over the positive divisors of `n`.","title":"Krizek's phitorial divisor-sum parity characterization","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/244e75b97a3e476289c39a068d78f03bf8694a104b9338882bafed05966243f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a284005-kurkov-lower-bit-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier anchors exactly the November 10, 2019 FORMULA in\n[OEIS A284005](https://oeis.org/A284005), attributed to Mikhail Kurkov:\n\n> Conjecture: a(n) = 2*a(f(n)) + Sum_{k=0..floor(log_2(n))-1} a(f(n) + 2^k*(1 - T(n,k))) for n > 1 with a(0) = 1, a(1) = 2, f(n) = A053645(n), T(n,k) = floor(n/2^k) mod 2. - _Mikhail Kurkov_, Nov 10 2019\n\nThe NAME defines `a(n)=(1+A000120(n))*a(floor(n/2))` for `n>1`.\nThe initial data begins `1,2,4,6,8,12,18,24`, including `a(1)=2`.\nHere `wt(n)=A000120(n)` is binary digit sum, `L=floor(log_2(n))`,\nand removing the leading binary one gives `f(n)=n-2^L`.\nAll operations in the formal statement are natural-number operations.\nEvery index `0 <= k < L` occurs: an existing one bit contributes\n`a(f(n))`, and these repeated contributions are retained.\n\nThe exact native result is:\n\n```lean\ntheorem result :\n a 0 = 1 ∧ a 1 = 2 ∧\n (∀ n : ℕ, 1 < n → a n = (1 + wt n) * a (n / 2)) ∧\n (∀ n : ℕ, 1 < n →\n a n = 2 * a (n - 2 ^ Nat.log 2 n) +\n ∑ k ∈ Finset.range (Nat.log 2 n),\n a (n - 2 ^ Nat.log 2 n + 2 ^ k * (1 - n / 2 ^ k % 2)))\n```\n\nThis includes `n=2` and every power of two. The lower-bit identity\nasserts nothing at zero or one. The April 24, 2023 two-parameter\nbinomial conjecture and the A329369 modulo-two binomial transform are\ndistinct questions. The divisor-count representation in the entry's\nNAME is outside this result.","title":"Kurkov's full lower-bit recurrence for A284005","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c51eb14dcdb3143f277ebf885f0eacef46af992d72d3a88c832cc6f066a70f3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a284005-kurkov-weight-product-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A284005, NAME (`%N`, verbatim):\n\n> a(0) = 1, and for n > 1, a(n) = (1 + A000120(n))*a(floor(n/2)); also a(n) = A000005(A283477(n)).\n\nThe settled FORMULA (`%F`, verbatim):\n\n> Conjecture: a(2^m*(2n+1)) = Sum_{k=0..m+1} binomial(m+1, k)*a(2^k*n) for m >= 0, n >= 0 with a(0) = 1. - _Mikhail Kurkov_, Apr 24 2023\n\nAUTHOR (`%A`, verbatim):\n\n> _Antti Karttunen_, Mar 18 2017\n\nHere `wt(n)=A000120(n)` counts ones in the binary expansion, represented\nby `(Nat.digits 2 n).sum`. The NAME specifies the recurrence for `n>1`;\nthe entry's data fixes `a(1)=2`. Accordingly, the formal definition is\n`a(0)=1` and `a(n+1)=(1+wt(n+1))*a((n+1)/2)` for every natural `n`,\nwhere `/` is natural floor division. This reproduces the initial terms\n`1,2,4,6,8,12,18,24,16,24,36,48,54,72,96,120`.\n\nThe exact Lean statement is:\n\n```lean\ntheorem result (m n : ℕ) :\n a (2 ^ m * (2 * n + 1)) =\n ∑ k ∈ Finset.range (m + 2), Nat.choose (m + 1) k * a (2 ^ k * n)\n```\n\nBoth quantifiers include zero. The sum runs from `k=0` through `k=m+1`,\nand every quantity is a natural number. Only the quoted binomial\nconjecture is asserted: the 2019 bit-flip recursion, the 2023 mod-2\nbinomial-transform conjecture for A329369, and the representation\n`A000005(A283477(n))` are not claimed.","title":"Kurkov's binomial identity for binary-weight products","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f3977b9fdb51733e1436347fee610254511fdd876be330d7fd564d2f6a827ae6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a290322-cyclotomic-five-residue-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For a modulus at least two, call a residue below it admissible when it is\ncoprime to the modulus and the value of `u^4 + u^3 + u^2 + u + 1` at it is\ncoprime to the modulus as well. Robert Israel conjectured on January 23, 2024\nthat the sum of the admissible residues is not divisible by the modulus\nwhenever five divides the modulus.","title":"The A290322 admissible residue sum conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a290322-cyclotomic-five-residue-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4e379349b27ee8b5da08d704a13beeef7808c99323f43642b28822df6cb141f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a292394-diagonal-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quoted directly from the seat-local file `oeis-A292394.src`\n(`seats/opB-diagidx2/oeis-A292394.src` in the supplied workspace):\n\n> NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^(n^2) ) = 0 for n>1.\n> COMMENT: a(n) is divisible by n^2 for n>=1 (conjecture): A292395(n) = a(n)/n^2.\n> AUTHOR: _Paul D. Hanna_, Sep 15 2017\n\nThe target is the quoted divisibility conjecture for this entry.\nThe formal object has `a(0)=a(1)=1` and exponent `e(n)=n^2`.\nThe formal vanishing condition is imposed for `n>1`, as in the quoted NAME.","title":"A292394: square-index divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a292394-diagonal-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b478bdc4c470df1451e7d4c634a6ffc3578d81ac43395de71bda2b6b336b6d35.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a299406-schulte-liouville-cube-mobius","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A299406, NAME (`%N`, verbatim):\n\n> Dirichlet g.f.: Sum_{n>0} a(n)/n^s = (zeta(s)*zeta(6*s))/(zeta(2*s)*zeta(3*s)).\n\nFORMULA (`%F`, verbatim):\n\n> Conjecture: a(n) = A008836(n) * A210826(n).\n\nAUTHOR (`%A`, verbatim):\n\n> _Werner Schulte_, Feb 20 2018\n\nThe offset (`%O`) is `1,1`. A008836, NAME (`%N`, verbatim):\n\n> Liouville's function lambda(n) = (-1)^k, where k is number of primes dividing n (counted with multiplicity).\n\nA210826, NAME (`%N`, verbatim):\n\n> G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^3).\n\nLet `zeta` be the arithmetic function equal to one on positive inputs, `mu`\nthe Möbius function, and `⋆` Dirichlet convolution. For positive `k`,\n`liftPow k f` has value `f(m)` at `m^k` and zero off exact k-th powers.\nThe total Lean definition uses the guard\n`(Nat.floorRoot k n)^k = n`. Here `Nat.floorRoot` divides prime-factor\nexponents by `k` using natural-number division; it returns zero if `k = 0`\nor `n = 0`. This is the root for divisibility, not a real-root approximation.\n\nThe coefficient reading of `zeta(6s)` is `liftPow 6 zeta`, and that of\n`1/zeta(ks)` is `liftPow k mu` for `k = 2, 3`. Hence set\n`A = zeta ⋆ liftPow 6 zeta ⋆ liftPow 2 mu ⋆ liftPow 3 mu`.\nThe Lambert-series reading is\n`sum_{d|n} B(d) = [n is a positive cube]`, where brackets are an indicator.\nMöbius inversion gives `B = mu ⋆ liftPow 3 zeta`, representing A210826.\nBoth functions take integer values and vanish at zero.\n\nThe exact formal claim is\n`forall n : Nat, 0 < n -> A n = ArithmeticFunction.liouville n * B n`.\nMathlib's `liouville_apply` identifies the named function with\n`(-1 : Int) ^ Omega(n)` for positive `n`, matching A008836's `%N`.\nOnly the quoted A299406 `%F` line is settled under these coefficient\ninterpretations. Analytic convergence, analytic uniqueness of the series,\nand other OEIS assertions are outside the claim. The hypothesis `0 < n`\nexcludes zero, as required by A299406's offset one.","title":"Schulte's Liouville and cube-Möbius product formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4e79d15cd6c6c6b7495617135aab13bd7da8febfd3a4459eeabba7de5d99336.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A300657 defines\n\n```text\na(n) = Sum_{d|n} sigma(d) mod d.\n```\n\nIts Comment line states verbatim:\n\n> a(n) >= A054024(n). Conjecture: a(n) = A054024(n) only for the noncomposite numbers A008578.\n\nThe entry's Author line states verbatim:\n\n> _Jaroslav Krizek_, Mar 10 2018\n\nHere `A054024(n) = sigma(n) mod n`, and A008578 consists of one and the\nprimes. With\n\n```text\na300657(n) = Sum_{d|n} (sigma(d) mod d),\n```\n\nthe proved equivalence, with all quantifiers explicit, is: for every natural\nnumber `n`, if `1 <= n`, then\n\n```text\na300657(n) = sigma(n) mod n if and only if n = 1 or n is prime.\n```","title":"Krizek's A300657 noncomposite characterization","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300657-krizek-divisor-sigma-mod-noncomposite/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb74be254b086d5cf245c19f86aca3e28424ac52d55f8224147b1dfc74deeeb6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a300732-diagonal-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quoted directly from the seat-local file `oeis-A300732.src`\n(`seats/opB-diagidx2/oeis-A300732.src` in the supplied workspace):\n\n> NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n) ) = 0 for n>=1.\n> COMMENT: Conjecture: n divides a(n) for n>=1.\n> AUTHOR: _Paul D. Hanna_, Mar 11 2018\n\nThe target is the quoted divisibility conjecture for this entry.\nThe formal object has `a(0)=a(1)=1` and exponent `e(n)=2*n`.\nThe printed degree-one boundary requires the correction disclosed under ASSUMED-UNVERIFIED.","title":"A300732: index divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300732-diagonal-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd2e6437e50a8f81a4c8f65332a570dde0c4500519be1e4c58b1bf0f9d2f22be.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a300733-diagonal-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quoted directly from the seat-local file `oeis-A300733.src`\n(`seats/opB-diagidx2/oeis-A300733.src` in the supplied workspace):\n\n> NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^(3*n) ) = 0 for n>=1.\n> COMMENT: Conjecture: n divides a(n) for n>=1.\n> AUTHOR: _Paul D. Hanna_, Mar 11 2018\n\nThe target is the quoted divisibility conjecture for this entry.\nThe formal object has `a(0)=a(1)=1` and exponent `e(n)=3*n`.\nThe printed degree-one boundary requires the correction disclosed under ASSUMED-UNVERIFIED.","title":"A300733: index divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300733-diagonal-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3daee4a2cb9c36e577ad6e009d55f8acd120d8deeeb264e4d79ba89554b5c90.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a300734-diagonal-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quoted directly from the seat-local file `oeis-A300734.src`\n(`seats/opB-diagidx2/oeis-A300734.src` in the supplied workspace):\n\n> NAME: G.f. A(x) satisfies: [x^n] A( x/A(x)^(2*n^2) ) = 0 for n>1.\n> COMMENT: Conjecture: n^2 divides a(n) for n>=1.\n> AUTHOR: _Paul D. Hanna_, Mar 11 2018\n\nThe target is the quoted divisibility conjecture for this entry.\nThe formal object has `a(0)=a(1)=1` and exponent `e(n)=2*n^2`.\nThe formal vanishing condition is imposed for `n>1`, as in the quoted NAME.","title":"A300734: square-index divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300734-diagonal-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dfa6f9346614575b4e1e7a58c3b9bc3ea7640e8afc527661e5f6f7eaca6c8d9b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a301420-positive-newton-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gregory Gerard Wojnar's [OEIS A301420](https://oeis.org/A301420), NAME and COMMENTS, defines the sequence by:\n\n> for a family of 5 data, the sum of the positive coefficients in these polynomials is a(n).\n\nThe FORMULA field states:\n\n> G.f.: (-x*(x+1)^4+1)/(x^6+4*x^5+5*x^4-5*x^2-6*x+1); this denominator equals (1-x)*(2-(x+1)^5) (conjectured).\n\nIt also states:\n\n> a(n+14) = 7*a(n+13) - a(n+12) - 6*a(n+11) + 2*a(n+10) - a(n+9) + 4*a(n+8) + a(n+7) + 4*a(n+5) + 2*a(n+4) - a(n+3) - 5*a(n+2) - 4*a(n+1) - a(n) (conjectured).\n\nThe source snapshot is A301420 revision #28; the full generating-function conjecture occurs in revision #2 and the recurrence is explicitly conjectural in revision #16. The published offset is 1,2. The shift and the universal recurrence domain $n\\ge1$ are explicit interpretations accepted in [#23723](https://github.com/the-omega-institute/trureturing/issues/23723), rather than printed source quantifiers. The [Library note](../Library/RecurrenceComposition/wojnar2018a301420.md) gives the source and coefficient normalization.\n\nFor $k\\ge1$ and $m=(m_1,\\ldots,m_5)\\in\\mathbb N_0^5$ of weighted degree $\\sum_{i=1}^5 i m_i=k$, put $r=\\sum_i m_i$ and define\n\n$$\nc_m=\\frac{k}{5r}(-1)^{k+r}\\frac{r!}{\\prod_i m_i!}\\prod_{i=1}^5\\binom5i^{m_i},\\qquad\na(k)=\\sum_{\\sum i m_i=k,\\ c_m>0}c_m.\n$$\n\nThese are definitions, using equation (2) of Wojnar, Wojnar and Brin, [arXiv:1706.08381v2](https://arxiv.org/abs/1706.08381v2), pp. 3–4. Each distinct monomial contributes separately, before substituting data values. No $a(0)$ is defined.\n\nThe exact target is the conjunction of the rational identity\n\n$$\nF(X)=\\sum_{k\\ge1}a(k)X^{k-1}\n=\\frac{1-X(1+X)^4}{(1-X)(2-(1+X)^5)}\\quad\\text{in }\\mathbb Q[[X]]\n$$\n\nand the displayed recurrence for every integer $n\\ge1$.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Five distinct variables and weights 1 through 5 | `Multiplicity`, `count`, `degree`, `multiplicities` | All nonnegative multiplicity vectors of a fixed weighted degree, each once. |\n| Normalized Girard–Waring coefficient | `literalCoefficient` | The displayed factorial, binomial and sign factors. |\n| Sum of positive coefficients | `positiveCoefficient` | Strict per-monomial positivity, before valuation. |\n| Offset-one generating function | `positiveSeries`, first conjunct of `result` | Coefficient at degree $j$ is $a(j+1)$. |\n| Printed recurrence | `recurrence`, second conjunct of `result` | Every positive index, with all signs and coefficients as printed. |","title":"A301420: positive monomial coefficients for five data","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad7bca8d1d7a535a27c3072898776ab6f2fd493c413e5c1b4f5414379beb0cb9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a301976-grid-no-leaf-mod-ten","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A301976, NAME (verbatim from `Library/Words/kagey2018a301976.md`):\n\n> Number of no-leaf subgraphs of the 3 X n grid.\n\nCOMMENT (verbatim; attributed there to Peter Kagey):\n\n> Conjecture: a(n) mod 10 = 3 for n > 2.\n\nThe literal object is the set of spanning edge-subgraphs of the `3 X n` grid\ngraph, with all `3n` vertices retained, in which every vertex has degree\ndifferent from one. Degree zero is allowed.","title":"Kagey's modulo-ten congruence for no-leaf grid subgraphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-mod-ten/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8973e635093fa86f94e4fb93d0e9ab68437926ff0a59ff5745f111e18a7c4fd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a301976-grid-no-leaf-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A301976, NAME (verbatim from `Library/Words/kagey2018a301976.md`):\n\n> Number of no-leaf subgraphs of the 3 X n grid.\n\nFORMULA (verbatim; attributed there to Colin Barker, Mar 30 2018):\n\n> a(n) = 12*a(n-1) - 6*a(n-2) - 20*a(n-3) - 5*a(n-4) for n>4.\n\nThe literal object is the set of spanning edge-subgraphs of the `3 X n` grid\ngraph, with all `3n` vertices retained, in which every vertex has degree\ndifferent from one. Degree zero is allowed.","title":"Barker's recurrence for no-leaf subgraphs of the 3 X n grid","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b5bdcb53ef93b3fab5f4d1c3abdae84bfc06c35e70504318ade17f156705e01.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a302975-divisor-count-power-ratio-denominator-square","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A302975, NAME (`%N`, verbatim):\n\n> a(n) = denominator of tau(n)^n / n^tau(n)\n\nCOMMENTS (`%C`, verbatim):\n\n> tau(n) = the number of the divisors of n (A000005).\n\n> Conjecture: all terms are squares.\n\nThe formal claim is the literal statement `∀ n ≥ 1, IsSquare (D n)`, where `D`\nis the reduced denominator of the displayed rational ratio. The definition is\ntotalized with `D 0 = 1`; that value is not a source claim.","title":"OEIS A302975 divisor-count power-ratio denominator square","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a302975-divisor-count-power-ratio-denominator-square/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/088394fe9e1a20b4abba37eb7a3c51262b0e81345a2e3df7e46b12a4ccd8e0eb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a305550-distinct-part-egf-totient-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A305550 NAME (entry by Ilya Gutkovskiy, Jun 15 2018):\n\n> Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).\n\nCOMMENT (Peter Bala, Jul 08 2022):\n\n> Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).\n\nIn the same comment:\n\n> More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series.","title":"Bala's OEIS A305550 totient-period conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d1f0656975b735bfb368270417068578684bfa6c1cf5fc5dce82a8f22b6e2f26.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a305550-integral-egf-substitution-totient-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A305550 NAME (entry by Ilya Gutkovskiy, Jun 15 2018):\n\n> Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k).\n\nCOMMENT (Peter Bala, Jul 08 2022):\n\n> Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k).\n\nIn the same comment:\n\n> More generally, we conjecture that the same property holds for integer sequences having an e.g.f. of the form G(exp(x) - 1), where G(x) is an integral power series.\n\nThe broader sentence appears verbatim on both A305550 and A004123. These are\ntwo source locations of **ONE conjecture settled once**, not two resolutions.","title":"Bala's general integral e.g.f. substitution conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7cb860a6a334c831cb663c20b69547c8e181e14755a454902474911310e80eb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A306270, NAME (`%N`, verbatim):\n\n> Composite numbers k such that b^(k(k-1)) == 1 (mod k^2) for every b coprime to k.\n\nThe offset is `%O 1,1`. The explanatory COMMENT (`%C`, verbatim) is:\n\n> These are composites k such that lambda(k^2) divides k(k-1), where lambda is the Carmichael function A002322.\n\nOrdowski's settled COMMENT (`%C`, verbatim) is:\n\n> Conjecture: all semiprimes > 4 in this sequence are in A190275. - _Thomas Ordowski_, Jul 19 2020\n\nThe following COMMENT records the finite verification:\n\n> The conjecture was verified up to 1063290841. - _Amiram Eldar_, Jul 19 2020\n\nOEIS A190275, NAME (`%N`, verbatim):\n\n> Semiprimes of the form p*(p^2 - p + 1).\n\nDefine `mem k` to mean that `k` is composite, `1 < k`, and for every natural\n`b` coprime to `k`, `b^(k*(k-1)) ≡ 1 [MOD k^2]`. The full-quantifier reading\nis: for every pair of natural primes `p` and `q` with `p ≤ q`, if `4 < p*q`\nand `mem (p*q)`, then `q = p^2-p+1`. Thus the semiprime is\n`p*(p^2-p+1)`, as required for A190275. All subtraction is truncated natural\nsubtraction. The scope wall is only Ordowski's Conjecture sentence for every\nsemiprime `k > 4`; no assertion about non-semiprimes or the finite verification\nbound is included.","title":"Ordowski's A306270 semiprime classification","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a925ed31f347e7b02603fb6aa40cec11bdcac80eac9ae6efd90cc7167117eef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A308090, NAME (`%N`, verbatim):\n\n> a(n) = gcd(2^n + n!, 3^n + n!, n+1).\n\nThe preceding COMMENT (`%C`, verbatim) is:\n\n> From observation: For n > 3, if n+1 is prime, then a(n) = n+1.\n\nThe settled COMMENT (`%C`, verbatim) is:\n\n> Conjecture: Conversely, if gcd(2^n + n!, 3^n + n!, n+1) = n+1, then n+1 is prime.\n\nThe offset is `%O 1,4`, and the AUTHOR line (`%A`, verbatim) is:\n\n> _Pedro Caceres_, May 11 2019\n\nThe full-quantifier reading is: for every `n : Nat`, if `0 < n` and\n`a n = n + 1`, then `(n + 1).Prime`, where\n`a n = gcd(gcd(2^n + n!, 3^n + n!), n+1)`. The scope wall is exactly the\nunsigned Conjecture sentence for every natural `n ≥ 1`. The positivity\nhypothesis is part of the statement: at `n = 0`, the gcd equality holds but\n`1` is not prime. The preceding one-way observation is context, not an\nadditional theorem in this resolution.","title":"Caceres' A308090 gcd-factorial-power primality criterion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308090-caceres-gcd-factorial-power-prime-criterion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e80d0ad82279f5277ee5c452541899ddc25195418d58d48a15fcc4e1bf1b299.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a308815-dfinite-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A308815, FORMULA section, states:\n\n> Conjecture: D-finite with recurrence: n*(2*n+1)*a(n) -32*(9*n-4)*(n-1)*a(n-1) +1024*(3*n^2-23*n+29)*a(n-2) +65536*(n-2)*(2*n-5)*a(n-3)=0. - _R. J. Mathar_, Jan 27 2020\n\nIts PARI definition is:\n\n> a(n) = 16^n*sum(i=0, n, (-2)^i*binomial(n, i)*C(2*n-i));\n\nHere $C_k$ is the ordinary Catalan number. Put\n\n$$a_n=16^n\\sum_{i=0}^n(-2)^i\\binom ni C_{2n-i}.$$\n\nFor every natural $n\\ge3$, the conjecture asserts\n\n$$n(2n+1)a_n-32(9n-4)(n-1)a_{n-1}\n+1024(3n^2-23n+29)a_{n-2}+65536(n-2)(2n-5)a_{n-3}=0.$$\n\nThe external tier-1 claim is preregistered in\nhttps://github.com/the-omega-institute/trureturing/issues/15713.\nThe signed PARI formula is the sequence convention; the NAME line's\nunsigned $2^i$ conflicts with the displayed data.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Natural sequence index and finite sum $0\\le i\\le n$ | `a308815`, `Finset.range (n+1)` | Exact natural indices; integer summands |\n| $16^n(-2)^i\\binom ni C_{2n-i}$ | `a308815`, Mathlib `Nat.choose` and `catalan` | Literal signed formula; natural subtraction at $2n-i$ |\n| Every $n\\ge3$ | `claim` | No extra hypothesis |\n| Four recurrence coefficients and lags | `claim` | Integer coefficient arithmetic; natural lagged arguments |\n| The recurrence equals zero | `result : claim` | Whole assertion proved |","title":"Mathar's D-finite recurrence for OEIS A308815","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308815-dfinite-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74d8b62f6d28378f4fcf93a54e164920c1df8e668112ae1527b3e643f08a87b3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A319197, NAME (`%N`, verbatim):\n\n> All entries from a(3) to a(n) appear in addition to 2^n as factors in the conjectured factorization of Fibonacci(2^(n-2)*3*m) for n >= 3 and all m >= 0.\n\nThe first COMMENT sentence (`%C`, verbatim), which is the settled clause:\n\n> It appears that Fibonacci(2^(n-2)*3*m)/(2^n) is a nonnegative integer for n >= 3 and all m >= 0.\n\nAUTHOR (`%A`, verbatim):\n\n> _Wolfdieter Lang_, Oct 09 2018\n\nThe exact Lean statement is\n`theorem result (n m : ℕ) (hn : 3 ≤ n) : 2 ^ n ∣ Nat.fib (2 ^ (n - 2) * 3 * m)`.\nThe natural-number domain includes `m=0`; `n-2` is truncated subtraction,\nwhich agrees with ordinary subtraction under `3 <= n`. The positive divisor\n`2^n` makes natural divisibility equivalent to the quoted\nnonnegative-integral quotient assertion, not a claim about truncated division.\n\nOnly that first COMMENT sentence is settled. The product\n`A(n) = Product_{j=3..n} a(j)`, the `I(n; m)` factorization conjecture, the\nspecific factors from A049660 and A253368, and the rest of the entry are not\nclaimed. The theorem does not assert an exact 2-adic valuation.","title":"Dyadic divisibility at Lang's Fibonacci indices","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cdfb850375f4dd42beaefd7a566d7c6c0202d271c62724005f6a68061984fe9a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A319927, NAME (verbatim):\n\n> Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k.\n\nCOMMENT conjecture line (verbatim; Ivan N. Ianakiev, Oct 02 2018):\n\n> Conjecture: For any nonnegative integer power p the sum of the p-th powers of the odd non-unitary divisors of a(n) divides the sum of the p-th powers of the non-unitary divisors of a(n).\n\nPeter Munn's COMMENT (verbatim):\n\n> The start of this sequence, including the 58 terms currently shown in the data section, is consistent with a definition \"nonsquarefree numbers not divisible by 4\", but some larger terms are divisible by 4: for example, a(1305) = 9216 = 2^10 * 3^2. - _Peter Munn_, Sep 21 2020\n\nThe PARI membership test (verbatim) is:\n\n> `isok(n) = my(suo = suo2(n)); if (suo, (su2(n) % suo) == 0);`\n\nThe AUTHOR line is:\n\n> _Ivan N. Ianakiev_, Oct 02 2018\n\nThe first terms printed in the DATA section are `9, 18, 25, 27, 45, 49, 50,\n54, 63, 75, ...`.\n\nFor natural numbers, define\n`nonunitaryDivisors(n) = Nat.divisors(n).filter(d => 1 < gcd(d,n/d))`,\nwhere `n/d` is natural-number division and is exact for every divisor in\n`Nat.divisors(n)`. Define `S(k,n)` as the sum of `d^k` over those divisors,\nand `O(k,n)` as the corresponding sum after the further filter\n`d % 2 = 1`. The source program's nonzero guard is retained in\n`member(n) := 0 < n and 0 < O(2,n) and O(2,n) divides S(2,n)`.\nThe literal refuted statement is\n`∀ n : ℕ, member n → ∀ k : ℕ, O k n ∣ S k n`.\n\nThe result does not assert the 2022 remark that `p*a(n)` is a term, and it\ndoes not assert anything about A034444 or A048105.","title":"Refutation of the A319927 odd non-unitary power-sum conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8c92483fc16249fd09f89a47546dda0f80f6c9e7dc10a9a36670fb53043d515.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A322327, NAME (`%N`, verbatim):\n\n> a(n) = A005361(n) * A034444(n).\n\nOFFSET (`%O`, verbatim):\n\n> 1,2\n\nAUTHOR (`%A`, verbatim):\n\n> _Werner Schulte_, Dec 03 2018\n\nCOMMENT (`%C`, verbatim):\n\n> Conjecture: Let k be some fixed integer and a_k(n) = A005361(n) * k^A001221(n) for n > 0 with 0^0 = 1. Then a_k(n) is multiplicative with a_k(p^e) = k*e for prime p and e > 0. For k = 0 see A000007 (offset 1), for k = 1 see A005361, for k = 2 see this sequence, for k = 3 see A226602 (offset 1), and for k = 4 see A322328.\n\nA005361, NAME (`%N`, verbatim):\n\n> Product of exponents of prime factorization of n.\n\nA001221, NAME (`%N`, verbatim):\n\n> Number of distinct primes dividing n (also called omega(n)).\n\nFor every integer `k`, define\n`a_k(n) = (product over prime divisors p of n of v_p(n)) * k^omega(n)`,\nwhere `v_p(n)` is the exponent of `p` in the prime factorization and the\nempty product is one. The full-quantifier reading asks that for all positive\nnaturals `m,n` with `gcd(m,n)=1`, `a_k(mn)=a_k(m)*a_k(n)`, and that for all\nnatural primes `p` and positive exponents `e`, `a_k(p^e)=k*e`. Lean's\nconvention gives `0^0=1`. The formal theorem establishes the same equations\nfor all natural `m,n` satisfying coprimality, which includes the stated\npositive domain.\n\nThe scope wall contains only those two assertions in the conjecture sentence\nfor every integer `k`. The trailing correspondences with A000007, A005361,\nA322327, A226602, and A322328 are not claimed.","title":"Schulte's exponent-product multiplicativity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/48532877c36b3085ba4cf6b8749eb0d844800efbd4da53ed2aa3e89391384122.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a326244-noncrossing-nonnesting-graph-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A326244, NAME:\n\n> Number of labeled n-vertex simple graphs without crossing or nesting edges.\n\nCOMMENTS:\n\n> Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b, and nesting if a < c < d < b or c < a < b < d.\n\nFORMULA:\n\n> Conjectures from _Colin Barker_, Jun 28 2019: (Start)\n>\n> G.f.: (1 - x)*(1 - 4*x) / (1 - 6*x + 8*x^2 - 4*x^3).\n>\n> a(n) = 6*a(n-1) - 8*a(n-2) + 4*a(n-3) for n>2.\n>\n> (End)\n\nThe entry line is `_Gus Wiseman_, Jun 20 2019`. The generating-function line\nis the same third-order recurrence together with initial values `1, 1, 2`; no\nseparate generating-function identity is claimed here.","title":"Barker's recurrence for crossing- and nesting-free labeled graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a326244-noncrossing-nonnesting-graph-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c126072cc8a27593779a02b3811f370d35184de5f16f9f4f6f4fd7ba3a22894c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a328959-divisor-count-prime-factor-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A328959, NAME (verbatim):\n\n> a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n), where sigma_0 = A000005, nu = A001221, omega = A001222.\n\nHere the entry's `omega` is A001222, the number of prime factors counted\nwith multiplicity, and its `nu` is A001221, the number of distinct prime\nfactors. In conventional notation these are respectively `Omega` and\n`omega`.\n\nCOMMENT (verbatim; Gus Wiseman, Nov 02 2019; idea from Mats Granvik):\n\n> Conjecture: All terms are nonnegative except for a(1) = -1.\n\nThe exception is exact: `a(1) = -1`. The target is therefore the\nnonnegativity assertion for every `n >= 2`.","title":"The divisor-count prime-factor bound in A328959","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/955ba20ef4e6b4dda1debbdbe9c78eafe0e5a43fbbccb1b4f1f06ff81e9217c2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a329278-triangular-residue-descent-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A329278, NAME (verbatim):\n\n> Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n).\n\nCOMMENT (verbatim; Peter Kagey, Nov 11 2019):\n\n> Conjecture: for n > 0, the n-th row has 2^(n-1)-1 descents.\n\nA descent is an index `k` with `0 <= k < 2^n - 1` and\n`T(n,k) > T(n,k+1)`. The target concerns every positive natural `n`.","title":"The triangular-residue descent count in A329278","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329278-triangular-residue-descent-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0763d73cf92964e732d22708a837150de8cce159d818aa71a3ac7714d156d55.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a329369-printed-row-sum-range","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Write a(n) for A329369 and T(n,k) for the A373183 row coefficients. Mikhail\nKurkov printed on June 05 2024 the assertion that\n\n a(2^m*n + q) = Sum over i from A001511(n+1) to A000120(n)+1 of\n T(n, i) * a(2^m*(2^(i-1)-1) + q)\n\nholds for n at least zero, m at least zero, and q at least zero. The same\nassertion appears in A373183 as its second conjecture. The question is whether\nthe printed range for q is correct.","title":"The printed index range of the sixth A329369 sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/78ee6b760ce7f6bbaa46c1403de70399efddcb845e7001304de18f7cf6e26d28.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a331474-hankel-generating-function","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A331473 defines, at offset zero,\n\n`s_n = Sum_{k=0..n} (-1)^(n-k) * binomial(2*k+2,k)`.\n\nOEIS A331474 is its literal Hankel transform\n\n`H_n = det(s_(i+j))_(0<=i,j<=n)`\n\nand Paul Barry's January 17, 2020 FORMULA line conjectures the full identity\n\n`Sum_{n>=0} H_n*x^n = (1 + 3*x + 16*x^2 - 8*x^3 + 36*x^4 - 12*x^5 + x^6 - x^7) / (1 + 7*x^2 + x^4)^2`.\n\nThe resolved statement is\n`D5/S3/Constants/Moments/A331474HankelGeneratingFunction.a331474_hankel_generating_function`.\nIt is an equality for `PowerSeries.mk H`, so the left side is the actual\ndeterminant sequence, not a recurrence-defined surrogate. The denominator has\nconstant coefficient 1 and the formal right side uses its unit inverse.","title":"Barry's A331474 Hankel generating-function conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3b7d06a1a85c4c7c09b945fd35140f7f6ccbe9989ed137b65244fcb559881db.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a334595-value-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Kagey's OEIS A334595, revision 18, gives the sequence name\n\"Binary interpretation of the right diagonal of the XOR-triangle with first\nrow generated from the binary expansion of n.\" Its fourth `%C` comment states:\n\n> Conjecture: a(n) = 1 if and only if n is a power of two.\n\nThe entry's definition uses the unpadded binary expansion of `n`. The literal\nformal claim is\n`forall n : Nat, 1 <= n -> (a n = 1 <-> exists k : Nat, n = 2 ^ k)`.\nThe exponent is allowed to be zero, so `n = 1` is included; `n = 0` is outside\nthe positive-index domain. The right edge is read from the upper-right corner\nto the apex with every position retained, including zero bits.","title":"The value-one terms of OEIS A334595 are exactly the powers of two","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a334595-value-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ef3e26f6ff7709db615cb21167ef495fff01ac448feeb35f25d77638845dfb7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a335901-self-referential-doubling-first-occurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A335901, %N (verbatim):\n\n> a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1.\n\n%C (verbatim):\n\n> Least k such that a(k) = 2^n are 1, 2, 5, 21, 169, 2705, ... (Conjecture: This sequence is A117261).\n\nThe comparison sequence A117261 is given by A117261(0) = 1 and\nA117261(r+1) = 2^r * A117261(r) + 1. The phrase \"least k such that\na(k) = 2^r\" is formalized as a hit together with minimality among positive\nindices.","title":"First Occurrences in OEIS A335901","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335901-self-referential-doubling-first-occurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72aa8631fdfe92e406ceaed33d51fc5ddd6cf05d4ddc17cb5fb9b5cf4c88e7d0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a335925-self-referential-quotient-first-occurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A335925, NAME (`%N`, verbatim):\n\n> a(n) = a(floor((n-1)/a(n-1))) + 1 with a(1) = 1.\n\nCOMMENT (`%C`, verbatim; Altug Alkan, Jun 30 2020):\n\n> Least k such that a(k) = n are 1, 2, 5, 16, 65, 326, 1957, ... (Conjecture: This sequence is A000522).\n\nHere A000522(0) = 1 and A000522(m) = m*A000522(m-1) + 1. The phrase\n\"least k such that a(k) = m\" is formalized as a hit at that index together\nwith minimality among every positive index carrying m.","title":"First occurrences in the A335925 self-referential quotient recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335925-self-referential-quotient-first-occurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b1a98f6f3ae2dddd4c53eaeeb17d3a6351f258b03e40d5adcd1a3de23b3c7c7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a338040-weighted-stirling-power-prime-period","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A338040, entry by Vaclav Kotesovec, Oct 08 2020, has the following\nNAME, quoted verbatim from `Library/Recurrence/bala2022a338040.md`:\n\n> E.g.f.: Sum_{j>=0} 4^j * (exp(j*x) - 1)^j.\n\nThe supplied attribution also identifies Paul D. Hanna, May 31 2022.\nThe COMMENT by Peter Bala, May 31 2022, states:\n\n> Conjecture: Let p be prime. The sequence obtained by reducing a(n) modulo p for n >= 1 is purely periodic with period p - 1.\n\nThe theorem proves periodicity with period p - 1 for every prime p and n >= 1.\nIt does not assert minimality of the period. The e.g.f. identity is read through\nthe entry's finite FORMULA quoted in Evidence; that identity is not separately\nformalized.","title":"Prime periods of the A338040 weighted Stirling power sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338040-weighted-stirling-power-prime-period/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3faae7d7421fb492d22f7c6f76de4bf403831ea35e2946d6fe84b4a591e1dc33.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a338193-egf-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The October 26, 2024 conjecture by Mikhail Kurkov in OEIS A338193 asserts\nthat a(n)=f(0,n-1) for every positive n. Here a(n)=n! times coefficient n\nof the rational formal series satisfying A(0)=1 and\nA=1+Integral ((x/A)' / (x/A^2)') dx, with integral constant zero.\nThe independent natural-valued recurrence has f(j,0)=1,\nf(0,m+1)=f(0,m)+(m+1)f(1,m), and\nf(j+1,m+1)=f(j,m+1)+(m+1)(f(j+1,m)+f(j+2,m)).","title":"The EGF coefficients of A338193 and Kurkov's recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338193-egf-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc76bfa3e3e5388f2a61c6838b8360905be6d8fecc215ece103312bf3b6470e2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a338636-continued-fraction-odd-square-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A338636 (Paul D. Hanna, Nov 04 2020) records the following NAME and\nFORMULA lines, quoted verbatim from `Library/Recurrence/hanna2020a338636.md`:\n\nNAME:\n> G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3^2*x/(A(x) - 5^2*x/(A(x) - 7^2*x/(A(x) - 9^2*x/(A(x) - ...))))), a continued fraction relation.\n\nFORMULA:\n> a(n) = 0 (mod 8) for n > 1 (conjecture).\n\nOnly this FORMULA conjecture is claimed proved. The entry's two further\nconjectures are outside this resolution:\n\n> For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1\n\n> For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2)","title":"Divisibility of the A338636 coefficients by eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338636-continued-fraction-odd-square-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55b67f7121af36d68a9cc91eb5a3973261ab7964d58134121963fdab6537cdcc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A339621 records these lines verbatim:\n\n```text\n%N Sum of Fibonacci divisors of n^2 + 1.\n%C A Fibonacci divisor of a number k is a Fibonacci number that divides k. (The divisor 1 is only counted once.)\n%C Conjecture: If the sum of the Fibonacci divisors of m^2 + 1 is a Fibonacci number, then this number belongs to the sequence A001906(n) = F(2n) where F(n) is the Fibonacci sequence.\n%A _Michel Lagneau_, Dec 10 2020\n```\n\nFormally, let `FibValues(N)` be the finite set of distinct positive Fibonacci\nvalues dividing `N`, so the value 1 occurs once. Every positive Fibonacci value\nhas a unique canonical index `r >= 2`. For every natural `m`, if\n`sum(FibValues(m^2 + 1))` is a Fibonacci value and `r >= 2` is its canonical\nindex, then there is a positive integer `q` such that\n`sum(FibValues(m^2 + 1)) = F_(2q)`.","title":"Fibonacci divisor sums at square-plus-one arguments have even index","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57581c9a9f89b0b42f387d33b98a1ff4fbdc45252dc1c0c3a9767d662af32cce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a339669-odd-lucas-square-fibonacci-divisors","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A339669, NAME (verbatim):\n\n> Number of Fibonacci divisors of Lucas(n)^2 + 1.\n\nCOMMENTS (verbatim):\n\n> Particular attention must be paid to the regularity properties of the number of divisors of\n> Lucas(n)^2 + 1 observed for n < 156, when a(n) = 1 or 2. From this observation, we propose two\n> conjectures verified for n < 156.\n\n> Conjecture 1: a(6*n+3) = 1.\n\n> Conjecture 2: a(6*n+1) = a(6*n+5) = 2.\n\nThe count is of distinct divisor values, not of indices. The entry's own EXAMPLE line settles\nthis (verbatim):\n\n> a(8) = 5 because the divisors of Lucas(8)^2 + 1 = 47^2 + 1 = 2210 are {1, 2, 5, 10, 13, 17,\n> 26, 34, 65, 85, 130, 170, 221, 442, 1105, 2210} with 5 Fibonacci divisors: 1, 2, 5, 13 and 34.\n\nThe value 1 appears once although `F_1` and `F_2` are both 1. The same line shows why the\nconjectures are confined to odd indices: at the even index 8 the count is 5, not 1 or 2.\n\nBoth indices `6n+1`, `6n+3` and `6n+5` are odd, so both conjectures are consequences of one\nstatement about odd indices:\n\n> For every odd `n`, the set of Fibonacci divisors of `L_n^2 + 1` is `{1}` when `3` divides `n`\n> and `{1, 2}` otherwise.","title":"Fibonacci Divisors of an Odd-Index Lucas Square Plus One","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339669-odd-lucas-square-fibonacci-divisors/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c6719b19991f402eee6b66eeb8cdab7f03a84313bd0f382c217a38134042db25.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a343502-divisor-count-primes-dequatre","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. H. R. Dequatre's OEIS A343502 is titled “Numbers k such that tau(tau(k)) and tau(k+1) are both prime, where tau is the number of divisors function.” Its comment states: “Conjecture: all members except 2 and 3 are composite.” The entry also asks whether there are finitely many odd terms.\n\nThe formal claim is that every natural number $k$ with $1 \\le k$, with both divisor-count conditions prime, and with $k \\ne 2$ and $k \\ne 3$, satisfies $1 < k$ and is not prime.\n\n### Statement echo\n\nThe membership predicate in `D5/S3/Arith/Divisors/OeisA343502DivisorCountPrimes.lean` expands tau literally as `k.divisors.card`, preserving the source convention. The settling declaration is `result : claim`; its Scribe formula expands the complete claim body.","title":"Dequatre's OEIS A343502 divisor-count prime conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343502-divisor-count-primes-dequatre/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/108ff926f8dfc820b75eeb8ee41c10eb2321125bbf1d9817cb3e58c343fbac58.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a343825-kagey-power-sequences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.\n\nThe source is P. Kagey's OEIS A343825, COMMENT, revision 22. The defining NAME and the linked A071537 NAME are quoted in [the source note](../Library/Factorization/kagey2021a343825.md). The preregistration is [#18687](https://github.com/the-omega-institute/trureturing/issues/18687).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| A nondecreasing sequence beginning at $k$ and ending at $m$ | `Valid n k m b`: option-valued first and last elements and pairwise natural order | Literal source convention |\n| Every multiplicity is below $n$; the product is an integer $n$th power | The count bound and integer-power conjuncts of `Valid` | All conjuncts retained |\n| $T(n,k)$ is the least such $m$ | `T` uses `Nat.find` | Zero extends the function outside the existence domain |\n| Integer exponents on $[k,m]$, with nonzero first exponent and product one | `Relation k m` | Integer powers are interpreted in the rationals |\n| A071537 is the least relation endpoint $m\\ge k$ | `A071537` uses `Nat.find` | Zero extends the function outside the existence domain |\n| Fixed $k$ and sufficiently large prime $p$ | `claim`: $\\forall k\\ge1,\\ \\exists N,\\ \\forall p,\\ p\\text{ prime}\\land N\\le p\\Rightarrow T(p,k)=A071537(k)$ | $k\\ge1$ is A071537's offset; the threshold depends only on $k$ |","title":"Kagey's eventual equality of power-sequence minima","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/144e870a3e886b27cf3499f88d0a511bac5abde1a4e236ffc4d2344a00b113d0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a344083-marcus-pythagorean-nonpolygonal-triple","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Michel Marcus's COMMENT in OEIS A344083 states:\n\n> Conjecture: there are no other Pythagorean triples that give this minimum. In other words, it is the only triple with 3 A090467 terms.\n\nOEIS A090467 consists of the natural numbers that are not polygonal numbers\nof order greater than two. In the formal statement, `nonpolygonal n` means\nthat no `k > 2` and `m > 2` satisfy\n`n = m + (k - 2) * Nat.choose m 2`.\n\nThe conjecture asserts both that `(3,4,5)` has three nonpolygonal sides and\nthat every positive Pythagorean triple with three nonpolygonal sides has\nunordered legs `{3,4}` and hypotenuse `5`.","title":"Marcus's A344083 nonpolygonal Pythagorean triple conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344083-marcus-pythagorean-nonpolygonal-triple/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2e60d313703d1d3a345b1dab07fc5695c642cef52cd6c73980c851bee176b9e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a344598-alternating-gcd-sum-pillai","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A344598 NAME (Seiichi Manyama, May 24 2021):\n\n> a(n) = Sum_{k=1..n} phi(k) * (floor(n/k)^2 - floor((n-1)/k)^2).\n\nFORMULA (Peter Bala, Jan 01 2024):\n\n> Conjecture: a(n) = Sum_{k = 1..2*n} (-1)^k * gcd(k, 4*n). Cf. A344372. - _Peter Bala_, Jan 01 2024\n\nBoth quotations are copied verbatim from the claim in\n`Library/Arith/bala2024a344598.md`. The target is the equality for every positive n.\n\n**范围与诚实边界**: This is a modest identity (tier 1). The two divisor-sum\nformulas `a(n) = Sum_{d|n} phi(d)*(2*n/d - 1)` and\n`a(n) = Sum_{d|n} (2*d*tau(d) - sigma(d))*mu(n/d)` were already proved elsewhere\nby Daniel Weber, as credited in Mikhail Kurkov's Mar 31 2026 comment. They are\nnot the target. The target is the alternating-gcd formula, which the search\nseat's 2026-09-08 reading recorded as still marked Conjecture. Natural division\nand truncated subtraction define a; the alternating sum and theorem subtraction\nare in the integers.","title":"Bala's alternating gcd-sum formula for A344598","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344598-alternating-gcd-sum-pillai/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c958924567b36a7215d610247a2ab14a81591d428aa3ab9d0b802c7c1ce50251.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A344695, revision 39, defines “a(n) = gcd(sigma(n), psi(n)), where sigma is the sum of divisors function, A000203, and psi is the Dedekind psi function, A001615.” Its comment states: “Conjecture: the set of primes that appear in the sequence is A065091 (the odd primes).” The assertion is preregistered in [#18688](https://github.com/the-omega-institute/trureturing/issues/18688).\n\nFor each natural prime $q$, the formal claim says that some natural $n\\ge1$ has $a(n)=q$ if and only if $q$ is odd.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A001615: “Dedekind psi function: n * Product_{p\\|n, p prime} (1 + 1/p).” | `psi`, using natural exact division by the product of distinct prime factors | The binding integer expression of #18688; positive indices make the division exact |\n| A344695 defines the gcd of sigma and psi | `a` | `ArithmeticFunction.sigma 1` and `Nat.gcd` retain the source operations |\n| “the set of primes that appear” | `claim` quantifies over every natural prime and an existential positive index | The guard excludes composite candidate values, and positivity matches the sequence's domain |\n| “is A065091 (the odd primes)” | The existential occurrence is equivalent to `Odd q` | Both directions, with no upper bound |\n\nThe public definitions are the binding expressions of #18688. The value at zero is outside the source sequence. The settling declaration `result` proves the complete `claim`.","title":"The Karttunen–Munn prime-values conjecture for OEIS A344695","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e30c78304903b5510de3d1fcf1f83f37c0417f43bc65dc3b8838694d124b158.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347285-de-vlieger-prime-power-descent","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. De Vlieger's conjecture in [OEIS A347285](https://oeis.org/A347285), COMMENTS (unpaginated), states: \"Conjecture: let S be the sum of the absolute values of the first differences of terms in row n. For all n > 0, n - S = 1.\" The NAME defines the rows as \"Irregular triangle T(n,k) starting with n followed by e_k = floor(log_p_k(p_(k-1)^e_(k-1))) such that e_k > 0.\" The FORMULA is \"T(n,1) = n; T(n,k) = floor(log_p_k(p_(k-1)^T(n,k-1))).\" The source text is [pinned at e7d0073e](https://raw.githubusercontent.com/oeis/oeisdata/e7d0073e/seq/A347/A347285.seq). Preregistration: [#18689](https://github.com/the-omega-institute/trureturing/issues/18689). Source note: [devlieger2021a347285](../Library/PrimeForms/devlieger2021a347285.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The primes are indexed from one | Existing `D5.S1.Recurrence.Parity.PrimePowerShiftLogDerivative.prime`, with value `Nat.nth Nat.Prime (k - 1)` | The literal preregistered enumeration is reused. |\n| Initial row entry is $n$ | `e n 0` | Equals $n$. |\n| Real logarithm and downward natural floor at the next prime | `e n (j + 1)` | The current exponent is raised on the current prime, and the logarithm base is the next prime. |\n| Keep entries greater than zero | `len n`, the first zero index, with zero fallback if none exists | `e_zero_exists` and `len_spec` establish existence and positivity before that index. |\n| Sum absolute first differences within the row | `S n` | Natural absolute value of each integer difference, over indices below `len n - 1`. |\n| Every $n>0$, with $n-S=1$ | `claim`, proved by `result` | Natural indices, strict positivity, and integer subtraction; no additional hypothesis. |\n\nThe source convention \"T(0,1) = 0 by convention; 0 is not allowed for n > 0\" leaves the separately listed row zero outside the conjecture. The recurrence extends through zero for proof purposes; only the positive prefix enters $S$.","title":"oeis a347285 de vlieger prime power descent","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347285-de-vlieger-prime-power-descent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/13c4133813aafa1492e5c1270c7535d3f0af00dbd7c961d38e14a2a64a2dac55.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A347293, NAME (`%N`, verbatim):\n\n> Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(1 + (i-1) * (k-1),n) for 1 <= k <= n.\n\nFORMULA (`%F`, verbatim):\n\n> Conjecture: Row sums equal Dirichlet convolution of A000290 and A127473.\n\nA000290 is the square function and A127473 is the square of Euler's totient.\nThus the full-quantifier statement is that for every positive natural `n`,\n\n`Sum_{x=0}^{n-1} Sum_{y=0}^{n-1} gcd(1+x*y,n)\n = Sum_{d|n} d^2 * phi(n/d)^2`.\n\nThe left side is the triangle row sum after the index changes `x=i-1` and\n`y=k-1`. The divisor sum is the pointwise definition of the stated\nDirichlet convolution.","title":"OEIS A347293 shifted-product gcd row-sum convolution","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff1628b2d7f6c18aad75a3521e29d80052dbc977a0926a1561bded7dbf04805d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347854-bala-half-index-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347854](https://oeis.org/A347854), states:\n\n> a(n) = (6*n)!/((3*n)!*(2*n)!) * (n/2)!/(3*n/2)!.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347854BalaHalfIndexIntegral.a` | Literal expression; natural factorials are cast to the reals. |\n| Fractional factorials use Gamma | `OeisA347854BalaHalfIndexIntegral.ffact` | $x! = \\Gamma(1+x)$; the natural-index factorials are `Nat.factorial`. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347854BalaHalfIndexIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347854BalaHalfIndexIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347854 fractional-factorial integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347854-bala-half-index-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66b60f6fad95a35384fe046a49b8448f8295c01af9c0a5a5cfad02c697546da1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347855-bala-aux-b-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347855](https://oeis.org/A347855), states:\n\n> a(3*n+1) = 3^(3*n+1) * b(n), where b(n) = (12*n+4)!/( (6*n+2)!*(3*n+1)!*Product_{k = 0..3*n} (3*n+3*k+4) ) appears to be integral.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxBIntegral.b` | Literal expression; natural factorials are cast to the rationals. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxBIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxBIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347855 auxiliary b integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-aux-b-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/215e3585a2a857311f27e7043a9fcdde9ff0751093577b46c00cb08662379af7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347855-bala-aux-c-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347855](https://oeis.org/A347855), states:\n\n> a(3*n+2) = 3^(3*n+2) * c(n), where c(n) = (12*n+8)!/( (6*n+4)!*(3*n+1)!*Product_{k = 0..3*n+2} (3*n+3*k+2) ) appears to be integral. (End)\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxCIntegral.c` | Literal expression; natural factorials are cast to the rationals. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxCIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347855BalaAuxCIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347855 auxiliary c integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-aux-c-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07ef6a311a727ac848734f36e41c987e824f6453f1b0480ed4d6995c7e43c621.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347855-bala-third-index-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347855](https://oeis.org/A347855), states:\n\n> a(n) = (4*n)!/((2*n)!*(n)!) * (n/3)!/(4*n/3)!.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347855BalaThirdIndexIntegral.a` | Literal expression; natural factorials are cast to the reals. |\n| Fractional factorials use Gamma | `OeisA347854BalaHalfIndexIntegral.ffact` | $x! = \\Gamma(1+x)$; the natural-index factorials are `Nat.factorial`. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347855BalaThirdIndexIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347855BalaThirdIndexIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347855 fractional-factorial integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-third-index-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/89d722ea7a244700b4cf67e6e8e37a6f82f7ab1ded0e11b3db40bafde736990e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347856-bala-half-index-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347856](https://oeis.org/A347856), states:\n\n> a(n) = (6*n)!/((4*n)!*n!) * (n/2)!/(3*n/2)!.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347856BalaHalfIndexIntegral.a` | Literal expression; natural factorials are cast to the reals. |\n| Fractional factorials use Gamma | `OeisA347854BalaHalfIndexIntegral.ffact` | $x! = \\Gamma(1+x)$; the natural-index factorials are `Nat.factorial`. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347856BalaHalfIndexIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347856BalaHalfIndexIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347856 fractional-factorial integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347856-bala-half-index-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/80a44d65bf15c3a115c1449900a3b171992fba4a8646216d26249355847f7856.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347857-bala-half-index-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347857](https://oeis.org/A347857), states:\n\n> a(n) = (6*n)!/((3*n)!*(2*n)!) * (3*n/2)!/(5*n/2)!.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347857BalaHalfIndexIntegral.a` | Literal expression; natural factorials are cast to the reals. |\n| Fractional factorials use Gamma | `OeisA347854BalaHalfIndexIntegral.ffact` | $x! = \\Gamma(1+x)$; the natural-index factorials are `Nat.factorial`. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347857BalaHalfIndexIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347857BalaHalfIndexIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347857 fractional-factorial integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347857-bala-half-index-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2209d05be8bc5f1c2c6025399a848e56c568715581eb687e29cdaa3870b182d4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a347858-bala-half-index-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Bala, OEIS [A347858](https://oeis.org/A347858), states:\n\n> a(n) = (9*n)!/((3*n)!*(2*n)!) * (n/2)!/(9*n/2)!.\n\nFractional factorials mean $x! = \\Gamma(1+x)$.\nThe integrality conjecture quantifies over every natural index, including zero.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The source's displayed ratio and product endpoints | `D5.S3.Arith.FactorialRatio.OeisA347858BalaHalfIndexIntegral.a` | Literal expression; natural factorials are cast to the reals. |\n| Fractional factorials use Gamma | `OeisA347854BalaHalfIndexIntegral.ffact` | $x! = \\Gamma(1+x)$; the natural-index factorials are `Nat.factorial`. |\n| Integral for all natural indices | `D5.S3.Arith.FactorialRatio.OeisA347858BalaHalfIndexIntegral.claim` | For every $n\\in\\mathbb N$ there exists $z\\in\\mathbb Z$ whose cast equals the ratio. |\n| The conjecture is true | `D5.S3.Arith.FactorialRatio.OeisA347858BalaHalfIndexIntegral.result` | Unconditional proof of the complete claim. |\n| Proof-only arithmetic parameters | Consumed floor, divisibility and Gamma helpers | Internal parameters support the proof; they add no assumption to the conjecture. |\n\nThe dropped or weakened source-clause set is empty.","title":"Bala's A347858 fractional-factorial integrality","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347858-bala-half-index-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/491eebfbf7fe27f35c2abdac531559873af8bab678c4f789c40e9c804ccc7ef0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A350900, NAME (`%N`, verbatim):\n\n> Triangle read by rows: T(n, k) = Sum_{i=1..n} gcd(i,n) / gcd(gcd(i,k),n) for 1 <= k <= n.\n\nThe settled FORMULA (`%F`, verbatim) is:\n\n> Conjecture: Row sums equal Dirichlet convolution of A002618 and A057660.\n\nThe full-quantifier reading is: for every positive natural `n`, define\n`rowSum(n) = Sum_{i=1..n} Sum_{k=1..n}\ngcd(i,n)/gcd(gcd(i,k),n)`; then\n\n`rowSum(n) = Sum_{d|n} d*phi(d) * (Sum_{e|(n/d)} e*phi(e))`.\n\nHere `phi` is Euler's totient, A002618(n) is `n*phi(n)`, and A057660(n)\nis `Sum_{d|n} d*phi(d)`. The identity\n`gcd(gcd(i,k),n) = gcd(gcd(i,n),k)` shows the denominator of each row\nsummand divides its numerator; these are exact natural quotients. For each\n`d|n` the complementary quotient `n/d` is also exact. Only the quoted\nrow-sum formula is settled. The `%C` conjecture for an arbitrary arithmetic\nfunction `f` is outside this claim.","title":"Schulte's gcd-quotient row-sum convolution","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3551eef5b29946e17469c04b447b47a3323f9e6fd626b76172454d68058334ae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a352700-hanna-catalan-congruences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A352700](https://oeis.org/A352700) (revision `#7`, March 31, 2022; unpaginated), NAME, defines the sequence by:\n\n> G.f.: Sum_{n>=0} binomial((n+1)*(2*n+1),n)/(2*n+1) * x^n / C(x)^(n*(2*n+1)+1), where C(x) = 1 + x*C(x)^2 is the Catalan function (A000108).\n\nIts COMMENTS state the six numbered conjectures:\n\n> Conjectures: given g.f. A(x), let C(x) = (1 - sqrt(1-4*x))/(2*x) be the Catalan power series (A000108), then\n> (1) A(x)^3 = A(x^3) (mod 3),\n> (2) A(x) = C(x) + x^2*C(x)^3 (mod 3) = (2 - x)*C(x) - 1 (mod 3),\n> (3) A(x) = C(x) (mod 2),\n> (4) a(n) = binomial(2*n+1,n)/(2*n+1) + 3*binomial(2*n-1,n-2)/(2*n-1) (mod 3) for n >= 0,\n> (5) a(n) = 2*A000108(n) - A000108(n-1) (mod 3) for n >= 1,\n> (6) a(n) = A000108(n) (mod 2) for n >= 0.\n\nThe published OFFSET is `0,3`. Its EXAMPLE field records the coefficient expansion of $C(x) + x^2 C(x)^3$, under which the second term $3\\binom{2n-1}{n-2}/(2n-1)$ vanishes for $n < 2$ by the negative-lower-index binomial convention $\\binom{2n-1}{n-2} = 0$, and its PARI PROG defines $a(n)$ by the finite degree-$n$ truncation over $0 \\le m \\le n$. The [Library note](../Library/RecurrenceComposition/hanna2022a352700.md) records the source and exact integer divisibility of each weight and Raney quotient. Preregistration: [#23834](https://github.com/the-omega-institute/trureturing/issues/23834).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Weight $\\binom{(m+1)(2m+1)}{m}/(2m+1)$ | `weight m` | Exact natural quotient; $(2m+1) \\mid \\binom{(m+1)(2m+1)}{m}$ proved for every $m \\in \\mathbb N$. |\n| Catalan generating series $C(x) = 1 + x C(x)^2$ and $C(x)^{-1}$ | `CatalanCompositionSquareParity.catalanUnit`, `catalanInvZ` | Integral formal series in $\\mathbb Z[[X]]$ with $[X^n]C(X) = \\operatorname{catalan}(n)$, constant term `CatalanCompositionSquareParity.catalan_unit_zero`, and multiplicative unit inverse. |\n| Defining series $A(x)$ and $a(n) = [x^n]A(x)$ | `coeffA`, `seriesA` | Finite $(X)$-adically exact sum over $0 \\le m \\le n$ in $\\mathbb Z$ and $\\mathbb Z[[X]]$. |\n| Reductions modulo $2$ and $3$ | `seriesMod p`, `catalanMod p` | Coefficientwise ring-homomorphism reductions in $(\\mathbb Z/p\\mathbb Z)[[X]]$. |\n| Second term of `(4)` under $\\binom{2n-1}{n-2}=0$ for $n < 2$ | `raneyTerm n` | Zero for $n < 2$ and exact natural quotient $(3\\binom{2n-1}{n-2})/(2n-1)$ for $n \\ge 2$. |\n| Conjectures `(1)`–`(6)` | Six conjuncts of `claim`, proved by `result` | Exact conjunction of `(1)` `(seriesMod 3)^3 = (seriesMod 3)(X^3)`, `(2)` both mod-$3$ series identities, `(3)` mod-$2$ series identity, `(4)` mod-$3$ Raney congruence for all $n \\ge 0$, `(5)` mod-$3$ Catalan difference congruence for all $n \\ge 1`, and `(6)` mod-$2$ Catalan congruence for all $n \\ge 0`. |","title":"Hanna's OEIS A352700 Catalan power-series and modular-coefficient conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11ee92eb821ffc0b8d4c0332691b0835130648060df5d22a05adf333a8980d9c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a355872-bilateral-theta-reversion-mod-sixteen","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A355872 (Paul D. Hanna, Aug 09 2022), as quoted in\n`Library/Recurrence/hanna2022a355872.md`, gives the NAME:\n\n> G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2).\n\nThe single conjecture anchor of this dossier is the COMMENT:\n\n> Conjecture: a(n) == 2 (mod 4) for n >= 1.\n\nThe entry also states two further conjectures:\n\n> Conjecture: a(2*n-1) == 2 (mod 8) for n >= 1.\n>\n> Conjecture: a(2*n) == 6 (mod 8) for n >= 1.\n\nThe latter two are settled by the companion `hanna_conjecture_mod_eight`;\nthey do not create additional resolution anchors in this dossier.","title":"Coefficient congruences for the A355872 bilateral theta reversion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a355872-bilateral-theta-reversion-mod-sixteen/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/96daea19f386b57ed8e7cac40d94c77d213a5505f9039494e94f5b6150490ab3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a356508-hanna-product-partition-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna, OEIS A356508, created 11 August 2022. The NAME field states:\n\n> G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x)).\n\nThe COMMENTS field states:\n\n> Conjecture: a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002), otherwise a(n) == 0 (mod 4) when n > 0.\n\nThe formal statement uses rational formal power series, takes the product coefficientwise through the finite window needed for each coefficient, and interprets the partition number as `Fintype.card (Nat.Partition k)`. It asserts existence of a solution and the quoted integer and residue properties for every positive coefficient.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| `G.f. A(x) satisfies: 2 = Product_{n>=1} (1 + x^n*A(x)) * (1 + x^(n-1)/A(x))` | `IsGF A` over `PowerSeries ℚ`; at degree `m` the product is `Finset.Icc 1 (m + 1)`, and `A⁻¹` is `invOfUnit A 1`. |\n| “a(n) == 2 (mod 4) at n = 2*k for all k > 0 that have an odd number of partitions (A052002)” | For `0 < n`, an integer `z` equal to `coeff n A` satisfies `z ≡ 2 [ZMOD 4]` whenever some `k > 0` has `n = 2 * k` and `Odd (Fintype.card (Nat.Partition k))`. |\n| “otherwise a(n) == 0 (mod 4) when n > 0” | For the same positive `n`, if no such `k` exists then `z ≡ 0 [ZMOD 4]`. |\n| “The terms are integers.” | The claim supplies an integer `z` equal to every positive coefficient. |","title":"Hanna's OEIS A356508 product-partition congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a356508-hanna-product-partition-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f4df1ce1ea3d9899741ec62785a89d3d65b96dbbaf1e517192d3c6f7a3ffbd7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a359482-positive-permutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Is this sequence a permutation of the integers > 0?\n\nOEIS A359482 defines the lexicographically earliest sequence of distinct\npositive terms for which a(n)+a(n+1) is a decimal substring of the\nconcatenation of a(n) and a(n+1). This dossier concerns exactly the\npermutation question. Hasler's July 3, 2023 comment conjectures a negative\nanswer and, more specifically, that the initial 1 is the only single-digit term.","title":"The permutation question for A359482","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a359482-positive-permutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd1271dbda9768a169adf15bce7531227e18f4c950e9ff6beeb56f65bb2f198e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a361552-literal-partition-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A361552 revision #14 states:\n\n> Conjecture: a(n)/2 == A000041(n-1) (mod 2) for n >= 1; that is, a(n) is even, and a(n)/2 has the same parity as the number of partitions of n-1, for n >= 1.\n\nThe original definition is:\n\n> G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following. (1) 2*x = Sum_{n=-oo..+oo} x^(n*(3*n+1)/2) * (A(x)^(3*n) - 1/A(x)^(3*n+1)).\n\nThe source, attribution, immutable locator and classical suppliers are in\n`Library/Recurrence/hanna2023a361552.md`. The preregistered exact question is\n.","title":"The literal A361552 equation and ordinary partition parity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a361552-literal-partition-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5836a4e3d8a3e8fd8f02aada840fff51cfd94b5d6e2e6172ceaa8ff2d34b432f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363104-bilateral-quarter-shift-theta-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A363104 (Paul D. Hanna, May 21 2023) gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2023a363104.md`:\n\n> Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-x)^n * (4*A(x) + x^(n-1))^(n+1).\n\n> Conjecture: g.f. A(x) == theta_3(x) (mod 4); a(n) == 2 (mod 4) iff n is a nonzero square and a(n) == 0 (mod 4) iff n is nonsquare.\n\nThe NAME is formalised coefficientwise with exact finite windows after the\ntwo Laurent terms cancel. `mod_four_identity` proves the generating-function\nclause; `hanna_conjecture` proves both coefficient biconditionals. This dossier\nhas one resolution claim, attached only to `hanna_conjecture`.","title":"The A363104 bilateral generating function modulo four","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363104-bilateral-quarter-shift-theta-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a0478a65bbc631fcda41675ce49f536c5c13a2b95a2114d8557f1dd3a8d63f5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363184-bilateral-quarter-double-shift-theta-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A363184 (Paul D. Hanna, May 20, 2023), quoted verbatim from\n`Library/Recurrence/hanna2023a363184.md`:\n\nNAME:\n\n> Expansion of g.f. A(x) satisfying 4 = Sum_{n=-oo..+oo} (-1)^n * x^n * (4*A(x) + x^(2*n-1))^(n+1).\n\nCOMMENT:\n\n> Conjecture: g.f. A(x) == theta_3(x^2) (mod 4); a(n) == 2 (mod 4) if n = 2*k^2 for integer k > 0, and a(n) == 0 (mod 4) if floor(n/2) is nonsquare.\n\nThe NAME is formalised coefficientwise with exact finite windows after\ncancelling the two Laurent terms. The g.f. clause is `mod_four_identity`, the\nfull coefficient classification is `hanna_conjecture`, and the COMMENT's\nweaker zero clause is `hanna_conjecture_floor`. This dossier has one resolution\nclaim, attached only to `hanna_conjecture`.","title":"The A363184 bilateral quarter-double-shift theta congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363184-bilateral-quarter-double-shift-theta-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba0ae70e059a813e0736312e4661d6a60d2da63c4a8f368aad7a012ea6986091.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363308-catalan-cubic-composition-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A363308 (Paul D. Hanna, May 28, 2023), quoted verbatim from\n`Library/Arith/hanna2023a363308.md`:\n\nNAME:\n\n> Expansion of g.f. C(x*C(x)^3), where C(x) = 1 + x*C(x)^2 is the g.f. of the Catalan numbers (A000108).\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n is a power of 2 or n = 0.","title":"Parity of the cubic Catalan composition A363308","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363308-catalan-cubic-composition-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c5c1308f06f24d152d7dede73b1b6658d728ae523fc0efee430478370160580.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363558-a363569-hanna-bilateral-power-and-prime-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363558](https://oeis.org/A363558) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).\n\n> a(2^n) = 4^(2^n) for n > 0 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18659](https://github.com/the-omega-institute/trureturing/issues/18659). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(2^n) = 4^(2^n) for n > 0 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363558 power-of-two coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-a363569-hanna-bilateral-power-and-prime-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/49c7167d1971d48d34746a1633475df814dc26ba1f96e61ccf7a50481437c288.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363558-hanna-bilateral-prime-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363558](https://oeis.org/A363558) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n).\n\n> a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18660](https://github.com/the-omega-institute/trureturing/issues/18660). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (2 + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(p) = p*2^(p-1) + 4^p for primes p > 3 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363558Prime.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363558Prime.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363558 prime-index coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-hanna-bilateral-prime-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb3ef3b6cf6ce179318e5694f6dd183c5d7e1bde421f6e11e8173a3f6173bffa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363559-hanna-bilateral-power-of-two-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363559](https://oeis.org/A363559) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).\n\n> a(2^n) = 9^(2^n) for n > 0 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18661](https://github.com/the-omega-institute/trureturing/issues/18661). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(2^n) = 9^(2^n) for n > 0 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363559PowerOfTwo.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363559PowerOfTwo.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363559 power-of-two coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-power-of-two-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ceaffc7d89f0225c2de5f43aefb3c7131eb5f444ae6fc6702229d6d6466cbc44.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363559-hanna-bilateral-prime-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363559](https://oeis.org/A363559) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n).\n\n> a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18662](https://github.com/the-omega-institute/trureturing/issues/18662). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (3 + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(p) = p*3^(p-1) + 9^p for primes p > 3 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363559Prime.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363559Prime.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363559 prime-index coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-prime-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4596e0d8cef4a211b99e20d17e9054250db437871f9a97a98cbc1bcbe3c8310d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363560-cubic-ninth-power-substitution-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A363560, Paul D. Hanna, Aug 12 2023, records the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2023a363560.md`:\n\n> Expansion of g.f. A(x) satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9).\n\n> Conjecture: a(n) == 0 (mod 3) for n > 0 except when n == 1 (mod 7).","title":"Divisibility of the A363560 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363560-cubic-ninth-power-substitution-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf7bf76f9d233c26acbdd1c74854038a7c7921a0861caec6183e5e2295e7bc10.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363561-hanna-bilateral-power-of-two-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363561](https://oeis.org/A363561) states:\n\n> G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).\n\n> a(2^n) = 2^(2^n) for n > 0 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18663](https://github.com/the-omega-institute/trureturing/issues/18663). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(2^n) = 2^(2^n) for n > 0 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363561PowerOfTwo.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363561PowerOfTwo.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363561 power-of-two coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-power-of-two-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31e1e9fe473cdf4bfa058a2fcf88528ed5bcb07ca5cb46847fc87da7c6663364.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363561-hanna-bilateral-prime-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363561](https://oeis.org/A363561) states:\n\n> G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n).\n\n> a(p) = p*2^((p-1)/2) + 2^p for primes p > 3 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18664](https://github.com/the-omega-institute/trureturing/issues/18664). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| G.f.: Sum_{n=-oo..+oo} x^n * (sqrt(2) + x^n)^(2*n). | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(p) = p*2^((p-1)/2) + 2^p for primes p > 3 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363561Prime.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363561Prime.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363561 prime-index coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-prime-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/535270fe8115c688f0fd334151f0bd47529fdd6ea24230687f8bd4d44945c33d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363569-hanna-bilateral-power-of-two-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363569](https://oeis.org/A363569) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.\n\n> a(2^n) = 1 for n > 0 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18665](https://github.com/the-omega-institute/trureturing/issues/18665). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1. | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(2^n) = 1 for n > 0 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363569PowerOfTwo.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363569PowerOfTwo.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363569 power-of-two coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363569-hanna-bilateral-power-of-two-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e639fcfe8d3491b71446184d7d99ae9ef065a15b81e3caf889c7f3585412f10.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363569-hanna-bilateral-prime-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A363569](https://oeis.org/A363569) states:\n\n> Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1.\n\n> a(p) = p*(-1)^((p-1)/2) - 1 for primes p > 3 (conjecture).\n\nThe binding coefficient convention and the exact claim are in [#18666](https://github.com/the-omega-institute/trureturing/issues/18666). Values are complex, and the bilateral expression is interpreted coefficientwise as finite sums of formal power-series coefficients.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Expansion of g.f. A(x) = Sum_{n=-oo..+oo} x^n * (i + x^n)^(2*n), where i^2 = -1. | `coeffA` in `D5.S1.Recurrence.HannaBilateral.OeisA363558PowerOfTwo`; the source sequence `a` | Literal preregistered finite coefficient expression; complex formal power series, natural coefficient degree. |\n| a(p) = p*(-1)^((p-1)/2) - 1 for primes p > 3 (conjecture). | `D5.S1.Recurrence.HannaBilateral.OeisA363569Prime.claim` and `D5.S1.Recurrence.HannaBilateral.OeisA363569Prime.result` | Every source quantifier and range is retained; `result` proves the entire claim. |\n| Sequence parameter | `a` | $c=2,3,\\sqrt{2},i$ for A363558, A363559, A363561, A363569, respectively. |\n| Complex codomain and natural arithmetic | The types in `claim` | Integer expressions are cast to complex numbers; the half exponent in the two relevant prime formulas uses natural division. This is the binding convention, not an extra hypothesis. |","title":"Hanna's A363569 prime-index coefficient formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363569-hanna-bilateral-prime-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/870954755be4264878453d3f078b7089490da5d87a4b6579491d1dba2db9a3f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a363956-positive-permutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> The sequence is conjectured to be a permutation of the positive integers.\n\nThe sequence begins with 1 and 2. Thereafter take the smallest positive unused\nmultiple of the omega-th prime, where omega counts distinct prime factors of\nthe previous term.","title":"The permutation conjecture for A363956","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363956-positive-permutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/141dd89771f03361947341045e5afa25fd0f612abde03cb14957831c9caa751a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a365095-perturbed-diagonal-square-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A365095, Paul D. Hanna, Sep 03 2023. NAME (the defining formula (1)):\n\n> Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n-1)*x*A(x)^2)^n / A(x)^n = 0 for n > 1.\n\nFORMULA (2), the target:\n\n> [x^(n-1)] (1 + (k*n-1)*x*A(x)^2)^n / A(x)^n is divisible by n^2 for n > 0 and all integer k (conjecture).\n\nThe normalization is `A(0) = 1`; the parameter ranges over all integers.","title":"Square divisibility of the A365095 perturbed diagonal","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365095-perturbed-diagonal-square-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ae759fc6dee0be383eb0535e0d6920f229e2a66da2ad1226eaabcb0de80fa05.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A365574](https://oeis.org/A365574) states:\n\n> Expansion of g.f. A(x) satisfying [x^(n-1)] (1 + (n+1)*x*A(x))^n / A(x)^n = n*(n+2)^(n-2) for n > 1.\n\n> Conjecture 1: a(k) is odd iff k = 2^n - 2 for n >= 1.\n\nThe literal binding conventions and source screen are in [#18829](https://github.com/the-omega-institute/trureturing/issues/18829). All series are formal; no analytic convergence is assumed.\n\n### Statement echo\n\nAll unqualified names below belong to `D5.S3.Combinatorics.PowerSeries.HannaLagrangeParity`.\n\n| Source clause or binding convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Normalization and every diagonal identity for $n>1$ | `IsGF` | Rational formal series, $A(0)=1$, and the literal coefficient expression are retained. |\n| Odd exactly at $k=2^n-2$, $n\\ge1$ | `claim`, `result` | Every natural $k$, including zero, has an integer coefficient witness and the biconditional. |\n| Existence and integrality | `claim`, `result` | These rule out vacuity and supply the integer meaning of oddness; they add no hypothesis. |","title":"Hanna's A365574 Lagrange parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365574-hanna-lagrange-parity-conjecture-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90eec41585590fb90139f78ee59e05445208912c3d71626896cc47531126b2cf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For n in N, let a(n) count subsets S of {1,...,n} for which no distinct\nx,y in S have x+y=|S|. The interval is empty at n=0; a(0)=1.\nEqual summands are permitted. Chai Wah Wu's November 21, 2023 conjecture\nin OEIS A367400 is\n\n$$ A(X)=\\sum_{n\\ge0}a(n)X^n\n =\\frac{1+X^2-X^3}{1-2X+X^2-2X^3+X^4}. $$\n\nThe exact retained statement is\n\n```lean\n(1 - 2*PowerSeries.X + PowerSeries.X^2 - 2*PowerSeries.X^3 +\n PowerSeries.X^4 : PowerSeries ℚ) * PowerSeries.mk (fun n => (a n : ℚ)) =\n 1 + PowerSeries.X^2 - PowerSeries.X^3\n```\n\nThe definition of a is the literal filtered powerset cardinality, with\nno recurrence, assumed count identity, or assumed bijection. Preregistration: [issue #11453](https://github.com/the-omega-institute/trureturing/issues/11453),\nfirst-tier external conjecture. The corresponding Library note gives the\nsource and bounded literature scope; neither establishes priority.","title":"Cardinality-sum-avoiding subset generating function","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/513de732b41a282d3f852e13013fa5989b92cebb7d708f2a0bce3415fba78694.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a369083-reflected-quadratic-quarter-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A369083, Paul D. Hanna, Jan 12 2024, gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2024a369083.md`:\n\n> Expansion of g.f. A(x) satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4.\n\n> Conjecture: a(n) == binomial(4*n+3,n) (mod 2) for n >= 0 (cf. A263133).","title":"Binomial parity of the A369083 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a369083-reflected-quadratic-quarter-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69f2ef2f54b850276a4e250381941de609be4c8628a9ebe0161d168fdb04599a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a370630-zeckendorf-greedy-permutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier anchors exactly the OEIS A370630 conjecture: “Conjecture: this\nsequence is a permutation of the positive integers.” The record title is\n“Lexicographically earliest sequence of distinct positive integers such that the\nZeckendorf expansions of two consecutive terms have exactly one common term.”\nRémy Sigrist is the record author; the conjecture comment has no separate signed\nauthor. The source version is revision 16, revision time\n2024-05-04T14:57:01-04:00, with offset 1,2.\n\nLet F(0)=0, F(1)=1 and F(k+2)=F(k+1)+F(k). For every integer m>=1, let Z(m)\nbe its unique finite set of nonconsecutive indices k>=2 whose Fibonacci weights\nsum to m. Set a(1)=1. At every integer n>=1, a(n+1) is the least integer m>=1\noutside the actual prefix {a(1),...,a(n)} with |Z(m) intersect Z(a(n))|=1.\nThe whole assertion includes existence of an eligible unused successor at every\nfinite positive stage, distinctness at positive indices, and, for every m>=1,\na finite n>=1 with a(n)=m, all for that same sequence.\n\nThe linked PARI definition is https://oeis.org/A370630/a370630.gp.txt.\nIts bit j encodes fibonacci(2+j); its candidate test is\n`!seen(v) && hammingweight(bitand(zp, tozeck(v)))==1`.\nOrdinary binary overlap, at-least-one overlap, zero overlap and finite checked\ncoverage are different assertions. The posted 1..10000 driver does not prove\nthe whole event.","title":"The one-common-Zeckendorf-term greedy sequence is a permutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f87b778b6d325c52b6e12194894848b08f62d9b006965e5c144ffc9123b3fbf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a372577-double-composition-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A372577, Paul D. Hanna, May 30, 2024:\n\nNAME:\n\n> Expansion of g.f. A(x) satisfying A(x)^2 = A(A( x*A(x) + x*A(x)^2 )).\n\nCOMMENT (target):\n\n> Conjecture: a(6*n - 3) == 3 (mod 4) and a(6*n - k) == 1 (mod 4) when k = {0,1,2,4,5} for n >= 1.\n\nSeparate COMMENT (a derived corollary, not a second resolution):\n\n> All terms appear to be odd.\n\nThe normalized generating series has constant coefficient zero and linear\ncoefficient one; `a(n) = [x^n]A(x)` for positive indices.","title":"Modulo-four classification of the A372577 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a372577-double-composition-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/63e327b2ee617347da4f375b1aae7ce00b4441eb1f68bb7c132d4279f938fb47.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a372991-factorial-quotient-linear-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A372991 NAME (Clark Kimberling, Jul 15 2024):\n\n> a(n) = (2n)!/(a(n-1)*a(n-2)), where a(0)=1, a(1) = 1.\n\nFORMULA (for n ≥ 3):\n\n> Conjecture D-finite with recurrence a(n) -2*n*(2*n-1)*a(n-3)=0. - R. J. Mathar, Jul 16 2024","title":"Mathar's linear recurrence for the A372991 factorial-quotient sequence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a372991-factorial-quotient-linear-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cff40bc2b7c1d5ca1a0e5b1e237b4c22f60a0d81a87526e77628c533f5a8e451.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a373308-cubed-binary-product-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A373308, Paul D. Hanna, Jun 20 2024, gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Arith/hanna2024a373308.md`.\n\nNAME:\n\n> Expansion of Product_{n>=0} (1 - x^(2^n))^3.\n\nCOMMENT:\n\n> Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.\n\nThe entry's separate Thue-Morse congruence for a(3n) and zero-index\ncharacterisation conjectures are context only; neither is claimed here.","title":"Divisibility modulo three of the A373308 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373308-cubed-binary-product-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dca3213296e9cba01095b27b36c589655802db5ee82ad2d849c10902c7701ec0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a373312-unit-reversion-square-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A373312, Paul D. Hanna, Jun 25 2024. The following NAME and COMMENT\nare copied verbatim from the claim in `Library/Arith/hanna2024a373312.md`.\n\nNAME:\n\n> Expansion of g.f. A(x) satisfying A(x)^2 = A( x*A(x)/(1 - A(x))^2 ).\n\nCOMMENT:\n\n> Conjecture: a(n) == 1 (mod 2) iff n = 2^k - 1 for k >= 1.\n\nThe formalization constructs the integer solution normalized by `A(0)=0`\nand `a(1)=1` and proves the parity assertion for every positive index.","title":"Square parity of the unit-reversion series A373312","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373312-unit-reversion-square-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4980f6c4648ecf97fb21179913315557eb08ad54ccbc4c20ac15681002c698e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a373561-gcd-bucketed-quadruple-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Three indices run over the interval from one to a modulus and contribute the\nquantity `x^2 + y^2 - z^2`. A fourth index selects, for each triple, only\nthose whose greatest common divisor with the modulus equals it. Mats Granvik\nconjectured on June 10, 2024 that the resulting fourfold sum equals the cube\nof the modulus times the sum of the first squares.","title":"The A373561 divisor-bucketed quadruple sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373561-gcd-bucketed-quadruple-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e7c8db892dee71a31060e21859532bdbbae55fab0f1621ef0dcae845abc47aa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a374568-catalan-double-composition-powers-of-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A374568 (Paul D. Hanna, Aug 14 2024) records the following NAME and\nCOMMENT, quoted verbatim from `Library/Arith/hanna2024a374568.md`:\n\nNAME:\n\n> Expansion of g.f. A(x) satisfying x = A(x - x^2) - A(x + x^2)^2.\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n = 4^k for k >= 0.","title":"Powers-of-four parity of the A374568 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374568-catalan-double-composition-powers-of-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2616737bc88eb688b7c85f9fe919250b2105e591d26728948ac614f9ef36d77b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a374570-catalan-composition-square-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A374570 (Paul D. Hanna, Jul 11 2024), quoted verbatim from\n`Library/Arith/hanna2024a374570.md`:\n\nNAME:\n> Expansion of g.f. A(x) satisfying A(x)^2 = A( A(x)*C(x) ), where C(x) = x + C(x)^2 is the Catalan function (A000108).\n\nCOMMENT:\n> Conjecture: for n > 1, a(n) is odd iff n = 2^k + 1 for k >= 0.\n\nThe normalization is `A(0) = 0`, `a(1) = 1`, and `C(0) = 0`.","title":"Parity of the Catalan-composition series A374570","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374570-catalan-composition-square-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eea88d07ec46ad98b116fda5ff25e4af2f2faa1c7ed143eacf473a5ae39371d5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a374571-fibbinary-square-substitution-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A374571 (Paul D. Hanna, Jul 11 2024), quoted verbatim from the claim in\n`Library/Arith/hanna2024a374571.md`:\n\nNAME:\n> Expansion of g.f. A(x) satisfying A(x) = A(x^2) - x*A(x^2)^2.\n\nCOMMENT:\n> Conjecture: for n > 0, a(n) is odd iff n = A003714(k) for some k > 0, where A003714 lists Fibbinary numbers whose binary representation contains no two adjacent 1's.\n\nThe formalization uses the normalization A(0) = 1 and the bitwise predicate\n`Fibbinary n := n &&& (n >>> 1) = 0` for no adjacent one bits.","title":"Fibbinary parity of the A374571 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374571-fibbinary-square-substitution-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1f55f5476734e38f3ce062bfa91556b2fd6a65b802c6354394e979ce1759b64.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a375439-cubic-ternary-substitution-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A375439 (Paul D. Hanna, Aug 21 2024) supplies the following statements,\ncopied from the claim in `Library/Arith/hanna2024a375439.md`:\n\nNAME \"Expansion of g.f. A(x) satisfying A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3.\"\n\nCOMMENT \"Conjecture: a(n) is odd iff n is in A038754, which consists of numbers of the form 3^k and 2*3^k.\"\n\nThe sequence starts at index one; the integer formal generating series is\nnormalized by its zero constant coefficient.","title":"Parity of the A375439 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375439-cubic-ternary-substitution-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aba1198fd1157a4926cb0402fe6f191f7e8330cacef7d140d9c6b8be2d8c3dbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a375449-hanna-diagonal-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna's OEIS A375449 asks about the normalized rational power series whose coefficients satisfy\n\n> Expansion of g.f. A(x) satisfying [x^n] x*A'(x) / (1 - n*A(x)) = n^(n+1) for n >= 1.\n\nIts Comments field states:\n\n> Conjecture: for n > 1, a(n) is odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.\n\nThe formal statement uses zero constant coefficient and the literal formal derivative and unit-denominator inverse. It asserts existence of a rational solution and integral coefficients with the stated parity support.\n\n### Statement echo\n\n| source clause | Lean counterpart | fidelity |\n| --- | --- | --- |\n| $[x^n]xA'(x)/(1-nA(x))=n^{n+1}$ for $n\\ge1$ | `D5.S1.Recurrence.Invariants.HannaDiagonalDerivativeParity.IsGF` | The formal derivative, coefficient operator, rational scalar, and inverse are literal. |\n| offset one and normalized solution | `constantCoeff A = 0` in `IsGF` | The constant coefficient condition is explicit. |\n| odd exactly at $2^k-2$ or $2^k-1$ for $n>1$ | `claim` and `result` | The quantifiers and natural subtraction are literal. |","title":"Hanna's OEIS A375449 diagonal derivative parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83962cead7992b8004f3a3445842c3ef7e1c8da2c77d45624b3c9cbe4f63f698.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a375452-hanna-diagonal-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. D. Hanna's OEIS A375452 asks about the normalized rational power series whose coefficients satisfy\n\n> Expansion of g.f. A(x) satisfying [x^n] x*A'(x) / (1 - n*A(x)) = n^n for n >= 1.\n\nIts Comments field states:\n\n> Conjecture: for n > 1, a(n) is odd iff n is of the form 2^k-2 or 2^k-1 for some k > 1.\n\nThe formal statement uses zero constant coefficient and the literal formal derivative and unit-denominator inverse. It asserts existence of a rational solution and integral coefficients with the stated parity support.\n\n### Statement echo\n\n| source clause | Lean counterpart | fidelity |\n| --- | --- | --- |\n| $[x^n]xA'(x)/(1-nA(x))=n^n$ for $n\\ge1$ | `D5.S1.Recurrence.Invariants.HannaDiagonalDerivativeParityA375452.IsGF` | The formal derivative, coefficient operator, rational scalar, and inverse are literal. |\n| offset one and normalized solution | `constantCoeff A = 0` in `IsGF` | The constant coefficient condition is explicit. |\n| odd exactly at $2^k-2$ or $2^k-1$ for $n>1$ | `claim` and `result` | The quantifiers and natural subtraction are literal. |","title":"Hanna's OEIS A375452 diagonal derivative parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aef845bfcef687f929611ca3a8cd2b10eebd9aefba66733a20748f9c8800cfd4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a375786-cuboid-minimum-volume-huber","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Huber's OEIS A375786 asks whether, among integer-sided cuboids with the\nsame surface $6n^2$ as a cube of edge $n$, the cuboid with the smallest first\nedge, breaking ties by the second edge, always has minimum volume. The\nformal claim is the non-strict comparison for every $n \\ge 1$, using ordered\npositive edges $x \\le y \\le z$ and the surface equation\n$xy+yz+xz=3n^2$.\n\n### Statement echo\n\nThe source's positive integer cuboids are the `cuboids` finset, whose three\ninclusive intervals are bounded by $3n^2$ and whose filter records ordering and\nthe surface equation. Its volume is `volume (x,y,z) = xyz`. The least-pair\nhypothesis is the lexicographic condition on `(p.1, p.2.1)`, and the conclusion\nis `volume p ≤ volume q` for every member `q` of the same finset. These are\nthe bodies of `cuboids`, `volume`, and `claim` in the settling module.","title":"Huber's minimum-volume cuboid conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375786-cuboid-minimum-volume-huber/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d89159ff4cc3dfe0f72f9efa3e3a5cf4a0049ef62edfa2f823f0d1422851ccb9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mikhail Kurkov's OEIS A376317, revision 25, states:\n\n> Conjecture: cases f(n) = n mod 2 and f(n) = [(n mod 3) > 0] both gives A006318.\n\nThe source array is the natural recursion with `P(0,m)=1`, the positive-row\nbranch `P(n,m)=P(n-1,m)` for `m<=n`, and\n`P(n,m)=P(n,m-1)+P(n-1,m+f(m-n))` otherwise. The zeroth-row clause has\npriority. The settled statement keeps both shifts in one proposition:\n\n```lean\n∀ n : ℕ,\n P f2 (n + 1) (n + 1) = Nat.largeSchroder n ∧\n P f3 (n + 1) (n + 1) = Nat.largeSchroder n\n```\n\nHere `f2 d = d % 2` and `f3 d = 0` exactly when `d % 3 = 0`, and\n`Nat.largeSchroder` is A006318. The statement includes `n=0`; no alternate\narray, truncation, integrality premise, or second count is substituted.\nA376388 repeats this same conjecture and is not a second resolution.","title":"Kurkov's periodic boundary recurrences and the large Schroder diagonal","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e2eb3f80673dc21b2515b7ce9f900e590bd2033e03370722f41f94f64026c0a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a376527-q-catalan-square-sum-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A376527 defines\n\n> a(n) = Sum_{k=0..n*(n-1)/2} A227543(n,k)^2.\n\nPaul D. Hanna's comment of October 11, 2024 states, verbatim:\n\n> Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.\n\nHere A227543 is the Carlitz-Riordan q-Catalan triangle defined by\n`A(x,q) = 1 + x*A(q*x,q)*A(x,q)`.","title":"Parity of q-Catalan row square sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376527-q-catalan-square-sum-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6226502b18345289423e06e6a876ff8b330ab3d07753652a43d24941f612357c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a377106-hanna-parity-reversion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: a(n) is odd iff n = 2^k for some k >= 0.\n\nThe entry defines $A^3=A(x^3+9xA^3)$, $A(0)=0$, $[x]A=1$. A normalized rational solution exists, and every such solution has integer coefficients at each $n\\geq1$, with odd coefficient exactly when $n=2^k$ for some natural $k$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| the displayed cubic equation and normalization | IsGF contains all three source conditions over rational power series | exact registered convention |\n| offset one | claim quantifies over every n at least one | exact registered convention |\n| odd iff a power of two | claim supplies an integer coefficient and its Odd equivalence | exact registered convention |\n| the entry describes a series | claim also asserts existence; uniqueness transfers the conclusion to every normalized solution | exact registered convention |","title":"Integrality and dyadic parity of OEIS A377106","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a377106-hanna-parity-reversion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/485cbc6e04343a7d90b0b8276005234ad16cba09e2e14cc197c42184c4372b5d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378252-running-sum-residue-zero","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fix integers with the start value coprime to the modulus. Let the first term\nbe the start value reduced modulo the modulus, and each later term be the\nstart value plus every earlier term, again reduced. Lechoslaw Ratajczak\nconjectured on October 13, 2025 that some term is zero exactly when the\nmodulus is a power of two, the exponent zero being allowed.","title":"The A378252 running-sum residue conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378252-running-sum-residue-zero/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0aec47aa742c464285383da3b75cbbff1450b3db02899bca2fb0ddf1b16359ed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378258-hanna-mod-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A378258 Comments state: \"Conjecture: a(n) is not congruent to 1 nor 5 (mod 6) for n > 1.\"\nThe Name states: \"G.f. satisfies A(x) = B(x) + 5*A(x)^2 where B(A(x) - A(x)^2) = x.\"\nThe separate preregistration is [#25601](https://github.com/the-omega-institute/trureturing/issues/25601).\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Rational formal series with offset one and a(1)=1 | The original `HannaA378258SquareRootParity.IsGF A`: constant coefficient zero, linear coefficient one. |\n| Both equations, including existential B | The same original predicate, without replacing either equation by a weaker consequence. |\n| Existence | The first conjunct of `claim`, reused from the frozen parity result. |\n| Every normalized solution | Universal rational `A` satisfying that original predicate. |\n| Every natural n>1 | Universal `n : ℕ` with `1 < n`; no cutoff. |\n| Integer coefficient excluding both residues | An integer `z` with `coeff n A = (z : ℚ)`, `z % 6 ≠ 1`, and `z % 6 ≠ 5`. |","title":"oeis a378258 hanna mod six","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378258-hanna-mod-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5733cb2847290dc53bb961fa7aaf8a0de90755b96795828567ac760dd300fefe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378258-hanna-square-root-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A378258 Name states: \"G.f. satisfies A(x) = B(x) + 5*A(x)^2 where B(A(x) - A(x)^2) = x.\" Its Comments state: \"Conjecture: a(n) is odd iff n is a power of 2.\"\n\nThe normalized rational series has constant coefficient zero and linear coefficient one, with offset one. The binding statement of [#18825](https://github.com/the-omega-institute/trureturing/issues/18825) asserts existence, integrality and dyadic parity for every normalized solution and every $n\\ge1$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $A=B+5A^2$ | `IsGF` existentially quantifies the rational series $B$ and includes this equality | Literal source equation. |\n| $B(A-A^2)=x$ | `IsGF` includes `PowerSeries.subst (A - A ^ 2) B = X` | The inner series is the first argument of substitution. |\n| Offset one and $a(1)=1$ | `IsGF` requires constant coefficient zero and linear coefficient one | Source normalization. |\n| A generating function exists | The first conjunct of `claim` is existence of an `IsGF` solution | Prevents vacuity. |\n| Integer sequence and parity conjecture | `claim` provides an integer representative of each coefficient, with oddness equivalent to $n=2^k$ for some natural $k$ | Every natural degree at least one; no finite cutoff. |\n| Every normalized solution | Universal quantification over rational $A$ satisfying `IsGF` | Uniqueness identifies all such solutions; no extra assumption. |","title":"oeis a378258 hanna square root parity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378258-hanna-square-root-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c784151d2d9d68fb258a2151e6bf26841d68a728a82d81fc34d27b3ef0dc6d6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378277-harmonic-triangle-alternating-row-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378277 records the following conjecture:\n\n> Conjecture: Alt. row sums of the harmonic triangle are Fibonacci(n-2) / Fibonacci(n+1), where Fibonacci(-1) = 1.\n\nWerner Schulte, Nov 21 2024.\n\nIts FORMULA gives the denominators:\n\n> T(n, k) = Fibonacci(n) * Fibonacci(n+1) if k = n, and Fibonacci(k) * Fibonacci(k+2) if 1 <= k < n.\n\nThe harmonic triangle has entries `1 / T(n, k)` for `1 <= k <= n`.\nWrite `F(n) = Nat.fib n` and\n`S(n) = sum_{k=1}^n (-1)^(k-1) / T(n, k)`, with the first term positive.","title":"Alternating row sums of the Fibonacci harmonic triangle","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378277-harmonic-triangle-alternating-row-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1d01624e17d3ab59d10111b4ccb81e4d1470eedc1adfd40a4313d458a30ca7f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378495-adjacent-difference","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378495, revision 70, FORMULA, states verbatim:\n\n> Conjecture: T(n,n-1) - T(n,n-2) = abs(A238474(n-4)) for n >= 4.\n\nOn the source-agreeing columns $2\\le k\\le n$, $T(n,k)$ counts derangements\nof $S_n$ having no $k$-cycle; the source explicitly sets $T(n,1)=0$.\nThe triangle has offset `1,5`. The dependent A238474 entry, revision 15,\nhas offset `0,1` and formula $b(m)=(-1)^m(m+3)!/(2(m+1))$ for every\nnatural $m$. This is interpreted in $\\mathbb Q$, with rational ordinary\nsubtraction between cast cardinalities and rational division in $b$.\n\n### Statement echo\n\n| Source clause | Exact formal counterpart |\n| --- | --- |\n| Every $n\\ge4$ | `claim` quantifies over every `n : Nat` with `4 <= n` |\n| Actual derangements with no $k$-cycle | Imported `TerminalDifference.T`, using permutations of `Fin n`, no fixed points, and cycle-factor support sizes |\n| Ordinary difference of adjacent columns | `(T n (n-1) : Q) - (T n (n-2) : Q)` |\n| Zero-based A238474 | `b (m : Nat) : Q` uses the source factorial quotient |\n| Literal source shift $n-4$ | Exactly `abs (b (n-4))`, with natural subtraction |\n| Universal assertion is false | `theorem result : Not claim`, a closed negation of the literal `Prop` definition |","title":"The literal adjacent difference of OEIS A378495","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-adjacent-difference/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/718884ba956dab535d92ee5bc028ec1a68048406b82f59220134eb9250477db9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378495-terminal-difference","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The FORMULA field of OEIS A378495, revision 70, states verbatim:\n\n> Conjecture: T(n,n-2) - T(n,n) = (n-3)!*(n-4)*(n-1)/2 for n >= 5.\n\nOn the source-agreeing columns $2\\le k\\le n$, $T(n,k)$ counts derangements\nin $S_n$ with no $k$-cycle; the source explicitly sets $T(n,1)=0$.\nThe triangle is read by rows, with offset `1,5`.\nBoth cardinalities are cast to $\\mathbb Q$ before ordinary subtraction;\nthe division by two is rational division.\n\n### Statement echo\n\n| Source clause | Exact formal counterpart |\n| --- | --- |\n| Every natural $n\\ge5$ | `result (n : Nat) (hn : 5 <= n)`, with no upper bound |\n| All permutations in $S_n$ | `Equiv.Perm (Fin n)`, relabelled from zero through $n-1$ |\n| No fixed point | `D n` filters by `forall x, p x != x` |\n| No $k$-cycle | `T n k` counts the filter by every cycle factor's support size differing from $k$ |\n| Cycle-factor interpretation | Consumed `noCycle_iff` connects this predicate to cycle-type membership |\n| Columns $n-2$ and $n$ | Exactly `T n (n-2)` and `T n n` |\n| $(n-3)!(n-4)(n-1)/2$ | Natural factorial and natural index subtractions, then rational casts and division |\n| No assumed counting identity | Only `5 <= n` is a hypothesis of `result` |","title":"The terminal difference of OEIS A378495","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-terminal-difference/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/58898c7007639f8ed1eab6d707f7907a6b7454005f5b2d82539ca07f186b9569.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378575-shifted-iterate-fixed-point-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378575, Paul D. Hanna, Dec 01 2024, states the following NAME and\nCOMMENT, reproduced verbatim from `Library/RecurrenceComposition/hanna2024a378575.md`:\n\n> G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(x))))), so that this sequence shifts left under the 5th self-COMPOSE.\n\n> Conjecture: a(n) == 1 (mod 4) for n >= 1.","title":"The A378575 shifted-iterate fixed-point congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378575-shifted-iterate-fixed-point-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d8301425ac685828525743e461da5202a1fbcce6f65c8698e9fff0b026fa889.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378576-shifted-iterate-fixed-point-mod-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378576, Paul D. Hanna, Dec 01 2024, states the following NAME and\nCOMMENT, reproduced verbatim from `Library/RecurrenceComposition/hanna2024a378576.md`:\n\n> G.f. A(x) satisfies A(x) = x + x*A(A(A(A(A(A(x)))))), so that this sequence shifts left under the 6th self-COMPOSE.\n\n> Conjecture: a(n) == 1 (mod 5) for n >= 1.","title":"The A378576 shifted-iterate fixed-point congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378576-shifted-iterate-fixed-point-mod-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c092cc5d0f3a2a9b1d43ec13aa6e1d70b5bb12342b0deec9647bd06fda75d33f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378578-strip-three-ternary-catalan-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378578 (Paul D. Hanna, Jan 03 2025) gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2025a378578.md`.\n\nNAME:\n\n> G.f. A(x) equals the series obtained by removing all factors of 3 from the coefficients in 1 + x*A(x)^3.\n\nCOMMENT:\n\n> Conjecture: a(n) == binomial(3*n,n)/(2*n+1) (mod 2) for n >= 0.\n\nThe quotient is exact: these are the ternary Catalan numbers. The Lean\nstatement uses natural-number division; `ternary_catalan_div` records the\nexact quotient identity for n > 0, with n = 0 handled directly.","title":"Strip-three ternary Catalan parity for A378578","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378578-strip-three-ternary-catalan-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa3e0269a7a1592ce3090cd9ae605421f9985f9218f0fa323816fcdb9bc91b8e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378580-quotient-theta-composition-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378580 (Paul D. Hanna, Jan 08 2025) gives the following NAME and\nCOMMENT, quoted verbatim from `Library/ArithSums/hanna2025a378580.md`.\n\nNAME:\n\n> G.f. A(x) satisfies: A(x/A(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.\n\nHere A has integer coefficients and constant coefficient one; x/A(x) means\nX times its formal unit inverse. The Lean construction and uniqueness theorem\nidentify the sequence defined by this quotient equation.","title":"Coefficients of the A378580 quotient theta composition modulo four","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378580-quotient-theta-composition-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97b642b000c61a3adf66a219898f3a042910a2c70bbb3c8f9f734be26be596c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a378581-theta-self-composition-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A378581 (Paul D. Hanna, Jan 08 2025), as quoted in\n`Library/Recurrence/hanna2025a378581.md`:\n\nNAME:\n> G.f. A(x) satisfies: A(x*A(x)) = theta_3(x) = 1 + 2*Sum_{n>=1} x^(n^2).\n\nCOMMENT:\n> Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is square, else a(n) is divisible by 4 if n is nonsquare.","title":"The theta self-composition coefficients modulo four","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378581-theta-self-composition-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65b459b8675e608f6a26629a880d2b020ca8a89ca4549a3de4e87e0c3c016c00.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a379204-bilateral-quartic-power-theta-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A379204 (Paul D. Hanna, Dec 20 2024) gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2024a379204.md`:\n\n> G.f. A(x) satisfies 1/x = Sum_{n=-oo..+oo} A(x)^n * (A(x)^n + 4)^(n+1).\n\n> Conjecture: a(n) is even for n > 1.\n\nThe evenness conjecture is this dossier's sole resolution claim, anchored by\n`hanna_conjecture`. The entry also states:\n\n> Conjecture: a(n) == 2 (mod 4) iff n = (k-1)^2 + 1 for some k > 1.\n\nThe companion `hanna_conjecture_mod_four` proves that second conjecture. The\nNAME is formalized after multiplication by x*A, using exact finite coefficient\nwindows; the source-to-formal interpretation is bounded as described below.","title":"Hanna's A379204 evenness conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379204-bilateral-quartic-power-theta-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/87d5caa142d89fbb2eba0b2d4ff9f5ad5ea718bdc79025e2ade5e7ff4dd66bb1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a379764-hanna-theta-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A379764](https://oeis.org/A379764) states:\n\n> G.f. A(x) satisfies Sum_{n=-oo..+oo} (4^n*A(x)^n - 4*A(x))^(n+1) = 2*theta_4(x) = 2*Sum_{n=-oo..+oo} (-x)^(n^2).\n\n> Conjecture: for n > 0, a(n) is odd iff n is a square.\n\n> Conjecture: for n > 0, a(n) == (-1)^(n-1) (mod 4) iff n is a square, otherwise a(n) is divisible by 4 if n is nonsquare.\n\nThe statement source and binding formal conventions are in [#18830](https://github.com/the-omega-institute/trureturing/issues/18830). The normalization is $A(0)=0$ and $a(1)=1$. Integer powers of the summand are taken in rational Laurent series. The bilateral equation means equality at coefficient $N$ for every symmetric window $[-K,K]$ with $N+1\\le K$; no analytic convergence assertion is made.\n\n### Statement echo\n\nAll names below belong to `D5.S1.Recurrence.Bilateral.HannaThetaFourModFour`.\n\n| Source or binding convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The sum defining $\\theta_4$ | `theta4` | Its coefficient at $N$ sums $(-1)^N$ at the integer roots of $j^2=N$ in $[-N,N]$. |\n| The Laurent summand with integer index | `term` | The source expression is retained literally after `HahnSeries.ofPowerSeries` embeds $A$. |\n| Normalization and every coefficient window | `IsGF` | Both normalization equalities and all $N,K$ with $N+1\\le K$ are required. |\n| Every positive index; odd coefficient iff square | `claim`, `result` | Every normalized rational solution has an integer coefficient witness and the odd-square biconditional. |\n| Signed residue iff square; divisibility otherwise | `claim`, `result` | The same coefficient witness satisfies the second biconditional and nonsquare divisibility. |\n| Existence of the normalized series | First conjunct of `claim` | This strengthens the source convention by ruling out a vacuous universal assertion. |\n| Rational-to-integer coefficient correspondence | Integer witness in `claim` | This proves integrality at every positive coefficient; it adds no hypothesis. |","title":"Hanna's A379764 theta-four coefficient conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f360a0fe6e78603a6e31677983204f9f9efc9570b15b553c08a9f117c0bd359.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380056-fourth-index-divisible-by-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A sequence is read off an exponential generating function whose numerator is\nthe exponential minus one and whose denominator is the cosine of twice the\nargument. Paul D. Hanna conjectured on January 28, 2025 that five divides\nevery term whose index is a multiple of four.","title":"The A380056 divisibility conjecture at multiples of four","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380056-fourth-index-divisible-by-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33124d01046859701f4305abc9ae788210b8252e72a16c61be981bbb843775c6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380066-hanna-bilateral-product-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A380066](https://oeis.org/A380066) states:\n\n> G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(n-1) * (A(x) + x^n)^(n-1).\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4).\n\nThe offset is zero and the listed coefficients begin $1,6,62,826,13562$. The binding convention in [#18805](https://github.com/the-omega-institute/trureturing/issues/18805) uses integer power series with $A(0)=1$. Each coefficient $N$ is read through every symmetric integer window $[-K,K]$ with $N\\le K$ in rational Laurent series. This is a formal coefficient statement, with no analytic convergence claim.\n\n### Statement echo\n\nAll new names belong to `D5.S1.Recurrence.Bilateral.BilateralProductModFourUnitExponent`.\n\n| Source clause or binding convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The displayed summand, with every integer index | Frozen `BilateralProductModFour.laurentTerm` at parameter one | The source expression is reused literally. |\n| Offset zero and constant coefficient one | `HannaA380066.IsGF`, first conjunct | The integer-series convention and normalization are explicit. |\n| The bilateral equation | `HannaA380066.IsGF`, every `N`, `K` and window hypothesis | Every coefficient and every radius at least its index are required. |\n| Every positive index has residue two modulo four | `HannaA380066.claim`, universal conjunct; `result` | Every normalized integral solution is covered. |\n| Existence | `HannaA380066.claim`, existential conjunct; `result` | This rules out a vacuous universal assertion and adds no hypothesis. |\n\nThe two public definitions retain the literal binding conventions. The proof uses the existing family's owners internally. There are no additional source assumptions.","title":"Hanna's A380066 bilateral product congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380066-hanna-bilateral-product-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1ee2e5d2232950ffb9384bbbfd90ddb2a75bc716138e47ada5518923d5ea773.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380558-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A380558, by Paul D. Hanna (February 13, 2025), defines the generating\nseries `A(x) = Sum_{n>=2} a(n)*x^n` through\n`A(x - A(x)) = x^2/(1 - x^2)`. Its comment states:\n\n> Conjecture: a(n) is odd iff n = 2*A004760(k) for some k > 1, where A004760 lists numbers whose binary expansion does not begin 10.\n\nThe formal target constructs an integral series with zero coefficients at\ndegrees zero and one, proves that source equation and uniqueness among all\nsimilarly normalized integral solutions, and proves for every natural `n`:\n\n`Odd ([x^n]A) <-> n = 2 or there exist m,j in N with n = 2*m and\n3*2^j <= m < 4*2^j`.\n\nThis is the source conjecture with its natural zero extension at degrees zero\nand one. A004760 has offset one: its first values are `0,1,3,6,7,...`.\nThe condition `k>1` retains `1`, giving `n=2`; all later values lie in the\nhalf-open dyadic intervals displayed above. Neither `m` nor `j` is the OEIS\nindex `k`.","title":"Parity of the A380558 square-reversion series","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b281aec84d4bc2d7c9a1bc62e1de263ad23ee73775bcb17613d0ae13e06ffcac.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380678-quadratic-square-reversion-dyadic-support-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A380678, February 2025. The Library note records the\nmonth and year but not the exact `%A` line date. The following NAME and\nCOMMENT are copied verbatim from `Library/Recurrence/hanna2025a380678.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A( x - A(x)^2/(1 - A(x)^2) ) = x.\n\nCOMMENT:\n\n> Conjecture: for n>= 1, a(n) is odd iff n is a term of A027383, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2.","title":"Dyadic support of the odd coefficients of A380678","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380678-quadratic-square-reversion-dyadic-support-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c0c82b919627f71d8208b047fc4de95dcada0a4c0aa953180ad19ef9932705f9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380680-hanna-bilateral-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A380680](https://oeis.org/A380680) states:\n\n> G.f. A(x) = Sum_{n=-oo..+oo} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4) iff n is a square, and a(n) is divisible by 4 when n is nonsquare.\n\nFormula (2) is:\n\n> A(x) = 1 + 2*Sum_{n>=1} x^(n^2) * (2 - x^n)^n/(1 - 2*x^n)^n.\n\nThe source's comment says each summand is invariant under sign reversal of\nits integer index. The zero summand is one. The citation is\n[the Library note](../Library/RecurrenceComposition/hanna2025a380680.md).\nPreregistration [#18667](https://github.com/the-omega-institute/trureturing/issues/18667)\nclassifies this explicitly published conjecture as tier 1.\n\n### Statement echo\n\nAll Lean names below are in `D5.S0.Certificates.OeisA380680BilateralModFourRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Bilateral generating function and formula (2) | `a` takes the rational power-series coefficient of the unilateral expression | The source's equal opposite summands and zero term supply the folding convention. |\n| Coefficient at degree $m$ | Finite sum over $1\\le n\\le m$ in `a m` | The factor $X^{n^2}$ makes larger indices irrelevant to that coefficient. |\n| Positive index | `claim` quantifies natural $n$ with $0 Conjecture: if F(x) satisfies p*x = Sum_{n=-oo..+oo} (-1)^n * x^(p*n) * (F(x) + x^n)^(p*n) with F(0) = 1 for some fixed integer p, then F(x) is an integer series in x iff p is prime. The g.f. A(x) of this sequence is the case when p = 2; other cases include A380682 (p=3), A380683 (p=5), A380684 (p=7), A380685 (p=11), and A380686 (p=13).\n\nPreregistration: [#18831](https://github.com/the-omega-institute/trureturing/issues/18831).\nThe [Library note](../Library/RecurrenceComposition/hanna2025a380681.md) records the source.\nThe formal parameter is $p\\in\\mathbb N$ with $p\\ge2$, and $F\\in\\mathbb Q[[X]]$.\nInteger powers are taken in $\\mathbb Q((X))$. At degree $N$, the bilateral sum\nmeans every symmetric window $[-K,K]$ with $K\\ge N+1$.\n\n### Statement echo\n\nThe Lean namespace is `D5.S1.Recurrence.Bilateral.HannaPrimeBilateralIntegrality`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $(-1)^n x^{pn}(F+x^n)^{pn}$, $n\\in\\mathbb Z$ | `term` | Literal integer Laurent powers, with $x=X$ and the embedded $F$. |\n| $F(0)=1$ | First conjunct of `IsSolution` | The rational constant coefficient is one. |\n| $px$ equals the bilateral sum | Remaining conjunct of `IsSolution` | Every coefficient in every sufficiently large symmetric window has the required value. |\n| Integer series | Integer witness for every coefficient in `claim` | Every rational coefficient is the cast of an integer. |\n| If and only if $p$ is prime | Equivalence in `claim` | Both directions hold for every normalized rational solution. |\n| Fixed integer parameter | Natural $p\\ge2$ in `claim` | The preregistered range; nonpositive parameters are outside the statement. |\n| A solution exists | Existential conjunct of `claim` | Additional conclusion for every $p\\ge2$, with no additional hypothesis. |\n\nNo source clause within the preregistered interpretation is dropped. Both the\n`claim` and `result` Blueprint nodes display the complete proposition.","title":"Hanna's prime bilateral integrality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b1541acbcb22dae36e71f7f09d8223d58fa20164fe3dd931d90718631818870.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A380708, Paul D. Hanna, Feb 08 2025, gives the following NAME and COMMENT,\nquoted verbatim from `Library/Recurrence/hanna2025a380708.md`:\n\n> G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^2.\n\n> Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for k >= 0, where A003714 are the fibbinary numbers.\n\nHere abs means coefficientwise absolute value of the reciprocal, taken before\nsquaring, as defined in the sibling entries A380709/A380710. A003714 enumerates\nthe nonnegative integers with no two adjacent 1s in their binary expansion.\nThe frozen public predicate is `Fibbinary f := f &&& (f >>> 1) = 0`.","title":"Fibbinary parity of the A380708 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77d662283a3a1ee62b3fb2794bb19b4116e990503291899dbae2fc9b2ac77467.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380709-absolute-reciprocal-cube-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A380709, Feb 09 2025. The following NAME and COMMENT\nare quoted verbatim from the claim in `Library/Recurrence/hanna2025a380709.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A(x) = 1 + x*abs( 1/A(x) )^3.\n\nCOMMENT:\n\n> Conjecture: a(n) == binomial(4*n-1, n) (mod 2) for n >= 0 (cf. A263132).\n\nHere abs is the series of coefficientwise absolute values, applied to the\nreciprocal before cubing, as defined by the entry's formulas and the sibling\nentries A380710/A383377. The positive-index assertion is `hanna_conjecture`.\nAt n = 0 the entry reads binomial(-1, 0) = 1; the companion `a_zero` proves\na(0) = 1. This reading does not use truncated natural subtraction to define\na generalized negative-upper-index binomial coefficient.","title":"Binomial parity of the A380709 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380709-absolute-reciprocal-cube-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/527a22b314d72543a15558956bfe9f5f4b06b58770d0cdf6532f15562699a684.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380710-absolute-reciprocal-square-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A380710, Feb 18 2025. The following NAME and COMMENT\nare copied verbatim from the claim in `Library/Arith/hanna2025a380710.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^2 ).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) is odd iff n is a power of 2.\n\nHere `abs` is the series of coefficientwise absolute values, as defined by\nthe entry's formulas and the sibling entry A383377. It is not a pointwise\nabsolute value of an analytic function.","title":"Absolute reciprocal squares and binary parity in A380710","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380710-absolute-reciprocal-square-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0493586622c32fd1bf5d506741bf411759e00480f9794f35c9004b2b38a2a61.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a380711-hanna-absolute-cubic-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A380711](https://oeis.org/A380711) states:\n\n> G.f. A(x) satisfies A(x) = 1 + x*A(x)*abs( 1/A(x)^3 ).\n\n> Conjecture: a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) for n >= 0.\n\nThe literal binding conventions and source screen are in [#18828](https://github.com/the-omega-institute/trureturing/issues/18828). All series are formal; no analytic convergence is assumed.\n\n### Statement echo\n\nAll unqualified names below belong to `D5.S1.Recurrence.Algebraic.HannaAbsoluteCubicCongruence`.\n\n| Source clause or binding convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Coefficientwise absolute value | `absSeries` | The coefficient of degree $n$ is the absolute value of the original rational coefficient. |\n| The generating equation | `IsGF` | The displayed source equation is retained literally over rational formal power series. |\n| Generalized binomial quotient | `quotient` | `Ring.choose` uses the integer top argument $3n-1$ and rational division; at zero the quotient is $-1$. |\n| Congruence for every $n\\ge0$ | `claim`, `result` | Integer witnesses represent both rational operands and satisfy `Int.ModEq 2`. |\n| Existence and integrality | `claim`, `result` | These strengthen the literal congruence's interpretation without adding hypotheses. |","title":"Hanna's A380711 absolute cubic congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380711-hanna-absolute-cubic-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/052052592f7f11eba9a6c990187c3f974992b131f333a6f00855da0d83c9680f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381190-trapezohedral-domination-series","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381190 is named “Number of connected minimal dominating sets in the n-trapezohedral graph.” Its conjectured formula is\n\n`-2*x^3*(4*x^5 + 8*x^4 + 4*x^3 - 9*x^2 - 8*x - 3)/(x^3 + x^2 - 1)^2`.\n\nThe resolved statement is the literal ordinary formal-power-series identity for every integer `n >= 3`. For each `n`, the graph has distinct labelled vertices `alpha`, `beta`, `a_0,...,a_(n-1)`, and `b_0,...,b_(n-1)`, the alternating rim cycle, alpha adjacent to every `a_i`, beta adjacent to every `b_i`, and no other edges. The coefficient counts actual subsets `S` whose closed neighbourhood is the full graph, every proper subset of `S` fails to dominate, and the induced graph on `S` is connected. Minimality ranges over all proper subsets independently of their connectivity; no quotient by rotations, reflections, or other automorphisms is taken. Coefficients below order three are zero.","title":"OEIS A381190 trapezohedral domination series","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3c2c0de81950700e660164dad16bdd26c132536965e59c137eb60dbfdcd8fcc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381362, Paul D. Hanna, Feb 21 2025:\n\nNAME:\n\n> G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(2*n-1) * (x^n + A(x))^(2*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4).\n\nOEIS A381363, Paul D. Hanna, Feb 21 2025:\n\nNAME:\n\n> G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(3*n-1) * (x^n + A(x))^(3*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4).\n\nThis dossier resolves A381362, the `c = 2` instance, and cites its own Library note\n`Library/ArithSums/hanna2025a381362.md`. The sibling quote records the shared family.\nThe generating series is normalized by `A(0) = 1`.","title":"The A381362 bilateral product coefficient congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04a8038535c5ce1b7b8597459875964641d3b291ebbfd494054a005c88ed30c9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381363-bilateral-product-cubic-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381362, Paul D. Hanna, Feb 21 2025:\n\nNAME:\n\n> G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(2*n-1) * (x^n + A(x))^(2*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4).\n\nOEIS A381363, Paul D. Hanna, Feb 21 2025:\n\nNAME:\n\n> G.f. A(x) satisfies 1/2 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + x)^(3*n-1) * (x^n + A(x))^(3*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 2 (mod 4).\n\nThis dossier resolves A381363, the `c = 3` instance, and cites its own Library note\n`Library/ArithSums/hanna2025a381363.md`. The sibling quote records the shared family.\nThe generating series is normalized by `A(0) = 1`.","title":"The A381363 bilateral product coefficient congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/567c642e3f04bac63b61c5a5e127c83a1109be22d2fd5999ea11b2681c07c346.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier cites its own Library note,\n`Library/ArithSums/hanna2025a381364.md`, for the A381364 conjecture.\nThe two entry statements describe instances of one family:\n\nOEIS A381364, Paul D. Hanna, Feb 21 2025, NAME:\n\n> G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(n-1) * (x^n + 2*A(x))^(n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 6 (mod 9).\n\nOEIS A381365, Paul D. Hanna, Feb 21 2025, NAME:\n\n> G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(2*n-1) * (x^n + 2*A(x))^(2*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 6 (mod 9).","title":"The A381364 scaled bilateral product congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9492a20391e0ffa8c1f89e6330c6fd398512b1b6d6579492c108cceae519cdee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier cites its own Library note,\n`Library/ArithSums/hanna2025a381365.md`, for the A381365 conjecture.\nThe two entry statements describe instances of one family:\n\nOEIS A381364, Paul D. Hanna, Feb 21 2025, NAME:\n\n> G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(n-1) * (x^n + 2*A(x))^(n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 6 (mod 9).\n\nOEIS A381365, Paul D. Hanna, Feb 21 2025, NAME:\n\n> G.f. A(x) satisfies 1/3 = Sum_{n=-oo..+oo} x^n*A(x)^n * (A(x)^n + 2*x)^(2*n-1) * (x^n + 2*A(x))^(2*n-1).\n\nCOMMENT:\n\n> Conjecture: for n > 0, a(n) == 6 (mod 9).","title":"The A381365 scaled bilateral product congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9d8c81358a27b9cb8d96235e6b7b0b5cfabb4a91dc4a096c80258cab9ccb453.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381423-hermite-differences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381423 revision 8, Mike Sheppard, has the independent conjecture clause:\n“Differences are either 1 or -3”. Its full comment continues with finer patterns,\nspacing and square-root growth. The independent target is preregistered at\n[#25634](https://github.com/the-omega-institute/trureturing/issues/25634).\n\nFor every natural n, including zero, let a(n) be the smallest j in zero through n\nmaximizing the signed coefficient of $x^j$ in the physicists Hermite polynomial\n$H_n$. The exact assertion is\n\n$$ (a(n+1):\\mathbb Z)-(a(n):\\mathbb Z)=1\\quad\\text{or}\\quad-3. $$\n\nThe source's Mathematica first PositionLargest and PARI upward scan establish\nthe tie convention. Its n=5 example selects exponent 1, distinguishing signed\norder from absolute coefficient order.\n\n### Statement echo\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| Physicists H_n | `physicistsHermite n`, the literal DLMF 18.5.13 finite factorial expansion over ℚ. |\n| Coefficient at exponent j | `polynomial_coeff` proves extraction equals `signedCoeff n j` throughout zero through n. |\n| Signed maximum | `IsSmallestMax` compares rational coefficients directly, with no absolute value. |\n| Smallest tied exponent | The last conjunct of `IsSmallestMax`; the proof chooses the largest positive-term index. |\n| Every n, including zero | Unrestricted `∀ n : ℕ` in both conjuncts of `claim`. |\n| Integer difference | Both values of `a` are cast to ℤ before subtraction. |\n| No uniqueness assumption | No assumption is present; all tied maximizers enter the selector predicate. |","title":"oeis a381423 hermite differences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381423-hermite-differences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cff07fabeb9b71bf4b3b070a24a799f89849c6c42a40b46ebdcc71428f291465.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381524-hermite-absolute-differences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381524, revision 18, by Mike Sheppard (2025), defines:\n\n> Smallest exponent of x of maximal coefficient (ignoring signs) in Hermite polynomial of order n.\n\nIts independently printed Comments assertion is:\n\n> Conjecture: Differences are either +1 or -1.\n\nThe offset is zero. Let $H_n$ be the physicists Hermite polynomial represented\nby the original `HermiteSignedMode.physicistsHermite`, and let $b(n)$ be the\nleast $j\\in\\{0,\\ldots,n\\}$ maximizing $|[x^j]H_n|$. The complete event is\n\n$$\n\\begin{aligned}\n\\forall n\\in\\mathbb N,\\quad &b(n)\\le n,\\\\\n&\\forall k\\le n,\\quad |[x^k]H_n|\\le |[x^{b(n)}]H_n|,\\\\\n&\\forall k\\le n,\\quad |[x^k]H_n|=|[x^{b(n)}]H_n|\\ \\Longrightarrow\\ b(n)\\le k,\\\\\n&b(n+1)-b(n)\\in\\{1,-1\\}\\quad\\text{in }\\mathbb Z.\n\\end{aligned}\n$$\n\nBoth selector values are cast to integers before subtraction. Every natural\norder, including zero, every exponent from zero through the order, parity\nzeros and all ties are included. The source's `HermiteH` convention and\nexamples fix physicists normalization: $b(5)=3$, and $b(8)=2$ chooses the\nsmaller of the tied exponents two and four. The claim is the one entire\nDifferences assertion with its selector certificate, without a uniqueness\npremise, finite cutoff or asymptotic qualification.","title":"Absolute Hermite coefficient modes: the complete A381524 difference assertion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381524-hermite-absolute-differences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1daca0de7c7ecc304ad5254fe2efa9b0b7b57d501753b6ef96163fce99e1a7e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381670-compositional-square-dyadic-denominators","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381670 (Thomas Scheuerle, Mar 03 2025), NAME, as quoted in\n`Library/ArithSums/scheuerle2025a381670.md`:\n\n> The function A(x) = x+(1/2)*x^2-(1/16)*x^4... = Sum_{k >= 0} x^k*A381669(k)/a(k) satisfies the functional equation: x*(A(x)+1) = A(A(x)).\n\nThe COMMENT that is the target, from the same Library note:\n\n> Conjecture: All terms are powers of two.\n\nThe brief separately supplied the purported comment:\n\n> All terms appear to be odd.\n\nThat sentence does not occur in the supplied Library note. It cannot describe\nthese denominators: the displayed coefficient of x^2 has denominator 2.\nIts attribution is unverified, and it is not a second resolution claim.","title":"Dyadic denominators of the A381670 compositional-square series","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381670-compositional-square-dyadic-denominators/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/635d64a82ac1ef9fafa560e61b8fad87a4b64871304443144ac8535fcfaeca3e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a381737-hermite-absolute-ties","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A381737, revision 28, by Mike Sheppard (2025), names the sequence:\n\n> Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.\n\nIts Formula field states:\n\n> Conjecture 1: a(n) = 2*n*(n + 2) + (n + 1)*(-1)^(n+1).\n\nThe offset is 1,1. Let $H_k$ be the physicists Hermite polynomial, with the\noriginal coefficients represented by `HermiteSignedMode.physicistsHermite`\nover the rationals. For natural $k$ and $j$, write\n\n$$\nM(k,j)\\iff j\\le k\\ \\land\\ \\forall i\\le k,\\quad\n|[x^i]H_k|\\le |[x^j]H_k|.\n$$\n\nExactly two maxima means that there exist $u0$ for every $n\\ge1$, that\n$a(n) Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.\n\nIts Formula field states:\n\n> Conjecture 3: G.f.: (8*x + 5*x^2 + 5*x^3 - x^4 - x^5) / ((1 - x)^3 * (1 + x)^2).\n\nThe offset is 1,1. Let $H_k$ denote the physicists Hermite polynomial over\n$\\mathbb Q$, represented by `HermiteSignedMode.physicistsHermite`. For natural\norders $k$ and exponents $j$, define\n\n$$\nM(k,j)\\iff j\\le k\\ \\land\\ \\forall i\\le k,\\quad\n|[x^i]H_k|\\le |[x^j]H_k|.\n$$\n\nWrite $T(k)$ when there exist natural exponents $u0$ for every natural $n\\ge1$.\n- $a(n) Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice.\n\nIts Formula field states:\n\n> Conjecture 2: a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5).\n\nThe offset is 1,1. Let $H_k$ be the physicists Hermite polynomial represented\nby `HermiteSignedMode.physicistsHermite` over the rationals. For every\nnatural order $k$ and exponent $j$, put\n\n$$\nM(k,j)\\iff j\\le k\\ \\land\\ \\forall i\\le k,\\quad\n|[x^i]H_k|\\le |[x^j]H_k|.\n$$\n\nLet $S$ consist of exactly the natural orders $k$ for which there are\nnatural exponents $u0$.\n- For all integers $1\\le n= 0.\"\n\nWrite $A(x)=\\exp(\\sum_{n\\ge1}\\sigma(n)\\sigma(2n)x^n/(3n))$ and $B(x)=\\exp(\\sum_{n\\ge1}\\sigma(n)\\sigma(2n)x^n/n)$, with $\\sigma(n)=\\sum_{d\\mid n}d$. The public rational-series definitions are exactly the preregistered expressions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every index $n\\ge0$ | `claim` quantifies over every natural index | Exact |\n| A382124 generating function | `HannaSigmaProductExpModThree.A` | Exact rational exponential substitution |\n| A382125 generating function | `HannaSigmaProductExpModThree.B` | Exact rational exponential substitution |\n| $a(n)\\equiv b(3n)\\pmod3$ | `claim`: `coeff n A = u`, `coeff (3*n) B = v`, `u ≡ v [ZMOD 3]` | Exact |\n| Integer coefficients | `∃ u v : ℤ` and the two rational coefficient equalities | Witnesses make congruence well typed; no extra hypothesis |","title":"Hanna's A382124 congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382124-hanna-sigma-exponential-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cbc76bed1fa50b39a6fafb3804bff0305c56670dd70af4145418b05e46ad93af.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a382125-hanna-sigma-exponential-mod-three-cube","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A382125 states: \"Conjectures: a(3*n) == A382124(n) (mod 3) for n >= 0; a(3*n+1) == 0 (mod 3) and a(3*n+2) == 0 (mod 3) for n >= 0.\"\n\nWrite $A(x)=\\exp(\\sum_{n\\ge1}\\sigma(n)\\sigma(2n)x^n/(3n))$ and $B(x)=\\exp(\\sum_{n\\ge1}\\sigma(n)\\sigma(2n)x^n/n)$, with $\\sigma(n)=\\sum_{d\\mid n}d$. The public rational-series definitions are exactly the preregistered expressions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every index $n\\ge0$ | `claim` quantifies over every natural index | Exact |\n| A382124 generating function | `HannaSigmaProductExpModThree.A` | Exact rational exponential substitution |\n| A382125 generating function | `HannaSigmaProductExpModThree.B` | Exact rational exponential substitution |\n| $b(3n)\\equiv a(n)\\pmod3$ | First conjunct: `coeff (3*n) B = u`, `coeff n A = v`, `u ≡ v [ZMOD 3]` | Exact |\n| $b(3n+1)\\equiv0\\pmod3$ | Second conjunct: `coeff (3*n+1) B = u` and `u ≡ 0 [ZMOD 3]` | Exact |\n| $b(3n+2)\\equiv0\\pmod3$ | Second conjunct: `coeff (3*n+2) B = v` and `v ≡ 0 [ZMOD 3]` | Exact |\n| Integer coefficients | Both conjuncts supply integer witnesses and rational coefficient equalities | Witnesses make congruence well typed; no extra hypothesis |","title":"Hanna's A382125 residue pattern","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382125-hanna-sigma-exponential-mod-three-cube/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/665978f43102a31e3efa9fa8fc01e9e15f64806cdc3f61a47db0afa8030023f7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a382357-valuation-step-permutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A382357 is the lexicographically earliest sequence of distinct positive\nintegers whose adjacent 2-adic valuations differ by exactly one. It starts at\n1, with offset 1. The supplied source states: “Conjecture: this sequence is a\npermutation of the positive integers.”","title":"The valuation-step greedy sequence is a permutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382357-valuation-step-permutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37dc3d325e61a9b83b334e12239a37f13e8b1b615435f935409c1ae197a75f9f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a382427-constant-blocks-distinct-run-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A382427 counts integer partitions of n that can be partitioned into\nconstant blocks with distinct sums. Its comment conjectures that the same\nnumber counts partitions having a permutation with all distinct run-sums.\nA381717, the complement, explicitly conjectures the pointwise equivalence.\nRuns are maximal consecutive constant subsequences, as defined by A382876.","title":"Constant Blocks and Distinct Run Sums","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382427-constant-blocks-distinct-run-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7d6e231beff35bf6a8567f459f45f2aa08dbe0422a8be01afd95aff6ae0563c7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a382814-fibonachos-score","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peter Kagey's OEIS A382814 comment states:\n\n> Conjecture:\n> a(n) = n/2 if and only if n is in {2, 8, 10, 32}.\n\n> Conjecture:\n> For n > 32, a(n) > n/2 if and only if F(m)-1 <= n <= F(m+1)-2 for some odd integer m, where F(n) = A000045(n).\n\nThe two players alternate taking successive Fibonacci numbers, beginning\nwith `F(1)=F(2)=1`. When the next number exceeds the remaining heap, the\nindex resets to one; the player's turn still alternates. The game stops\nat the empty heap, and `a(n)` is the first player's total.\nFor every positive integer `n`, the tie condition is `2a(n)=n`.\nFor every `n>32`, first-player majority is `n<2a(n)` and the block\ncondition has an existential natural index `m` with `Odd m`.\n\nThe formal interval is `F(m)<=n+1` and `n+2<=F(m+1)`. In this range the\nupper condition forces a positive, nontruncated upper endpoint, and\nthese are equivalent to the displayed source inequalities. Multiplying\nthe score comparison by two avoids natural-division rounding.","title":"Fibonachos ties and first-player majority","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e252e26873501e1b16d25b133190cd9f141618c2b9b60db0bb6f8d7503e13248.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a383165-scaled-log-column-periodicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The entry-specific Library note `Library/ArithSums/bala2026a383165.md`\nrecords the NAME and conjecture verbatim as follows:\n\n> Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^2 / 2.\n\n> Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic.\n\nThe note attributes the entry to Seiichi Manyama (April 18, 2025) and the\nconjecture to Peter Bala (February 17, 2026). This dossier claims only this\nentry's eventual-periodicity conjecture, not a divisibility clause of a\nnegative-power diagonal family. The claimed conclusion is existence of an\nonset and a positive period, without a specified or minimal period.","title":"Eventual periodicity of OEIS A383165","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/593e3f1c2abd1a31b9336edf513ce6ee305c030759e3d9e47e4435f8b06e89c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a383166-scaled-log-column-periodicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The entry-specific Library note `Library/ArithSums/bala2026a383166.md`\nrecords the NAME and conjecture verbatim as follows:\n\n> Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^3 / 6.\n\n> Conjecture: the sequence {a(n)} reduced modulo a positive integer k is eventually periodic.\n\nThe note attributes the entry to Seiichi Manyama (April 18, 2025) and the\nconjecture to Peter Bala (February 17, 2026). This dossier claims only this\nentry's eventual-periodicity conjecture, not a divisibility clause of a\nnegative-power diagonal family. The claimed conclusion is existence of an\nonset and a positive period, without a specified or minimal period.","title":"Eventual periodicity of OEIS A383166","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29bb884840291fb5b08016be3a0b0feefc309e9e961715482dd4f9884e5a8d16.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a383377-absolute-reciprocal-geometric-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A383377, May 15 2025. The following quotations are\nverbatim from `Library/Arith/hanna2025a383377.md`.\n\nNAME:\n\n> G.f. satisfies A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n), where abs(F(x)) equals the series expansion formed by the unsigned coefficients in F(x).\n\nCOMMENT:\n\n> Conjecture: a(n) is even for n > 1.","title":"Parity of the A383377 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383377-absolute-reciprocal-geometric-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95da1ee39baeb35ac0cfd26874b378da431d90ab01ec1372aaf6339e6654e026.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a384222-two-dense-divisor-blocks-palindrome","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture 1: row n is a palindromic composition of A000005(n).\n\n— Omar E. Pol (entry Jun 03 2025), OEIS A384222.\n\nThe defining comments supplied with the conjecture are:\n\n> The 2-dense sublists of divisors of n are the maximal sublists whose terms increase by a factor of at most 2.\n\n> In a sublist of divisors of n the terms are in increasing order and two adjacent terms are the same two adjacent terms in the list of divisors of n.\n\nFor positive n, the row lists the lengths of these maximal consecutive blocks;\nA000005(n) is the number of positive divisors, τ(n).","title":"Palindromic composition of the A384222 rows","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384222-two-dense-divisor-blocks-palindrome/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08a892e4d0d8a8f896b55f9aeeb77ea24820710db83d944a79007d67bc791e6f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a384267-absolute-reciprocal-square-shift-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A384267, Paul D. Hanna, Jun 19 2025. The NAME and COMMENT below are\nverbatim excerpts of the claim in `Library/Recurrence/hanna2025a384267.md`.\n\nNAME:\n\n> a(n) = abs([x^n] 1 + x/A(x)^2), where A(x) is the g.f.\n\nCOMMENT:\n\n> (C.1) a(n) == binomial(3*n-1,n)/(3*n-1) (mod 2) (cf. A006013).\n\nThis dossier anchors exactly C.1 for n >= 1. The entry lists three conjectures;\nC.2 (sign pattern) and C.3 (asymptotic ratios) are untouched. The companion\n`a_zero` gives a(0) = 1. Here abs means coefficientwise absolute value of the\nseries `1 + x*A(x)^(-2)`, represented by the frozen `absSeries` applied to\n`1 + X * invOfUnit (A ^ 2) 1`.","title":"Parity conjecture C.1 of OEIS A384267","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384267-absolute-reciprocal-square-shift-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8890a9abc1e3f86b63734bc7c77785003488b4f26d6b27aed52da1844c8e7cc1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a384270-quartic-cubic-catalan-quotient-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A384270, Paul D. Hanna, Jul 10 2025. The following NAME and COMMENT\nare copied verbatim from the claim in `Library/Arith/hanna2025a384270.md`.\n\nNAME:\n\n> G.f. satisfies A(x) = A(x^4 + 4*x*A(x)^4) / A(x^3 + 3*x*A(x)^3).\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n = 2^k for k >= 0.\n\n**范围与诚实边界**: The module formalizes the cross-multiplied identity\n`A·A(x³+3xA³) = A(x⁴+4xA⁴)`, with `A(0)=0` and `a(1)=1`. The displayed\ndenominator has zero constant term, so it is not a unit in the integer\nformal power-series ring. The public parity theorem covers every positive\nnatural index; the constant coefficient is zero by `generating_equation`.","title":"Parity of the A384270 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384270-quartic-cubic-catalan-quotient-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af82a5ac54e59aa51be52b027b50ed3e4e5cc893ef5c8f02d5f40b75610d5a92.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a385590-alternating-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The single statement considered here is the formula-field conjecture in\nWerner Schulte's OEIS A385590, dated 2025-07-03. For every integer n >= 1,\nlet i > 1 be the unique index with F(i) <= n < F(i+1), with F(0)=0 and F(1)=1.\nPut T(n,k) = F(i-1)^2 + 1 - ((i-1) mod 2) + (n-F(i))*F(i-2)\n+ (k-1)*F(i-1). Is the sum from k=1 to n of\n(-1)^k * binomial(n-1,k-1) * T(n,k) equal to (-1)^n when n<3,\nand zero otherwise? All triangle values and the sum are integers.\nThe entry's separate permutation conjecture is outside this question.","title":"oeis a385590 alternating binomial","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a385590-alternating-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7dc03f44d037456ce20eb21179997b2e858dab8480c714d07f577562d71da931.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Preregistration: [#18668](https://github.com/the-omega-institute/trureturing/issues/18668).\nPaul D. Hanna's OEIS A386655 NAME reads: \"E.g.f.: Sum_{n>=0} (2^n*x + LambertW(x))^n / n!.\"\nIts COMMENTS sentence is: \"Conjecture: for n >= 6, a(n) (mod 6) equals [1, 0, 3, 0, 3, 2] repeating.\"\nThe corresponding entries A386656, A386657 and A386658 replace $2$ by $3$, $4$ and $5$, respectively.\nTheir COMMENTS sentences specify $[4,3,2,1,0,1]$, $[1,2,3,2,3,4]$ and $[4,3,0,3,0,5]$.\nAll four statements include integrality and the index $(n-6)\\bmod6$ for every natural $n\\ge6$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Principal formal Lambert W | `OeisA386655LambertWEgfModSix.W` | Constant coefficient zero and positive coefficient $(-m)^{m-1}/m!$. |\n| Exponential generating coefficient | `egf q m` | $m!$ times the coefficient of $x^m$; summation through $m$ is exact because each later summand has order greater than $m$. |\n| Bases $q=2,3,4,5$ | Each module's `a` | The respective `egf` specialization. |\n| Every $n\\ge6$ | Each module's `claim` and `result` | Universal natural index and the literal lower bound. |\n| Integral remainder with the published pattern | Each module's `claim` and `result` | An integer representative with rational cast equal to `a n`, and integer remainder equal to the zero-based `List.getD` entry. |\n| Finite-sum endpoint | `closedZ q n` | The source's last summand is $q^{n^2}$; all remaining exponents are natural numbers. |\n\nThe public definitions retain the preregistered expressions. The formal series interpretation makes no assertion about analytic convergence or values outside that series.","title":"Hanna's Lambert W coefficient congruences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386655-hanna-lambert-w-egf-mod-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/441a04607d20bdee185bd14986a613e1fd1a81b3696fb9e76f33ac9edcf52645.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a386666-signed-catalan-cubic-substitution-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A386666 (Paul D. Hanna, Sep 07 2025) states:\n\n> G.f. A(x) satisfies: A(x)^2 = A( x^2 + 4*A(x)^3 ).\n>\n> Conjecture: a(3*n) (mod 3) == 2*a(3*n+1) (mod 3) == a(3*n+2) (mod 3) == 2*A039966(n) for n > 0, where A039966(n) equals the number of partitions of n into distinct powers of 3.\n\nThe conjecture is formalized with `distinctPowersOfThree n` standing for A039966(n).","title":"Signed Catalan cubic substitution modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386666-signed-catalan-cubic-substitution-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac45a951b68f10495c0ca5d07f9a8777d3b7532222c6e3656dea5ed5b77296cd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a387319-multiple-of-three-coincidence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A387319, revision 69 (November 10, 2025), conjectures that every\ncomposite k except 4, 8, 10, and 25 has a coincidence between m=3 and\nanother multiple of 3 among the rational fractions (2k mod m)/m for\n1≤m≤k, whereas no prime does. The exact predicate is\n`∃ m : ℕ, 6 ≤ m ∧ m ≤ k ∧ 3 ∣ m ∧ 3*((2*k)%m) = m*((2*k)%3)`.\nThe theorem states that this is equivalent to\n`1 < k ∧ ¬Nat.Prime k ∧ k ∉ ({4,8,10,25} : Finset ℕ)` for every natural k.","title":"Multiple-of-Three Residue Coincidences","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a387319-multiple-of-three-coincidence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/91b5e10f6b4cc88d378b26b222c9e4961fdf2609cde76f8a3264effb15b0acd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a388711-choosable-initial-intervals","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A388711 counts partitions of n into k positive parts whose initial\nintervals admit distinct representatives. Its comment conjectures that the\nsame number counts superdiagonal reversed partitions: the parts in increasing\norder satisfy y_i >= i at every one-based position.","title":"Choosable Initial Intervals and Reversed Partitions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388711-choosable-initial-intervals/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e3a6e5ba3d4a30faa57b19d0c301ca882be6ab82d6d601a3020ed89228db5bc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a388986-euler-form-divisor-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Does every number in A228058 belong to A388986? Explicitly, let p be prime,\np congruent to 1 modulo 4, a a natural number, and r an odd integer greater\nthan one with gcd(p,r)=1. Put N=p^(4a+1)r^2. The question asks whether the sum\nof the unitary divisors of N plus the sum of its squarefree divisors is\nstrictly less than 2N.","title":"The A388986 Euler-form inclusion question","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/685d7e1996ee78ede75c89c0af72caa5d93cac072a55ee8cbce9f55af5be1b43.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389000-digit-sum-square-least-witness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: a(9m+5) = 210^(2*m+1)-1 for m >= 0.\n\n— Chai Wah Wu, Oct 01 2025. The OEIS rendering means\n`2·10^(2m+1) − 1`. Here `a(n)` is the least positive integer `k` such that\n`n` divides both the base-10 digit sum of `k` and that of `k^2`.\nThe entry was created Sep 22 2025.","title":"The least simultaneous digit-sum witness in A389000","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389000-digit-sum-square-least-witness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8bfbf8700e38c21d3320e4dbdaf67c6e26632d305305c593c2fa6db2e12bbab3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389472-square-cube-substitution-odd-index-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A389472, Paul D. Hanna, Oct 15 2025, gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Arith/hanna2025a389472.md`:\n\n> G.f. satisfies A(x) = A(x^2 + x^3)/x^2 - 1.\n\n> Conjecture: a(2*n-1) = 0 (mod 2) for n > 1.\n\n范围与诚实边界: The separate comment \"Conjecture: a(3*n-1) = 0 (mod 3)\nfor n > 1.\" is not resolved here. The generating equation alone leaves a(2)\nfree. The module fixes a(0)=0, a(1)=a(2)=1, matching the OEIS PARI program\nand DATA, and formalizes the cross-multiplied identity\n`x²(A+1) = A(x²+x³)` over integer formal power series. Only the first,\nodd-index parity conjecture for that normalized sequence is claimed.","title":"Odd-index parity of the A389472 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389472-square-cube-substitution-odd-index-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b775c97c437a3c58a666cd0b551dea0281bee02ff00f745889f864197f8147d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389476-quadratic-reversion-dyadic-support-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A389476 (Paul D. Hanna, Oct 05 2025) gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Arith/hanna2025a389476.md`.\n\nNAME:\n\n> G.f. A(x) satisfies: A( x - A(x)^2/(1 - A(x))^2 ) = x.\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n is a term of A027383, where A027383(2*m) = 3*2^m - 2 and A027383(2*m+1) = 4*2^m - 2 for m >= 0.","title":"Dyadic parity support of the A389476 quadratic reversion","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389476-quadratic-reversion-dyadic-support-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0c6c8a0f909745249fac63aa4a1c4ebac479aacbebb7002cbf753d2fa02cf5b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389536-substitution-square-cube-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A389536, Paul D. Hanna, Oct 28 2025. The following verbatim NAME and\nCOMMENT quotations are recorded in `Library/Arith/hanna2025a389536.md`:\n\nNAME:\n\n> G.f. A(x) satisfies: A(x) = A(x^2 + 2*x^3) / x.\n\nCOMMENT:\n\n> Conjecture: for n > 1, a(n) == 2 (mod 4) when n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4.","title":"The A389536 square-cube substitution conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389536-substitution-square-cube-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b82fb09f5182c55386f0d0110e2cf04be70d030de3964f4f79ba320c57757d86.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389537-alternating-dyadic-reversion-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A389537, Nov 03 2025, gives the following NAME and\nCOMMENT. These quotations are copied verbatim from the claim in\n`Library/Arith/hanna2025a389537.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A( A(x) - x ) = A(x)^2.\n\nCOMMENT:\n\n> Conjecture: for n > 2, a(n) == 2 (mod 4) when n = 3*2^k for k >= 0 otherwise a(n) is divisible by 4.\n\nThe formal target uses the normalized integral formal power series with\n`A(0) = 0` and `a(1) = 1`.","title":"Alternating dyadic reversion modulo four for A389537","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389537-alternating-dyadic-reversion-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/26997fcc91f36af0a63f2656053a6565efd12ce064c0b8466f322accafd12dce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389540-half-scaled-reflection-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> NAME \"G.f. A(x) satisfies A(x)^2 = A(2*x - 2*A(x)) / 2.\"\n>\n> COMMENT \"Conjecture: a(n) is odd iff n = 2^k for k >= 0.\"","title":"G.f. A(x) satisfies A(x)^2 = A(2*x - 2*A(x)) / 2.","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389540-half-scaled-reflection-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c5dfb371b2dd7971a43843b4b42d1f497821dce54c5acc787a08ac67a5a47dbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a389542-quadratic-square-reversion-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A389542, Paul D. Hanna, Nov 01 2025, supplies the following NAME\nand COMMENT, quoted verbatim from `Library/Recurrence/hanna2025a389542.md`:\n\n> G.f. A(x) satisfies A( A(x)^2 - x^2 ) = 4*A(x)^3.\n\n> Conjecture: for n > 1, a(n) == 2 (mod 4) if n = 2^k + 1 (for some k >= 0) otherwise a(n) is divisible by 4.","title":"Quadratic square reversion modulo four for A389542","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389542-quadratic-square-reversion-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/507c5aed1e01c8dbc991047b2c33d1474567f50895d76946126328386fbeefeb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a390088-prime-digit-base-existence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For each number, ask whether some base above one writes it with all digits\nprime. Felix Huber conjectured on October 29, 2025 that exactly five numbers\nadmit no such base.","title":"The A390088 exception set","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a390088-prime-digit-base-existence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d779e55deebc0d2029ae9e9266d9cc50a3f785d34484cfae6098727386114f7a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391620-first-sums-partition-characterization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A391620 (Gus Wiseman, Dec 30 2025) states:\n\nNAME \"Number of integer partitions of n that are not the first sums of any composition with all parts > 1.\"\n\nCOMMENT \"Conjecture: These are integer partitions of n whose least part is < 4.\"\n\nThe NAME line is copied verbatim from `Library/Words/wiseman2025a391620.md`.\nThe reading used by Lean writes a partition q as a weakly decreasing,\nnonempty list of positive parts. It is the first sums of a composition\nx = (x_1, ..., x_{r+1}), with every x_j >= 2, exactly when\nq_i = x_i + x_{i+1} for each i. Equality preserves this order; it is not\nequality after arbitrarily permuting the adjacent sums. The conjecture is\nequivalent to `(not first sums) iff min q < 4`. For a nonempty partition,\nthe latter condition is equivalent to the existence of a part below four.","title":"First sums of decreasing partitions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391620-first-sums-partition-characterization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73ce3e59e140a11e85cde443aae58b9dd6def1f149f476fc62fff8e9bd795feb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391632-andrasfai-mutual-visibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A391632, revision 27, attributes this assertion to Andrew Howroyd:\n\n> Conjectured g.f.: x*(4 - 19*x + 33*x^2 - 40*x^3 + 8*x^4)/(1 - 10*x + 29*x^2 - 32*x^3 + 8*x^4).\n\nThe entry's name is \"Number of mutual-visibility sets in the n-Andrásfai\ngraph\" and its offset is 1,1. For every integer $n\\ge1$, let $G_n$ be the\nsimple undirected Cayley graph on the labelled vertices\n$\\mathbb Z/(3n-1)\\mathbb Z$, with connection set\n$\\{1,4,\\ldots,3n-2\\}$. A subset $S$ is mutual-visible when every distinct\npair $u,v\\in S$ admits a shortest $u$-$v$ path in the original host $G_n$\nwhose internal vertices avoid $S$. Different pairs may choose different\npaths. Let $a(n)$ count every such labelled subset, including the empty set\nand singletons. The entire question is the identity in $\\mathbb Q[[X]]$\n\n$$\n\\sum_{n\\ge1}a(n)X^n\n=\\frac{X(4-19X+33X^2-40X^3+8X^4)}{1-10X+29X^2-32X^3+8X^4}.\n$$\n\nThe series has constant coefficient zero; no graph of index zero is\nintroduced. There is no quotient by graph symmetries, maximality,\nindependence or induced-host restriction. The quotient denotes\nmultiplication by the formal inverse of a series with constant coefficient\none, without an analytic convergence premise.\n\n### Statement correspondence\n\nPut $m=n-1$ and $N=3m+2=3n-1$. `Andrasfai.host m` uses `ZMod N`, with\nadjacency determined by the canonical representative of $v-u$ being one\nmodulo three. `Andrasfai.count m` is the cardinality of the original\n`Configuration (host m)` subtype of actual finite vertex subsets.\n`AndrasfaiActual.generatingSeries` is defined directly as\n$X\\operatorname{mk}(m\\mapsto(\\operatorname{count}(m):\\mathbb Q))$.\n`AndrasfaiActual.result` has no hypotheses and states the displayed\nidentity for that series.","title":"Howroyd's full Andrasfai mutual-visibility generating function","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391632-andrasfai-mutual-visibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3110470f519454b0299d3f27e51c100be54666d9c3625c4b4edec53f00fec0e2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391810-alternating-jacobsthal","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A391810 formula (5), in its original positive-index range, is\n\n$$\n-\\sum_{k=0}^{\\lfloor(n+1)/2\\rfloor^2}(-1)^k T(n,k)\n=2\\operatorname{round}(2^n/3)=2\\,\\mathrm{A001045}(n),\\qquad n\\ge1.\n$$\n\nHere T(n,k) is the signed coefficient of x to the power -k in the formal\nexpansion at zero of the literal bilateral sum\n`Sum_{j=-oo..oo} x^j*(1-x^j)^(n+j)`. Integer powers with negative exponent\nuse the Laurent-field inverse. The range includes k=0 and its upper endpoint.\nRounding acts on the exact rational number, with nearest-integer ties toward\npositive infinity. A001045 has offset zero and recurrence\n`J(0)=0, J(1)=1, J(n+2)=J(n+1)+2*J(n)`.","title":"The full alternating Laurent-row formula in A391810","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-alternating-jacobsthal/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb4892a2f4acb421714cb76e4b7fdcc76a1d241654545b78af4554d73a1d97b3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391810-full-laurent-row-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A391810 formula (2) states:\n\n> (2) Sum_{k=0..floor((n+1)/2)^2} T(n,k) = 0 for n >= 1 (conjecture).\n\nThe defining NAME is:\n\n> Irregular triangle, read by rows, where row n lists the coefficients of nonpositive powers of x in the doubly infinite series Sum_{k=-oo..oo} x^k * (1 - x^k)^(n+k) for n >= 1.\n\nFormula (1) identifies the truncated Laurent expansion, including the constant\nterm, with `Sum_{k=0..floor((n+1)/2)^2} T(n,k)*x^(-k)`.\nAll rows with positive integer n are included. Expansion takes place at zero;\nnegative integer exponents mean Laurent-field inverses.","title":"The full Laurent rows in A391810 sum to zero","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17189743f84da3e26352ecc52508d353d51d80e8753daf4ea0e89ebf83bc8eae.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391810-laurent-first-moment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A391810, revision 39, formula (3), states:\n\n> (3) Sum_{k=0..floor((n+1)/2)^2} k * T(n,k) = 1 for n >= 1 (conjecture).\n\nThe original definition is:\n\n> Irregular triangle, read by rows, where row n lists the coefficients of nonpositive powers of x in the doubly infinite series Sum_{k=-oo..oo} x^k * (1 - x^k)^(n+k) for n >= 1.\n\nThus, for every positive natural n, expand\n`S_n(x)=Sum_{j in Z} x^j*(1-x^j)^(n+j)` formally at zero and let\n`T(n,k)=[x^(-k)]S_n(x)`. The outer sum includes every integer j;\nnegative integer powers use Laurent-field inverses. The row includes its\nconstant coefficient at k=0 and its exact inclusive final index\n`floor((n+1)/2)^2`. Coefficients and the weighted row sum are signed integers.\nThe literal Laurent realization uses rational coefficients and the exact\ninteger-to-rational coefficient bridge. The shared library note's claim\nquotation concerns formula (2); the quoted formula (3) is the distinct target here.","title":"The first weighted moment of A391810","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc0e88ce52a010ccb32a84a1870cf97e286d1a55ebf91eb83cf4c94efbcb78da.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a391810-second-moment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A391810 revision 39 formula (4) states:\n\n> (4) Sum_{k=0..floor((n+1)/2)^2} k^2 * T(n,k) = -(2*n-5) for n >= 2 (conjecture).\n\nThe source defines T(n,k) as the coefficient at zero of x^k times the\nbilateral series `Sum_{j=-oo..oo} x^j*(1-x^j)^(n+j)`, expanded formally\nat zero. Equivalently it is the series coefficient at -k. All integer\nouter indices, the constant coefficient, and both finite-sum endpoints are\nincluded. Negative integer exponents are Laurent-field inverses. The bound\nis natural division `(n+1)/2` followed by squaring. The right side uses\nsigned integer subtraction and negation, then a rational cast.","title":"The full signed Laurent second moment in A391810","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-second-moment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5517f61600e9b66466773ae493bc623e0839bc079181efc8c954a83cc0eec097.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a392095-antidiagonal-source-series","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mikhail Kurkov's A392095 (December 30, 2025) states:\n\n> Array read by downward antidiagonals: A(n,k) = A(n-1,k+1) + Sum_{j=0..k} A(n-1,j)*A(k-j,0) with A(0,k) = 1.\n\n> Conjecture: A(n,0) = A088713(n+1).\n\nPaul D. Hanna's A088713 (October 12, 2003) states:\n\n> G.f. A(x) satisfies A(x/A(x)) = 1/(1-x).\n\nBoth sequences start at index zero. A088713 has constant coefficient one.\nThe exact sources, author lines, immutable locators and licenses are in\n`Library/Recurrence/kurkov2025a392095.md` and\n`Library/Recurrence/hanna2003a088713.md`.\n\nThe full assertion concerns a total array T : N -> N -> N with\nT(0,k)=1 for every k and\nT(n+1,k)=T(n,k+1)+sum(j=0..k) T(n,j)*T(k-j,0) for every n,k.\nThere is a unique such T. Independently there is a unique F in Q[[X]]\nwith F(0)=1 and F(X*F^(-1))=(1-X)^(-1). There is a unique b : N -> N\nwhose rational coefficient series satisfies that equation; b(0)=1,\n[X^m]F=b(m) for every m, and T(n,0)=b(n+1) for every n, including zero.\nInverses here are multiplicative unit inverses, and the inner substitution\nseries has zero constant coefficient. No extra source equation is assumed.","title":"The A392095 column and A088713 source series","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392095-antidiagonal-source-series/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/88a73c37cb56794e47f8211013e1681a001bc92033e382fc81ccc3e4166ce726.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a392525-cubic-fifteen-substitution-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392525 (Paul D. Hanna, Jan 15 2026) gives the following NAME,\nFORMULA (1), and COMMENT. The quotations are copied from the claim in\n`Library/Arith/hanna2026a392525.md`; the attribution and date were supplied\nby the orchestrator.\n\nNAME:\n\n> G.f. satisfies: A(x) = A( x^3 + 15*x*A(x)^3 )^(1/3), with A(0)=0, A'(0)=1.\n\nFORMULA (1):\n\n> (1) A(x)^3 = A( x^3 + 15*x*A(x)^3 ).\n\nCOMMENT:\n\n> Conjecture: a(n) is odd iff n = 2^k for k >= 0.\n\n### 范围与诚实边界\n\nThe module formalizes FORMULA (1) over integer formal power series with\n`A(0)=0` and `A'(0)=1` (constant coefficient zero and linear coefficient one).\nIt does not formalize the cube-root operation in the NAME. The public parity\ntheorem is stated for every `n ≥ 1`; at `n = 0` the normalization gives `a(0)=0`,\nand zero is not a power of two. The separate second comment,\n`a(n) ≡ 5 mod 10 iff n = 2^k, k ≥ 1`, is not resolved here.","title":"Parity of the A392525 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392525-cubic-fifteen-substitution-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86a5262960b83494442425dd845fbc3b9de46f5ef17f619ee168d72ca522c561.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a392694-zero-prepended-first-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392694, revision 7 (January 25, 2026), states:\n\n> Conjecture: Reversed integer partitions of this type (with weakly increasing 0-based partial alternating sums) are counted by A000009.\n\nA reversed partition is a weakly increasing positive integer list y.\nThe alternating recurrence is s0=0, sj=yj-s(j-1), and the condition is that\n0,s1,...,sk is weakly increasing. A000009(n) counts distinct-part partitions\nof n. The claim has no shift.","title":"Zero-Prepended First Sums and Odd Parts","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18aad8e67b4696c06f45c05d0d38b2a007b91b35161cbf9069c5254cfc44b7f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a392698-trimmed-alternating-partitions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For an ordinary partition q of n, list the positive parts in weakly\ndecreasing order. Set s_0=0 and s_j=q_j-s_(j-1) in the integers. Count\npartitions for which s_1,...,s_k are pairwise distinct, excluding s_0.\nThe corrected conjecture asserts that this count is the number of\npartitions of n+1 into distinct positive parts, for every n including zero.\nThe unshifted A000009(n) sentence on A392698 is incorrect; the source note\nrecords the correction and the equivalent conjectural A392703 comment.","title":"Trimmed Alternating Sums of Ordinary Partitions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392698-trimmed-alternating-partitions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecc9646080446a5da825f23e96e9b2b668892c9cba5b27ba63e9c5fa7eb1e853.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392707, revision 10, states:\n\n> Conjecture: Reversed integer partitions of this type (with strictly increasing 0-based partial alternating sums) are counted by A053251.\n\nThe formal target here is the self-contained combinatorial bridge: for every\nn, the number of partitions with a strictly increasing positive preimage s\nunder firstSums(0::s) equals the number of partitions whose support is exactly\n{1,3,...,2k-1} for some k. The two classes include the empty partition.\nThe further equality with mock theta coefficients is not formalized here.","title":"Strict First Sums and Gapfree Odd Partitions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/14d2b4752eda7b17b8e9be2431582578416897c0d11d88db87514defbf0695ca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393172-catalan-shift-square-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A393172, Paul D. Hanna, Feb 04 2026. The following NAME and COMMENT\nare copied verbatim from the claim in `Library/Arith/hanna2026a393172.md`.\n\nNAME:\n> G.f. A(x) satisfies A(x-x^2) = x + A(x)^2.\n\nCOMMENT:\n> Conjecture: for n > 1, a(n) == 2 (mod 4) iff n is a power of 2, otherwise a(n) is divisible by 4.","title":"The mod-four coefficients of the Catalan-shift square series","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393172-catalan-shift-square-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/800124755d6e3b523e84f4b372816036d54122d2d32dd5abd45d349e31f3b2f7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393173-hanna-power-of-two-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's [OEIS A393173](https://oeis.org/A393173) states:\n\n> G.f. A(x) satisfies [x^n] 1/(A(x) - n^2*x) = 0 for n >= 1.\n\n> Conjecture: for n > 0, a(n) is odd iff n is a power of 2.\n\nThe offset-zero data specify $a(0)=1$.\n[Preregistration #18824](https://github.com/the-omega-institute/trureturing/issues/18824)\nclassifies the published conjecture as tier 1. The source is documented in\n[the Library note](../Library/RecurrenceComposition/hanna2026a393173.md).\n\n### Statement echo\n\nNames below belong to `D5.S1.Recurrence.Invariants.HannaA393173PowerTwoParity`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Offset zero, $a(0)=1$ | `IsGF` constant coefficient | Literal normalization. |\n| $[x^n](A-n^2x)^{-1}=0$ for $n\\ge1$ | `IsGF` inverse coefficient, natural index cast into $\\mathbb Q$ | Literal source expression. |\n| Positive indices | `claim`, $n>0$ | Zero is excluded. |\n| $a(n)$ is odd iff $n=2^k$ | `claim`, integer coefficient witness and both directions of the equivalence | Exponent $k$ is natural. |\n| Existence and integrality implicit in the sequence | `claim`, an existing solution and an integer witness for every coefficient of every solution | Explicit strengthened clauses from the preregistration; these prevent vacuity. |","title":"Hanna's power-of-two parity conjecture for OEIS A393173","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bf340f6c531703a5580f636357e9347b6b2eeded0d7ed17015ca2ceb2925c06.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393434-factorial-heraclitus","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A393434 starts at a(0) = 0. Each subsequent term is the unused integer\nwith smallest absolute value at factorial distance k!, k >= 1, from the previous\nterm. Positive values are preferred when absolute values tie. The source states:\n“It is conjectured that every integer appears in this sequence.”","title":"Every integer occurs in the factorial Heraclitus transform","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393434-factorial-heraclitus/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ccca7ce115f6628f012d995033888a9fe6c850d0e079062e356573779c16db9c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393856-scaled-reversion-mod-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A393856, Paul D. Hanna, Mar 11 2026. The following NAME and COMMENT\nare quoted verbatim from the claim in `Library/Arith/hanna2026a393856.md`.\n\nNAME:\n\n> G.f. A(x) satisfies: A( x - x*A(4*x)/4 ) = x.\n\nCOMMENT:\n\n> Conjecture: a(n) == 1 (mod 5) for n >= 1.","title":"Coefficients of A393856 are one modulo five","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393856-scaled-reversion-mod-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c152ef4a79bb7128b87c27caddb096554c8f25317a9afed9ca93b68ae55bee77.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393857-scaled-reversion-mod-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A393857, Paul D. Hanna, Mar 11 2026. The following NAME and COMMENT\nare quoted verbatim from the claim in `Library/Arith/hanna2026a393857.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A( x - x*A(5*x)/5 ) = x.\n\nCOMMENT:\n\n> Conjecture: a(n) == 1 (mod 6) for n >= 1.","title":"Coefficients of A393857 are one modulo six","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393857-scaled-reversion-mod-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b860bcc51360fa1b08754583975201addcad1893e0aea126d35d57aeddac0357.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393867-all-terms-odd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The second OEIS A393867 comment is exactly: \"Conjecture: all terms are odd.\"\nThe target is `Odd (a393867 n)` for every `n >= 1`, using the definition in\n`PrimePowerShiftLogDerivative`. This is the first-tier 2026 conjecture; the\nprime-divisibility comment and A393866's shifted printed formula are distinct.","title":"A393867: every logarithmic-derivative coefficient is odd","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-all-terms-odd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd922b8819dd770b68342c6acc31aadc5a1a8327f40f2ca410bd918e2953c850.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a393867-prime-power-shift-log-derivative","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS entries A393866 and A393867, Mar 29 2026, specify:\n\nA393866 NAME (quoted verbatim in `Library/Recurrence/hanna2026a393867.md`):\n\n> G.f. A(x) satisfies [x^n] A(x)^prime(n) = prime(n) * [x^(n-1)] A(x)^prime(n) for n >= 1.\n\nA393867 NAME (quoted verbatim in the same note):\n\n> Logarithmic derivative of A393866.\n\nWith constant coefficient 1 for A, this means\n`a(n) = [x^(n-1)] A'(x)/A(x)`, with offset 1. This dossier concerns A393867.\nIts target COMMENT (quoted verbatim in the same note) is:\n\n> Conjecture: a(n) is divisible by prime(n) for n > 1.\n\nThe second A393867 comment, \"Conjecture: all terms are odd.\", is addressed\nseparately in `Problems/oeis-a393867-all-terms-odd.md` by the new companion\nmodule `PrimePowerShiftLogDerivativeOdd`; the original module is unchanged.\nA393866 FORMULA (2), \"[x^n] A'(x)/A(x) == 0 (mod prime(n)) for n > 1\n(conjecture - see A393867)\", is refuted as printed by the companion theorem;\nit is not a second resolution claim in this dossier.","title":"Prime divisibility of the A393867 logarithmic-derivative terms","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-prime-power-shift-log-derivative/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0d8cadc5fedcdaecd076065f9739448dde16f2c43b91b6b70854c4ef575c7535.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a394399-twin-prime-sigma-gcd-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every natural number k, if k-1 and k+1 are prime and gcd(k, sigma(k))=3, then there is a natural number m with k=2m². Here sigma(k) is the sum of positive divisors.","title":"Twin-prime sigma-gcd three forces twice a square","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394399-twin-prime-sigma-gcd-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a3f0fcc8aa464c84dce66879f61579d7729b797c94dbd0043087550a0c022a4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a394407-totient-tau-primes-huber","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Felix Huber's OEIS A394407 has the Name:\n\n> Positive integers k such that there is no m < k with phi(m)/tau(m) = phi(k)/tau(k).\n\nIts Comment states:\n\n> Conjecture: All primes p != 3 are in the sequence; equivalently, for every prime p != 3 there is no m < p with phi(m)/tau(m) = phi(p)/tau(p) = (p - 1)/2; equivalently, the sequence is a supersequence of A045344.\n\nThe source is an explicitly stated conjecture in an approved OEIS entry. The\nsettled proposition is the first form of that conjecture, preregistered in\n[#18591](https://github.com/the-omega-institute/trureturing/issues/18591).\nFor positive integers, write $r(m)=\\varphi(m)/\\tau(m)$ with rational division.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Quotient phi(m)/tau(m) | `ratio m` | `Nat.totient m` divided by `m.divisors.card`, with both cast to rational numbers. |\n| Positive integers k | First conjunct of `IsTerm k` | Zero is excluded. |\n| No smaller m has the same ratio | Second conjunct of `IsTerm k` | Every positive natural m less than k is compared; no search bound is imposed. |\n| All primes p != 3 | `claim` | Universal natural p, `Nat.Prime p`, and p different from 3. |\n| Membership in the sequence | Conclusion `IsTerm p` of `result` | The first-occurrence predicate is unchanged. |\n\nThe positive domain for m follows the sequence's positive-integer convention.\nLean totalizes the ratio at zero to zero; no zero input enters the claim.","title":"Huber's prime conjecture for OEIS A394407","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394407-totient-tau-primes-huber/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba95ff387afd41adb4db620c0c47f4af46b44bb142496fe23c778d87abb50917.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a394578-interpolation-ratio-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The selected source COMMENT is:\n\n> Conjecture: a(n+1)/a(n) -> 1/4.\n\nIts complete native definition is quoted in the corresponding Library\nnote. For every natural n, p_n is the rational Lagrange polynomial\nthrough (1,0) and (k,1) for all integers k from 2 through n+1. Define\n$a_n=(4^n/2)p_n(n+1/2)$, including zero. The exact settling declaration\nis `OEISA394578RatioRefutation.result : Not claim`, where `claim : Prop`\nis convergence of the real consecutive ratio at natural `atTop` to 1/4.\nThe degree and node conditions have no extra hypothesis; no shifted\nsequence or recurrence replaces the interpolation definition.","title":"The ratio-limit assertion for A394578","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394578-interpolation-ratio-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9588751b0e6db3843e52fa73d27579a00be9c15bfdae8cb83d34ba52f96d642b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a394757-sigma-gcd-two-or-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every natural number k, if k-1 and k+1 are prime and gcd(k, sigma(k)) is prime, then gcd(k, sigma(k))=2 or gcd(k, sigma(k))=3.","title":"Prime twin-center sigma-gcd values are two or three","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394757-sigma-gcd-two-or-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b804ba6a67800417b30b04e7c7b6197b7c5616456d9137df3ed8e21e12c9372c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a394757-twin-prime-sigma-gcd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A394757 lists the centers `k` of twin-prime pairs for which the greatest\ncommon divisor of `k` and the sum-of-divisors function `sigma(k)` is prime. The\nentry states that every such term is divisible by 18.","title":"Divisibility of the A394757 terms","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394757-twin-prime-sigma-gcd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd7f0b66a4654d44b9e8f3007faa03bb2f6166431b86919fd344e662fb1c4052.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a395531-greedy-brick-all-index","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A395531, revision 30, Aleksei Udovenko:\n\n> Conjecture: when n = a(m) for some m, we have a(n) = n(n+3)/2 - m. Verified experimentally up to m=1172, n=688491.\n\nRows are numbered from one. The value a(m) is the width of the first brick\nin row m under the literal highest-supported-row placement rule. The exact\nassertion is a(a(m)) = a(m)(a(m)+3)/2-m for every integer m >= 1.","title":"The greedy-brick all-index identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395531-greedy-brick-all-index/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5fb336de87be6752b37324770206e89243b09ff6ab26e4d51a5ca7787bac5cbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a395833-diagonal-exponent-self-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The Library note quotes the NAME and divisibility conjecture verbatim:\n\n> G.f. A(x) satisfies [x^n] A( x/A(x)^(2*n-1) ) = 0 for n > 1. Conjecture: (2*n-1) divides a(n) for n >= 1.\n\nThe supplied local OEIS source identifies the entry and its separate conjecture:\n\n```text\n%N A395833 G.f. A(x) satisfies [x^n] A( x/A(x)^(2*n-1) ) = 0 for n > 1.\n%C A395833 Conjecture: (2*n-1) divides a(n) for n >= 1.\n%C A395833 Conjecture: a(n) == 0 (mod 3) for n >= 2.\n%A A395833 _Paul D. Hanna_, May 07 2026\n```\n\nThe problem resolved here is the divisibility clause only: `(2*n-1)` divides\n`a(n)` for every `n >= 1`. The separate modulo-three clause is outside this claim.\nThe sequence is the unique normalized integer series with constant and linear\ncoefficients one satisfying the displayed vanishing-diagonal equation.","title":"OEIS A395833: diagonal-exponent self-divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-diagonal-exponent-self-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9517878dd9d519fc539493ec5da0039af2dda079e46bd001d38bb3cf2f6e7f92.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a395833-negative-power-diagonal-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"From [Library/ArithSums/hanna2026a395833.md](../Library/ArithSums/hanna2026a395833.md),\nOEIS A395833, Paul D. Hanna, May 07 2026:\n\nNAME:\n\n> G.f. A(x) satisfies [x^n] A( x/A(x)^(2*n-1) ) = 0 for n > 1.\n\nCOMMENT:\n\n> Conjecture: a(n) == 0 (mod 3) for n >= 2.\n\nThe normalization is `a(0) = a(1) = 1`. Only this congruence clause is claimed.","title":"A395833: the mod-three congruence clause","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07133ab7625504e471b3ca48361e79964c0741cfcc036db84ecfd9aa01468694.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396099-iterate-product-two-three-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396099 (Paul D. Hanna, Jun 02 2026) gives the NAME:\n\n> G.f. satisfies A(x) = x + A^2(x)*A^3(x).\n\nHere A^k denotes the k-fold compositional iterate, as the entry's formulas\nand PARI program make explicit: A^2(x) = A(A(x)) and A^3(x) = A(A(A(x))).\nThe single resolution anchor of this dossier is the COMMENT:\n\n> Conjecture: a(n) = [1,3,3,1] repeating (mod 4) for n > 2.\n\nThe entry also states three companion conjectures:\n\n> Conjecture: all terms are odd.\n\n> Conjecture: [x^n] A(A(x)) == 0 (mod 4) for n > 2.\n\n> Conjecture: [x^n] ( A(x) - x*A(A(A(x))) ) == 2 (mod 4) for n > 2.\n\nThese are settled by `all_odd`, `iterate_two_mod_four`, and `shift_mod_four`,\nrespectively, without adding a second resolution claim.","title":"The A396099 period-four coefficient pattern","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396099-iterate-product-two-three-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9331201b39428f71b9bad4a0f2379aff0373703f77748325a6bc89d91384188a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396102-triple-iterate-shift-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"NAME \"G.f. A(x) satisfies A(A(A(x))) = (1+x) * A(A(x)).\"\n\nCOMMENT \"Conjecture: a(n) == 1 (mod 3) for n >= 1.\"\n\nThese are the verbatim NAME and COMMENT lines supplied for OEIS A396102 by Paul D. Hanna, Jun 06 2026.","title":"G.f. A(x) satisfies A(A(A(x))) = (1+x) * A(A(x))","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396102-triple-iterate-shift-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fba99a170e611765839b51c72e01731481772a43314069b4cbfe99cd0054175.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396596-divisor-power-union","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hal M. Switkay, OEIS A396596, contribution August 31, 2026, revision 65:\n\n> Natural numbers k such that the sum of the x-th powers of the divisors of k achieves a record, for some x <= 0.\n\nThe selected COMMENTS assertion is:\n\n> We conjecture that the present sequence can be constructed simply as a union of highly composite numbers and deeply composite numbers.\n\nThe primary URL in this dossier is byte-for-byte the URL in the source note's\n`Verified locator`. The source note is\n`Library/notes/switkay2026divisorpowerunion.md`, referenced as\n`D5/L/switkay2026divisorpowerunion`.\n\nThe complete statement mirror, including the two needed definitions and the\nclosed proposition that the single result negates, is:\n\n```lean\nnoncomputable def DivisorPowerSum (n : ℕ) (x : ℝ) : ℝ :=\n ∑ d ∈ n.divisors, (d : ℝ) ^ x\n\ndef StrictDivisorPowerRecord (n : ℕ) (x : ℝ) : Prop :=\n 0 < n ∧ ∀ m : ℕ, 0 < m → m < n →\n DivisorPowerSum m x < DivisorPowerSum n x\n\ndef claim : Prop :=\n ∀ n : ℕ, 0 < n →\n ((∃ x : ℝ, x ≤ 0 ∧ StrictDivisorPowerRecord n x) ↔\n (StrictDivisorPowerRecord n 0 ∨\n ∃ B : ℝ, B < 0 ∧ ∀ x : ℝ, x ≤ B →\n StrictDivisorPowerRecord n x))\n\ntheorem result : ¬ claim\n```\n\nThe last line is a type display; the proof remains solely in the Lean module.\nEvery positive divisor is summed, and every smaller positive natural is\ncompared strictly at one common real exponent. The empty predecessor case\nat 1 is allowed; zero candidates are excluded. The right-hand eventual\nquantifier ranges over all real exponents below one negative real threshold,\nfollowing A095848's NAME, “all sufficiently low (i.e., negative) values of k,”\nread in the real-parameter setting of A396596. It is not a test of finitely\nmany exponents or an assumed lexicographic characterization.","title":"A396596 divisor-power union: a counterexample","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396596-divisor-power-union/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/546333b6cdeb798cb162cce6290a40b0812d3d8b0a55de7e3158a25276f93afc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396659-edge-collision-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every n at least two and every fixed starting vertex v on the complete\ndirected graph on n labeled vertices, including loops, count length-k walks\nthat visit every vertex, have distinct first k-1 directed edges, and repeat\none of these edges at the final step. With T(n,k) denoting this count, prove\nor refute\n\n$$\\frac{T(n,n^2)}{T(n,n^2+1)}=\\frac{2n-1}{n}.$$\n\nThe denominator must be positive. The sequences retain a distinguished start\nand are not identified under rotation. The source is OEIS A396659 revision 16.","title":"The A396659 universal edge-collision ratio","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396659-edge-collision-ratio/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22b56c8c62127ada5978ba5c8403cbdcd3173af1399791e671dd205187ada065.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396793-iterate-product-nine-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396793, Paul D. Hanna, Jun 07 2026, states:\n\nNAME:\n> G.f. A(x) satisfies A(x) * A(A(x)) = x^2 + 9*x^3.\n\nCOMMENT:\n> Conjecture: a(n) == 0 (mod 3) for n > 1.\n\nThese verbatim quotations are recorded in `Library/Recurrence/hanna2026a396793.md`.\nHere A is an integer formal power series normalized by A(0) = 0 and a(1) = 1,\nand a(n) is its coefficient of x^n.","title":"Divisibility of the A396793 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396793-iterate-product-nine-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd5334ea96830ddbe580e9a6da22d2e6ba1b79c30e61d54da38d5165e684f803.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396794-triple-iterate-product-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396794, Paul D. Hanna, Jun 07 2026. The following exact NAME and\nCOMMENT quotations are recorded in `Library/Arith/hanna2026a396794.md`.\n\nNAME:\n\n> G.f. A(x) satisfies A(x) * A(A(A(x))) = x^2 + 16*x^3.\n\nCOMMENT:\n\n> Conjecture: a(n) == 0 (mod 4) for n > 1.\n\nThe formal target uses the normalized integer series with constant coefficient\nzero and degree-one coefficient one; `a(n)` is its degree-n coefficient.","title":"Divisibility of the A396794 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396794-triple-iterate-product-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e485ce45d1b0a1ca30649979e493fc26aa88856a8b65bd1c4d32f822137292b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396795, Paul D. Hanna, Jun 06 2026, gives the following NAME and COMMENT,\nquoted verbatim from `Library/Recurrence/hanna2026a396795.md`:\n\n> G.f. A(x) satisfies A^2(x) * A^3(x) = x^2 + 25*x^3, where A^n(x) is the n-th iteration of A(x).\n\n> Conjecture: a(n) == 0 (mod 5) for n > 1.","title":"Divisibility of the A396795 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396795-iterate-product-twenty-five-mod-five/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c35e0ec7492fa83bbabad2e5afab30305047f8fb163e9c8ece084ddc3a4c4cf9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396797-three-four-iterate-product-mod-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396797, Paul D. Hanna, Jun 15 2026. The following NAME and COMMENT\nquotes are supplied by the orchestrator and recorded in\n`Library/Arith/hanna2026a396797.md`:\n\nNAME:\n\n> G.f. satisfies A(x) = x + A^3(x)*A^4(x).\n\nHere `A^n(x)` denotes the n-th compositional iteration, not an ordinary power.\n\nCOMMENT:\n\n> Conjecture: a(n) = 1 (mod 6) for n >= 1.\n\n### 范围与诚实边界\n\nThe resolution concerns the first conjecture for every positive index of the\nunique zero-constant integer formal power series satisfying the NAME equation.\nThe second comment, asserting `[x^n] A^k(x) ≡ k^(n−1) (mod 6)` for all k,\nis not resolved here. Existence and uniqueness identify the constructed series\nwith the defining equation; no coefficient congruence is assumed.","title":"Coefficients of the A396797 three-four iterate product series modulo six","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396797-three-four-iterate-product-mod-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff68bc8089b34aedcf9fe1d3e42b5b834b05f0947eefc88e0cf18fcb4a08adee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396798-eighth-iterate-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396798, revision 12, defines an ordinary generating\nseries by \"G.f. satisfies A(x) = x + A^4(x)*A^5(x).\" Its formula field says\nthat A^k denotes the k-th iteration. The eighth comment states:\n\n> Conjecture: [x^n] A^8(x) == 0 (mod 8) for n > 1.\n\nLet I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant\ninteger series satisfying A=X+I4(A)*I5(A), prove\n`forall n:Nat, 1 (8:Int) divides coeff n (I8(A))`.\nThe product is ordinary multiplication and the coefficients have no factorial\nnormalization. The source object is constructed, not assumed to exist.","title":"The eighth compositional iterate of A396798 modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17df59e3fb4f4db60b063d72e39503ed5a1e2e069e23aab99be020598af73791.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396798-fifth-iterate-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396798, revision 12, defines the ordinary generating\nseries by \"G.f. satisfies A(x) = x + A^4(x)*A^5(x).\" Its formula field says\nthat A^k denotes the k-th iteration. The fifth comment states:\n\n> Conjecture: [x^n] A^5(x) == [5,1,1,5] repeating (mod 8) for n > 1.\n\nLet I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant\ninteger series satisfying A=X+I4(A)*I5(A), the exact target is\n`forall n:Nat, 1 8 divides coeff n (I5(A)) - r(n)`, where\n`r(n)=if (n-2)%4=0 or (n-2)%4=3 then 5 else 1`.\nThe four-term period begins at n=2. Multiplication is ordinary series\nmultiplication, with no factorial normalization or convergence assumption.\nThe source is the existing\n`D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries`.","title":"The fifth compositional iterate of A396798 modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d254eeba96f878901f92b808cbfacab58f635336c23cc8d2865d192b77246fce.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396798-fourth-iterate-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396798, revision 12, defines the ordinary generating\nseries by \"G.f. satisfies A(x) = x + A^4(x)*A^5(x).\" Its formula field says\nthat A^k denotes the k-th iteration. The fourth comment states:\n\n> Conjecture: [x^n] A^4(x) == 0 (mod 8) for n > 2.\n\nLet I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant\ninteger series satisfying A=X+I4(A)*I5(A), prove\n`forall n:Nat, 2 (8:Int) divides coeff n (I4(A))`.\nMultiplication is ordinary multiplication of formal series. There is no\nfactorial normalization or analytic convergence hypothesis.\n\nThe source is the existing\n`D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries`.\nThe fourth-iterate theorem uses precisely this definition.","title":"The fourth compositional iterate of A396798 modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fbf6b9e1380cefdb639db076d6f6ba9f0a5ecc8fa74d0235961c267b297abaaf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396798-seventh-iterate-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396798, revision 12, defines the ordinary generating\nseries by \"G.f. satisfies A(x) = x + A^4(x)*A^5(x).\" Its formula field says:\n\n> G.f. A(x) = Sum_{n>=1} a(n)*x^n has the following properties, where A^n(x) denotes the n-th iteration of A(x).\n\nThe seventh comment states exactly:\n\n> Conjecture: [x^n] A^7(x) == [7,1,3,5] repeating (mod 8) for n > 1.\n\nLet I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant\ninteger ordinary series satisfying A=X+I4(A)*I5(A), the result is:\n\n```lean\nforall n : Nat, 1 < n ->\n (8 : Int) ∣ PowerSeries.coeff n\n (CompositionalIterateCongruence.iterate\n IterateProductFourFiveEighthModEight.generatingSeries 7) -\n (if (n - 2) % 4 = 0 then (7 : Int)\n else if (n - 2) % 4 = 1 then 1\n else if (n - 2) % 4 = 2 then 3 else 5)\n```\n\nEvery natural n>1 is included, with no upper bound or extra premise.\nResidues 0,1,2,3 of (n-2)%4 select 7,1,3,5, starting at n=2. A^7 is\ncompositional iteration; products are ordinary multiplication, without\nfactorial normalization or convergence premises. The existing source is\n`D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries`,\ndeclaration identity\n`sha256:ec9535ab3a9b0a1e907e9e393ae6d5cccfb485ff6209d712252354e4035bdfbc`.","title":"The seventh compositional iterate of A396798 modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/62ef2fb641c9b3403df70747b3a57d9d28c7d03c82530425daa0c05d21b79a68.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396798-third-iterate-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396798, revision 12, defines the ordinary generating\nseries by \"G.f. satisfies A(x) = x + A^4(x)*A^5(x).\" Its formula field says\nthat A^k denotes the k-th iteration. The third comment states:\n\n> Conjecture: [x^n] A^3(x) == [3,1,7,5] repeating (mod 8) for n > 1.\n\nLet I0(A)=X and I(k+1)(A)=Ik(A) composed with A. For the unique zero-constant\ninteger series satisfying A=X+I4(A)*I5(A), the target is\n`forall n:Nat, 1 8 divides coeff n (I3(A)) - r(n)`, where\n`r(n)=if (n-2)%4=0 then 3 else if (n-2)%4=1 then 1 else if (n-2)%4=2 then 7 else 5`.\nThe period starts at n=2. Multiplication is ordinary series multiplication,\nwith no factorial normalization or convergence assumption. The source is\n`D5/S1/Recurrence/Residue/IterateProductFourFiveEighthModEight.generatingSeries`.","title":"The third compositional iterate of A396798 modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2b347ca0b2010e26a2d64cd28155cd4420865db6c9fa5fc1a05208b9659741b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396803-iterate-exponential-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396803 entry, dated June 8, 2026, states:\n\n```text\n%N A396803 E.g.f. satisfies A(x) = x*exp( A^3(x) ).\n%C A396803 Conjecture: a(n) == [1,2,0] repeating (mod 3) for n >= 1.\n```\n\nThe formula field defines `A^3` as the third compositional iterate.\nFor the unique zero-constant rational formal solution of\n`A=X*exp(A composed with A composed with A)`, put `a(n)=n![X^n]A`.\nThe assertion is `forall n >= 1, a(n) = n (mod 3)`.","title":"A396803 coefficients modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f05dd7287c3dbe6ab20143a3684ed71994ca32f9e61bc253d4f8053a06f8a597.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396803-iterate-exponential-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and every conjecture COMMENT, quoted verbatim from the\nentry snapshot:\n\n```text\n%N A396803 E.g.f. satisfies A(x) = x*exp( A^3(x) ).\n%C A396803 Conjecture: a(n) is odd iff n is odd for n >= 1.\n%C A396803 Conjecture: a(n) == [1,2,0] repeating (mod 3) for n >= 1.\n%A A396803 Paul D. Hanna, Jun 08 2026\n```\n\nIn this entry `A^3(x)` denotes the 3-fold COMPOSITIONAL ITERATE of `A`, not\nthe 3-th power; the entry's formula field states that convention explicitly.\nThe offset is 1 and the first terms are 1, 2, 21, 472, 17165, 885696, 60160723.\n\nThis dossier concerns the parity conjecture. The modulo-three conjecture\nis proved separately by `IterateExponentialModThree.result`; see\n[the modulo-three dossier](oeis-a396803-iterate-exponential-mod-three.md).\nIt is not a theorem of the parity module.","title":"Parity of the A396803 iterated-exponential coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b0ad6191e00f15c52da6b3dd51a0c489eae024edfae483fb5cd06a58e4935e1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396804-quartic-egf-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396804 (2026-06-09) defines the exponential generating\nfunction by `A(x)=x exp(A^4(x))`, explicitly specifying that superscripts mean\ncompositional iteration. Its independent modulo-four conjecture is:\n`a(n) == [1,2,3,0] repeating (mod 4) for n >= 1`.\n\nPrecisely: A is the unique rational formal power series with constant term\nzero and `A=X*(exp Rat).subst(iterateComp A 4)`, where the zeroth iterate is X.\nThe independently established natural sequence satisfies\n`(a(n):Rat)=n!*[x^n]A`; the target is `Nat.ModEq 4 (a n) n` for every `n≥1`.\nThe old, refuted modulo-three pattern is outside this dossier.","title":"OEIS A396804: the fourth-iterate EGF modulo four","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396804-quartic-egf-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6187e1b2b69f88c0d28467b571cd47df39ae53cb747f469dbc85a23a4cd5f3f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396805-iterate-exponential-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396805, dated June 9, 2026, states:\n\n> E.g.f. satisfies A(x) = x*exp( A^5(x) ).\n> Conjecture: a(n) == [0,1,0] repeating (mod 3) for n >= 3.\n\nThe formula field defines A^5 as the fifth compositional iterate and specifies\nA(x)=Sum_{n>=1} a(n)*x^n/n!. For the unique zero-constant rational formal\nsolution A=X*exp(A^{[5]}), the exact conclusion is\nforall n : Nat, 3 <= n -> Nat.ModEq 3 (IterateExponentialParity.aK 5 n)\n(if n % 3 = 1 then 1 else 0).\nThe residues 0,1,0 begin at n=3. The source value a(2)=2 excludes extension\nof this formula to all positive indices.","title":"A396805 coefficients modulo three from index three","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa447d9ad60f1ea8b8a0079a54e1cd587056acdbb6ae4bdfd23cd75f36e3281e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396805-iterate-exponential-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and every conjecture COMMENT, quoted verbatim from the\nentry snapshot:\n\n```text\n%N A396805 E.g.f. satisfies A(x) = x*exp( A^5(x) ).\n%C A396805 Conjecture: a(n) is odd iff n is odd for n >= 1.\n%C A396805 Conjecture: a(n) == [0,1,0] repeating (mod 3) for n >= 3.\n%C A396805 Conjecture: a(n) == [1,2,3,4,0] repeating (mod 5) for n >= 1.\n%A A396805 Paul D. Hanna, Jun 09 2026\n```\n\nIn this entry `A^5(x)` denotes the 5-fold COMPOSITIONAL ITERATE of `A`, not\nthe 5-th power; the entry's formula field states that convention explicitly.\nThe offset is 1 and the first terms are 1, 2, 33, 1264, 80505, 7365456, 893006317.\n\nThis dossier settles the parity conjecture. The separate\n[modulo-three dossier](oeis-a396805-iterate-exponential-mod-three.md) binds\n`IterateExponentialFiveModThree.result`, which proves the pattern from n=3.\nThe modulo-five assertion has the non-kernel counterexamples recorded under\nTriage; the parity theorem does not formalize that refutation.","title":"Parity of the A396805 iterated-exponential coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8c391bd5ca549e80411e9c63eb4854eb468c5320bfed4328efda031ed95a89a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396806-iterate-exponential-mod-six","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A396806 entry, dated June 9, 2026, states:\n\n> E.g.f. satisfies A(x) = x*exp( A^6(x) ).\n> Conjecture: a(n) == [1,2,0] repeating (mod 3) for n >= 1.\n> Conjecture: a(n) == [1,2,3,4,5,0] repeating (mod 6) for n >= 1.\n\nThe formula field defines A^6 as the sixth compositional iterate and gives\nA(x)=Sum_{n>=1} a(n)*x^n/n!. For the unique zero-constant rational formal\nsolution A=X*exp(A^{[6]}), the assertion is\nforall n : Nat, 1 <= n -> Nat.ModEq 6 (IterateExponentialParity.aK 6 n) n.\nIt entails the quoted modulo-three clause because three divides six.","title":"A396806 coefficients modulo six","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-mod-six/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/258da425969bb13a2381947a54a1c1223f5aab4b7f0d2214d999031744727be3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396806-iterate-exponential-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and every conjecture COMMENT, quoted verbatim from the\nentry snapshot:\n\n```text\n%N A396806 E.g.f. satisfies A(x) = x*exp( A^6(x) ).\n%C A396806 Conjecture: a(n) is odd iff n is odd for n >= 1.\n%C A396806 Conjecture: a(n) == [1,2,0] repeating (mod 3) for n >= 1.\n%C A396806 Conjecture: a(n) == [1,2,3,4,5,0] repeating (mod 6) for n >= 1.\n%A A396806 Paul D. Hanna, Jun 09 2026\n```\n\nIn this entry `A^6(x)` denotes the 6-fold COMPOSITIONAL ITERATE of `A`, not\nthe 6-th power; the entry's formula field states that convention explicitly.\nThe offset is 1 and the first terms are 1, 2, 39, 1804, 139625, 15563526, 2301954109.\n\nThis dossier settles the parity conjecture through parity_iterate_six.\nThe independent result IterateExponentialModSix.result settles the\nmodulo-six assertion and consequently the modulo-three assertion; see\nProblems/oeis-a396806-iterate-exponential-mod-six.md. These residue\nconclusions are not additional statements of the parity module.","title":"Parity of the A396806 iterated-exponential coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecb071c0bb6c2ad87edb126418785396dc39d2ce295a8035ebad34dc9b864f39.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396807-compositional-iterate-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"NAME \"G.f. satisfies A(x) = x + A^5(x)*A^6(x).\" where A^n(x) denotes the n-th iteration (compositional power) of A(x).\n\nCOMMENT \"Conjecture: a(n) = 1 (mod 10) for n >= 1.\" — Paul D. Hanna, Jun 16 2026.","title":"Coefficient congruence for OEIS A396807","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396807-compositional-iterate-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/35c50005892f47a2005dcb014ec581b9a697a0970aabd6f4a3be749801bdccaf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396843-compositional-square-all-odd-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396843 defines `A(x) = Sum_{n>=1} a(n)*x^n` by\n\n> G.f. A(x) satisfies A( x*A(x) - 3*x*A(x)^2 ) = x^2.\n\nPaul D. Hanna's comment of July 1, 2026 states, verbatim:\n\n> Conjecture: all terms are odd.","title":"OEIS A396843: all coefficients are odd","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396843-compositional-square-all-odd-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/806417c27a319baaf1fd83130dbe954166342d50424a748aaf7ffdfde04bf4d3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396844-scaled-quadratic-reversion-mod-sixteen","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A396844, Paul D. Hanna, Jul 03 2026, gives the following NAME and\nCOMMENT, quoted verbatim from `Library/Recurrence/hanna2026a396844.md`:\n\n> G.f. A(x) satisfies A( x*A(x) - 4*x*A(x)^2 ) = x^2.\n\n> Conjecture: for n > 8, a(n) == 12 (mod 16) iff n is of the form 2*4^k+1 otherwise a(n) is divisible by 16.\n\nThe conjecture is refuted at n = 17. The orchestrator reports\n`a(17) = 113101684573952804`, congruent to 4 modulo 16. Since 17 is not\nof the form `2*4^k+1`, the \"otherwise divisible by 16\" clause is false.\nThe supplied numeric-check locator is `results/verify-a396844-mod16.out`;\nthat file was not available at this worktree path during Stage B.","title":"Refutation of the A396844 mod-sixteen conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396844-scaled-quadratic-reversion-mod-sixteen/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/709a8fa0b717b3fd9ceb9be5b93345ed7f1085c9aeb07cf20f9f3b878b2fc6ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a396846-logarithmic-quarter-normalization-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna, OEIS A396846, Jun 24 2026. The following NAME and COMMENT\nare copied verbatim from the quotation in `Library/Recurrence/hanna2026a396846.md`:\n\n> L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 4*n/(4*n^2-1) * a(n)*x^n ).\n\n> Conjecture: a(n) == [1,7,5,7] repeating (mod 8) for n >= 1.\n\nThis dossier resolves only the mod-8 conjecture. The entry's first conjecture,\n\"a(n) == 1 (mod 3) iff n is a power of 3, otherwise a(n) is divisible by 3\",\nis not resolved here.","title":"The A396846 logarithmic coefficients modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396846-logarithmic-quarter-normalization-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/111ca1756621a0aa8b589df5fe8185e94ccf1246e1e7b6bed30bdd0d9c8adaab.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397241-diagonal-power-ratio-all-odd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397241, Paul D. Hanna, Jun 19 2026:\n\n> G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1.\n\nThe entry seeds are `a(0) = a(1) = 1`. Its first conjecture is:\n\n> Conjecture: all terms are odd.\n\nThese NAME and COMMENT sentences are reproduced verbatim from the claim in\n`Library/Recurrence/hanna2026a397241b.md`. The two further entry comments,\n\"Conjecture: a(n) == 2 (mod 3) for n > 2.\" and\n\"Conjecture: a(n) == 3 (mod 4) for n > 2.\", are not resolved by this dossier.\nThis dossier records one claim, on `hanna_conjecture` only.","title":"Oddness of all A397241 terms","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-diagonal-power-ratio-all-odd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3dd11435aec34150be99dd49e8df823c43459daa87e07278e3e10b835435f091.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397241-mod-four","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397241 (Paul D. Hanna, Jun 19 2026) records the following NAME:\n\n> G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1.\n\nThe COMMENT targeted by this dossier is:\n\n> Conjecture: a(n) == 3 (mod 4) for n > 2.\n\nThe entry's third COMMENT, \"Conjecture: all terms are odd\", was already proved\nand frozen in this repository as\n`D5/S1/Recurrence/Residue/DiagonalPowerRatioAllOdd.hanna_conjecture`.\nIt is NOT re-claimed here. This dossier claims only the modulo-4 conjecture.","title":"OEIS A397241 coefficients modulo 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-mod-four/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/486fa01b25dd39dc08fedb8dc2fdaae174bd573eb631ab691b3d17f22314e16b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397241-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397241 (Paul D. Hanna, Jun 19 2026) records the following NAME:\n\n> G.f. A(x) satisfies n * [x^n] A(x)^n = (n-1) * [x^n] A(x)^(n+1) for n > 1.\n\nThe COMMENT targeted by this dossier is:\n\n> Conjecture: a(n) == 2 (mod 3) for n > 2.\n\nThe entry's third COMMENT, \"Conjecture: all terms are odd\", was already proved\nand frozen in this repository as\n`D5/S1/Recurrence/Residue/DiagonalPowerRatioAllOdd.hanna_conjecture`.\nIt is NOT re-claimed here. This dossier claims only the modulo-3 conjecture.","title":"OEIS A397241 coefficients modulo 3","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6820ba94e26a060f5d0d6655b53e8bf0e9b567d4a0c2d99781f7089120dfd81.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397242-exponential-square-weight-catalan-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397242, Paul D. Hanna, Jun 19 2026, gives the following NAME and first\nCOMMENT, quoted from `Library/Recurrence/hanna2026a397242b.md`:\n\n> O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)*x^n / n^2 ).\n\n> Conjecture: a(n) is odd iff n+1 is a power of 2.\n\nThe formal reading is `A(0)=1` and `X*A' = M*A`, where\n`M = X + sum_{n>=2} (n^2-1) a(n) X^n / n`. The integral normalization\n`d(n) = a(n)/n` for positive n realizes these weights. The entry's two other\ncomments conjecture classifications modulo 3; neither is resolved here.\nThis dossier makes one resolution claim, on `hanna_conjecture` alone.","title":"Parity of the A397242 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-exponential-square-weight-catalan-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9ffc92f01535e3c1832c57d9b7770b4981096482a2f8f084d30637fd072cb0a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397242-mod-three-powers","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397242 (Paul D. Hanna, Jun 19 2026), NAME:\n\n> O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)*x^n / n^2 ).\n\nCOMMENT (the sole resolution target of this dossier):\n\n> Conjecture: for n > 0, a(n) == 1 (mod 3) iff n+2 is of the form 3^k or 2*3^k.\n\nThe entry's parity COMMENT was already proved and frozen here as\n`D5/S1/Recurrence/Residue/ExponentialSquareWeightCatalanParity.hanna_conjecture`.\nIt is NOT re-claimed. This module imports that frozen module and reuses its\nsequence rather than redefining it. The other mod-three COMMENT has its own dossier.","title":"The residue-one conjecture of OEIS A397242","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-powers/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7153910d14817930742a6be28a4d6de2fcc297023e8d3096708a7c51c8ee86c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397242-mod-three-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397242 (Paul D. Hanna, Jun 19 2026), NAME:\n\n> O.g.f. satisfies A(x) = exp( x + Sum_{n>=2} (n^2-1) * a(n)*x^n / n^2 ).\n\nCOMMENT (the sole resolution target of this dossier):\n\n> Conjecture: for n > 0, a(n) == 2 (mod 3) iff n+2 is a sum of 2 distinct powers of 3 (A038464).\n\nThe entry's parity COMMENT was already proved and frozen here as\n`D5/S1/Recurrence/Residue/ExponentialSquareWeightCatalanParity.hanna_conjecture`.\nIt is NOT re-claimed. This module imports that frozen module and reuses its\nsequence rather than redefining it. The other mod-three COMMENT has its own dossier.","title":"The residue-two conjecture of OEIS A397242","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad6ccd8d3adc49adc8404fea0f09cddc750b808dea7d11b9b995314ce5b46fa6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A397244, Paul D. Hanna, gives the following NAME and Fibbinary COMMENT,\nquoted verbatim from `Library/ArithSums/hanna2026a397244.md` and matched to\n`oeis-A397244.src` supplied by the orchestrator:\n\n> G.f. A(x) satisfies (2*n) * [x^n] A(x)^(2*n) = (2*n-1) * [x^n] A(x)^(2*n+1) for n > 1.\n\n> Conjecture: for n > 0, a(n) is odd iff n = 2*A003714(k) + 1 for some k >= 0, where A003714 lists the Fibbinary numbers.\n\nThe initial values are a(0)=a(1)=1. A003714 enumerates the nonnegative\nintegers with no adjacent binary ones; the frozen public predicate is\n`Fibbinary f := f &&& (f >>> 1) = 0`.","title":"Fibbinary parity of the A397244 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b7b7f7d7580cf2a93071007340cc1860e2944ce6fdd816c8588e707a40d7a69.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397315-three-row-blackout","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Arjun Pemmasani's Three-Row conjecture states that the maximum number of\npoints that can be blacked out in a three-row grid with m columns, while\nevery lattice rectangle remains identifiable, is m+2 for every m>=3.\nThe exact source definitions and complete target are:\n\n```lean\ndef IsRectangle {m : ℕ} (C : Finset (Fin m × Fin 3)) : Prop :=\n C.card = 4 ∧\n ∃ a b c d : Fin m × Fin 3,\n C = {a, b, c, d} ∧\n (((b.1.val : ℤ) - (a.1.val : ℤ)) *\n ((c.1.val : ℤ) - (a.1.val : ℤ)) +\n ((b.2.val : ℤ) - (a.2.val : ℤ)) *\n ((c.2.val : ℤ) - (a.2.val : ℤ)) = 0) ∧\n (a.1.val : ℤ) + (d.1.val : ℤ) =\n (b.1.val : ℤ) + (c.1.val : ℤ) ∧\n (a.2.val : ℤ) + (d.2.val : ℤ) =\n (b.2.val : ℤ) + (c.2.val : ℤ)\n\ndef ValidBlackout {m : ℕ} (S : Finset (Fin m × Fin 3)) : Prop :=\n ∀ C D : Finset (Fin m × Fin 3),\n IsRectangle C → IsRectangle D → C \\ S = D \\ S → C = D\n\ntheorem result :\n ∀ m : ℕ, 3 ≤ m →\n (∃ S : Finset (Fin m × Fin 3),\n ValidBlackout S ∧ S.card = m + 2) ∧\n (∀ S : Finset (Fin m × Fin 3),\n ValidBlackout S → S.card ≤ m + 2)\n```\n\nThe first coordinate is column and the second row. Translating both\ncoordinates by one identifies this grid with the source convention.\nThe parallelogram equations and perpendicular adjacent vectors describe\nEuclidean rectangles; four distinct corners exclude degeneracy. Squares\nand tilted rectangles are included. Rectangle identity is equality of\nunordered corner sets. Empty presentations are allowed. The result gives\nboth an attaining valid blackout and the matching bound on every valid\nblackout, for every natural width at least three.","title":"OEIS A397315: the Three-Row blackout conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1e82b1a33766f1474c2ff28c9e939215b66b06298d5373fe59992a3a8b0ca78.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397345-square-weight-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A397345 NAME is the formal equation\nA(x)=exp(x+Sum_{n>=2}(5*n^2-1)*a(n)*x^n/(5*n^2)).\nIts first COMMENT is the single target of this dossier:\n\n> Conjecture: a(n) is odd iff n+1 is a power of 2.","title":"A397345: square-weight congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397345-square-weight-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f64f826b52af286bc134ea2a7a817b2a0f3462214300013e5f5cdb4c67e8a177.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397346-square-weight-mod-eight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A397346 NAME is the formal equation\nA(x)=exp(x+Sum_{n>=2}(2*n^2-1)*a(n)*x^n/(2*n^2)).\nIts first COMMENT is the single target of this dossier:\n\n> Conjecture: a(n) == [4,2,0,2] repeating (mod 8) for n >= 2.","title":"A397346: square-weight congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397346-square-weight-mod-eight/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0d0e19cba549e548b9bb1567d42648cd7b573f921a428766de1cb32b21c4dfc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397348-square-weight-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A397348 NAME is the formal equation\nA(x)=exp(x+Sum_{n>=2}(3*n^2-1)*a(n)*x^n/(3*n^2)).\nIts first COMMENT is the single target of this dossier:\n\n> Conjecture: a(n) is odd iff n+1 is a power of 2.","title":"A397348: square-weight congruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397348-square-weight-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3459bac0e258fdfdeb443d74931ad6d71024ebdec2a2ccc74471b0d09f321ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397349-logarithmic-weight-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and BOTH conjecture COMMENT lines are quoted verbatim\nfrom the orchestrator-supplied `oeis-A397349.src`:\n\n```text\n%N A397349 L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 3*n/(3*n^2 - 1) * a(n)*x^n ).\n%C A397349 Conjecture: a(n) is odd iff n is a power of 2.\n%C A397349 Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.\n```\n\nA397349 carries two conjectures. This lane settles ONLY the modulo-three\nalternation for every positive index. The separate assertion that a(n) is\nodd exactly at powers of two remains open and is not claimed as proved.\nThe module retains it solely as the unproved definition `parity_conjecture : Prop`.","title":"Alternation modulo three of the A397349 coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397349-logarithmic-weight-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b956c457182a5b22f20a258250012f9c408d77f3d57304e7c2fd09e9cf00a6b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397349-logarithmic-weight-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and both conjecture COMMENT lines, quoted verbatim:\n\n```text\n%N A397349 L.g.f. Sum_{n>=1} a(n)*x^n/n = log(1+x + Sum_{n>=2} 3*n/(3*n^2 - 1) * a(n)*x^n ).\n%C A397349 Conjecture: a(n) is odd iff n is a power of 2.\n%C A397349 Conjecture: a(n) = [1,2] repeating (mod 3) for n >= 1.\n```\n\nA397349 carries two conjectures. The modulo-three alternation is already settled\nin this repository by `LogarithmicWeightCatalanParity.hanna_conjecture_a397349_mod_three`.\n**This lane settles the other one, the parity clause**, which that module deliberately\nleft as the bare proposition `parity_conjecture` carrying no proof.","title":"Parity of the A397349 logarithmic-weight coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397349-logarithmic-weight-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a463913f6413adbd28bb1f1444598e9e3ae8b36d2fd7652b9c8ee2f86753a1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-mod-three","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and BOTH conjecture COMMENT lines, quoted verbatim from the entry\nsnapshot `oeis-A397356.src`:\n\n```text\n%N A397356 G.f. A(x) satisfies: [x^n] 1/A(x)^(n^2) = [x^n] 1/A(x)^(n^2-1) for n > 1 with A(0) = A'(0) = 1.\n%C A397356 Conjecture: for n >= 0, a(n) is odd iff n+1 is a power of 2.\n%C A397356 Conjecture: for n >= 0, a(n) is not divisible by 3 iff 2*(n+1) is a sum of two powers of 3 (A055235).\n%A A397356 Paul D. Hanna, Jul 03 2026\n```\n\nA397356 carries two conjectures. The parity classification was settled earlier in\n`D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalParity`. This lane settles the\nsecond one, the characterisation of the indices at which three does not divide `a n`.","title":"Ternary support of the A397356 reciprocal square-exponent diagonals","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-mod-three/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b8cb52c0be5682938550795a0b8074701eb442bde5007dbd1f609c29993dd1c7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's NAME and BOTH conjecture COMMENT lines, quoted verbatim from\nthe entry snapshot `oeis-A397356.src`:\n\n```text\n%N A397356 G.f. A(x) satisfies: [x^n] 1/A(x)^(n^2) = [x^n] 1/A(x)^(n^2-1) for n > 1 with A(0) = A'(0) = 1.\n%C A397356 Conjecture: for n >= 0, a(n) is odd iff n+1 is a power of 2.\n%C A397356 Conjecture: for n >= 0, a(n) is not divisible by 3 iff 2*(n+1) is a sum of two powers of 3 (A055235).\n%A A397356 Paul D. Hanna, Jul 03 2026\n```\n\nA397356 carries two conjectures. This lane settles ONLY the parity\nclassification; the assertion about divisibility by three is not claimed\nhere and appears nowhere in this module as a theorem or as a definition.\nThat second conjecture is settled separately in\n`D5/S1/Recurrence/Residue/ReciprocalSquareExponentDiagonalModThree`.","title":"Parity of the A397356 reciprocal square-exponent diagonals","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/506ab169f53aa8fb1c7d6f369e06cbecc820cfcf8b182d2d4eeb73319a7410f6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397436-general-supercongruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The selected external assertion is OEIS A397436 revision 44, Conjecture 3,\nquoted in Library/Certificates/oeis2026a397436.md. It asserts the following\ncomplete same-r event, with arbitrary signed m and inclusive endpoints:\n\n```lean\nnoncomputable def U {r : ℕ} (m : Fin r → ℤ) (M n : ℕ) : ℤ :=\n ∑ k ∈ Finset.range (M * n + 1), ∏ i : Fin r, Ring.choose (m i * (n : ℤ)) k\n\ndef claim : Prop :=\n ∀ (r : ℕ), 0 < r → ∀ (m : Fin r → ℤ) (M : ℕ), 0 < M →\n ∀ (p : ℕ), Nat.Prime p → 5 ≤ p →\n (p : ℤ) ^ (3 * r) ∣ U m M (p ^ r) - U m M (p ^ (r - 1))\n\ntheorem result : ¬ claim\n```\n\nThe last line displays the theorem type; its proof lives in D5. Natural\nsubtraction r-1 agrees with the source exponent because r is positive.","title":"A397436: the unrestricted product supercongruence","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397436-general-supercongruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1b86514f92433728c9f09574087ef407750f9e45f927da59a16d9fef4e7bfbf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397578-sigma-neighbour-peaks","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ratushnyak's OEIS A397578 defines a(n) as the least k satisfying sigma(k) > n sigma(k-1) and sigma(k) > n sigma(k+1), and states: “Conjecture: a(n) exists for all n.” The exact assertion is preregistered in [#14990](https://github.com/the-omega-institute/trureturing/issues/14990).\n\nWriting $\\sigma(k)=\\sum_{d\\mid k}d$, the closed claim is\n\n$$\\forall n\\in\\mathbb N,\\ \\exists k\\in\\mathbb N,\\quad k\\ge2\\ \\land\\ n\\sigma(k-1)<\\sigma(k)\\ \\land\\ n\\sigma(k+1)<\\sigma(k).$$\n\nThe natural domain includes zero. Existence of a witness entails existence of a least witness by well-ordering.","title":"oeis a397578 sigma neighbour peaks","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397578-sigma-neighbour-peaks/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc596e317ac5e33883bc819940ca02cbe397eb9ab1ba476742550cafdf436cd7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a397592-dyadic-parity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Paul D. Hanna's OEIS A397592 (July 10, 2026), revision 18, has NAME:\n\n> G.f. A(x) such that the sum of the first n coefficients in A(x/n)^n equals 2^(n-1) for n >= 1.\n\nIts sole conjecture is:\n\n> Conjecture: for n > 3, a(n) is odd iff n is of the form 2^k-1 or 2^k+1 (k > 1).\n\nThe FORMULA states A(x)=Sum_{n>=0} a(n)*x^n; the original offset is 0\nand a(0)=1. For A in Z[[X]], A_Q is its coefficientwise image in Q[[X]],\nnamely A.map (Int.castRingHom Rat). The exact source and single result are:\n\n```lean\ndef SourceA397592 (A : PowerSeries Int) : Prop :=\n PowerSeries.constantCoeff A = 1 ∧\n ∀ m : Nat, 0 < m →\n (Finset.range m).sum (fun j => PowerSeries.coeff j\n ((PowerSeries.rescale ((m : Rat)⁻¹)\n (A.map (Int.castRingHom Rat))) ^ m)) = (2 : Rat) ^ (m - 1)\n\ntheorem result :\n (∃! A : PowerSeries Int, SourceA397592 A) ∧\n (∀ A : PowerSeries Int, SourceA397592 A → ∀ n : Nat, 3 < n →\n (Odd (PowerSeries.coeff n A) ↔\n ∃ k : Nat, 1 < k ∧ (n = 2 ^ k - 1 ∨ n = 2 ^ k + 1)))\n```\n\nThese are ordinary formal powers, not compositional iterates; there is no\nconvergence or factorial normalization. The sum includes j=0,...,m-1 for\nevery natural m>0. The separate integer-existence-and-uniqueness conjunct\nexcludes an unproved integrality premise.","title":"The integer source and dyadic parity of A397592","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b4decc860933c76b5e434d898908727c51ac5594bcf632d6175e85fe74c9375.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a398542-fixed-bottom-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Charles Cornell Norton, OEIS A398542, August 1, 2026; revision 18,\nAugust 30, 2026, 18:45:04, exact selected %F:\n\n> Conjecture: for every m >= 1 and every b in S(m), d(b,k) = p(k)*binomial(2*k,k) + q(k)*4^k for polynomials p and q depending on b, with deg p <= m-1 and deg q <= m-2. Verified for m <= 6 and k <= 12.\n\nS(m)=Av_m(132). For every m>=1 and b in S(m), choose p_b,q_b in Q[X]\nindependently of k such that the displayed identity holds for every k>=0.\nThe degree bounds are m-1 and m-2 for m>=2, with q_b=0 when m=1.\nHere d(b,k) counts actual permutations of 1,...,m+k avoiding 1324 whose\nlower-value subsequence is exactly b and whose standardized upper-value\nsubsequence avoids 213. The value cut is fixed, the empty upper cell is\nallowed, and d(b,0)=1. Rational coefficients are an explicit sufficient\nchoice beyond the source's unspecified coefficient field.","title":"A398542 fixed-bottom polynomial conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398542-fixed-bottom-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8b98ca88bc0741cf1a62d2dc1a9936d9f94855e63ef3d18804ae9d940d46c0ea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a398589-all-row-eventual-periodicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: All of the rows in this sequence are eventually periodic.\n\nFor every k in Nat, let a_k(0)=k. At time t>0 choose the least natural x>=k\nsuch that every prior occurrence a_k(s)=x, s0 such that\nfor every t>=N, a_k(t+p)=a_k(t). The row is its infinite continuation;\nOEIS display truncation is not a premise.","title":"Eventual periodicity of every A398589 row","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398589-all-row-eventual-periodicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/197d69061027ef9670a860c9b7d378ad2f92bebefa39020b0205ba6d93de0763.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a398690-palindromic","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A398690 defines `A(r,q)` by its signed finite sine sum. For every\n`n >= 0`, let `F_n(z) = sum_{q>=0} A(n+3,q) z^q`. Its comment says that\n`F_n` has denominator `(1-z)^(n+1)` and conjectures that its numerator\nis palindromic. The formal result proves that `(1-z)^(n+1) F_n(z)` is a\npolynomial of degree exactly `n` and that its coefficients at `j` and\n`n-j` agree for every `0 <= j <= n`.","title":"Palindromic numerators of OEIS A398690","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398690-palindromic/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd612476f6018985beabaf7bad5bd0af47003b6f3f8124cfc81b5c36ebf0e0dd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a398726-distinct-cycle-size-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vaclav Kotesovec's August 14, 2026 FORMULA line in OEIS A398726 states:\n\n> Conjecture: a(n) ~ n * n!.\n\nThe sequence, entered by Alois P. Heinz on August 8, 2026, is the sum of the\ndistinct cycle sizes over all permutations of [n]. The exact source and full\nstatement, preregistered in issue #8059, are:\n\n```lean\nnoncomputable def DistinctCycleSizes {n : ℕ}\n (σ : Equiv.Perm (Fin n)) : Finset ℕ :=\n σ.partition.parts.toFinset\n\nnoncomputable def CycleSizeSum (n : ℕ) : ℕ :=\n ∑ σ : Equiv.Perm (Fin n), ∑ k ∈ DistinctCycleSizes σ, k\n\ntheorem result :\n Filter.Tendsto\n (fun n : ℕ => (CycleSizeSum n : ℝ) /\n ((n : ℝ) * (Nat.factorial n : ℝ)))\n Filter.atTop (nhds (1 : ℝ))\n```\n\nPartition parts include fixed points as lengths one; the finite set removes\nmultiplicities. Each distinct length contributes its size. Both sums range\nover their entire finite domains. The empty permutation contributes zero.\nFin n is a relabeling of [n], and the real normalization is positive for n>0.\nNo sequence, counting identity or asymptotic estimate is an assumed premise.","title":"OEIS A398726: the distinct cycle size limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/21be83d628a9426ac3d270b8ca5975b43f127d5bdd5cfd607d61196f2b1fd2a6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a399155-value-finiteness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dilkhush's OEIS A399155 defines $a(n)=f(n)-g(n)$ using the smallest- and largest-prime-factor subtraction step counts. Its empirical observation is that each positive value is attained only finitely often. The exact assertion is preregistered in [#14993](https://github.com/the-omega-institute/trureturing/issues/14993): for every integer $v>0$, the set $\\{n\\in\\mathbb N: n\\ge2\\text{ and }a(n)=v\\}$ is finite.\n\nThe two walk definitions and their integer-valued difference are defined together in the finiteness module, reusing `CenteredReducedResidueProgressions.GreatestPrimeFactor`.","title":"oeis a399155 value finiteness","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399155-value-finiteness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a81ef7d99600728efc6ac9cc25eeab0ae9d0c61b640b8aa37fbf4b5923d45e5d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a399232-radical-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rui Ferreira's first COMMENT in OEIS A399232 is:\n\n> a(n) appears to satisfy rad(2*n+1) = (2*n+1)^2 / ((2*n+1)^2 - a(n)).\n\nIts NAME is:\n\n> a(n) = Sum_{k=1..4*n+2} (k^(4*n+1) mod (2*n+1)).\n\nOFFSET `1,1` gives the domain `n >= 1`. The literal finite sum uses least\nnonnegative natural remainders. The radical is the product of distinct\nprime divisors. The full intended statement proves, for every such `n`,\nstrict positivity of the signed rational deficit `D=(2*n+1)^2-a(n)` and\nthe rational equality `rad(2*n+1)=(2*n+1)^2/D`. Natural truncated\nsubtraction and integer division are not the meaning of the quotient.\nThere is no squarefree or coprime restriction.\n\nThe immutable source is revision 23, Sep 01 2026, of the official export\nat `2ce625d68f85e35e26e15c0d9681f69280c095e7`. Its author line is\n`_Rui Ferreira_, Aug 23 2026`. Source quotations and adapted description\nare attributed to Rui Ferreira and the OEIS Foundation Inc., under\n[CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/).\nSource provenance is distinct from the repository proof.","title":"OEIS A399232 radical identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399232-radical-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a7d0f72e1f878dc365148189a9fab5828ce494defb20e5d66e6ea99988a71620.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a399755-pell-class-successor","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The single external named target is Chai Wah Wu's A399755 conjecture,\nattributed in the entry to September 10, 2026:\n\n> Conjecture: A399491(k) < 4*k^3+3*k if and only if k is a term of this sequence.\n\nThe A399755 name is:\n\n> Numbers k such that the generalized Pell equation x^2-(k^2+1)*(y^2+1) = 0 has more than one fundamental solution in (x, y).\n\nThe companion A399491 name is:\n\n> The smallest integer m > n such that (n^2+1)*(m^2+1) is a perfect square.\n\nFor every natural k, define D = Discriminant(k) = (k : Int)^2 + 1.\nAll generalized Pell coordinates and all unit coordinates below are integers.\nThe exact definitions and sole target are:\n\n```text\nDiscriminant(k : Nat) : Int = (k : Int)^2 + 1\nSol(k : Nat, x y : Int) := x^2 - Discriminant(k)*y^2 = Discriminant(k)\nSameClass(k : Nat, x y r s : Int) :=\n exists u v : Int,\n u^2 - Discriminant(k)*v^2 = 1 AND\n x = r*u + Discriminant(k)*s*v AND y = r*v + s*u\nMultipleClasses(k : Nat) :=\n exists x y r s : Int,\n Sol(k,x,y) AND Sol(k,r,s) AND NOT SameClass(k,x,y,r,s)\nNextSquarePartner(k m : Nat) :=\n k < m AND IsSquare((k^2+1)*(m^2+1)) AND\n forall n : Nat,\n k < n -> IsSquare((k^2+1)*(n^2+1)) -> m <= n\nresult : forall k : Nat, 0 < k ->\n exists m : Nat, NextSquarePartner(k,m) AND\n (m < 4*k^3+3*k IFF MultipleClasses(k))\n```\n\nIsSquare here is the predicate on natural numbers. The existential result\nincludes totality and the full minimum property, not only a bound on an\nassumed partner. Membership refers to k itself, not to the kth value listed\nin A399755. The source convention selects one fundamental representative\nper generalized solution class; it is distinct from the unique fundamental\npositive solution of the norm-one equation.\n\nSameClass is Robertson's integral norm-one relation, including the unit -1\nand hence simultaneous negation of both coordinates. It does not impose\npositive coordinates, primitivity, or squarefree D, and it does not use\nfractional units of a maximal order. Any representative-based formulation\nmust be proved equivalent to this predicate on arbitrary integral pairs.","title":"A399755: generalized Pell classes and the least square partner","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8bcca171ba153681f23abdb47e1f908b0f0ef95f4dfbdd94896da5815e562a36.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400168-primorial-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Antti Karttunen, OEIS A400168, revision 8, September 23, 2026:\n\n> Question: Are there terms greater than 1?\n\nThe NAME is `a(n) = A235224(A003415(n)) - A400164(n)`, with offset one.\nLet $P_k$ be the product of the first $k$ primes and $P_0=1$.\nSet $L(0)=0$ and, for $m>0$, let $L(m)$ be the largest $k\\ge1$ with\n$P_{k-1}\\le m$, equivalently the least $k\\ge0$ with $mM(n)+1$.\n\nThe closed `claim` is the universal negative answer\n$\\forall n\\ge1,\\ L(D(n))\\le M(n)+1$.\nKarttunen asked the existence question; he did not assert this bound.\n`result : Not claim` answers the original question positively.","title":"A400168 has a term greater than one","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/861b4090711f8322f8a4f34344a50cbe8ba2f2e0058dbfa081b32dccd3006b81.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400194-two-dense-divisor-block-maximum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"> Conjecture: a(n) = A001511(n) * A400195(n).\n\n— Omar E. Pol, OEIS A400194 revision 18.\n\nFor every positive natural n, split its complete strictly increasing list of\npositive divisors into maximal contiguous blocks, cutting precisely at\nb>2a. Let M(n) be the maximum block length and O(n) the maximum odd count\nover the SAME actual block family. The exact assertion is\nM(n)=(v₂(n)+1)O(n). Equality b=2a and singleton blocks are retained.","title":"Maximum length of a two-dense divisor block","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69023806389eaee56921556c92eb7865fd0d4ef7fdaf587dba724a8fd40d0397.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400216-a400218-tetrahelix-generating-functions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A400216, A400217 and A400218, Peter Kagey (2026), conjecture the\ncoordinate generating functions of one Boerdijk-Coxeter tetrahelix. The\nordered initial vertices are (-1,-1,-1), (-1,1,1), (1,-1,1), (1,1,-1).\nAt each step the oldest vertex is reflected across the face through the next\nthree. The coordinates are scaled by 1 through index 3 and by 3^(n-3)\nthereafter. In the order x, y, z, the proposed series are\n\n- (54t^6-84t^5-66t^4+23t^3+9t^2+t-1)/((3t-1)^2(9t^2+4t+1));\n- (-54t^6+84t^5-90t^4+15t^3+3t^2+3t-1)/((3t-1)^2(9t^2+4t+1));\n- (18t^5+26t^4-24t^3-5t^2+1)/(27t^3+3t^2-t-1).\n\nThe denominators are units in rational formal power series, with respective\nconstant coefficients 1, 1 and -1. Preregistration is issue #10182.","title":"Three tetrahelix coordinate generating functions","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400216-a400218-tetrahelix-generating-functions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a7d0fc442b80f45f0f7639b42817b6c2e1d09b615d5c01ef3b78cb6fa0b71e78.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400361-second-order-pillai-floor-gcd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Write P(n)=sum_{k=1}^n gcd(n,k), F(n)=sum_{k=1}^n gcd(n,floor(n/k)),\nand a(n)=P(n)-F(n), using signed subtraction. The COMMENT conjecture in\nrevision 17 asks, along all positive integers N, for\n\n$$\n\\sum_{n=1}^{N}a(n)=\\frac{3}{\\pi^2}N^2\\log N+CN^2+o(N^2),\n$$\n\nwith the literal constant\n\n$$\nC=\\frac{3}{\\pi^2}\\left(-\\frac12+2\\gamma-\n\\frac{6\\zeta'(2)}{\\pi^2}\\right)-\\frac{\\pi^2}{12}\n-\\frac12\\sum_{d\\ge1}\\frac{\\varphi(d)\n(\\psi(1+1/d)+\\gamma)}{d^2}.\n$$\n\nThe full literal source and attribution are in\n[oeis2026a400361](../Library/Asymptotics/oeis2026a400361.md).\nThe preregistration is [issue 18775](https://github.com/the-omega-institute/trureturing/issues/18775).\nThe target is first tier, an external OEIS conjecture. Its recent source\nversion does not imply novelty; the bounded literature scope and remaining\nBu-Xu full-text uncertainty are recorded in the source dossier.\n\n### Native statement\n\n`D5.S3.ArithSums.A400361SecondTerm.originalClaim : Prop` conjoins absolute\nsummability of the literal real coefficient sequence with little-o of the\nexact remainder against N^2 at the natural atTop filter. The theorem\n`D5.S3.ArithSums.A400361SecondTerm.result : originalClaim` has no hypotheses.\nThe definitions preserve the positive source range, integer subtraction,\nreal Gamma logarithmic derivative, real zeta restriction, and every term\nof the source constant. Normalizing the remainder by N^2 is proved\nequivalent to the stated little-o. No subsequence or finite proxy is used.","title":"OEIS A400361: the exact second-order cumulative formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400361-second-order-pillai-floor-gcd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b403b4c91a2a2635151d5f3d77dcfffc0524d7fb14a9a3ee6b08bd6ff45f154d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400429-semi-meander-second-diagonal","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A400429 revision 16 states verbatim: \"Conjecture: D_2(n) =\n(n^2 + 2*n + n mod 2 - 20)/2, for n >= 4.\" Its diagonal is\n`D_j(n)=T(n,floor(n/2)-j+1)`, where `T(n,k)` counts one-loop semi-meanders\nwith `n` crossings and winding `2*(k-1)+(n mod 2)`. The source's equivalent\narch model fixes a lower rainbow pairing of `2*n` vertices and counts\nnoncrossing upper perfect matchings whose union with it is one loop. The\nsecond diagonal has exactly `n-4` upper arches crossing the midpoint.","title":"OEIS A400429 second semi-meander diagonal","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400429-semi-meander-second-diagonal/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/35df0c80afe286d4e2b1e394e6b092810cacf50c8a7e8d9675588a6d04605919.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400585-odd-cofactor-asymptotic","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A400585, revision 22, defines\n\n> a(n) = Sum_{d|n, n/d odd} binomial(d, 2).\n\nIts FORMULA field states:\n\n> Conjecture: Sum_{k=1..n} a(k) ~ 7*zeta(3)*n^3/48. - _Vaclav Kotesovec_, Oct 01 2026\n\nThe delivered statement keeps the source's full natural range. It defines the\nnatural divisor sum `a`, the real positive-prefix sum `S`, and the real series\n`Z3 = sum_{q >= 0} 1/(q+1)^3`, then proves\n\n```lean\nTendsto (fun N : ℕ => S N / ((7 * Z3 / 48) * (N : ℝ)^3))\n atTop (𝓝 1)\n```\n\nThere is no parity, primality, or finite-range restriction. The natural value\nat zero is harmless because the limit is at `atTop`.","title":"OEIS A400585: the odd-cofactor summatory asymptotic","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400585-odd-cofactor-asymptotic/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b20f3d59a24895609a5a4f52bf15138a49155dafd57c554cc906674ad8dc347.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oeis-a400644-rounded-three-term-norm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A400644 revision 28 has the name:\n\n> a(n+1) = round(sqrt(a(n)^2 + a(n-1)^2 + a(n-2)^2)).\n\nIts conjectural comment is:\n\n> Numerical evidence suggests that the ratio of consecutive terms (a(n+1)/a(n)) tends to 1.356203065626295... which is the largest positive real root of x^6-x^4-x^2-1=0. Setting x^2=t gives t^3-t^2-t-1 which yields the tribonacci constant (A058265) as the largest positive root.\n\nThe Mathematica source initializes `a(1)=a(2)=0`, `a(3)=1` and uses\n`Round[Sqrt[a[n]^2+a[n-1]^2+a[n-2]^2]]`. The Python source starts\n`[0,0,1]`, computes `s` as the sum of the last three squares, puts\n`r=isqrt(s)`, and appends `r+(s>r*r+r)`.\n\nThe target quantifies over the unique sequence with values in the\nnonnegative integers on every positive index and over every recurrence\nindex `n>=3`. It asks for the full infinite consecutive-ratio limit.","title":"OEIS A400644 rounded three-term norm ratio","url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400644-rounded-three-term-norm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a97ebfec461fd43f669fb11900b6f0643dc8e02fc6cdedc44af53362f06aa361.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:oh-scrimshaw-2018-two-shifted-q-motzkin-hankel-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Oh and Scrimshaw consider Cigler's `q`-Motzkin numbers\n\n> M̃†_{n+1}(q) = M̃†_n(q) + Σ_{k=0}^{n−1} q^{k+1} M̃†_k(q) M̃†_{n−k−1}(q),\n> M̃†_0(q) = 1,\n\nand state, based on numerical computations, the first conjecture labelled\n`conj:factored_motzkin_2shifted` in their appendix on other `q`-determinants:\n\n> Define f_n(q) := Σ_{1 ≤ k ≤ n, k ≢ 1 mod 3} q^k if n ≡ 0 mod 3, and\n> (q+1)(Σ_{k=0}^{⌊n/3⌋} q^{3k}) otherwise. Then we have\n> (det[M̃†_{i+j+2}(q)]_{i,j=0}^{n−1})_{n=1}^∞ = (q^{c_n} f_n(q))_{n=1}^∞\n> for some c_n ∈ ℤ_{≥0}.\n\nIssue #10046 fixes the readings: the polynomials and the determinant live in\n`ℤ[q]`; \"for some `c_n`\" means that for each `n ≥ 1` some natural number\n`c_n` makes the polynomial identity hold; the second conjecture carrying the\nsame label, on the three-shifted determinants, is not addressed.","title":"Oh and Scrimshaw's two-shifted q-Motzkin Hankel conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/oh-scrimshaw-2018-two-shifted-q-motzkin-hankel-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24c9f52ccdee58b504411366ac9d0ea32409555a12f0e6216d3ea7fee5be54c6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:okada-2020-premerged-determinant-nonpalindromic","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"So Okada, *Merged-log-concavity of rational functions, almost strictly unimodal sequences, and phase transitions of ideal boson-fermion gases*, arXiv:2003.02112v2 (2020 preprint, revised 2024), states the following in the subsection “On the almost palindromicity”, conjecture labelled `conj:palindromic-width-three`, `phases.tex`, lines 8770–8779:\n\n> Let $\\mu\\in\\Z_{\\geq 3}$. Then, $\\dt(U)_{a}^{b}(\\mu,\\iota^{\\mu}(1),\\iota^{\\mu}(1), 0, \\iota^{\\mu}(q))$ is not an almost palindromic $q$-polynomial, provided $\\tp{\\mu,a,b}$ is almost strictly pre-fitting.\n\nThe quotation retains the source macros: `\\Z` denotes $\\mathbb Z$, `\\dt` denotes $\\tilde d$, and `\\tp` prints a tuple. The source defines almost strictly pre-fitting by $0\\le a_1<\\cdots There are no perfect square that are term of the sequence with length of convergence equal to 3.\n\nSection 1.2 defines `F(n)` to be the square of the product of the nonzero\ndecimal digits of `n`. A positive integer is a sequence member when some\nfinite iterate reaches one. With `F^[0](n) = n`, exact convergence length\nthree means `F^[3](n) = 1` and `F^[k](n) != 1` for every natural `k < 3`.\n\nThe closed proposition `claim` is exactly:\n\n```text\n∀ n : ℕ, 0 < n → (∃ r : ℕ, n = r ^ 2) →\n SequenceMember n → ¬ FirstHit n 3\n```\n\n`SequenceMember n` is `∃ t : ℕ, step^[t] n = 1`, and `FirstHit n t` is\n`step^[t] n = 1 ∧ ∀ k : ℕ, k < t → step^[k] n ≠ 1`. The square root is\nan arbitrary natural number. There is no smoothness, image, digit-alphabet,\nroot-bound or exponent-variation condition.","title":"Refutation of Onnis's square convergence prohibition","url":"https://the-omega-institute.github.io/trureturing-pages/research/onnis-2022-square-three-step-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7afdfd21b4f866a006d8b65f7ae4a188b3a1fcbefa5359f6d787fbf99d781c22.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ordered-zeckendorf-long-game-strategy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A state is an ordered list of Fibonacci numbers. The game starts with `n` copies\nof `F_1`. Legal adjacent moves are `(F_i, F_{i+1}) -> F_{i+2}`;\n`(F_1, F_1) -> F_2`; `(F_i, F_i) -> (F_{i-2}, F_{i+1})` for `i > 2`;\n`(F_2, F_2) -> (F_1, F_3)`; and switching an inversion `(F_i, F_j) -> (F_j, F_i)`\nwhen `i > j`.\n\nThe Long Game Strategy uses the priority: all switch moves in any order, combine\nadjacent ones from the left, split from the right, then merge from the left.\n\nConjecture 1.7, quoted from arXiv:2508.20222v2:\n\n> “The LGS has the longest game length.”\n\nStatus statement, quoted from the same version:\n\n> “The longest-game strategy described in Conjecture 1.7 is supported by\n> empirical simulations. However, a rigorous proof establishing its optimality\n> remains an open problem.”\n\nFormal statement, with the five moves represented exactly:\n\n```text\n∀ n > 0, ∃ g, CompleteLGSRun n g\n∀ n > 0, ∀ g, CompleteLGSRun n g ->\n ∀ h, LegalTerminalRun n h -> h.length ≤ g.length\n```\n\nEvery permitted switch ordering is retained in the proved universal comparison.\nPriorities restart after every move, including a switch.\n\nThe paper says the conjecture is backed by exhaustive simulations rather than a\nproof. In its broader exact game-tree analysis it also states:\n\n> “Due to the combinatorial explosion in the number of game states and legal\n> move sequences, a full resolution of the winner for \\(n>25\\) was\n> computationally infeasible.”\n\nThat second quote concerns winner computation, not directly Conjecture 1.7, but\nit identifies the same state-space obstacle. For length, the paper proves only\nupper and lower asymptotics and a structural lemma about repetitions under LGS.","title":"Optimality of the Ordered Zeckendorf Long Game Strategy","url":"https://the-omega-institute.github.io/trureturing-pages/research/ordered-zeckendorf-long-game-strategy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f69dbe32b5dcacfc7f667dad6ac4d401595b6e8443965812cca93588102d1829.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:orthey-augusiak-2025-steering-power-preservation-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. C. Orthey Jr and R. Augusiak, *Certifying classes of $d$-outcome measurements with quantum steering*, New J. Phys. 27, 064501 (2025), arXiv:2410.20477v2, Appendix “Robust self-testing”, Conjecture 1:\n\n> For any prime $d$, we have that $\\widetilde{B}_k^{(n)}=\\widetilde{B}_k^n$, for every $n\\in\\{1,\\ldots,d-1\\}$. Note that this statement is true for $d=3$.\n\nThis quotation is from the downloaded v2 source `main2.tex`, line 1272, MD5 `b3c5b40a34f721352981dcf58659087e`. Its projective setting is explicit at line 1263:\n\n> For simplicity, we will assume that the state is pure and the measurements performed by Bob are projective.\n\nWrite $\\omega=e^{2\\pi\\mathrm i/d}$ and let $\\oplus$ denote addition modulo $d$. The coefficients and substitution of lines 204–206 and 246–249 are\n\n$\\gamma_{yk}^{(n)}=\\frac1d\\omega^{kn(n-1)/2}\\sum_{l=0}^{d-1}e^{-\\mathrm i\\sum_{m=1}^n\\phi_{l\\oplus m,y}}\\omega^{knl}$,\n\n$\\widetilde B_k^{(n)}=\\sum_y\\gamma_{yk}^{(n)}B_y^n$, with $\\widetilde B_k=\\widetilde B_k^{(1)}$ in the projective setting of lines 293–305.\n\nIssue [#21069](https://github.com/the-omega-institute/trureturing/issues/21069) fixes the following reading. For every prime $d$, every real phase array $\\phi$ satisfying $\\sum_j\\phi_{j,y}=0$ for every $y$ and\n\n$\\sum_k\\overline{\\gamma_{y'k}^{(n)}}\\gamma_{yk}^{(n)}=\\delta_{yy'}$\n\nfor every $1\\le n\\le d-1$ and every $y,y'$, every family of unitary $d\\times d$ complex matrices $B_y$ with $B_y^d=I$, every $k\\in\\{0,\\ldots,d-1\\}$ and every $1\\le n\\le d-1$, the identity $\\widetilde B_k^{(n)}=(\\widetilde B_k^{(1)})^n$ holds. This is `claim`; `result` refutes it. The definitions and source locators are also given in [the literature note](../Library/QuantumContext/ortheyaugusiak2025steering.md).","title":"The steering substitution need not preserve powers","url":"https://the-omega-institute.github.io/trureturing-pages/research/orthey-augusiak-2025-steering-power-preservation-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3db29cdd14afd07fc4ed9309a610fd807a88376c3d0bc6ac6359d02481a7e97.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:orthey-philip-kondra-streltsov-2026-four-copy-concentration-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. C. Orthey, A. Philip, T. V. Kondra and A. Streltsov, *Optimality of universal conclusive entanglement concentration protocols*, Phys. Rev. A 113, 062404 (2026), arXiv:2509.09423v2, study LOCC protocols that turn copies of an unknown two-qubit pure state into the Bell state ϕ+.\n- **Universality.** Definition 1: an n-to-1 LOCC map E1 is a universal conclusive protocol if ⟨ϕ+|E1(ρ⊗n)|ϕ+⟩ = Tr E1(ρ⊗n) for every input in the set, here every pure state |ψ⟩ = c1|00⟩ + c2|01⟩ + c3|10⟩ + c4|11⟩ (Eq. (1)).\n- **The bound.** Theorem 3 bounds the four-copy success probability of concatenated protocols based on two-qubit operations by B(c) = 2|c2⁴c3⁴| + 2|c1⁴c4⁴| − 4Re[c1²c4²(c2*)²(c3*)²] (Eq. (8)).\n- **The conjecture.** §VIII: \"protocols that process four copies directly without intermediate Schmidt-state conversion warrant investigation. We conjecture that such protocols cannot surpass the success probability bound in Eq. (8)\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/ortheyphilipkondrastreltsov2026optimality.md).\n\nIssue [#17404](https://github.com/the-omega-institute/trureturing/issues/17404) fixes the reading: for every universal conclusive LOCC map on four copies and every normalized ψ, Tr E1(ψ⊗4) ≤ B(c).\n\nThe conjecture is refuted.","title":"Direct four-copy universal entanglement concentration beats the bound of Eq. (8)","url":"https://the-omega-institute.github.io/trureturing-pages/research/orthey-philip-kondra-streltsov-2026-four-copy-concentration-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/586928a727410feb194fd33104f272ac789e3a589ee76d1b778e15dd8dcca3ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ostrovskii-shcherbakov-conjecture-62-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Ostrovskii and Shcherbakov, *Amplitude maximization in stable systems, Schur\npositivity, and some conjectures on polynomial interpolation*, arXiv:2508.13554v2\n[math.CV], state on page 15:\n\n> For all $0 \\le k < n \\le t$ and self-conjugate $z_{1:n} \\in \\mathbb{D}^n$,\n> $|Q_{t,n,k}(z_{1:n} + 1_n)| \\le 1$. Moreover, if the grid additionally\n> satisfies $\\Re(z_{1:n}) \\in \\mathbb{R}^n_+$, then\n> $|s_{(t-n|n-k-1)}(z_{1:n})| \\le \\binom{t}{n}\n> |e_{n-k}(z_{1:n})|$.\n\nThe same paper labels this Conjecture 6.2 equivalent to Conjecture 6.1.\nThe formal claim retains both clauses, the self-conjugacy conditions, the closed\nunit disk, and the natural inequalities k **Conjecture.** Similarly to the classical Sudoku there are no $4\\times4$ SudoQ with 3 clues that are uniquely solvable.\n\nG. Rajchel-Mieldzioć, *Quantum mappings and designs*, arXiv:2204.13008,\nChapter 7, restates it:\n\n> Uniquely solvable grids of $4 \\times 4$ SudoQ admit at least 4 clues.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| A quantum Latin square has orthonormal rows and columns. | `SudoQClueGeometry.IsSudoQ` | Four vectors in $\\mathbb C^4$ in each row and column. |\n| Every disjoint block is an orthonormal basis. | `IsSudoQ` and `blockCell` | Four $2\\times2$ blocks; zero-based coordinates $(2b_1+k_1,2b_2+k_2)$. |\n| A quantum grid has blank entries; its clues are the nonblank entries. | `C` and `v` in `IsSolution` | A finite set of cells with literal prescribed vectors; values outside `C` impose no constraint. |\n| Solvability includes existence of a completion. | `UniquelySolvable` | Its existential quantifier supplies an `IsSolution`. |\n| Solutions agree up to phase. | `SameRays` | A separate complex scalar of norm one for each entry. Clues remain literal vector equalities. |\n| No uniquely solvable grid with three clues. | `SudoQMinimumClues.claim` and `result` | At most three clues, hence also exactly three; arbitrary clue values, including unsolvable grids. |\n\n`RayEq` and `HasRepeatedRay` are auxiliary predicates for a pair of vectors\nand for one ray occurring once in every unit. They add no hypothesis to the\nsettled claim. The four-dimensional specialization is the source's $N=2$\nSudoQ, not the $N=4$ Sudoku parameter.","title":"Four clues are necessary for unique four by four SudoQ solvability","url":"https://the-omega-institute.github.io/trureturing-pages/research/paczos-et-al-2021-sudoq-three-clues/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44b369e3ed9c95074ba835ae4efb7283ed1cc3d29619efb082cdd16dcd6381a9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pahuja-2026-hankel-minimal-matrices","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nimisha Pahuja, *Minimal Inversions in Integer Matrices of Fixed RSK Shape*,\narXiv:2602.14931v1, Conjecture 1.1, asserts that every minimal nonnegative\ninteger $n\\times n$ matrix whose shape is a partition with exactly $n$\npositive parts is symmetric and Hankel. Minimality compares the inversion\ncount $I(M)=\\sum_{il}m_{i,j}m_{k,l}$ against all matrices of that shape.\nThe Hankel clause asks for integers $s_2,\\ldots,s_{2n}$ with $m_{i,j}=s_{i+j}$.\nThe shape is that of the insertion tableau of the lexicographic matrix\nbiword, with the first strictly larger entry bumped during row insertion.","title":"Minimal matrices of fixed RSK shape need not be Hankel","url":"https://the-omega-institute.github.io/trureturing-pages/research/pahuja-2026-hankel-minimal-matrices/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4c97b49bc6ad406c1c140942de60ad9c75f9797fae654c3486fc8135a6c8291.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pal-toth-bene-vertesi-2021-second-family-all-dimensions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. F. Pál, G. Tóth, E. Bene and T. Vértesi, *Bound entangled singlet-like states for quantum metrology*, Phys. Rev. Research 3, 023101 (2021), arXiv:2002.12409v4, section “Second family of states and its quantum Fisher information,” state:\n\n> For this family of states we have constructed analytical examples only for $d=3$ and $2^n.$ However, we believe that such states exist for all $d$.\n\nThis quotation is from line 674 of the v4 TeX source `metroboundent53.tex`. Definition 2 and its accompanying requirements are at lines 556–640; the source's analytic constructions are in the appendix “Special properties of orthogonal matrices,” lines 1196–1308. See [the literature note](../Library/Estimation/paltothbenevertesi2021singlet.md).\n\nIssue [#25511](https://github.com/the-omega-institute/trureturing/issues/25511) fixes the question as existence of the data required by Definition 2 for every natural $d\\ge2$. In the module, `claim` asserts that there exist real $d\\times d$ matrices $Q^k$, real vectors $\\bar q_m\\in\\mathbb R^d$, labels $\\mu_{ij}\\in\\mathrm{Fin}\\,d$, real coefficients $a_{ij}$ and complex vectors $s_m\\in\\mathbb C^{d\\times d}$ satisfying all six conditions below. All indices range over $\\mathrm{Fin}\\,d$; the coefficient $a_{ij}$ is the source's $S_{ij}$.\n\nWrite $N_m=\\#\\{(i,j):a_{ij}\\ne0,\\ \\mu_{ij}=m\\}$, the module's `multiplicity`, and define the module's `tVec` by $t_m(i,j)=a_{ij}/\\sqrt d$ when $a_{ij}\\ne0$ and $\\mu_{ij}=m$, and $t_m(i,j)=0$ otherwise. The conditions in `IsSecondFamilyData` are:\n\n1. $(Q^k)^{\\mathsf T}Q^k=I_d$ for every $k$.\n2. $\\sum_k\\bar q_m(k)\\bar q_n(k)=\\delta_{mn}$ for every $m,n$.\n3. $Q^k_{ij}=a_{ij}\\bar q_{\\mu_{ij}}(k)$ for every $i,j,k$.\n4. $a_{ij}\\ne0$ implies $a_{ij}^2=d/N_{\\mu_{ij}}$ for every $i,j$.\n5. $\\sum_{i,j}\\overline{s_m(i,j)}s_n(i,j)=\\delta_{mn}$ for every $m,n$.\n6. $(\\sum_m s_ms_m^\\dagger)^\\Gamma=\\sum_m t_mt_m^\\dagger$, where $M^\\Gamma_{(i,j),(i',j')}=M_{(i,j'),(i',j)}$.\n\nThe last operation is `AbsolutePPT.Bipartite.partialTranspose`. The fixed $|01\\rangle_{AB}$ factor of the source's shield vectors is suppressed in these coordinates. The theorem is `result : claim`.","title":"Second-family singlet-like data exist in every dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/pal-toth-bene-vertesi-2021-second-family-all-dimensions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/637e25ea81480ea80bcbd5c9885ca096db803ace83185354d103b6393d687065.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pandey-parity-conjecture-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rohan Pandey, arXiv:2601.03293v1, Conjecture 4.1 (Parity Conjecture),\nPDF page 4:\n\n> For all integers n ≥ 2k+1, the independence polynomial I(GP(n,k),x) has only real roots if and only if k is even.\n\nDefinition 2.1 supplies `n >= 3` and `1 <= k < n/2`. Precisely, for every\n`n,k : Nat` with `3 <= n`, `1 <= k`, and `2*k < n`, the assertion is\n`(forall z : Complex, eval₂ (Int.castRingHom Complex) z\n(independencePolynomial (gp n k) univ) = 0 -> z.im = 0) <-> Even k`.\nThe polynomial is the existing integer independent-set generating\npolynomial on all vertices. No experimental-range restriction is added.","title":"Pandey's Parity Conjecture 4.1","url":"https://the-omega-institute.github.io/trureturing-pages/research/pandey-parity-conjecture-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ff3e5efdbfa94999b50c14e81ecec7f0a97749cfe7fe25fa8f776a30680ac55.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pandey-patra-2021-subgraph-core","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Dinesh Pandey and Kamal Lochan Patra, *Some new central parts of connected\ngraphs*, arXiv:2110.00738v1, Conjecture 3.25, page 9:\n\n> The subgraph core of a graph G is contained in a block of G.\n\nThe native domain is every finite simple connected undirected graph. The\nsubgraph number counts all actual connected subgraphs containing a vertex,\nincluding singleton subgraphs and distinct edge choices. The conclusion is one\nnative maximal no-cut block containing the entire set of maximizing vertices.\nSingleton, bridge-edge, empty-deletion, and unbounded-order cases remain in\nscope.","title":"Pandey–Patra subgraph core conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/pandey-patra-2021-subgraph-core/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec4af3baa54cd0660402c1237251dca854a5c3c4e9e87429d84e6c85bdec7f86.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:parisse-stirling-fibonacci-alternating-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Parisse, *Integers* **24** (2024), Article A70, Section 4, states two conjectures.\nThe first, after Equation (4.7):\n\n> Conjecture 1. For all ℓ ∈ N_0, we have\n>\n> ∑_{m=0}^{ℓ} (-1)^m m! S(ℓ+1, m+1) F_{2(m+1)} = (-1)^ℓ ∑_{m=0}^{ℓ} m! S(ℓ, m) F_{m+2} .\n\nThe second, after Equation (4.9):\n\n> Conjecture 2. For all ℓ ∈ N_0, we have\n>\n> ∑_{m=0}^{ℓ} (-1)^m m! S(ℓ+1, m+1) L_{2(m+1)} = (-1)^ℓ ∑_{m=0}^{ℓ} m! S(ℓ, m) L_{m+2} .\n\nHere `S(j, m)` is the Stirling number of the second kind, `F` the Fibonacci\nnumbers with `F_1 = F_2 = 1`, and `L` the Lucas numbers with `L_0 = 2`, `L_1 = 1`.\nBoth are the constant term of the article's own formula for the weighted sums\n`∑_{k=0}^{n} k^ℓ F_k` and `∑_{k=0}^{n} k^ℓ L_k`, rewritten through Equation (3.14),\n`c_{ℓ,m}(0) = (-1)^m m! S(ℓ+1, m+1)`.\n\nThe article's remarks fix the reading of the stacked bracket symbols. The\nright-hand side of Conjecture 1 is named there as A000557, whose recorded formula\nis `a(n) = ∑_{k=0..n} k! Stirling2(n, k) Fibonacci(k+2)`; the right-hand side of\nConjecture 2, times `-1`, is named as A263968, whose recorded formula is\n`a(n) = (-1)^{n+1} ∑_{k=0..n} k! Lucas(k+2) Stirling2(n, k)`. Both readings are\ntherefore Stirling numbers of the second kind, not binomial coefficients.","title":"The Alternating Stirling Weighted Sum of Even-Index Fibonacci and Lucas Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/639d4d0d7d5b5ef7cdee66a40a4bf9caebd4d3027782b66be0bcbcf87a086552.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:parry-string-attractor-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For k >= 2, let c be a length-k vector of natural coefficients with\nc(0) >= 1 and c(k-1) >= 1. Set d to c with the last entry decreased by\none and assume d is lexicographically at least every cyclic rotation,\nallowing equality. The original substitution sends a < k-1 to c(a)\nzeros followed by a+1, and sends k-1 to c(k-1) zeros.\n\nLet u be its fixed point starting with zero and U(n) the length of the\nn-th iterate of that singleton. Conjecture 42 states that for every m >= 1,\nthe prefix of length m has minimum attractor size i+1 when\n0 <= i <= k-2 and U(i) <= m < U(i+1), and size k when m >= U(k-1).\nThe minimum ranges over every position subset; each nonempty factor needs\nan equal occurrence wholly inside the same prefix crossing a selected\noriginal position.","title":"Original cyclic-morphism string-attractor minimum","url":"https://the-omega-institute.github.io/trureturing-pages/research/parry-string-attractor-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/afdb5be40df2b66f50fed4fc660dfd99b7fc4bd9d4b9c206aaf3af1c9f2ec418.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:partition-weighted-sum-reversal","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A362559 counts the integer partitions of `n` whose one-based weighted sum is divisible\nby `n`, and fixes the reading in its own comment:\n\n> The (one-based) weighted sum of a sequence (y_1,...,y_k) is Sum_{i=1..k} i*y_i. This is\n> also the sum of partial sums of the reverse.\n\nThe next comment line, at revision #18 of Apr 29 2023, is the question:\n\n> Conjecture: A partition of n has weighted sum divisible by n iff its reverse has weighted\n> sum divisible by n.\n\nThe identical line stands on A362560, which counts the complementary partitions, at\nrevision #6 of the same date. Both are by Gus Wiseman, Apr 24 2023, and both still carry\nthe word `Conjecture`.","title":"Reversal and the Weighted Sum of a Partition","url":"https://the-omega-institute.github.io/trureturing-pages/research/partition-weighted-sum-reversal/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/837de7caa2fd51d897d28f8768d0c57d21ad78bc397adb7bc76c781752afe23c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:parzygnat-2021-two-qubit-conditional-domain-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. J. Parzygnat, *Conditional Distributions for Quantum Systems* (QPL 2021, EPTCS 343, 1–13; arXiv:2102.01529), constructs quantum conditionals by categorical Bayesian inversion. For a density matrix ν on M_m ⊗ M_n with invertible marginals ρ = tr_B ν and σ = tr_A ν (Corollary 4.5), the conditional domains are:\n- B_ν = {B : tr_B(ν(1 ⊗ B)) commutes with ρ};\n- A_ν = {A : tr_A((A ⊗ 1)ν) commutes with σ} (Definition 5.5).\n\nThey are the largest operator systems on which the conditionals are positive (Lemma 5.8, Corollary 5.13).\n- After Example 5.10 the author asks whether the domains are always C*-subalgebras, and answers no with an unrecorded 9 × 9 density matrix of rank 2.\n- Footnote 13 adds: \"Also, I could not find a 4 × 4 density matrix ν for which the conditional domains are not C∗-subalgebras, and I suspect that this may always be the case. I hope to resolve this in future work.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/parzygnat2021conditionaldistributions.md).\n\nIssue [#18167](https://github.com/the-omega-institute/trureturing/issues/18167) fixes the reading: for every two-qubit density matrix with invertible marginals, both conditional domains are C*-subalgebras of M₂(ℂ). This suspicion is refuted.","title":"Two-qubit conditional domains need not be C*-subalgebras","url":"https://the-omega-institute.github.io/trureturing-pages/research/parzygnat-2021-two-qubit-conditional-domain-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5857617204d4fd9c739469240afb0b7fa912f1951cd6ca8f00dd7c1805a9cc3a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:parzygnat-takayanagi-taki-wei-2023-two-qubit-svd-entropy-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. J. Parzygnat, T. Takayanagi, Y. Taki and Z. Wei, *SVD entanglement entropy*, JHEP 12 (2023) 123, arXiv:2307.06531v2, define an entropy for a pair of pure states.\n- **The SVD entropy.** For |ψ¹⟩, |ψ²⟩ in H_A ⊗ H_B, the reduced transition matrix is τ_A = Tr_B(|ψ¹⟩⟨ψ²|)/⟨ψ²|ψ¹⟩, the SVD density matrix is ρ^{1|2}_A = √(τ_A†τ_A)/Tr √(τ_A†τ_A), and the SVD entanglement entropy is its von Neumann entropy (§2). It lies between 0 and log d_A.\n- **The open question.** \"In Figure 8, we provide numerical evidence suggesting that a non-trivial lower bound of min{S(ρ¹_A), S(ρ²_A)} might hold for qubits, fails in general for qudits … Rigorous proofs of these observations for qubits … are still lacking\" (§3.3). The authors prove it only when the second state is a local unitary image of the first (Eq. (3.37)), \"a restricted version of S(ρ^{1|2}_B) ?≥ min{S(ρ¹_B), S(ρ²_B)} (3.38)\".\n\nThe verbatim statements are in [the literature note](../Library/Entropy/parzygnat2023svdentropy.md).\n\nIssue [#16821](https://github.com/the-omega-institute/trureturing/issues/16821) fixes the reading: two-qubit pure states as coefficient matrices of Frobenius norm 1, the unnormalized transition Tr_B|ψ⟩⟨φ| (the factor 1/⟨φ|ψ⟩ cancels in the SVD density matrix, so orthogonal pairs with nonzero transition are included), and natural-logarithm entropies. The bound is proved for every such pair.","title":"For two qubits the SVD entanglement entropy is at least the smaller of the two entanglement entropies","url":"https://the-omega-institute.github.io/trureturing-pages/research/parzygnat-takayanagi-taki-wei-2023-two-qubit-svd-entropy-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/626219125fb366b5746634d0e237d8e8f0078aa8e2f1e9d63b98b33017d45eb5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pathak-2026-kicked-ising-negativity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Pathak, *Mixed-State Entanglement in a Minimal Model of Quantum Chaos*,\narXiv:2603.14292v1, Conjecture 1, p. 3, states:\n\n> 2𝓔(t) = I_{A:B}^{(α)}(t), hold for generic states at all times t.\n\nThe periodic kicked Ising chain has $J=π/4$, $b=-π/4$ and arbitrary real\nlongitudinal fields, with $U=\\exp(-iH_K)\\exp(-iH_I)$.\nIts initial state is the source's product of Bloch-angle qubit states.\n“Generic” excludes both the all-transverse class and the all-longitudinal class.\nThe target in #13296 is the exact finite-chain equality at $α=1/2$ for every\nchain length, field, generic product state, contiguous nonempty tripartition\nand integer time. Refuting this specialization refutes the equality asserted\nfor all Rényi orders; it does not refute each order separately.","title":"Pathak's generic-state negativity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/pathak-2026-kicked-ising-negativity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb8d856bf085291cd3e379e32cd541bcd6ffbc352dd94f8e48520f1c81ae17ef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pauwels-guhne-2026-commuting-encoding-reflection-class","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Pauwels and O. Gühne, *The Entanglement Content of Quantum Measurement Bases*, arXiv:2608.21185v1 (2026), Section VI, asks:\n\n> Which other encoding classes define natural state classes, and what do these sets look like?\n\nOne of the proposed classes is:\n\n> commuting families whose local factors are not Hermitian, the smallest step beyond our theorem\n\nThe paragraph states:\n\n> For none of these is a characterization known, although $M_4$ now proves that the last class is strictly larger than commuting reflections.\n\nThese quotations are from the downloaded source `main.tex`, lines 1539–1547, md5 `217f1dae7e17966b2c9016fa75ef756b`. In the last quotation, “the last class” means tables of local Clifford unitaries. The source's local-encoding definition is at lines 328–340, and its encoder-structures definition is at lines 357–368; see also [the literature note](../Library/QuantumStates/pauwelsguhne2026measurementbases.md).\n\nIssue [#20897](https://github.com/the-omega-institute/trureturing/issues/20897) fixes the following reading. For every $n\\ge1$, put $N=2^n$ and let $\\psi:\\{0,1\\}^n\\to\\mathbb C$ satisfy $\\sum_x|\\psi(x)|^2=1$. A commuting product encoding is a family of exactly $N$ operators $U_a=\\bigotimes_{j=1}^n U_{a,j}$, with every $U_{a,j}$ a unitary $2\\times2$ matrix, such that $U_aU_b=U_bU_a$ and $\\langle U_a\\psi,U_b\\psi\\rangle=\\delta_{ab}$ for every pair of labels. A commuting-reflection encoding additionally requires $U_{a,j}^\\dagger=U_{a,j}$ and $U_{a,j}^2=I$ for every label and site.\n\nThe claim is that $\\psi$ admits a commuting product encoding if and only if it admits a commuting-reflection encoding. This equality of state classes is proved.","title":"Commuting product encoders define exactly the commuting-reflection qubit state class","url":"https://the-omega-institute.github.io/trureturing-pages/research/pauwels-guhne-2026-commuting-encoding-reflection-class/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7362214a08e321895f0137a7e39343b3a5bda7e1b7f34a4e64ec1af96ea30f59.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pekala-online-majority-four-colours","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 11 of Pękała (2026) asks whether Algorithm can guarantee an online\nmajority edge-colouring with at most four colours for each of `n=5,6,7`\nvertices when Presenter must finish with minimum degree **exactly** two.\nPresenter chooses each next edge after seeing previous colours. Algorithm\nmust colour it immediately and cannot recolour old edges.","title":"Four colours in small online majority edge-colouring games","url":"https://the-omega-institute.github.io/trureturing-pages/research/pekala-online-majority-four-colours/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/092332c1bc25f4cb7e60c7c0cd1d47ff3a88703020f11f39e1752782a721e3c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:peled-teeni-carmi-cohen-2021-correlation-minor-norm-fnf","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Y. Peled, A. Te'eni, A. Carmi and E. Cohen, *Correlation minor norms, entanglement detection and discord*, arXiv:2005.12079v2, Scientific Reports 11 (2021) 2849, states in Theorem 3 that a separable state in filter normal form satisfies the $p=\\infty$ correlation-minor bound $\\mathcal{M}_{h,p=\\infty} \\leq \\frac{1}{\\sqrt{Dd}}\\left[\\frac{D-1}{D(h-1)}\\frac{d-1}{d(h-1)}\\right]^{(h-1)/2}$. The source then asks whether the filter-normal-form hypothesis can be removed. In the source TeX file `CMN_16_12_20_arXiv.tex`, line 281, it says:\n\n> As in Theorem \\ref{thm_bound_p_1}, we conjecture this theorem still holds without the assumption that $\\rho$ is in FNF.\n\nThe compiled Lean `claim` is the universal assertion over natural numbers $da$, $db$, and $h$ that, when $2 \\leq da$, $2 \\leq db$, $\\max(da,db)\\min(da,db) \\leq h^2$, and $h \\leq \\min(da,db)^2$, every separable matrix $\\rho$ on the product index space and every pair of `IsHermBasis` families satisfies $\\operatorname{cmnInf}(h,\\operatorname{corr}(\\rho,a,b))\\leq\\operatorname{bound}(da,db,h)$. Here `IsHermBasis`, `corr`, `compound`, `cmnInf`, and `bound` are the definitions in the module, so the formal assertion has no filter-normal-form assumption.","title":"The correlation-minor bound fails without filter normal form","url":"https://the-omega-institute.github.io/trureturing-pages/research/peled-teeni-carmi-cohen-2021-correlation-minor-norm-fnf/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0ff17ae43b393cd7be183689065b9888211575fad2daf0b3e52d82ca729cc0dc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pelekis-2018-circular-orientation-entropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pelekis, Le Matematiche 74(1), pp. 128–129, Section 4:\n“We conjecture that circular hypergraphs are such that H(X_{C_{n,r}}) ≥ H(X_H),\nfor all H ∈ D_{n,n,r}.” The preceding definition assumes $r\\ge3$ and $n>r$;\n$C_{n,r}$ has all $n$ cyclic paths of $r$ vertices as edges. The competitor\nclass consists of multi-hypergraphs and permits repeated edges and isolated\nvertices. Every edge independently selects one uniformly random vertex, and\n$X$ counts the selected vertices. The title's complementary zero-in-degree\ncount has the same entropy.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $r\\ge3$ and $n>r$ | `claim` quantifies over `r n`, with `3 ≤ r` and `r < n` | No additional hypothesis. |\n| All cyclic paths of size $r$ | `circular n r` maps each starting vertex to the image of natural residues $(i+j)\\bmod n$, $0\\le j0$ from the actual index `i : Fin n`; for $n=0$ there is no index. |\n| $D_{n,n,r}$ | `E : Fin n → Finset (Fin n)` with all cardinalities equal to `r` | Exactly $n$ labelled edges; multiplicities retained. |\n| Uniform independent orientations and Shannon entropy | Shared `X` and `entropyX r E` from OrientationEntropyBalance | Literal filter, denominator $(r:\\mathbb R)^n$, base-two logarithm and zero-probability convention. |\n| Circular entropy at least every competitor | `entropyX r E ≤ entropyX r (circular n r)` for every uniform `E` | No restriction of the competitor class. |\n| Refutation | `result : ¬ claim` | Unconditional negation; the numbered asymptotic Conjecture 4.1 is a separate statement. |","title":"Circular hypergraphs do not maximise orientation-count entropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/pelekis-2018-circular-orientation-entropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/13b5ea2d7b7494d21394c2dd89e19b9c05b6dbfc5abebcbadb79ed3e0e6603b0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pelekis-2018-orientation-entropy-balance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pelekis, Le Matematiche 74(1), pp. 119–130, Conjecture 1.3, p. 122:\n“Let the positive integers m,n,r be such that m ≥ n > r ≥ 2. A hypergraph\nH = (V,ℰ) from D_{n,m,r} for which it holds H(X_H) ≥ H(X_G), for all\nG ∈ D_{n,m,r} is such that |deg_H(v₁) − deg_H(v₂)| ≤ 1, for all v₁,v₂ ∈ V.”\nHere $D_{n,m,r}$ includes repeated edges and isolated vertices. Each edge\nindependently selects a uniformly random vertex, and $X$ counts the vertices\nselected at least once. Entropy uses logarithms of base two and $0\\log 0=0$.\nThe title's zero-in-degree count is $n-X$; that bijection preserves entropy.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $m\\ge n>r\\ge2$ | `claim` quantifies over `n m r`, with `n ≤ m`, `r < n`, `2 ≤ r` | All parameters are positive under these hypotheses. |\n| $r$-uniform multi-hypergraphs | `E : Fin m → Finset (Fin n)` and every edge has cardinality `r` | Ordered edges retain multiplicities; isolated vertices remain. |\n| Uniform independent orientations | The literal orientation filter inside `entropyX r E`, with denominator $(r:\\mathbb R)^m$ | Mathlib `Fintype.piFinset` supplies the public correspondence. |\n| $X$ is the number with nonzero in-degree | `X E o` counts the image of `o` | The edge argument labels the family; the value depends on `o`. |\n| Shannon entropy | `entropyX r E` sums $-p\\log_2p$ over $k=0,\\ldots,n$ | `r` is explicit; zero-probability outcomes are included. |\n| Entropy at least every competitor | Every uniform `F` satisfies `entropyX r F ≤ entropyX r E` | Full competitor class, with no sampling restriction. |\n| Absolute degree difference at most one | Every ordered pair satisfies the integer difference bound | Both orders give the two signs of the absolute-value bound. |\n| Refutation | `result : ¬ claim` | Unconditional negation of the complete conjecture. |","title":"Entropy maximisers need not have balanced degrees","url":"https://the-omega-institute.github.io/trureturing-pages/research/pelekis-2018-orientation-entropy-balance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/96b57acb36cb336e5a74ceab69a31ca03e73c3700cbd0f62bce9a679cdc26c99.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:peltomaki-2020-silver-slope-abelian-periods","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Peltomäki, *Abelian periods of factors of Sturmian words*, Journal of Number Theory 214 (2020),\n251–285, arXiv:1905.06138v3, Conjecture on page 283:\n\n> Let $\\alpha = [0; \\overline{2}]$. The abelian period set of a Sturmian word of slope $\\alpha$ is $\\mathcal{Q}^+_\\alpha \\cup \\mathcal{M}_\\alpha$.\n\nAn abelian period set comprises the minimum abelian periods of all nonempty factors.\nFor $\\alpha=\\sqrt{2}-1$, let $q_0=1$, $q_1=2$ and $q_{k+1}=2q_k+q_{k-1}$.\nThe asserted set is $\\{q_k:k\\geq0\\}\\cup\\{2q_k:k\\geq0\\}\\cup\\{q_k+q_{k-1}:k\\geq1\\}$.","title":"Peltomäki's silver-slope abelian period set conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/peltomaki-2020-silver-slope-abelian-periods/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2486f8b17cd00d04cd1c908995045a0a16aaef41a3b9218782744351e3f8378.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:peltomaki-2024-episturmian-diophantine-exponent","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Peltomäki, *Initial nonrepetitive complexity of regular episturmian words and their Diophantine exponents*, European Journal of Combinatorics 118 (2024), 103942, DOI 10.1016/j.ejc.2024.103942, arXiv:2103.08351v3, Section 8, Conjecture 8.4:\n\n> Let $\\mathbf t$ be an aperiodic episturmian word. Then $\\mathrm{dio}(\\mathbf t)<\\infty$ if and only if $\\mathbf t$ has bounded partial quotients.\n\nThe source proves the regular case in Theorem 6.6. The unrestricted statement is the target of [#15858](https://github.com/the-omega-institute/trureturing/issues/15858).\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| A finite alphabet | Every natural $d$, with letters in `Fin d` |\n| A directive word $\\Delta$ | Every stream $\\mathbb N\\to\\mathrm{Fin}\\,d$ |\n| An episturmian word with directive $\\Delta$ | `Lang t = Lang (cDelta Delta)` |\n| Aperiodic | The literal negation of an eventually periodic natural tail, expanded in `claim` |\n| Bounded partial quotients | `BoundedPQ Delta`, a uniform bound on the distance to the next different directive letter |\n| Finite Diophantine exponent | `dio t < ⊤` |\n| If and only if | Both implications of `DiophantineExponentCriterion.result` |\n\nFinite factors use the frozen operator `AbelianBorderQuestionDefs.factor` at start zero. Its value is the literal list of the first $n$ letters. Directive prepending uses Mathlib's `Stream'.cons`, and finite powers use `Powers.wordPower`. No alphabet size, regularity, intercept or recurrence hypothesis is added.\n\nThe source's run convention and `BoundedPQ` agree: if every maximal run has length at most $K$, the next different letter occurs within $K+1$ positions; conversely a witness $B$ bounds each run by $B-1$. An eventually constant directive has neither property. This is a finite run-length argument, proved in prose.","title":"Diophantine exponents of aperiodic episturmian words","url":"https://the-omega-institute.github.io/trureturing-pages/research/peltomaki-2024-episturmian-diophantine-exponent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9aaf22f36895ee4aca94a1e8b820a264f8afb2ff675202fb4a7b873a0ba76981.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:peng-2026-zero-dimensional-shadowing-extension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Peng's Question 5.1 asks whether every compact Hausdorff shadowing system\n(X,f) is a continuous onto equivariant factor of a compact zero-dimensional\nshadowing system (Y,g). Time is natural, including zero. Neither f nor g is\nassumed invertible or onto; no metrizability or countable-weight condition\nis imposed. Empty X has the solution Y empty. The general nonempty question\nis unresolved here.","title":"Zero-dimensional shadowing extensions","url":"https://the-omega-institute.github.io/trureturing-pages/research/peng-2026-zero-dimensional-shadowing-extension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57d08c8dc43c40bb510b341250d5cfc2eda38ed2101239dea1dec895c170a0e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:perito-2026-tsirelson-bound-id","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"I. Perito, R. D'Avino, M. Jung, P. Mironowicz, A. Acín and R. Augusiak, *Bell inequalities tailored\nto optimal global randomness certification*, arXiv:2606.21362v3, §III.2, define for $d$-outcome\nmeasurements in unitary notation ($A_x=\\sum_a\\omega^a\\Pi_{a|x}$, $\\omega=e^{2\\pi i/d}$)\n$\\mathcal I_d=\\frac12\\sum_{x<2}\\sum_{y **Conjecture 1.** The Tsirelson bound of $\\mathcal{I}_d$ is given by Eq. (tsir_bound) [$2/\\sin(\\pi/(2d))$]\n> and is attained by taking $A_0=Z$, $A_1=X$; the state $\\ket{\\psi_d} = \\frac{1}{\\sqrt d} \\sum_k \\ket{kk}$;\n> and $B_y$ defined by Eq. (optbob).\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/perito2026bell.md).\nIssue [#14106](https://github.com/the-omega-institute/trureturing/issues/14106) reads it for every\n$d\\ge2$, every pair of finite dimensions, every density matrix and every projective strategy\n($A_x$, $B_y$ unitary with $A_x^d=B_y^d=I$), together with the validity and the value of the stated\nstrategy.","title":"The Tsirelson bound of the d-outcome Bell functional of Perito et al.","url":"https://the-omega-institute.github.io/trureturing-pages/research/perito-2026-tsirelson-bound-id/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e4bf3ca207775ea31e4f63edef5973f23afe9ece497eb91168dec5cd759f9a2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:perrier-multidimensional-continued-fraction-generating-functions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rachel Perrier, *Multidimensional Continued Fractions and Riordan Arrays*,\nEnumerative Combinatorics and Applications 6:4 (2026), Section 5, printed\npp. 14–15, gives the period-one recurrence with initial values\n$r_1^{(-1)}=1$ and $r_{j+1}^{(-1)}=a_j$ for $1\\le j\\le k-1$:\n\n$$\nr_1^{(n+1)}=r_k^{(n)},\\qquad\nr_{j+1}^{(n+1)}=r_j^{(n)}+m_jr_k^{(n)}.\n$$\n\nThe source states on printed p. 15:\n\n> We conjecture that solving these recurrences yields\n\n$$\nR_1(t)=\\frac{1+\\sum_{j=1}^{k-1}(a_j-m_j)t^{k-j}}\n{1-\\sum_{j=1}^{k-1}m_jt^{k-j}-t^k},\\qquad\nR_k(t)=\\frac{t^{k-1}+\\sum_{j=1}^{k-1}a_jt^{k-1-j}}\n{1-\\sum_{j=1}^{k-1}m_jt^{k-j}-t^k}.\n$$\n\nThe finite sums reproduce the two printed numerators and their common\ndenominator. The shifted convention is\n$R_i(t)=\\sum_{n\\ge0}r_i^{(n-1)}t^n$, explicitly defined on printed pp. 8\nand 12. Section 5 introduces higher dimensions with the sentence\n‘The same method extends to higher dimensions.’ It does not repeat the\ndefinition of the generating functions; retaining that shift is the\ncontextual reading used here.\n\nSet $k=d+1$. Lean time $n$ represents source time $n-1$, coordinate\n$j:\\operatorname{Fin}(d+1)$ represents source coordinate $j+1$, and\n$m,a:\\operatorname{Fin}(d)\\to K$ likewise use zero-based indices.\nFor every commutative ring $K$, every natural $d$, every parameter pair,\nand every sequence satisfying the initial values and both recurrence\nequations, the formal result is:\n\n```lean\ntheorem result {K : Type*} [CommRing K] {d : ℕ}\n (m a : Fin d → K) (r : ℕ → Fin (d + 1) → K)\n (h : Recurrence m a r) :\n R r 0 * D m = P m a ∧ R r (Fin.last d) * D m = Q a\n```\n\nHere `R` is the power series with coefficients `r`; `D`, `P`, and `Q`\nare exactly the denominator and two numerators above, with constants\nembedded in $K[[X]]$. All finite sums are empty when $d=0$; the result\nincludes this case. The initial vector and next-vector rule recursively\ndetermine a unique sequence for every parameter pair.","title":"Perrier's period-one generating functions","url":"https://the-omega-institute.github.io/trureturing-pages/research/perrier-multidimensional-continued-fraction-generating-functions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3c01c6ace9ea4a92b548f9297cb23e898abc44bc42c455447d830e196416855.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:petr-turek-2025-wreath-eigenvalue-coincidence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jan Petr and Pavel Turek, *The wreath matrix*, arXiv:2501.07269v2,\nQuestion 5.5, page 16, ask:\n\n> Are the eigenvalues $\\lambda_1,\\lambda_2,\\ldots,\\lambda_k$ from Theorem 1.5 all distinct?\n\nThe source and its formulas appear in the [Library note](../Library/GraphInvariants/petrturek2025wreath.md).\nThe preregistration is [#19058](https://github.com/the-omega-institute/trureturing/issues/19058).\nThe answer is negative: at $(n,k)=(14,5)$, the values indexed by $3$ and $5$\nare both $17107200$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Positive integers $k There exist values of ρ and q for which the following specialization (which we call the alternative q-Charlier) is Fibonacci positive:\n\n$$\nx_k=\\frac{q^{k-1}(1+\\rho q^{k-2}+\\rho q^{k-1}-\\rho q^{2k-2})}{(1+\\rho q^{2k-3})(1+\\rho q^{2k-1})},\\qquad\n y_k=\\frac{\\rho q^{3k-2}(1-q^k)(1+\\rho q^{k-1})}{(1+\\rho q^{2k})(1+\\rho q^{2k-1})^2(1+\\rho q^{2k-2})},\\qquad k\\ge1.\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (2.10): first tridiagonal determinant, including the empty determinant | `A` | Every matrix entry is literal; size is any natural number. |\n| Equation (2.10): second determinant, first row $y_1,x_1y_2$, then diagonal $x_3,x_4,\\ldots$ | `B` | `B k` has size $k+1$. |\n| Definition 2.5: leading ones, then a two and a shifted remaining word | `sAux`, `s` | Shift is the inline function $k\\mapsto u(k+\\lvert w\\rvert)$; the weight is `List.sum`. The extension to other natural letters does not alter the predicate on Fibonacci words. |\n| Definition 4.1: strict positivity for every word over $\\{1,2\\}$ | `FibonacciPositive` | Includes the empty word and arbitrary weight; no finite cutoff. |\n| Conjecture 6.3: the displayed specializations for every $k\\ge1$ | `xAQC`, `yAQC` | Integer powers retain negative exponents at $k=1$. Values at index zero are outside the source domain and unused by the word determinants. |\n| There exist values of $\\rho,q$ | `claim`, `result` | Both range over $\\mathbb R$ without extra assumptions; the proof chooses $\\rho=q=1/4$. |\n\nEvery public definition has its full body in the Blueprint. No source clause is dropped. The private stronger induction applies to the extension on all natural-letter lists; the public result retains exactly the source's Fibonacci-word restriction.","title":"Alternative q-Charlier Fibonacci positivity: Petrov–Scott Conjecture 6.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/petrov-scott-2026-alternative-q-charlier-fibonacci-positivity-conjecture-6-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5baf8d8d2caee891d55612ec3506cfd1524f078a048b14b1b8ae75f919f0aa2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pham-vu-2026-well-covered-powers-diameter","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"My Hanh Pham and Thanh Vu, *On graphs whose powers are well-covered*,\narXiv:2610.09300v1 (7 October 2026), Question 3.9:\n\n> Question 3.9. Let G be a simple connected graph such that G^d is well-covered for all d ≥ 1. Is it true that diam(G) ≤ 3? Furthermore, if diam(G) = 3, are there only finitely many such graphs without true twins?\n\nDefinition 2.2 puts an edge between distinct vertices of G^d precisely when\ntheir G-distance is at most d. Definition 2.3 defines well-coveredness by equal\ncardinalities of minimal vertex covers. Complementation identifies these\nwith maximal independent sets. Corollary 3.5 defines true twins as adjacent\nvertices with equal closed neighbourhoods.\n\nThis dossier settles the first clause. The closed Lean `claim` quantifies over\nfinite vertex types, simple graphs, their decidable adjacency, connectedness,\nand every positive power, and asserts diameter at most three. The settling\nstatement is literally `result : ¬ claim`.","title":"Pham–Vu Question 3.9: diameter of graphs with well-covered powers","url":"https://the-omega-institute.github.io/trureturing-pages/research/pham-vu-2026-well-covered-powers-diameter/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ade1053412f5125de9739710ac42e442e408f115c5955542442e4e99c2f1437d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:phunphayap-noppakaew-pongsriiam-2023-digit-sum-arcs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Phunphayap, Noppakaew and Pongsriiam, *Directed graphs of integers with arcs\ndetermined by an arithmetic function*, arXiv:2303.13017v1, Question 3\n(Question 3.3 in Utilitas Mathematica 124, DOI 10.61091/um124-06), ask:\n\n> We obtain in Theorem 2 that Out(s_b, n) contains every multiple of s_b(n). Does Out(s_b, n) contain an infinite number of positive integers that are not divisible by s_b(n)? When d = 1 and s_b(n) ≥ 2, the answer is yes. Can we determine a simple necessary and sufficient condition for the infinitely many elements in A ∩ Out(s_b, n) where A is the set of positive integers that are not divisible by s_b(n)? In that case, does Out(s_b, n) has a natural density?\n\nFor $b\\ge2$ and $n\\ge1$, write $s=s_b(n)$ and $d=\\gcd(b-1,n)$.\nThe answer is that the difference set is infinite exactly when $d `\\item $\\{0,1,2\\}$ if $b \\neq \\pm1$ and $|a|-|b| = 1$,`\n\nSection 4 defines the objects: `F_0 = 0`, `F_1 = 1`, `F_n = a F_{n-1} + b F_{n-2}`. Section 1\ndefines the order: \"The number of zeros in a Pisano period is the order of `m`, denoted\n`omega(m)`.\" Section 5 fixes the degenerate convention: \"define the order to be zero if the\nsequence is eventually periodic modulo `m` and this period contains no multiples of `m`.\"\n\nWritten out, the assertion is that for all integers `a`, `b` and all `m > 1`, if `b ≠ ±1` and\n`|a| - |b| = 1`, then `omega_(a,b)(m) ∈ {0,1,2}`.","title":"The Order of a Pisano Period Is Not Confined to Two","url":"https://the-omega-institute.github.io/trureturing-pages/research/pisano-order-range-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/942d607cba20f694be382a3f11b567de818ed99fd735956d65c7e7c154a84039.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:polettini-pinned-tree-span","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Polettini, Harunari, Dal Cengio and Lecomte, *Coplanarity of rooted\nspanning-tree vectors*, arXiv:2407.16093v1, Introduction, conjecture:\n\n> Finally, we conjecture that the 3ⁿ-vectors representing the deletion-constriction through n reversible edges span a n+1 dimensional space.\n\nThe source is recorded in the [Library note](../Library/StatisticalMechanics/polettini2024coplanarity.md).\nThe preregistration is [#19157](https://github.com/the-omega-institute/trureturing/issues/19157).\nThe universal assertion is refuted by unit-rate $K_4$ with the reversible\nedges $(0,1),(1,2),(2,0)$ pinned.\n\n### Statement echo\n\n| Source clause or encoding | Lean counterpart | Fidelity and scope |\n| --- | --- | --- |\n| Irreducibility: a path of nonzero weight between any two vertices | `Irreducible` | Reflexive transitive closure of positive-rate arcs; all rates are nonnegative |\n| A spanning tree directs every edge toward its root | `IsRootedTree` | The parent fixes the root; every non-root edge has positive rate; every vertex reaches the root |\n| An arc belongs to the tree | `Contains` | Parent-map encoding excludes the auxiliary root loop |\n| Deletion-constriction through $n$ reversible edges | `HasStatus` | Neither arc, forward only, or backward only, giving $3^n$ coordinates |\n| Definition 1: sum of products over the conditioned trees | `tau` | The literal sum over parent maps with the selected statuses; products exclude the root |\n| Deletion of all pins | `deletePins` | Both orientations of every pin receive rate zero |\n| Dimension $n+1$ | `claim` | Real finrank of the span of the root-indexed vectors |\n| Distinct reversible edges | Positive rates in both directions, distinct endpoints and injectivity through `Sym2.mk` | No parallel duplicate pins |\n| Irreducibility after deletion | Last hypothesis of `claim` | A stronger admissibility condition, satisfied by the counterexample; it does not weaken the refutation |\n| Universal parameters | Natural $N,n$, rates on `Fin N`, pins indexed by `Fin n` | Finite labelled graphs; aliases are expanded without changing the source convention |\n\nThe root loop encodes a fixed root and contributes no weight. An empty\nproduct is one. The defining formulas and all parameters appear in the\nBlueprint mirror; `result` displays the entire negated universal assertion.","title":"A pinned triangle refutes the spanning-tree dimension conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/polettini-pinned-tree-span/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e4d03a6af9e623dfb60f7bba84f2d23e620d7e8b7f6b6c6b56bda29ae81c10d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:poncini-rasmussen-2022-transfer-expansion-conjecture-c12","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Poncini and Rasmussen, *Integrability of planar-algebraic models*, arXiv:2206.14462v2, Appendix C, page 47, state Conjecture C.12:\n\n> For every n ∈ ℕ and all u ∈ F, we have Ť_n(u) = 2 Σ_{k=0}^{n−1} (1 + u²)^{n−k} (2u)^k Σ_{j=0}^{⌊k/2⌋} [n − k − 2 + 2j over 2j]!! (−1)^j m_{k−2j}(h_n)/(k − 2j)!\n\nThe domain is $n\\ge1$ and an algebraically closed field $F$ of characteristic zero. The source's coefficients and ordered operators are (C.9)–(C.15), the Hamiltonian is (C.40), the central factorial polynomials are (C.44), and the double factorial quotient is (C.65). The source note is `D5/L/StatisticalMechanics/poncini2023planar`. Preregistration: [#19637](https://github.com/the-omega-institute/trureturing/issues/19637).\n\n### Statement echo\n\nThe settling namespace is `D5.S3.StatisticalMechanics.VertexModels.PonciniRasmussenTransferExpansion`; the scalar namespace is `D5.S1.Recurrence.CentralFactorialTransferExpansion`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Algebraically closed characteristic-zero field $F$ | `claim` quantifies over `F`, `Field F`, `IsAlgClosed F`, `CharZero F` | Full source domain. |\n| Every positive length and every spectral parameter | `claim` quantifies over `n`, its lower bound, and `u` | No parameter is specialized. |\n| Two-dimensional tensor basis | Matrices indexed by functions from `Fin n` to `Fin 2` | Coordinates start at zero. |\n| Four cases of (C.15) | `Rcoef` | The same equalities, inequalities and coefficients. |\n| $K^k_j=\\bar K^k_j=\\delta_{jk}$ | `Kcoef`, `oneSite` | The two boundary operators are identity coefficients on their sites. |\n| Two-site matrix units, (C.9)–(C.10) | `twoSite` | Row and column entries use $(k,j)=(x(a),y(a))$ and $(m,l)=(x(b),y(b))$; other coordinates agree. |\n| Ordered double row, (C.12) | `preTrace` | Descending first row, first boundary, ascending second row, auxiliary boundary. |\n| Auxiliary partial trace, (C.13) | `transfer` | Sum over the same appended auxiliary bit in row and column. |\n| Adjacent flips, (C.40) | `hamiltonian` | The bounded next-index proof only supplies the final adjacent site. |\n| $m_r(x)=\\prod_{a=0}^{r-1}(x-(r-1-2a))$ | scalar `mPoly` | The full product, including $m_0=1$. |\n| $n_1!!/(n_2!!(n_1-n_2)!!)$, (C.65) | scalar `dfBinom` | Uses `Int.toNat` and `Nat.doubleFactorial`, giving $(-1)!!=0!!=1$. Only $-1$ can occur negatively in the claim. |\n| Both finite sums and all factors of (C.66) | `claim` | Natural half-index division and truncated subtraction; the summation bounds ensure the intended nonnegative indices. |\n| The universal equality | `result` | Proves the whole claim; its Scribe formula expands it in full. |","title":"The Poncini–Rasmussen central factorial transfer expansion","url":"https://the-omega-institute.github.io/trureturing-pages/research/poncini-rasmussen-2022-transfer-expansion-conjecture-c12/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c7386dbccf44ecea06399c084a052fe51335619bb538480d9ff538697865e3c4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:popoli-shallit-stipulanti-2024-valuation-question-31","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Popoli, Shallit and Stipulanti, *Additive word complexity and Walnut*,\narXiv:2410.02409v1, Section 5, Question 31, PDF page 15, ask:\n\n> Given an alphabet A, is there a valuation such that the additive complexity\n> of a given sequence is equal to its abelian complexity?\n\nThe preceding definition is:\n\n> A map v:A→ℕ is called a valuation over an alphabet A.\n\nThe identical question is HTML Section 4, Question 30, element\n`Thmtheorem30`, and official FSTTCS 2024 Question 31, printed page 32:14,\nDOI 10.4230/LIPIcs.FSTTCS.2024.32. The full general assertion was\npreregistered in [#23272](https://github.com/the-omega-institute/trureturing/issues/23272).\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Every finite alphabet and each given infinite sequence | Closed `claim`: every `A : Type` with `[Fintype A]` and every `w : ℕ → A`. |\n| One-sided infinite word, indexed from zero | The function `w`; the witness `universalWord : ℕ → Fin 3` is defined at every natural index. |\n| Ordinary contiguous length-n factors | Reused `MorseHedlund.wordFactor` and `wordFactorSet`, with every natural starting index. |\n| Abelian equivalence of original symbols | `parikh` counts every original letter; `parikhSet` takes its finite image. `List.perm_iff_count` identifies equal counts with permutation. |\n| A valuation into the natural numbers | `v : A → ℕ`, including zero, equal and repeated weights; the alphabet is retained. |\n| Additive equivalence | `sumSet` is the finite image of actual factors under the sum of their letter values. |\n| Equality of complexities at all lengths | `∀ n : ℕ, additiveComplexity w v n = abelianComplexity w n`; length zero is included. |\n| General existence order | `∀ A, ∀ w, ∃ v, ∀ n`; the witness word is fixed before any valuation is extracted. |\n| Negative answer | The only public theorem is literally `result : ¬ claim`. |\n\nNo recurrence, automaticity, morphicity or balancedness restriction is\npresent in the original question. Its neighboring positive results have\ntheir own hypotheses.","title":"Natural valuations cannot preserve every abelian complexity function","url":"https://the-omega-institute.github.io/trureturing-pages/research/popoli-shallit-stipulanti-2024-valuation-question-31/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f37f267f4b455cfdf698f8bda78bfe0bd2a924a604a42fec9c526b9fb4cb9c1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:porto-et-al-2024-n-cycle-nonlocality-contextuality-tradeoff","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. E. A. Porto, G. Ruffolo, R. Rabelo, M. Terra Cunha and P. Kurzyński, “Trade-off relations between Bell nonlocality and local Kochen–Specker contextuality in generalized Bell scenarios”, *New Journal of Physics* 26, 083028 (2024), arXiv:2405.02964v2, states the following conjecture in `final_version.tex`, lines 373–375:\n\n> Consider a bipartite generalized Bell scenario where Bob has an $n$-cycle contextuality setup. For any normalized generalized-Bell inequality $S_{GBell}$ and any normalized noncontextuality inequality $S_{NC}$ of Bob's local experiment, every $\\mathcal{NSND}$ behavior satisfies expression (12).\n\nThe compiled formalization fixes `Behavior n` to `Fin 2 → Fin n → Bool → Bool → Bool → ℝ`, `BobBehavior n` to `Fin n → Bool → Bool → ℝ`, and `AliceResponse` to `Fin 2 → Bool → ℝ`. `IsNSND p` imposes nonnegativity, normalization in every Alice setting and cyclic Bob context, equality of the shared Alice–Bob marginal in consecutive contexts, independence of Bob's marginal from Alice's setting, and independence of Alice's marginal from Bob's context. `IsGeneralizedLocal` is the real convex hull of products of a row-stochastic Alice response with an `IsND` Bob behavior, and `IsNoncontextual` is the real convex hull of deterministic global Boolean assignments. The formal claim is: for every $n$ with $3 \\le n$, every $f$ of type `((Behavior n →ₗ[ℝ] ℝ) × ℝ)` and every $t$ of type `((BobBehavior n →ₗ[ℝ] ℝ) × ℝ)` satisfying `IsNormalizedGBell f` and `IsNormalizedNC t`, and every $p$ satisfying `IsNSND p`,\n$$\nf.1(p)+f.2+\\bigl(t.1(\\operatorname{bobMarginal}(p))+t.2\\bigr)\\le 1.\n$$","title":"The normalized nonlocality–contextuality trade-off holds for every binary n-cycle","url":"https://the-omega-institute.github.io/trureturing-pages/research/porto-et-al-2024-n-cycle-nonlocality-contextuality-tradeoff/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4be3cbe16f7d81bfa5b2bb809dd992a785e9178d18eac92866bf00e45e263a4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:posske-2026-a392387-xx-chain-nullspace-2p","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A392387 (Thore Posske, 2026) is the nullspace dimension of the periodic\nspin-1/2 XX Heisenberg chain on `n > 1` sites. Its COMMENTS give the count\n\n> Also, the number of subsets K of {1,...,n} such that the sum of cosines of\n> the angles in {(2j + (1 + (-1)^|K|)/2 )*Pi/n | j in K} is zero.\n\nand its FORMULA section records\n\n> Conjecture: a(2*p) = 2*(6^((p-1)/2)+1) for odd prime p.\n\nIssue #10063 fixes the readings: the formal object is this subset count, the\ncombinatorial description common to the entry and to Hu, Gerken and Posske\n(arXiv:2602.15098, appendix on the XX model); its identification with the\ndimension of the zero-energy subspace is the source's Jordan–Wigner reading and\nis not formalized; the empty set is counted; `p` ranges over odd primes and\n`n = 2p`.","title":"The nullspace of the periodic XX chain on twice a prime number of sites","url":"https://the-omega-institute.github.io/trureturing-pages/research/posske-2026-a392387-xx-chain-nullspace-2p/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29b44de64a34a768201e6ef70097404c3ac0e027706910b38c4307f7148a6aec.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pozsgay-wanless-2024-nonlinear-perfect-tensor-factors","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Pozsgay and I. M. Wanless, *Tensor network decompositions for absolutely maximally entangled states*, Quantum 8 (2024) 1339, arXiv:2308.07042v3, ask whether the linear factorization result extends to nonlinear classical maps. In `factoringtriunit.tex`, the source defines orthogonal columns and orthogonal arrays in lines 891–902, associates a map from three input symbols to three output symbols with an array in lines 929–937, and gives the six-leg factorization\n$u^{(123)}=\\mathcal C^{(23)}\\mathcal B^{(13)}\\mathcal A^{(12)}$\nwith classical two-wire functions in lines 945–964. After the linear calculation, lines 1196–1200 state:\n\n> In other words, if $G$ represents a linear perfect tensor with 6 legs, and it has a $\\triangleright$-decomposition into linear tensors with 4 legs, then those factors will themselves be perfect. It is an open question whether this statement is true in the nonlinear case for $k=6$.\n\nThe settled classical statement uses a finite type $x$ and arbitrary maps $a,b,c:x\\times x\\to x\\times x$. The circuit applies $a$ to the first two wires, $b$ to the first output of $a$ and the third wire, and $c$ to the second output of $a$ and the second output of $b$. If its six-column graph is an index-one orthogonal array of strength $3$, the question is whether each four-column gate graph is an index-one orthogonal array of strength $2$.","title":"Every classical six-leg perfect tensor factorization has perfect two-wire factors","url":"https://the-omega-institute.github.io/trureturing-pages/research/pozsgay-wanless-2024-nonlinear-perfect-tensor-factors/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d9d3c73a3e16614e966ae1fd6c8aeda36a8f646e7e36164489b44551c33ecf4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pronko-2025-fredkin-anti-adjoint-expansion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pronko fixes the one-site Pauli matrices in section 2.1, printed p. 3:\n\n> The basis in End(ℂ²) is provided by the Pauli matrices\n> `σ⁺ = [[0, 1], [0, 0]], σ⁻ = [[0, 0], [1, 0]], σᶻ = [[1, 0], [0, −1]]`.\n\nSection 2.1, printed p. 3, normalizes the total spin operators in an unnumbered\ndisplay:\n\n> `S^± = Σ_{j=1}^N σ_j^±`, `S^z = ½ Σ_{j=1}^N σ_j^z`.\n\nTheorem 2, equation (3.1), printed pp. 6-7, defines\n\n> `Σ^± = Σ_{r₁,...,r_N ∈ {−1,0,1}, r₁+⋯+r_N = ±1} σ₁^{r₁} ⋯ σ_N^{r_N}`,\n> where `σ_i^0 = 1` and `σ_i^{±1} = σ_i^±`.\n\nConjecture 2, printed p. 7, is:\n\n> For the operators Σ^± there exists the representation\n> `Σ^± = Σ_{k=1}^{⌈N/2⌉} γ_k (ãd S^± ãd S^∓)^{k−1} S^±`,\n> where γ_k are some coefficients and ãd denotes the anti-adjoint action,\n> `(ãd a) b ≡ {a, b} = ab + ba`.\n\nThe paper gives `Σ^± = −¼ S^± + ⅛ {S^±, {S^∓, S^±}}` for `N = 3` and the\ncoefficients for `N ≤ 10` in Table 1, printed p. 8.\n\nIssue #9982 fixes the readings: (i) `Σ^±` is (3.1) literally; (ii) `S^±` has\nthe section 2.1 normalization; (iii) `(ãd S^± ãd S^∓)^{k−1} S^±` is the map\n`X ↦ {S^±, {S^∓, X}}` iterated `k − 1` times on `S^±`; (iv) one family of\ncomplex coefficients serves both signs; (v) every natural number `N`.","title":"Pronko's anti-adjoint expansion conjecture for the periodic Fredkin chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/pronko-2025-fredkin-anti-adjoint-expansion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/faf79084fe22f582c0b26ea740d1628657854579143be5d53c2deb8c8d13ce18.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pronko-2025-fredkin-noncyclic-annihilation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pronko fixes the one-site basis in section 2.1, printed p. 3:\n\n> We denote basis vectors in ℂ² by\n> `|↑⟩ = (1, 0)ᵀ, |↓⟩ = (0, 1)ᵀ`.\n>\n> The basis in End(ℂ²) is provided by the Pauli matrices\n> `σ⁺ = [[0, 1], [0, 0]], σ⁻ = [[0, 0], [1, 0]], σᶻ = [[1, 0], [0, −1]]`.\n>\n> We will also use projectors onto the spin-up and spin-down states:\n> `n↑ = ½(1 + σᶻ), n↓ = ½(1 − σᶻ)`.\n\nThe same page defines the adjacent exchange and antisymmetrizer:\n\n> Consider two adjacent copies of ℂ², say, the jth and (j + 1)th. Important\n> objects are the permutation operator `P_{j,j+1}`, which exchanges these copies,\n> and the projector `Π_{j,j+1}` onto the spin-0 state, which is essentially the\n> antisymmetrizer,\n> `Π_{j,j+1} = ½(1 − P_{j,j+1})`.\n\nIt then prints the Hamiltonian density and periodic Hamiltonian:\n\n> `F_{j,j+1,j+2} = n↑_j Π_{j+1,j+2} + Π_{j,j+1} n↓_{j+2}`. (2.2)\n>\n> The periodic boundary conditions mean that the Hamiltonian is the cyclic sum,\n> `H = Σ_{j=1}^{N} F_{j,j+1,j+2}`. (2.3)\n>\n> where identification of sites by modulus N is assumed.\n\nAcross printed pp. 3-4 the source gives the ordered product and its action:\n\n> The operator `C = P_{1,2} ⋯ P_{N−2,N−1} P_{N−1,N}` is the cyclic shift\n> operator C, acting on sites as `i ↦ i + 1`.\n\nTheorem 2, equation (3.1), printed pp. 6-7, states:\n\n> The operators\n> `Σ^± = Σ_{r₁,...,r_N ∈ {−1,0,1}, r₁+⋯+r_N = ±1}\n> σ₁^{r₁} ⋯ σ_N^{r_N}`, (3.1)\n> where `σ_i^0 = 1` and `σ_i^{±1} = σ_i^±`.\n\nConjecture 1, printed p. 7, is:\n\n> The operators Σ^± annihilate all non-cyclic invariant (C ≠ 1)\n> eigenstates of the Hamiltonian.\n\nIssue #9359 fixes four readings: (i) an eigenstate is a nonzero `H`-eigenvector\nthat is also a `C`-eigenvector; (ii) non-cyclic means its `C`-eigenvalue `c` is\nnot one; (iii) (3.1) is represented literally, with `Fin 3` values `0,1,2`\nencoding exponents `-1,0,1`; and (iv) sites are modulo `N`, `N >= 3`, over\ncomplex scalars.","title":"Pronko's non-cyclic Fredkin eigenstate annihilation conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/pronko-2025-fredkin-noncyclic-annihilation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f2e6d95a79c084391285cea7075cb5f2606ca3894d74de4852f596048b16c6c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pronko-2025-fredkin-xi-hamiltonian-commutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pronko reads spin words as lattice paths in section 2.2, printed p. 4:\n\n> A Dyck path of length `N` starts at `(x,y)=(a,0)` and ends at `(x,y)=(b,N)`,\n> at each step `Δx = 1` and `Δy = ±1`, with the condition that at least once the\n> path hits the `x`-axis, i.e., `min y = 0` along the path. … we will call all\n> such paths as belonging to the equivalence class `C_{a,b}(N)`.\n\nA letter `↑` is a step up and a letter `↓` a step down. Section 4.1, printed\npp. 9-10, defines for even `N`\n\n> `Ξ = Σ_{k=0}^{N/2} (−1)^k Σ_{ℓ₁ℓ₂…ℓ_N ∈ C_{k,k}(N)} n_1^{ℓ₁} n_2^{ℓ₂} ⋯ n_N^{ℓ_N}`, (4.1)\n\nwith `n^{ℓ}` the spin-up and spin-down projections, and states on printed\np. 10:\n\n> It commutes with the Hamiltonian and anti-commutes with the cyclic shift\n> operator: `[Ξ, H] = 0`, `C Ξ C⁻¹ = −Ξ`. The first relation may require a proof\n> though we find it satisfied in all checks; the second one is obvious from (4.1).\n\n`H` is the periodic Fredkin Hamiltonian (2.3) with density (2.2),\n`F_{j,j+1,j+2} = n↑_j Π_{j+1,j+2} + Π_{j,j+1} n↓_{j+2}`, sites modulo `N`.\n\nIssue #9996 fixes the readings: (i) `ℓ ∈ C_{a,b}(N)` when the path started at\nheight `a` stays at height `≥ 0`, touches `0`, and ends at `b`; (ii) the\nprojections and `H` are those of the frozen module\n`D5/S3/Quantum/Dynamics/PronkoFredkinNonCyclicAnnihilation`; (iii) `N` is even\nand `N ≥ 3`; (iv) only the first relation `[Ξ, H] = 0` is settled.","title":"Pronko's commutation of the Dyck-class operator Ξ with the periodic Fredkin Hamiltonian","url":"https://the-omega-institute.github.io/trureturing-pages/research/pronko-2025-fredkin-xi-hamiltonian-commutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/787d2128097343956029a3d344b6fe0a24a6f3bccc2936d0ecaad97c5b3122a3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pronko-2025-motzkin-raising-lowering","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pronko (arXiv:2504.00835v3, Nucl. Phys. B 1017 (2025) 116963) studies the\nperiodic Motzkin spin-1 chain\n`H^periodic = Σ_{i=1}^{N-1} Π_{i,i+1} + Π_{N,1}` with `Π = U + D + F` and\nconjectures:\n\n> There exist raising and lowering operators satisfying \\eqref{Sigmavec} and\n> \\eqref{SpmH}, and they are given by\n> $\\Sigma^\\pm=\\sum_{r_1,\\dots,r_N\\in\\{-2,-1,0,1,2\\},\\ r_1+\\dots+r_N=\\pm 1} s_1^{r_1}\\cdots s_N^{r_N}$\n\nwhere \\eqref{Sigmavec} asks `Σ^± v_{S^z} = c_±(S^z) v_{S^z ± 1}` with\n`c_±(S^z) ≠ 0` for `S^z ≠ ±N` and `Σ^± v_{±N} = 0`, and \\eqref{SpmH} asks\n`[Σ^±, H^periodic] = 0`. Issue #10569 fixes the readings: the ground state\n`v_m` is the sum of the basis words of height `m` (`u`, `f`, `d` count\n`+1`, `0`, `−1`), as in Conjecture 1; `Π_{i,j}` acts with its first factor\non site `i`; `N ≥ 2`.","title":"Pronko's raising and lowering operators of the periodic Motzkin chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/pronko-2025-motzkin-raising-lowering/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7bb61a03ee0f6cdef1aad7832dc9a5a79cfb0518fcfd2817ca2a15b9c37d1023.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:psw-all-split-permanental-inequality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sihong Pan, Mark Skandera and Jiayuan Wang, *Permanental Inequalities and Unit Interval Orders*,\nEPTCS 445 (2026), pp. 139–147, DOI 10.4204/EPTCS.445.17, arXiv:2606.13162v1, abstract:\n\n> We also conjecture the inequalities (∗) to hold for all TNN matrices and all h = 1, …, n−1.\n\nSection 1 states:\n\n> We conjecture the inequalities to hold for all totally nonnegative matrices and I = [h].\n\nFor every real totally nonnegative matrix `A` of order `n ≥ 2` and every `1 ≤ h ≤ n−1`, the target is\n\n$$\n\\operatorname{per}(A_{E,E})\\operatorname{per}(A_{O,O})\n\\leq \\operatorname{per}(A_{[h],[h]})\n\\operatorname{per}(A_{[n]\\setminus[h],[n]\\setminus[h]}).\n$$\n\nHere `[h] = {1,…,h}`, and `E` and `O` are the even and odd one-based indices in `[1,n]`.\nTotal nonnegativity means that every square minor selected in increasing row and column order\nhas nonnegative determinant. The empty-block permanent is one. Lean's `Fin n` indices start\nat zero; `evenIndices n` tests `(i.val + 1) % 2 = 0`, and `prefixIndices n h` tests `i.val < h`.\nThe source's introductory word “negative” conflicts with its displayed `det(A_{I,J}) ≥ 0`;\nthe predicate follows that displayed inequality.","title":"Pan–Skandera–Wang's Permanental Inequality at Every Initial Split","url":"https://the-omega-institute.github.io/trureturing-pages/research/psw-all-split-permanental-inequality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a6f6a778e4c5de83c61df96e0503d8b0b486b6a22d53f6674bc823f7d725190.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:psw-bruhat-increasing-bijection","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sihong Pan, Mark Skandera and Jiayuan Wang, *Permanental Inequalities and Unit Interval Orders*,\nEPTCS 445 (2026), arXiv:2606.13162v1, Section 7:\n\n> Conjecture 7.8. For all n and all w ∈ A_n we have w ≤ f_n(w) in the Bruhat order.\n> A proof of Conjecture 7.8 would extend Theorem 5.1 to all totally nonnegative matrices.\n\nHere `A_n` is the set of permutations whose first `⌊n/2⌋` entries permute `{1, …, ⌊n/2⌋}`, and\n`f_n` is defined by Algorithm 7.5 from `f_4` through the insertion maps `ins_p`, `inss_q` and the\nreverse-complement conjugation. Theorem 7.7 of the paper states the case `n ≤ 13`, with proof omitted.","title":"The Pan–Skandera–Wang Map Is Bruhat-Increasing for Every n","url":"https://the-omega-institute.github.io/trureturing-pages/research/psw-bruhat-increasing-bijection/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00bf9d7576713f477ea4e4bb5c0a6accbc43429f9649a8f2352338c18cb2ac94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ptaszynski-2022-unicyclic-skewness-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ptaszyński, *Bounds on skewness and kurtosis of steady state currents*, Phys. Rev. E 106, 024119 (2022), arXiv:2204.08421v4, Appendix, `main.tex` line 792:\n\n> As discussed in Sec.~\\ref{subsec:unmark}, it was numerically conjectured that the minimum and maximum values of $\\mathcal{S}$, $\\mathcal{K}$, $\\mathcal{S}+\\mathcal{K}$, $\\mathcal{S}-\\mathcal{K}$ and $\\mathcal{S} \\times \\mathcal{K}$ are always obtained for a specific type of network topology, with all rates $k_{ij}$ equal to $k$ apart from one (here $k_{21}$) equal to $ak$, with the parameter $a$ depending on the optimized quantity and the number of states $N$.\n\nThe Appendix, `main.tex` line 804, specifies the skewness minimum:\n\n> The minimum value $\\min (\\mathcal{S})$ is found for $a=1/(N-1)$; it is given by Eq.~\\eqref{skewunidet}.\n\nThe reading fixed in issue [#22468](https://github.com/the-omega-institute/trureturing/issues/22468) concerns only the minimum skewness for unidirectional cycles. For every integer $N\\ge2$ and positive clockwise rates $k_i$, put\n\n$R_j=\\sum_{i=0}^{N-1}k_i^{-j},\\qquad \\mathcal{S}(k)=3R_2^2/R_1^4-2R_3/R_1^3,\\qquad g(N)=-\\frac{8-8N+N^2}{16(N-1)^2}.$\n\nThe claim is that $g(N)$ is the least value of $\\mathcal{S}(k)$ over all such positive rate vectors, including attainment. The source's candidate rates are $(1/(N-1),1,\\ldots,1)$, up to an overall positive scale. This minimum claim is proved.\n\nThe source defines $\\mathcal{S}$ as the third scaled current cumulant divided by the first, rather than the standardized third central moment. Its definition is at `main.tex` line 53; the Appendix's current and waiting-time cumulant formulas at lines 772–789 give the expression above. The finite-$N$ candidate is displayed in Section III E 1, lines 417–423. See [the literature note](../Library/StatisticalMechanics/ptaszynski2022skewness.md).","title":"The minimum skewness of a unidirectional unicyclic current","url":"https://the-omega-institute.github.io/trureturing-pages/research/ptaszynski-2022-unicyclic-skewness-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77815067ddebf93a99b083e132d39bc6a420fcade36fe407174313d62aa271e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ptaszynski-2022-unidirectional-kurtosis-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ptaszyński, *Bounds on skewness and kurtosis of steady-state currents*, Phys. Rev. E 106, 024119 (2022), arXiv:2204.08421v4, Appendix, `main.tex` line 792:\n\n> As discussed in Sec.~\\ref{subsec:unmark}, it was numerically conjectured that the minimum and maximum values of $\\mathcal{S}$, $\\mathcal{K}$, $\\mathcal{S}+\\mathcal{K}$, $\\mathcal{S}-\\mathcal{K}$ and $\\mathcal{S} \\times \\mathcal{K}$ are always obtained for a specific type of network topology, with all rates $k_{ij}$ equal to $k$ apart from one (here $k_{21}$) equal to $ak$, with the parameter $a$ depending on the optimized quantity and the number of states $N$.\n\nSection III E 1, `main.tex` lines 424–428, gives the kurtosis lower endpoint $C=-(1+\\sqrt5)/10$ in the limit $N\\to\\infty$ along rates $(A/N,1,\\ldots,1)$, where $A=(3-\\sqrt5)/2$. The asymptotic family supplies a candidate endpoint; the conjectured extremal topology leaves a universal bound over arbitrary positive rates to establish.\n\nIssue [#25526](https://github.com/the-omega-institute/trureturing/issues/25526) fixes the reading. For each natural $n\\ge2$ and strictly positive real rate vector $k:\\mathrm{Fin}\\,n\\to\\mathbb R$, put\n\n$R_j=\\sum_{i=0}^{n-1}k_i^{-j},\\qquad \\mathcal K(k)=\\frac{15R_2^2}{R_1^4}-\\frac{20R_3}{R_1^3}+\\frac{6R_4}{R_1^2R_2},\\qquad C=-\\frac{1+\\sqrt5}{10}.$\n\nThe claim is that $\\mathcal K(k)>C$ for every such $n$ and $k$, and that $C$ is the greatest lower bound of the set of all these values, allowing $n$ to vary. This is proved. The endpoint is an infimum over state counts, with no attainment by a finite positive cycle of at least two states.\n\nThe source defines $\\mathcal K$ as the fourth scaled current cumulant divided by the second (`main.tex` line 54). Substituting its current-cumulant formulas at lines 773–775 and waiting-time cumulants at lines 787–790 gives the displayed expression. This is the quantity encoded by `kurtosis`, rather than the standardized fourth central moment. See [the literature note](../Library/StatisticalMechanics/ptaszynski2022skewness.md).","title":"The sharp lower bound on the kurtosis of a unidirectional unicyclic current","url":"https://the-omega-institute.github.io/trureturing-pages/research/ptaszynski-2022-unidirectional-kurtosis-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1ca216f75684740bf6c8fb56abec619c2a86c9e74a89ee658c6f99e9d8c48b4e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ptaszynski-2022-unidirectional-skewness-minus-kurtosis-upper-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ptaszyński, *Bounds on skewness and kurtosis of steady-state currents*, Phys. Rev. E 106, 024119 (2022), arXiv:2204.08421v4, Section III E 1, `main.tex` line 432, gives\n\n$$\n\\max(\\mathcal S-\\mathcal K)=B=\\frac{3(107+51\\sqrt{17})}{2048},\n\\qquad A=\\frac{5-\\sqrt{17}}4.\n$$\n\nLine 424 specifies the asymptotic family: the exceptional clockwise rate is $ak$, the other rates are $k$, and $a=A/N$ as $N\\to\\infty$. Appendix line 792 states that the extremal-topology assertion was numerically conjectured. Thus the analytic endpoint on the proposed topology is source input; its universal validity for arbitrary positive rate vectors needs proof.\n\n**Source sign discrepancy.** The detailed display at line 432 and Figure 6 concern $\\mathcal S-\\mathcal K$, but the summary at line 133 prints $\\mathcal K-\\mathcal S\\in[-0.15,0.465]$. The two signs conflict. This problem follows the detailed display and Figure 6; it does not identify the reversed-sign summary with that claim.\n\nThe exact reading is preregistered in [#25765](https://github.com/the-omega-institute/trureturing/issues/25765) and its [unchanged endpoint comment](https://github.com/the-omega-institute/trureturing/issues/25765#issuecomment-6585311119). For every natural $N\\ge2$ and every $k:\\mathrm{Fin}\\,N\\to\\mathbb R$ with $k_i>0$ for all $i$, put\n\n$$\nR_j=\\sum_{i:\\mathrm{Fin}\\,N}k_i^{-j},\\qquad\n\\mathcal S(k)=\\frac{3R_2^2}{R_1^4}-\\frac{2R_3}{R_1^3},\\qquad\n\\mathcal K(k)=\\frac{15R_2^2}{R_1^4}-\\frac{20R_3}{R_1^3}\n+\\frac{6R_4}{R_1^2R_2}.\n$$\n\nDefine the set of actual admissible values\n\n$$\nV=\\{s\\in\\mathbb R\\mid \\exists N\\in\\mathbb N,\\ N\\ge2\\ \\land\\\n\\exists k:\\mathrm{Fin}\\,N\\to\\mathbb R,\\ (\\forall i,\\ k_i>0)\\ \\land\\\ns=\\mathcal S(k)-\\mathcal K(k)\\}.\n$$\n\nThe proved claim is the conjunction of $\\mathcal S(k)-\\mathcal K(k)\\le B$ for every such $N,k$ and $\\operatorname{IsLUB}(V,B)$. Both ratios use the same vector, repeated rates retain their multiplicities, the coordinates are arbitrary positive reals, and $N$ is unbounded. The claim includes $N=2$ and assumes no extremal topology. Sharpness concerns this full actual-value set, not a finite prefix or a relaxation to independently chosen moments.\n\nThe source's $\\mathcal S$ and $\\mathcal K$ are third-to-first and fourth-to-second **scaled current-cumulant ratios** (`main.tex` lines 53–54), not standardized central moments. The inverse-rate formulas follow by substitution in the source's winding-number and waiting-time cumulant expressions (Appendix lines 772–775 and 787–790). Their original Lean definitions are `UnicyclicSkewnessMinimum.skewness` and `UnicyclicKurtosisLowerBound.kurtosis`. See [the source note](../Library/StatisticalMechanics/ptaszynski2022skewness.md).","title":"The sharp upper bound on skewness minus kurtosis of a unidirectional unicyclic current","url":"https://the-omega-institute.github.io/trureturing-pages/research/ptaszynski-2022-unidirectional-skewness-minus-kurtosis-upper-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9c130df250e651301c365302a8ecec62ff8feaa2bd6a9f24a939b12161fbf8e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ptaszynski-2022-unidirectional-skewness-plus-kurtosis-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ptaszyński, *Bounds on skewness and kurtosis of steady state currents*, Phys. Rev. E 106, 024119 (2022), arXiv:2204.08421v4, Appendix, `main.tex` line 792:\n\n> As discussed in Sec.~\\ref{subsec:unmark}, it was numerically conjectured that the minimum and maximum values of $\\mathcal{S}$, $\\mathcal{K}$, $\\mathcal{S}+\\mathcal{K}$, $\\mathcal{S}-\\mathcal{K}$ and $\\mathcal{S} \\times \\mathcal{K}$ are always obtained for a specific type of network topology, with all rates $k_{ij}$ equal to $k$ apart from one (here $k_{21}$) equal to $ak$, with the parameter $a$ depending on the optimized quantity and the number of states $N$.\n\nSection III E 1, `main.tex` line 424, specifies the limiting interpretation:\n\n> The other bounds have been derived in the assymptotic limit $N \\rightarrow \\infty$ in which the saturating value of $a$ scales as $a=A/N$.\n\nThe lower endpoint displayed at lines 434–437 is $\\min(\\mathcal S+\\mathcal K)=-8/27$ for $A=1/2$.\n\nThe reading fixed in issue [#25636](https://github.com/the-omega-institute/trureturing/issues/25636) concerns this lower endpoint for positive unidirectional cycles. For $n\\ge2$ and $k:\\mathrm{Fin}\\,n\\to\\mathbb R$ with every $k_i>0$, put\n$R_j=\\sum_i k_i^{-j}$ and\n\n$$\n\\mathcal S(k)=\\frac{3R_2^2}{R_1^4}-\\frac{2R_3}{R_1^3},\\qquad\n\\mathcal K(k)=\\frac{15R_2^2}{R_1^4}-\\frac{20R_3}{R_1^3}+\\frac{6R_4}{R_1^2R_2}.\n$$\n\nThe claim is that $\\mathcal S(k)+\\mathcal K(k)>-8/27$ for every such $n,k$, and that $-8/27$ is the greatest lower bound of the set $V$ of all these values as $n$ and $k$ vary. This is proved.\n\nThe source defines $\\mathcal S$ and $\\mathcal K$ as the third-to-first and fourth-to-second scaled current-cumulant ratios (`main.tex` lines 53–54). Substitution of its winding-number cumulants at lines 772–775 and waiting-time cumulants at lines 787–790 gives the displayed expressions. These are the reused Lean definitions `UnicyclicSkewnessMinimum.skewness` and `UnicyclicKurtosisLowerBound.kurtosis`; they are not standardized central moments. See [the literature note](../Library/StatisticalMechanics/ptaszynski2022skewness.md).","title":"The sharp lower bound on skewness plus kurtosis of a unidirectional unicyclic current","url":"https://the-omega-institute.github.io/trureturing-pages/research/ptaszynski-2022-unidirectional-skewness-plus-kurtosis-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e2b4c435c1195051c5829920f70e3263c84ed64d84590133bb81570d950580cf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ptaszynski-2022-unidirectional-skewness-times-kurtosis-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ptaszyński, *Bounds on skewness and kurtosis of steady-state currents*, Phys. Rev. E 106, 024119 (2022), arXiv:2204.08421v4, gives the following numerical conjecture in the Appendix (`main.tex`, line 792):\n\n> it was numerically conjectured that the minimum and maximum values of $\\mathcal{S}$, $\\mathcal{K}$, $\\mathcal{S}+\\mathcal{K}$, $\\mathcal{S}-\\mathcal{K}$ and $\\mathcal{S} \\times \\mathcal{K}$ are always obtained for a specific type of network topology, with all rates $k_{ij}$ equal to $k$ apart from one (here $k_{21}$) equal to $ak$, with the parameter $a$ depending on the optimized quantity and the number of states $N$.\n\nFor the product's lower endpoint, the subsection “Unidirectional networks” gives (`main.tex`, line 445):\n\n> $\\min(\\mathcal{S} \\times \\mathcal{K})=\\min_A \\frac{1-10A+22A^2-12A^3}{(1+A)^8}$.\n\nLine 424 specifies the limit $N\\to\\infty$ with $a=A/N$; line 439 gives the approximate minimum $-0.054$ at $A\\approx0.23$. The summary's Eq. `produni` states $\\mathcal{K}\\times\\mathcal{S}\\in[-0.054,1]$ (lines 129–137), explicitly treating decimals as approximate. These quotations were checked against the v4 source member `main.tex`, MD5 `5b72a7d5de1e5c34c76db21badb56492`.\n\nIssue [#25992](https://github.com/the-omega-institute/trureturing/issues/25992) fixes the reading to the sharp lower endpoint for unidirectional cycles. Write $R_j=\\sum_{i=1}^N k_i^{-j}$ for a strictly positive real rate vector $k:\\mathrm{Fin}\\,N\\to\\mathbb R$, and use the existing definitions\n\n$\\mathcal S(k)=3R_2^2/R_1^4-2R_3/R_1^3$,\n\n$\\mathcal K(k)=15R_2^2/R_1^4-20R_3/R_1^3+6R_4/(R_1^2R_2)$.\n\nThese are `UnicyclicSkewnessMinimum.skewness` and `UnicyclicKurtosisLowerBound.kurtosis`, respectively. They match the source's current-cumulant ratios (lines 53–54) upon substitution of its waiting-time formulas (Appendix, lines 772–790); they are not the usual standardized third and fourth moments of a probability distribution.\n\nDefine $E(A)=(1-10A+22A^2-12A^3)/(1+A)^8$ and $C=\\inf\\{E(A):A>0\\}$, the module's `lowerBound`. The statement is that $\\mathcal S(k)\\mathcal K(k)>C$ for every $N\\ge2$ and every strictly positive real rate vector, and that $C$ is the greatest lower bound of all these products, allowing $N$ to vary. Both factors use the same vector; repeated rates are allowed. This is proved.","title":"The sharp lower bound on skewness times kurtosis for unidirectional unicyclic currents","url":"https://the-omega-institute.github.io/trureturing-pages/research/ptaszynski-2022-unidirectional-skewness-times-kurtosis-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/138c8d646e8bedd85aa8e7171a3f0c392633748b8486be61cf920d8fd8c336a4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:pudelko-fibonacci-minimum-limit-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pudelko, *Modular Periodicity of Random Initialized Recurrences*,\narXiv:2510.24882v5, Conjecture 8, proposes that the probability of a\nFibonacci sequence having its absolute minimum at position zero is one\nquarter. Equation (14) gives the bounded-initialization interpretation:\n\n> P_N(n) = P(n) + epsilon(N), epsilon(N) -> 0 as N -> infinity.\n\nFor integer initial values `(x,y)` in the inclusive square `[-N,N]^2`,\n`IsBilateralFibonacci x y a` specifies a sequence by `a 0 = x`, `a 1 = y`,\nand `a (k + 2) = a (k + 1) + a k` for every integer index `k`.\nThe predicate `min0 x y` requires the absolute value at position zero to be\nno greater than the absolute value at every integer index for every sequence\nsatisfying those initial values and the bilateral recurrence. Ties are\ntherefore included. The definition `boundedMin0Probability N` counts those\npairs and divides by `(2N+1)^2`. The resolved clause, `claim`, is the\nquantified convergence assertion\n\n```text\nforall epsilon : Rat, 0 < epsilon -> exists N0 : Nat,\n forall N : Nat, N0 <= N -> abs (P_N(0) - 1/4) < epsilon.\n```","title":"Refutation of Pudelko Conjecture 8 at position zero","url":"https://the-omega-institute.github.io/trureturing-pages/research/pudelko-fibonacci-minimum-limit-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a4a79d7f7b63ecc7a3ef49c9037c4821670569429351a1498bb88d00579d0d9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:qian-wang-2025-nonlocal-nonstabilizerness-haar-mean","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Qian and J. Wang, *Quantum Non-Local Nonstabilizerness*, arXiv:2502.06393v4, Phys. Rev. A **111**, 052443 (2025), define the non-local nonstabilizerness (NN) of a state as the stabilizer Rényi entropy minimized over local unitaries. For two-qubit pure states it depends only on the Schmidt weight x = cos²θ (Eq. (C1)): M(x) = ln(1/(1 + 4x(x − 1)(1 − 2x)²)). For Haar-random states x has density 3(1 − 2x)², and Appendix C derives the density (C5) of y = M(x) on 0 < y < ln(4/3).\n\n§III states that \"a closed-form expression for the average value of M … remains unclear\" and reports approximately 0.192. Appendix C writes: \"we have not been able to find a simple closed-form expression for the average amount of NN, ȳ ≡ ∫ yP(y)dy, for the Haar random state\", and gives ȳ ≈ 0.1917 by numerical integration. The verbatim statements are in [the literature note](../Library/QuantumStates/qian2025nonlocalnonstabilizerness.md).\n\nIssue [#16544](https://github.com/the-omega-institute/trureturing/issues/16544) fixes the reading. With P the literal (C5) on (0, ln(4/3)) and 0 elsewhere, y·P(y) is integrable and ȳ = ∫ y P(y) dy = 10/3 − π. It is proved.","title":"The Haar mean of two-qubit non-local nonstabilizerness is 10/3 − π","url":"https://the-omega-institute.github.io/trureturing-pages/research/qian-wang-2025-nonlocal-nonstabilizerness-haar-mean/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e33e46f5e4b4df184966e2a70a6fa101c6d7786d431228821558ff3243647a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:qian-wang-2025-separable-states-stab-local-unitaries","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Qian and Wang, *Quantum non-local nonstabilizerness*, arXiv:2502.06393v4, Appendix F, conjecture that some separable multi-qubit states cannot be transformed into `STAB` using only local unitary transformations. Their statement is that they are unaware of a proof that every separable state can be so transformed and conjecture that this is false.\n\nThe settled two-qubit claim is the preregistered QW-F statement from GitHub issue #14115: for real parameters $0 < p < 1$, $c > 0$, $s > 0$, $c^2+s^2=1$, and $c^2 \\ne 1/2$, let $b=c|0\\rangle+s|1\\rangle$ and\n$$\n\\rho=p|00\\rangle\\langle00|+(1-p)|bb\\rangle\\langle bb|.\n$$\nThen no $U_A\\otimes U_B$ with single-qubit local unitaries sends $\\rho$ into the full convex stabilizer hull `STAB`.","title":"qian wang 2025 separable states stab local unitaries","url":"https://the-omega-institute.github.io/trureturing-pages/research/qian-wang-2025-separable-states-stab-local-unitaries/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/91c212295149c6d9562ee564b3f70ee8c3005a17324ea3679b9493cdb5f42b53.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:qiu-2025-fredkin-entangling-power","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Qiu, Z. Song and L. Chen (arXiv:2410.15253, quant-ph; Phys. Rev. A 111,\n022407 (2025)) define the entanglement generation `K_{L:L^c}(U)` of a\nmultipartite gate as the supremum, over pure inputs `ψ_i` of each party with a\nlocal auxiliary system `R_i`, of the von Neumann entropy in ebits of the\nreduced state on `L` of `U(ψ_1 ⊗ ⋯ ⊗ ψ_n)`. For the four-qubit Fredkin gate\n`F_4 = (|00⟩⟨00| + |01⟩⟨01| + |10⟩⟨10|)_{AB} ⊗ I_{CD} + |11⟩⟨11|_{AB} ⊗ (S_2)_{CD}`\nthey prove `K_{AD:BC}(F_4) ∈ [2, log_2 5]` and state:\n\n> Further we conjecture that $K_{AD:BC}(F_4)\\leq 2$ by numerical results and\n> thus have the following. The entanglement generation $K_{AD:BC}(F_4)=2$.\n> Hence the entangling power of a four-qubit Fredkin gate $F_4$ is equal to\n> two ebits.\n\nIssue #11460 fixes the reading. Each auxiliary system has any finite\ndimension `d_i ≥ 1`, the inputs are unit vectors, `F_4` acts as the identity on\nthe auxiliary systems, and the entropy is taken on `A R_A D R_D`. The formal\n`claim` is the first sentence, `K_{AD:BC}(F_4) = 2`; the result refutes it.","title":"The four-qubit Fredkin gate generates more than two ebits across AD:BC","url":"https://the-omega-institute.github.io/trureturing-pages/research/qiu-2025-fredkin-entangling-power/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e17f37f3616e0784414bd77f198e0d0363874c73041f4eaae8cd561c946d6b4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:qiu-ma-xiang-2024-zero-balanced-gauss-turan","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Qiu, Ma and Xiang, *Turán-type inequalities for Gaussian hypergeometric\nfunctions, and Baricz's conjecture*, arXiv:2408.15723v1, §6,\nConjecture 6.2:\n\n> The inequality (1.6) holds for all r ∈ (0, 1) and a ∈ (0, ∞) if and only if b ∈ (0, √(a(a+1))], and the inequality (1.7) holds for all r ∈ (0, 1) if and only if a ∈ [3/7, 1).\n\nWrite $F(a,b;c;r)={}_2F_1(a,b;c;r)$. Inequality (1.6) is\n\n$$\\frac{\\partial}{\\partial b}\n\\left[\\frac{F(a+1,b+1;a+b+1;r)}{F(a,b;a+b;r)}\\right]<0,$$\n\nand inequality (1.7) is\n\n$$\\frac{F(a+1,2-a;2;r)}{F(a,1-a;1;r)}\n>\\frac{F(a+1,5/2-a;5/2;r)}{F(a,3/2-a;3/2;r)}.$$\n\nThe first classification is read pointwise in the positive parameters:\nfor each $a,b>0$, the derivative is negative at every $r\\in(0,1)$\nif and only if $b\\le\\sqrt{a(a+1)}$.\nThe second classification assumes $00$; negativity of `deriv (fun β => quot a β x) b` for every $0 Question 19. Suppose `f` and `g` are quasi-injective. Is the composition\n> `f ∘ g` quasi-injective?\n\nThe same item goes on to ask for a classification of the pairs whose composite is\nquasi-injective, and records that \"An obvious sufficient condition for `f ∘ g` to\nbe quasi-injective is that `g` is both surjective and completely multiplicative,\nbut there may be a weaker condition.\" Only the displayed question is settled here.","title":"Quasi-Injective Composition Refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/quasi-injective-composition-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2b9ec8eae8dce8d483562136ef4d84900b01d9c33dc8146067dcd225ae4602e4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:quasi-injective-order-separation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Pongsriiam, *Journal of Integer Sequences* 24 (2021), Article 21.10.1,\nDefinition 1, reads: \"We call a function `f : N → C` a quasi-injective function\nif for all `a, b ∈ N`, the condition `f(an) = f(bn)` for all `n ∈ N` implies\n`a = b`. In addition, if `f : N → N` and `ℓ ∈ N`, then we say that `f` is\nquasi-injective of order `ℓ` if `f, f^(2), f^(3), …, f^(ℓ)` are quasi-injective,\nthat is, for any `a, b, k ∈ N` with `1 ≤ k ≤ ℓ`, if `f^(k)(an) = f^(k)(bn)` for\nall `n ∈ N`, then `a = b`.\" Section 5 asks:\n\n> Question 17. For each `m ≥ 2`, is there a function `f : N → N` such that `f` is\n> quasi-injective of order `m − 1` but not of order `m`?","title":"Quasi-Injective Order Separation","url":"https://the-omega-institute.github.io/trureturing-pages/research/quasi-injective-order-separation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5692c2853d8c736a932acfa1a4f56c090f3aa47cf7f00f1983f7ca414526a666.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:quesne-2023-kepler-coulomb-qes-coefficient-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Quesne, *Quasi-exactly solvable extensions of the Kepler-Coulomb potential on the sphere*, Ann. Phys. 451, 169265 (2023), arXiv:2206.01033v3, builds extensions of the Kepler–Coulomb potential on a sphere of curvature κ with 2m additional terms and known ground and first excited states.\n- **The system.** The generating-function method (§4.1, Eq. (4.10)) expresses the first two superpotentials through m + 1 constants a_0, …, a_m, which must solve the linear system (4.13)–(4.17) in the rescaled coupling Q and κ (§4.2).\n- **The conjecture.** From the solutions for m ≤ 7: \"Conjecture For any m > 1, the solution of the system of equations (4.13)–(4.17) can be written as\" the closed forms (4.19)–(4.21) for m = 2µ + 1 and (4.22)–(4.24) for m = 2µ. They share the denominator Q^{2µ} + b_1κQ^{2µ−2} + ⋯ + b_µκ^µ, and the odd-index numerators contain brackets with constants c_j^{(k)}, \"where b_1, b_2, …, b_µ, and c^{(k)}_1, c^{(k)}_2, …, c^{(k)}_{µ−k} or c^{(k)}_{µ−k−1} are some positive constants.\"\n- **The open problem.** §5: \"The problem that remains open is to prove our conjecture by finding the explicit values of the constants appearing …\"\n\nThe verbatim statements are in [the literature note](../Library/Eigenstructure/quesne2023keplercoulomb.md).\n\nIssue [#17076](https://github.com/the-omega-institute/trureturing/issues/17076) fixes the reading: the constants are independent of Q and κ, and for all Q, κ > 0 the system has a solution, every solution has the closed forms, and solutions agree on 0, …, m. The conjecture is proved for every m ≥ 2, with explicit constants.","title":"The coefficient system of Quesne's Kepler–Coulomb extensions on the sphere has the conjectured closed forms for every m","url":"https://the-omega-institute.github.io/trureturing-pages/research/quesne-2023-kepler-coulomb-qes-coefficient-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3852257c9b3a81231c743802583f6e99a45440998842ded46b73b1973b0e75a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:quinta-andre-burchardt-zyczkowski-2019-m-resistant-states","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Quinta, André, Burchardt and Życzkowski, *Cut-resistant links and multipartite entanglement resistant to particle loss*, Phys. Rev. A 100, 062329 (2019), arXiv:1908.10390v1, Section V, state:\n\n> **Conjecture.** For any $N$ and $m$, there exists an $m$-resistant $N$-qudit state in some local dimension $d$.\n\nTheir Section III definition says that an entangled state remains entangled after tracing out any $m$ subsystems and becomes separable after tracing out any $m+1$ subsystems. This dossier uses finite full separability for the separable predicate and the admissible range $N\\geq3$, $m+2\\leq N$.\n\n### Statement echo\n\nThe Lean `claim` quantifies all natural $N,m$ with $3\\leq N$ and $m+2\\leq N$, then supplies a natural local dimension $d$, a pure amplitude function on $(\\mathrm{Fin}\\,N)\\to\\mathrm{Fin}\\,d$, unit norm by the finite sum of `Complex.normSq`, and `IsMResistant m ψ`. `IsMResistant` is the conjunction of initial failure of full separability, failure after every loss set of cardinality $m$, and full separability after every loss set of cardinality $m+1$. The pure-state convention is the frozen `rankOneDensity` declaration.","title":"M-resistant qudit states exist for every admissible loss parameter","url":"https://the-omega-institute.github.io/trureturing-pages/research/quinta-andre-burchardt-zyczkowski-2019-m-resistant-states/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2f4dca78d10c3d56ea493731cc1042ae1099e626542e5553f7b22fd7c3cec3c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:radsarar-anando-2026-apollonian-second-eigenvalue","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rad Sarar Anando's MathOverflow question 516405, revision 1,\nGUID `8C9D6CD5-E3A3-4B25-BDB6-FF5A9A3D3A2B`, asks whether the second largest\nordinary adjacency eigenvalue, counted with multiplicity, satisfies\n\n$$\\lim_{m\\to\\infty}\\theta_2(G_m)/2^{m/2}=\\sqrt2.$$\n\nHere $G_0$ is plane $K_4$ with outer triangle $abc$ and centre $o$.\nEvery step inserts one vertex in each inner triangular face and joins it\nexactly to the three corners; the outer triangle is never subdivided.\nThe [source note](../Library/Eigenstructure/radsararanando2026apollonian.md)\nattributes the quotation and adapted exposition to the author under CC BY-SA 4.0.","title":"Second adjacency eigenvalue of the inner-face Apollonian stages","url":"https://the-omega-institute.github.io/trureturing-pages/research/radsarar-anando-2026-apollonian-second-eigenvalue/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09ce44c46bdd082c9818593c69c1bb9160e7b288e706c0f0c3e0c977218d4d91.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rajaei-qubit-generalized-fidelity-dpi","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rajaei's abstract says: \"In dimension two, although we do not settle the DPI\nin full generality,\". The exact residual assertion is that every trace-one\npositive semidefinite $P,Q\\in M_2(\\mathbb C)$, every positive-definite\ntrace-one $R$, and every completely positive trace-preserving channel\n$\\Phi:M_2(\\mathbb C)\\to M_2(\\mathbb C)$ with positive-definite $\\Phi(R)$ obey\n\n$$B_{\\Phi(R)}(\\Phi(P),\\Phi(Q))\\le B_R(P,Q).$$\n\nHere $F_R(P,Q)$ is the ordered matrix-root trace in the source, and\n$B_R(P,Q)=\\operatorname{Re}\\operatorname{tr}(P+Q)-2\\operatorname{Re}F_R(P,Q)$.\nSingular input states are permitted. Positivity only on three chosen states\ndoes not establish complete positivity.","title":"Unrestricted qubit generalized-fidelity data processing","url":"https://the-omega-institute.github.io/trureturing-pages/research/rajaei-qubit-generalized-fidelity-dpi/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b011bc1b231311d9fbc94f9ae2ed93d8549cc08ee45103cbdc75d98ec6ad72d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rajchel-gasiorowski-zyczkowski-2018-strongly-complementary-permutations","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Rajchel, A. Gąsiorowski and K. Życzkowski, *Robust Hadamard matrices, unistochastic rays in Birkhoff polytope and equi-entangled bases in composite spaces*, Mathematics in Computer Science 12, 473–490 (2018), arXiv:1804.10715v2, Section 4, ask about sets of $n\\times n$ permutation matrices whose distinct members are pairwise strongly complementary. The v2 source `unisto9.tex`, lines 686–690, states:\n\n> Due to the definition of strong complementarity each pair of matrices has non-zero entries in different places, so that $k \\leq n$. We are not able to determine how the maximal number $k$ of such matrices depends on the dimension $n$, nor whether the bistochastic matrices belonging to the interior of this polytope are unistochastic.\n\nDefinitions 4.1 and 4.3 (`unisto9.tex`, lines 606–612 and 623–627) specify the relations:\n\n- Permutation matrices $P,Q$ are complementary if, for every $i,j,h,k$, the equalities $P_{ij}=P_{hk}=Q_{ik}=1$ imply $Q_{hj}=1$.\n- They are strongly complementary if they are complementary and, for every $i,j$, $P_{ij}=1$ implies $Q_{ij}=0$.\n\nThe definitions and source locators are recorded in [the literature note](../Library/FiniteGroups/rajchelgasiorowskizyczkowski2018robust.md).\n\nIssue [#23851](https://github.com/the-omega-institute/trureturing/issues/23851) fixes the cardinality clause: for every natural $n\\ge1$, the maximum is $k(n)=2^{v_2(n)}$, where $v_2(n)$ is the exponent of two in $n$. This identity answers the cardinality question. The question about unistochasticity is separate.","title":"The maximum number of pairwise strongly complementary permutation matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/rajchel-gasiorowski-zyczkowski-2018-strongly-complementary-permutations/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f0db9bf4739027efb1b7145f160ef1ae1de0add7521ce323de41a0f2ccdd7f6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rajchel-korzekwa-puchala-zyczkowski-2021-circulant-bracelet-sufficiency","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Rajchel-Mieldzioć, K. Korzekwa, Z. Puchała and K. Życzkowski, *Algebraic and geometric structures inside the Birkhoff polytope*, arXiv:2101.11288, Section V, asks:\n\n> Another observation that holds for d ≤ 4, and should be further investigated for d > 4, is whether the bracelet condition is sufficient for unistochasticity if we restrict the analysis to the set of circulant matrices.\n\nThe universal sufficiency assertion is refuted. Preregistration: [#16299](https://github.com/the-omega-institute/trureturing/issues/16299). The source locator and quoted definitions are in [the Library note](../Library/QuantumBounds/rajchelkorzekwapuchalazyczkowski2021birkhoff.md).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Definition 5: circulant, with indices modulo $d$ | `IsCirculant` expresses existence of a vector whose Mathlib circulant is the matrix | Mathlib uses $v(i-j)$; reversing the first row gives the source's class |\n| Definition 4: entries are squared moduli of a complex unitary matrix | `IsUnistochastic` uses `Matrix.unitaryGroup` and the frozen `RHLinalg.normSqMatrix` | The unitary may be arbitrary; no circulant-unitary restriction is imposed |\n| Definition 7: bistochasticity and column and row bracelet inequalities | `IsBracelet` uses `doublyStochastic` and both quantified inequalities | Pairs are distinct: $k\\ne l$ for columns, $j\\ne k$ for rows. The source counts $d(d-1)$ conditions; ordered pairs repeat the same inequalities |\n| Section V: all $d>4$, all circulant bracelet matrices | `claim` quantifies over natural $d$, $44$ ensures nonempty index sets |\n| A negative answer to the universal assertion | `result` proves the literal negation of `claim` | A counterexample at $d=12$ suffices |\n\nAll parameters and conjuncts in these definitions implement the source clauses. There is no additional hypothesis on a unitary realizing the matrix.","title":"Circulant bracelet sufficiency fails in dimension twelve","url":"https://the-omega-institute.github.io/trureturing-pages/research/rajchel-korzekwa-puchala-zyczkowski-2021-circulant-bracelet-sufficiency/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fdccb85a0c25f36a5fbe2094e3a8eb26db1f42cfd8c9f5678700f0297ce45ec1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rajchel-korzekwa-puchala-zyczkowski-2021-conjecture19-circulant-lift","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Rajchel-Mieldzioć, K. Korzekwa, Z. Puchała and K. Życzkowski, *Algebraic and geometric structures inside the Birkhoff polytope*, arXiv:2101.11288, J. Math. Phys. 63, 012202, DOI 10.1063/5.0046581, Section IV, states:\n\n> Conjecture 19. The sets Cd ∩ Ud and Cd² coincide for all d, i.e., for every circulant unistochastic matrix it is possible to find a corresponding circulant unitary matrix.\n\nThe universal assertion is **Refuted** at dimension ten. [Preregistration #16445](https://github.com/the-omega-institute/trureturing/issues/16445) fixes the source, quantifiers and Lean conventions. Definition 6 defines doubly circulant unistochastic matrices as those having a circulant unitary realization. The source quotations and locator are in [the existing Library note](../Library/QuantumBounds/rajchelkorzekwapuchalazyczkowski2021birkhoff.md).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every dimension $d$ | `claim` quantifies over natural $d$ | No restriction to even or positive dimensions in the assertion |\n| Every circulant matrix $B$ | Every real `Matrix (Fin d) (Fin d)` with the reused `IsCirculant` | `Matrix.circulant` uses $v(i-j)$ with modular indices; the source's reversed convention describes the same class |\n| Unistochastic $B$ | Reused `IsUnistochastic`, using `Matrix.unitaryGroup` and `RHLinalg.normSqMatrix` | Allows every complex unitary realization, including a real orthogonal one |\n| A corresponding circulant unitary matrix | `HasCirculantLift` quantifies over a complex vector $z$, requires its circulant to be unitary, and requires entrywise squared norms to equal $B$ | The full source condition, with no certificate restriction |\n| Equality of the two sets | `claim` gives the inclusion from circulant unistochastic matrices to matrices with a circulant unitary lift | The reverse inclusion follows from the source definitions |\n| Refutation of the all-dimensional claim | `result` proves the literal negation of `claim` | One instance at $d=10$ suffices |\n\nAll binders and conjuncts implement these clauses. There is no additional hypothesis on $B$ or on a realizing unitary. `HasCirculantLift` mirrors Definition 6 verbatim: “The set of d × d doubly circulant unistochastic matrices Cd² consists of these circulant unistochastic matrices for which there exists the corresponding unitary matrix that is circulant.” `claim` mirrors the quoted Conjecture 19. The two other source predicates are reused from `CirculantBraceletNonUnistochastic`, without aliases.","title":"A circulant unistochastic matrix without a circulant unitary lift","url":"https://the-omega-institute.github.io/trureturing-pages/research/rajchel-korzekwa-puchala-zyczkowski-2021-conjecture19-circulant-lift/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6766675d8c6a8191bad6807bc2860b195a1ece5eedbb47d17f0dfea9dd11e8df.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ramane-kitturmath-2023-regular-graph-energy-conjecture-3-11","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. S. Ramane and D. V. Kitturmath, *On the conjecture of Sombor energy of a graph*, Examples and Counterexamples 3 (2023), 100115, Conjecture 3.11, page 8:\n\n> There is no regular graph with adjacency energy 2k√2, where k∈Z⁺.\n\nThe quotation is the search seat's source reading recorded in preregistration #18578; independent verification of the publisher text and the scope of the corrigendum, DOI 10.1016/j.exco.2024.100135, is `ASSUMED-UNVERIFIED`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Regular graph | A finite type `V`, `G : SimpleGraph V`, and `G.IsRegularOfDegree r` for `r : ℕ` | Includes disconnected graphs, empty vertex types and degree zero. |\n| Adjacency energy | Imported `graphEnergy G MatrixKind.adjacency` | Sum of absolute adjacency eigenvalues, with multiplicities and center zero. |\n| $k\\in\\mathbb Z^+$ | `k : ℕ` with `0 < k` | Exactly the positive integers. |\n| There is no such graph | Universal inequality in `claim`; `result : claim` | Positive settlement. |\n| Finite enumeration and decidability instances | Anonymous `Fintype`, `DecidableEq` and `DecidableRel` instances | Represent finite simple graphs without adding a mathematical restriction. |\n\nThe only public definition is `claim`. It states the source proposition through the definitions explicitly fixed in #18578; it introduces no replacement notion of energy or regularity.","title":"Ramane–Kitturmath's regular graph energy conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/ramane-kitturmath-2023-regular-graph-energy-conjecture-3-11/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bee4aee067ceaa86d01a8f73c784ea834703e969986af76de126628f181fc0ca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rana-parashar-lewenstein-2016-l1-relative-entropy-coherence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Rana, P. Parashar and M. Lewenstein, *Trace-distance measure of coherence*,\nPhys. Rev. A 93, 012110, arXiv:1511.01854v2, Section III.2, Conjecture 7:\n\n> Conjecture 7. For all states ρ, C_{l1}(ρ) ≥ C_r(ρ).\n\nThe resolution is **Proved** for every finite dimension and every complex density\nmatrix, including singular states. [Preregistration #16965](https://github.com/the-omega-institute/trureturing/issues/16965)\nfixes the conventions. [The source note](../Library/Divergence/rana2016coherence.md)\nlocates the conjecture and the definitions.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every finite-dimensional state | `claim` quantifies over every natural `n` and every complex matrix on `Fin n`; `IsState` is positive semidefiniteness and trace one | No invertibility or rank hypothesis. In dimension zero the state premise is impossible. |\n| Section II, Eq. (3b), off-diagonal absolute-value sum | `D5.S3.Quantum.Information.CorrelatedCoherenceMonogamyRefutation.l1Coherence` | The complex norm gives the absolute value; both indices range over the full basis. |\n| Section III after Eq. (13), base-two entropy | `vonNeumann2` | Negative real trace of continuous functional calculus for $x\\log_2x$, with its value zero at zero. |\n| $C_r(\\rho)=S_2(\\Delta\\rho)-S_2(\\rho)$ | `coherenceRel` | The diagonal map retains exactly the diagonal entries. |\n| $C_{l1}(\\rho)\\ge C_r(\\rho)$ | `claim`, proved by `result` | Literal preregistered inequality. |\n\n`IsState` and `vonNeumann2` have the unchanged defining bodies shared with\n`QuantumTriangularJensenShannonRefutation`; the latter imports their general owner.\nThe claim uses the frozen `D5.S3.Quantum.Information.CorrelatedCoherenceMonogamyRefutation.l1Coherence`\nfor $\\sum_{i\\ne j}|\\rho_{ij}|$; exchanging the branches of its diagonal test\ngives the same off-diagonal sum term by term. The state and entropy definitions\nretain the source conventions.","title":"Relative entropy coherence is bounded by l1 coherence","url":"https://the-omega-institute.github.io/trureturing-pages/research/rana-parashar-lewenstein-2016-l1-relative-entropy-coherence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e09764650d6b9557fb20ff2fb9084989c6c7eea387c588162a2b92feb04adc7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:random-zeckendorf-game-gaussianity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The unordered Zeckendorf game starts from `N` copies of `F_1`; at each step it\ncombines consecutive Fibonacci terms or splits a repeated term until the\nZeckendorf decomposition is reached. Let `L_N` be the number of moves.\n\nConjecture 1.7, quoted from arXiv:2210.11038v1:\n\n> “In the limit \\(N\\to\\infty\\), the distribution of the number of moves in a\n> random Zeckendorf game on input \\(N\\) converges to a Gaussian, with\n> expectation and variance approximately \\(0.215N\\).”\n\nThe paper studies both the uniform measure `mu_N` on complete games and the path\nmeasure `P_N` obtained by choosing uniformly among legal moves at each state.\nBecause \"approximately\" and the measure are not fixed inside the conjecture\nsentence, the first declaration-ready target should be the paper's exact mixing\nQuestion 6.5 for the locally uniform path measure.\n\nLet `d_N` be the longest game length and let `X_{N,i}` be the Bernoulli\nindicator that move `i` is a split, padded by zero after termination. With\n`alpha_N(k)` the strong-mixing coefficient between the sigma-algebras before `m`\nand after `m+k`, the paper asks whether there exist `delta, C > 0` such that for\nall `N`, the sum over `k >= 0` of `(k+1)^2 alpha_N(k)^(delta/(4+delta))` is less\nthan `C`.\n\nIt states that an affirmative answer immediately yields the Gaussianity\nconjecture via the cited mixing central limit theorem. Question 6.6 proposes the\nstronger pointwise bound `|P_N(A ∩ B) - P_N(A) P_N(B)| <= C/k^(2+epsilon)`.\n\nThe paper states the difficulty:\n\n> “The scope of this result is admittedly restricted...”\n\n> “...it is unclear how to extend this to the entirety of \\(\\Omega_N\\).”\n\nThe paper obtains Gaussianity only on certain natural partition components. Its\nfull-space obstacle is dependence between separated move indicators; the open\nsection explicitly turns that into Questions 6.5 and 6.6.","title":"Gaussianity of random Zeckendorf game lengths via mixing","url":"https://the-omega-institute.github.io/trureturing-pages/research/random-zeckendorf-game-gaussianity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b4f290a7225899dec6d5a540b0196016fcbb3b3964234f7ef998d827df957228.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:raney-conjecture-7-1-zero-run-record-families","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Eu, Huang, and Kao, arXiv:2609.25742v1, Conjecture 7.1, state:\n\n> Fix k, r and a prime p not dividing kr. The left-to-right record values of\n> the zero-run sequence of R_(k,r)(n) mod p belong to a finite union of\n> families of the form (a p^m + b)/c, where a,b,c depend only on k,r,p.\n\nThe source quantifies over positive `k,r` and prime `p` with `p` not dividing\n`k*r`. Its Raney numbers are\n\n`R_(k,r)(n) = r * binomial(k*n+r,n) / (k*n+r)`.\n\nThe source's left-to-right records allow ties and come from maximal zero runs\nin the full infinite sequence. The formal target was preregistered in issue\n`#9544` with these parameters, the literal natural quotient, tied records, and\nthe source-to-formal implication. It does not ask for a converse saying that\nevery term of every family occurs.","title":"Raney Conjecture 7.1: zero-run record families","url":"https://the-omega-institute.github.io/trureturing-pages/research/raney-conjecture-7-1-zero-run-record-families/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6753f179a9a16962ebedb79dcc316b18a0a5ff1141558da4b298e9f82dc0ad37.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rather-2026-book-domination-root-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bilal Ahmad Rather, arXiv:2604.08998v1, *Root geometry of domination\npolynomials for friendship and book graphs*, and v2, *On roots of domination\npolynomials for friendship and book graphs*, Section 5, printed page 12,\nConjecture 5.4:\n\n> The numerical data suggest the following sharper parity-dependent statement.\n> (i) If n is even, then D(B_n,x) has exactly four real roots counting multiplicity:\n> 0 with multiplicity 2, one simple root in (-infinity,-2), and one simple root in (-1/2,0).\n> (ii) If n is odd, then D(B_n,x) has exactly four real roots counting multiplicity:\n> 0 with multiplicity 2 and two simple roots in (-1/2,0).\n\nSection 2, printed page 3, defines the domain and graph:\n\n> For n >= 1, the book graph B_n is formed by identifying a common edge in n copies of C_4, see Figure 1(b).\n\nThe source uses finite simple undirected graphs and ordinary domination:\n$S$ dominates $G$ when every vertex is in $S$ or has a neighbor in $S$,\nequivalently $N[S]=V(G)$. Its polynomial counts each dominating subset once:\n\n$$D(G,X)=\\sum_{S\\text{ dominates }G}X^{|S|}.$$\n\nThe complete closed claim quantifies over every natural $n\\geq1$. For even\n$n$ there are distinct nonzero $a,b\\in\\mathbb R$ with $a<-2$ and\n$-1/2 For no nontrivial family is it currently known whether ψ(G) is always real-rooted. Even partial real-rootedness results for chordal graphs, bipartite graphs, or claw-free graphs (in analogy with [8, 13]) would be significant.\n\nSection 1 says: “All graphs considered are finite, simple and undirected.”\nSection 2, page 4, defines a general position set by the absence of three\ndistinct selected vertices $x,y,z$ with $d(x,z)=d(x,y)+d(y,z)$, and says:\n\n> For each k ≥ 0, let α_k(G) denote the number of general position sets of size k.\n\n> The general position polynomial of G is ψ(G) = ∑_{k≥0} α_k(G)x^k.\n\nThe settlement is **Proved**: for every nonempty finite connected simple\ngraph, $\\psi(G)$ is real-rooted if and only if $G$ is complete.\n\n### Statement echo\n\n| Source clause or scope | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Finite, simple, undirected graphs | `V : Type`, `Fintype V`, `SimpleGraph V` | Exact finite simple-graph carrier. |\n| Selected vertices avoid the distance equality | Frozen `CylinderSevenTen.IsGeneralPosition` and `OnGeodesic` | On connected graphs `SimpleGraph.dist` is shortest-path distance. |\n| Number of general position sets of size $k$ | `alpha` | Cardinality of the complete finite-subset filter; each subset occurs once. |\n| $\\psi(G)=\\sum_{k\\geq0}\\alpha_k(G)x^k$ | `psi` | Integer coefficients; the finite range ends at $|V|$, since larger subsets do not exist. |\n| Real-rootedness | `IsRealRooted` | Every complex zero has imaginary part zero. |\n| Classify connected graphs | `claim`, `result` | The nonempty connected scope is explicit; the conclusion is equivalence with `G = ⊤`. |\n| Decidable equality | `DecidableEq V` | Finite-set infrastructure; it places no additional mathematical restriction on finite graphs. |\n\nThe four public definitions are the literal preregistered conventions.\n`IsGeneralPosition` and `OnGeodesic` are reused from their frozen owner.\nNo statement about disconnected graphs under raw `SimpleGraph.dist` is part\nof the formal classification.","title":"Rather's real-rootedness question for general position polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/research/rather-2026-general-position-real-rooted-complete/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/309c5a45aabcb6bc95ff76c19073116de0ec33df19a730d46038b314c697eec2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ray-2025-local-perception-chsh-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. K. Ray, *Local perception operators and classicality: new tools for old tests*, arXiv:2511.00314v1, replaces each correlator C_xy = Tr(ρ A_x ⊗ B_y) of a Bell functional by its \"locally perceived\" version a_x b_y C_xy. Here a_x = Tr(ρ_A A_x) and b_y = Tr(ρ_B B_y) are the local means of dichotomic observables (Eqs. (10), (11) and (35)). For CHSH (Eq. (19)) this gives the symmetric witness W^s_LPO(ρ) = sup over settings of a₀b₀C₀₀ + a₀b₁C₀₁ + a₁b₀C₁₀ − a₁b₁C₁₁ (Eq. (21)).\n- Theorem 4 proves W^s_LPO ≤ 2 when each party's settings are mutually orthogonal Bloch directions (Eq. (31)).\n- The author then writes: \"we conjecture that for all other conceivable measurement settings, the upper-bound of W^s_LPO(ρ) saturates at 2.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/ray2025localperception.md).\n\nIssue [#16628](https://github.com/the-omega-institute/trureturing/issues/16628) fixes the reading.\n- The observables are the source's dichotomic observables \"(POVM, or projective)\": Hermitian A with 1 ± A ⪰ 0.\n- The witness is the termwise sum. The first line of Eq. (21), Tr(ρ (B)^A_ρ ⊗ (B)^B_ρ), mixes different settings and is not equal to that sum.\n\nThe conjecture is proved: the sum is at most 2 for every two-qubit state and all dichotomic settings, and 2 is attained.","title":"The termwise local-perception CHSH witness is at most 2 for every measurement setting","url":"https://the-omega-institute.github.io/trureturing-pages/research/ray-2025-local-perception-chsh-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c83bb9f44af97646446c8a0d04f7d6fea4861029941328de57aee63ec1fb9dfd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rectangle-rowmotion-max-minus-min-homomesy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Elder, Lafrenière, McNicholas, Striker and Welch, *Toggling, rowmotion, and\nhomomesy on interval-closed sets*, arXiv:2307.08520v2, Conjecture 4.9, states\nthat for every product of two finite chains the number of maximal elements\nminus the number of minimal elements is 0-mesic under rowmotion on\ninterval-closed sets. Lafrenière, Lewis, McNicholas, Striker and Welch,\n*Interval-closed set rowmotion and homomesy on products of two chains*,\narXiv:2505.04000v1, restates this target as Conjecture 4.2.\n\nThe exact formal target keeps all source quantifiers: positive `m,n`, every\n`N` and equivalence `e : Fin N ≃ Fin m × Fin n` satisfying the reverse\nlinear-extension condition, and every order-convex initial set `I`, including\nthe empty set. `literalOrbit e I` is the finite set of distinct states reached\nby iterating the literal successive toggle trace. `maxMinusMin` is the integer\ncardinality difference of global maxima and minima in the actual state. The\nconclusion is the sum of this statistic over the actual distinct orbit states.","title":"Max-minus-min homomesy for rectangle interval-closed rowmotion","url":"https://the-omega-institute.github.io/trureturing-pages/research/rectangle-rowmotion-max-minus-min-homomesy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb2e65e32815ce7d8c95fb65f98d3e48f7098232f25335c68ce7bbff66fc7d12.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:reda-sacchi-macchiavello-guarnieri-2026-dynamic-cooling-tie-uniqueness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Reda, M. Sacchi, C. Macchiavello and G. Guarnieri, *Optimal Dynamic Cooling of Multiple Qubits*, arXiv:2607.23641v1, cool the first M of N identical thermal qubits, in the state ρ = τ^{⊗N} with τ = diag(1 − P₁, P₁), by one global unitary. The aim is the lowest common local temperature: every target must end with the same diagonal one-qubit marginal. The optimum is p_glob(N, M; P₁) = p★(N, M; P₁).\n\n§IX compares it with M disjoint groups of n = ⌊N/M⌋ qubits, each running the optimal n-qubit single-target protocol: p_par(N, M; P₁) = p★(n, 1; P₁) (Eqs. (88)–(89)). The source proves p_glob ≤ p_par, strictness for M + 2 ≤ N ≤ 3M − 1 (Eq. (91)), and the tie p_glob(6, 2) = p_par(6, 2) = 3P₁² − 2P₁³ (Eq. (92)). It then writes: \"Exact symbolic checks over the finite range investigated revealed no other equality, although the uniqueness of this tie remains a conjecture.\" The verbatim statements are in [the literature note](../Library/QuantumChannels/reda2026dynamiccooling.md).\n\nIssue [#16483](https://github.com/the-omega-institute/trureturing/issues/16483) fixes the reading: for all M ≥ 2, N ≥ M + 2 with (N, M) ≠ (6, 2), and every 0 < P₁ < 1/2, p_glob(N, M; P₁) < p_par(N, M; P₁). It is proved. The tie (92) is the source's and is not re-proved here.","title":"Global dynamic cooling strictly beats equal parallel groups except at (N, M) = (6, 2)","url":"https://the-omega-institute.github.io/trureturing-pages/research/reda-sacchi-macchiavello-guarnieri-2026-dynamic-cooling-tie-uniqueness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc80264efc67273d6856fbddbc85c2e119e89e821ecf7ad7d014a182a653b2f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rexiti-memarzadeh-mancini-2022-positive-kraus-side-entanglement","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Rexiti, L. Memarzadeh and S. Mancini, *Discrimination of dephasing channels*, J. Phys. A 55, 245301 (2022), arXiv:2201.00388v2, Section 7, conjecture:\n\n> In contrast, if all Kraus operators have simply the identity as unitary part in the polar decomposition, there is no benefit from side entanglement.\n\nThe complete source passage is in the [Library note](../Library/QuantumChannels/rexiti2022dephasing.md). Issue [#16535](https://github.com/the-omega-institute/trureturing/issues/16535) preregisters the negative clause, its representation-dependent reading and the exact finite-dimensional conventions.\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart and scope |\n| --- | --- |\n| Kraus operators have identity polar unitary | `claim` assumes every Kraus operator is strictly positive definite, which makes that unitary uniquely the identity; this stronger hypothesis suffices to refute the source statement. |\n| Channels and side entanglement | Both channels act on complex qubit matrices; the ancilla is a qubit and the first tensor factor. These are instances of the source's unrestricted systems. |\n| Trace-preserving Kraus representations | Both right Gram sums equal the identity, with the same finite number of operators. |\n| No benefit | For every assisted density and binary effect, some unassisted density and binary effect achieve at least the same success probability, with equal priors (`successAs` and `successUn`). |\n| Density and effect | Frozen `GHZMeasureBiseparableBound.IsDensity` is the positive-semidefinite, trace-one predicate. The preregistered effect condition and the two Kraus sums are inlined literally; proofs use the frozen Kraus and Bell owners. |\n| Negative clause fails | `result : ¬ claim`, with four operators per channel. |","title":"Positive Kraus operators do not exclude a side-entanglement advantage","url":"https://the-omega-institute.github.io/trureturing-pages/research/rexiti-memarzadeh-mancini-2022-positive-kraus-side-entanglement/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39d111c739100f42db67a9d08ba40a1107be9e9dadd4c62d1e3b8ed407aa1c04.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rexiti-oskouei-mancini-2024-werner-holevo-bayesian-feedback","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Rexiti, S. Khabbazi Oskouei and S. Mancini, *On the effectiveness of Bayesian discrete feedback for quantum information reclaiming*, arXiv:2402.18941v1, Quantum Information Processing 23, 258 (2024), Section 5, Conclusion, ask:\n\n> There are a number of problems that this study poses in future perspective.\n> First, find other examples of channels for which Bayesian feedback is more effective than Marokovian one.\n> Then, for these channels see if by increasing the dimension the advantage increases.\n\nThis quotation is from the downloaded v1 source, `EffOfBayesianFB.tex`, lines 910–912, md5 `aeee6c3398ec2e49e99d0f461ed53290`; the spelling “Marokovian” is the source's. The [literature note](../Library/HilbertOperators/rexitioskoueimancini2024bayesianfeedback.md) gives the feedback definitions and source locators.\n\nIssue [#23263](https://github.com/the-omega-institute/trureturing/issues/23263) specifies an answer to the first request using the qutrit Werner–Holevo channel\n$\\Phi(X)=\\tfrac12(\\operatorname{tr}(X)I_3-X^{\\mathsf T})$.\nFor a finite family $K=(K_a)_{a\\in\\mathrm{Fin}\\,m}$ of complex $3\\times3$ matrices, `IsKrausDecomposition K` means\n$\\sum_a K_a X K_a^\\dagger=\\Phi(X)$ for every complex $3\\times3$ matrix $X$.\n\nThe module specializes the source's two-step formulas in Section 2, lines 245–253, to dimension three:\n\n$F_{\\mathrm M}(K,L)=\\tfrac19\\sum_{a,b}\\bigl(\\operatorname{Re}\\operatorname{tr}(|L_b|\\,|K_a|)\\bigr)^2$,\n\n$F_{\\mathrm B}(K,L)=\\tfrac19\\sum_{a,b}\\|L_b|K_a|\\|_1^2$,\n\nwhere $|T|=\\sqrt{T^\\dagger T}$ and $\\|M\\|_1=\\operatorname{Re}\\operatorname{tr}\\sqrt{M^\\dagger M}$.\nThe settled statement is that every pair of finite Kraus decompositions has $F_{\\mathrm M}\\le\\tfrac12$, while some pair of three-operator Kraus decompositions has $F_{\\mathrm B}>\\tfrac12$.","title":"A strict Bayesian feedback advantage for the qutrit Werner–Holevo channel","url":"https://the-omega-institute.github.io/trureturing-pages/research/rexiti-oskouei-mancini-2024-werner-holevo-bayesian-feedback/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/edd0a685171b7fbb6e793e8a03bf73986d42bafa3cc0476b76d0095149b5ef40.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:reyes-bustos-wakayama-2026-two-photon-rabi-constraint-polynomials","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Reyes-Bustos and Wakayama define, for the two-photon asymmetric quantum Rabi\nmodel with parity `ρ ∈ {0, 1}`, bias `ε`, `x = (2g)^2` and `y = Δ^2`, the\npolynomials\n\n> P_0 = 1, P_1 = y + 2x(4N + 2ρ + 2ε − 1) − 4(1 + ε),\n> P_k = (y + 2xk(4N + 2ρ − 2k + 2ε + 1) − 4k(k + ε)) P_{k−1}\n> − 4k(k − 1)(2(N − k + 1) + ρ)(2(N − k + 1) + ρ − 1) x P_{k−2},\n\n(Definition 4.1 of arXiv:2609.00750v1) and state Conjecture 4.5:\n\n> We have P_N^{(N,ρ,ε)}(1, y) = ∏_{n=1}^{N} (y + 2n(2n + 2ρ − 1)). Moreover,\n> for x > 1, the polynomial P_N^{(N,ρ,ε)}(x, y) has positive coefficients,\n> and therefore, no positive roots for y.\n\nIssue #10148 fixes the readings: `P_k` is a real polynomial in `y` with real\nparameters `x` and `ε`; the first statement is taken for every real `ε`; the\nsecond statement is taken for `ε ≥ 0` and means that the coefficients of\n`y^0, …, y^N` are all positive and that `P_N^{(N,ρ,ε)}(x, y) ≠ 0` for every\n`y > 0`. The source derives its constraint condition\nfor `ε ≥ 0` and writes `P_N^{(N,ρ,±ε)}` for the two signs, while the\nconjecture writes `+ε` only.","title":"The constraint polynomials of the two-photon Rabi model at and beyond the critical coupling","url":"https://the-omega-institute.github.io/trureturing-pages/research/reyes-bustos-wakayama-2026-two-photon-rabi-constraint-polynomials/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7094453e3acc7c6e36031fda588a771c5d45d326045512893be03b9ec97d0502.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:reyes-dalfo-fiol-2024-new-laplacian-layer-maxima","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. A. Reyes, C. Dalfó and M. À. Fiol, *On the spectra and spectral radii\nof token graphs*, arXiv:2310.16929v1, Section 3, Definition 3.1 and\nConjecture 3.3; journal version, *Boletín de la Sociedad Matemática\nMexicana*, volume 30, article 11 (2024), DOI\n10.1007/s40590-023-00583-3, Section 3:\n\n> Conjecture 3.3. For any graph G, ρ₁(G) ≤ ρ₂(G) ≤ ⋯ ≤ ρ⌊n/2⌋(G).\n\nHere G is a finite simple unweighted undirected graph of order n. The source\nuses the ordinary real Laplacian D−A. The vertices of F_j(G) are all\nj-element subsets of V(G), with an edge precisely when their symmetric\ndifference is an edge of G. F_0(G) is the singleton empty configuration.\nWriting Sp_j for the full Laplacian spectral **multiset**, Definition 3.1\nsets Δ_j = Sp_j − Sp_(j−1) by multiset subtraction and ρ_j = max Δ_j,\nfor 1 ≤ j ≤ floor(n/2). Thus the conjecture quantifies every\n1 ≤ k < floor(n/2) and asserts ρ_k ≤ ρ_(k+1).\nThe source's connected-graph convention is satisfied by the counterexample.","title":"Successive new Laplacian layer maxima of token graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/reyes-dalfo-fiol-2024-new-laplacian-layer-maxima/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bdcf27d969a0fe726f13a68923201026e57abdcb921aabb2d67d0b2415964fbb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:richman-2023-a053871-signed-congruence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A053871 is the sequence `a(n) = 2(n − 1)(a(n − 1) + a(n − 2))` with\n`a(0) = 1`, `a(1) = 0`. Its FORMULA section states\n\n> Conjecture: if m == n (mod q) for q odd, then (-1)^m*a(m) == (-1)^n*a(n)\n> (mod q). - _Harry Richman_, Aug 29 2023\n\nIssue #10354 fixes the reading: `a` is the sequence of the recurrence in the\nentry's name, `q` is a positive odd integer, and the congruence is in the\nintegers. The statement settled is that for every odd `q` and all `m ≡ n\n(mod q)`, `(−1)^m a(m) ≡ (−1)^n a(n) (mod q)`.","title":"Richman's congruence for the deranged-matching numbers","url":"https://the-omega-institute.github.io/trureturing-pages/research/richman-2023-a053871-signed-congruence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85a216f771b4ca5828dd37e871c0f5dbca4a9e044bf162e369b6510e41a66aca.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rico-zyczkowski-2024-nonlinear-majorization-monotone","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Rico and K. Życzkowski, *Discrete dynamics in the set of quantum\nmeasurements*, J. Phys. A 57 (2024) 435302, arXiv:2308.05835v2,\nAppendix C.1, Conjecture 1, asks whether blockwise bistochastic dynamics\npreserve the centered-prefix Hilbert–Schmidt norm in the following sense:\n\n> Let $P\\in\\Delta_{n,d}$ and $Q\\in\\Delta_{n,d}$ be blockwise probability\n> vectors. If there exists a blockwise bistochastsic matrix $B\\in B_{n,d}$\n> such that $Q=B\\ast P$, then for any ordering of $\\{Q_i\\}$ there exists\n> an ordering of $\\{P_i\\}$ such that\n> $\\left\\|\\sum_{i=1}^{k}(P_i-1/n)\\right\\|_2\\geq\n> \\left\\|\\sum_{i=1}^{k}(Q_i-1/n)\\right\\|_2$\n> for all $1\\leq k\\leq n$, where\n> $\\|A\\|_2=\\sqrt{\\operatorname{tr}(A^\\dagger A)}$ is the 2-norm.\n\nDefinitions 1, 3 (Eq. (19)) and 5 give positive semidefinite effects\nsumming to $I_d$, positive semidefinite blocks with $d\\geq2$ and both\nblock row and column sums $I_d$, and the principal-square-root product\n$Q_i=\\sum_j\\sqrt{P_j}B_{ij}\\sqrt{P_j}$. Here $1/n$ in the centered\nprefix denotes $(1/n)I_d$. Issue #14670 preregisters this tier-1 question.","title":"A commuting obstruction to the nonlinear measurement monotone","url":"https://the-omega-institute.github.io/trureturing-pages/research/rico-zyczkowski-2024-nonlinear-majorization-monotone/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1b74fdb1003c626855e4ee07cdeef6b36788ee9e7f3db739cca9b026b7267f3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rivin-2026-cycle-geodesic-midpoint-formula","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Igor Rivin, “Permanents of matrix ensembles: computation, distribution, and\ngeometry”, arXiv:2602.10141v3, Section 8.4, Open Problem 2, states:\n\n> Prove the midpoint formula $\\operatorname{perm}(\\gamma(\\frac12))=(-1)^{(n-1)/2}\\cdot 2e^{-n}(1+\\frac{1}{3n}+O(n^{-2}))$.\n\nThe literal Section 4.1 matrix is\n\n$$\\gamma_n(t)_{jl}=\\frac{e^{2\\pi it}-1}{n(e^{2\\pi i(l-j+t)/n}-1)},\\qquad j,l\\in\\{0,\\ldots,n-1\\}.$$\n\nObservation 5 also states the even-dimensional zero clause. Dimension is\npositive: $n\\ge1$. The empty permanent at $n=0$ is $1$.","title":"Rivin's cycle-geodesic midpoint formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/rivin-2026-cycle-geodesic-midpoint-formula/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8081210b3c798bdfcef0e379771963b99d8334220e830b44a160b774b636d9e4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rivin-2026-cycle-geodesic-universal-function","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Igor Rivin, “Permanents of matrix ensembles: computation, distribution, and\ngeometry”, arXiv:2602.10141v3, Section 8.4, Open Problem 3, states:\n\n> Find a closed form for the universal function $f(t)$.\n\nObservation 4 defines the finite-dimensional quantity as\n$-n^{-1}\\log|\\operatorname{perm}\\gamma_n(t)|$ for the literal matrix\n\n$$\\gamma_n(t)_{jl}=\\frac{e^{2\\pi it}-1}{n(e^{2\\pi i(l-j+t)/n}-1)},\\qquad j,l\\in\\{0,\\ldots,n-1\\}.$$","title":"Rivin's cycle-geodesic universal function","url":"https://the-omega-institute.github.io/trureturing-pages/research/rivin-2026-cycle-geodesic-universal-function/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61d20d8e5db9283c99ddfec123e53b444370db52f434d55504b09d3c96a3c15b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:roch-i-carceller-tavakoli-2025-energy-restricted-unitary-encoding","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Roch i Carceller and A. Tavakoli, *The role of entanglement in energy-restricted communication and randomness generation*, arXiv:2510.27473v1, Quantum Sci. Technol. 11, 025020 (2026), study prepare-and-measure communication whose only restriction is the vacuum component of the message, ⟨0|τ_x^C|0⟩ ≥ 1 − ω (Eq. (3)).\n- **Result 4.** For the probabilistic transmission of a bit, unitary encodings of a shared pure entangled state give no advantage over unassisted quantum communication with shared randomness.\n- **The question** (§VI, p. 9): \"Whether unitary encoding remains resourceless for general tasks is an interesting open problem.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/rochicarcellertavakoli2025energyentanglement.md).\n\nIssue [#17895](https://github.com/the-omega-institute/trureturing/issues/17895) fixes the reading:\n- resourceless for general tasks means that, for every linear task and every energy 0 ≤ ω < 1/2, each unitary-encoded entanglement-assisted value is matched by an unassisted value;\n- at ω ≥ 1/2 the question is settled by dense coding with a maximally entangled pair.\n\nThis statement is refuted: unitary encoding is a resource for a four-message task at ω = 288/625.","title":"Unitary entanglement-assisted encoding beats every unassisted strategy in an energy-restricted four-message task","url":"https://the-omega-institute.github.io/trureturing-pages/research/roch-i-carceller-tavakoli-2025-energy-restricted-unitary-encoding/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86dc69c76e6b76b5bc9f841eb8e5843da2e07e11bb9c754ea92c7d8c105a2ead.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rodrigues-kevrekidis-dobson-2019-n-break-instability-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. S. Rodrigues, P. G. Kevrekidis and M. Dobson, *N-break states in a chain of nonlinear oscillators*, Phys. Rev. E 99, 022201 (2019), arXiv:1802.03477v2, state in the abstract:\n\n> Each of these higher ``excited state'' solutions possesses an additional unstable pair of eigenvalues. We thus conjecture that $k$ break solutions will possess at least $k-1$ (and at most $k$) pairs of unstable eigenvalues.\n\nThis quotation is from `oscillatorsNN_resubmit.tex`, lines 55–59, in the v2 source, whose MD5 is `f0e3a2d64045c090b53b217d75e86a05`. The model and terminology are documented in [the literature note](../Library/StatisticalMechanics/rodrigueskevrekidisdobson2019nbreak.md).\n\nThe module uses $m$ free particles, displacements $u_0,\\ldots,u_{m+1}$, and zero-based bond indices $j=0,\\ldots,m$. Its definitions are\n\n$\\phi(r)=r^{-12}-2r^{-6}$, $r_i=(13/7)^{1/6}$, and $r_j=d+u_{j+1}-u_j$.\n\nThe force at free particle $n+1$, indexed by $n=0,\\ldots,m-1$, is $F_n=\\phi'(r_{n+1})-\\phi'(r_n)$. With $c_j=\\phi''(r_j)$, the defined matrix $J$ has entries $J_{nn}=-c_n-c_{n+1}$, $J_{n,n+1}=J_{n+1,n}=c_{n+1}$, and zero elsewhere.\n\nIssue [#23865](https://github.com/the-omega-institute/trureturing/issues/23865) fixes the reading: for every $m\\ge1$, $d>0$, and displacement vector with $u_0=u_{m+1}=0$, $F_n=0$ at every free particle, and all $r_j>0$, if exactly $k\\ge1$ bonds satisfy $r_j>r_i$, then, for every Hermitian witness $h$ for $J$, the number $U=\\#\\{i\\in\\mathrm{Fin}\\,m:0 Definition 3.1. We say that a mapping F : F_{2^n} → F_{2^n} breaks a flat A of F_{2^n} if F(A) is not a flat of F_{2^n}. If F breaks all k-dimensional flats, F is said to be k-breaking.\n\n> Conjecture 3.1. Any APN function F over F_{2^n} is k-breaking for ⌊n/2⌋ + 1 ≤ k ≤ n − 1.\n\nA k-flat is a coset a+V of a k-dimensional binary subspace. APN means\nthat F(x+a)+F(x)=b has at most two solutions for every nonzero a and every b.\nThe Lean `claim : Prop` quantifies over all finite fields K of characteristic\ntwo and cardinality 2^n, every such APN function, every permitted k,\nevery offset and every submodule of binary dimension k. Its conclusion\nexcludes an image equal to a coset of a subspace of any dimension.\nCharacteristic two provides the canonical `ZMod 2`-algebra and module.\nA decidable equality instance enables finite fiber counting and places\nno restriction on the mathematical fields.","title":"Conjecture 3.1: APN functions and large binary flats","url":"https://the-omega-institute.github.io/trureturing-pages/research/rodriguez-aldama-et-al-2026-apn-breaking-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cfde35828a89f1b997d28497d94c6a340b8f77a3e75b9f2b575c0c44e4621611.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rognone-nicosia-2023-square-cluster-exit-time-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Rognone and V. Nicosia, *Clusters in randomly-coloured spatial networks*, arXiv:2308.07880v1 (J. Phys. Complexity, 2026), section on the mean exit time from a cluster:\n\n> Characterising clusters with maximal exit time at each fixed $N$ seems to be a harder quest. Here we conjecture that the maximal exit time is obtained for squares with $N$ nodes.\n\nHere:\n- **Cluster:** a set of nodes of the square lattice, connected through nearest neighbours; the walk moves to each of the four neighbours with probability 1/4.\n- **Mean exit time** $\\tau(S)$: the average over the starting nodes of $S$ of the expected first time the walk is outside $S$.\n\nThe definitions are quoted in [the literature note](../Library/StatisticalMechanics/rognonenicosia2023clusters.md). The source's displayed formula averages node-to-node hitting times, while its prose and every worked first-step system use the first-exit time; the settlement uses the first-exit time.\n\nIssue [#20034](https://github.com/the-omega-institute/trureturing/issues/20034) fixes the reading: for every $m\\ge1$ and every connected cluster $S$ of $m^2$ nodes, $\\tau(S)\\le\\tau(\\{0,\\dots,m-1\\}^2)$. This is refuted.","title":"Squares do not maximize the mean exit time of lattice clusters","url":"https://the-omega-institute.github.io/trureturing-pages/research/rognone-nicosia-2023-square-cluster-exit-time-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b47c96a97edd676489bd8c6f96009b78d8d048db28ceab4a0e844f243e741cb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:roldan-2026-integral-circulant-energy-p2q3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. G. Roldán, *Graph Energy Maximisation for Integral Circulant Graphs of Order n = p^2q^3*, arXiv:2604.09491v1, Section 1, Conjecture 1.3 (Universality):\n\n> For every pair of distinct odd primes (p, q), the set D* = {1, p^2, pq, q^2, p^2q^2, pq^3} is the unique maximiser of E(p^2q^3, D) over all nonempty D ⊆ div(p^2q^3) ∖ {p^2q^3}.\n\nThe graph has vertices $\\mathbb Z_n$ and adjacency condition $\\gcd(a-b,n)\\in D$. Its energy is the sum of the absolute values of its adjacency eigenvalues, with multiplicity. Source quotations and locators are in [the literature note](../Library/GraphInvariants/roldan2026integralcirculant.md). Preregistration: [#19144](https://github.com/the-omega-institute/trureturing/issues/19144).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Vertices $\\mathbb Z_n$, modular difference, gcd adjacency | `adjacency n D`, on `Fin n` | Literal preregistered expression; subtraction is modulo $n$. |\n| Energy sums $|\\lambda_i|$ with multiplicity | `energy n D`, using the multiset roots of the real characteristic polynomial | The proof identifies all roots through Fourier diagonalization of the real symmetric adjacency. |\n| Distinct odd primes $p,q$ | `claim` quantifies $p,q$ with both primality, inequality and oddness hypotheses | No ordering of $p,q$ and no restriction $q\\ge5$. |\n| $D^*=\\{1,p^2,pq,q^2,p^2q^2,pq^3\\}$ | `Dstar p q` | Literal six-term finite set. |\n| Allowed, nonempty proper-divisor sets | First conjunct of `claim` and the guards on its quantified `D` | The first conjunct proves admissibility of $D^*$; $1\\in D^*$ gives nonemptiness. |\n| Unique maximizer | Strict energy inequality for every different allowed nonempty `D` | Full proposition, proved by `result : claim`. |\n\nNo additional premise restricts the published conjecture. The spectral argument proves the comparison even when the competing divisor set is empty.","title":"Unique integral circulant energy maximum at order $p^2q^3$","url":"https://the-omega-institute.github.io/trureturing-pages/research/roldan-2026-integral-circulant-energy-p2q3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd68ca3c19a6ffd6c3089ce6ef4bb28a9c0b2f2e02396a1aa8dc54ac1aaab4b2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:roofeh-karimipour-2024-werner-holevo-lindblad-embedding","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Shayan Roofeh and Vahid Karimipour, *The noisy Werner-Holevo channel and its properties*, Phys. Rev. A 109, 052620 (2024), arXiv:2310.15353v7, subsection “Spectral properties of the channel,” ask:\n\n> The question remains whether, within the range of $0\\leq x\\leq \\frac{2}{3}$, the channel can be described as the exponential of Lindbladian dynamics. We will explore this question in our future work.\n\nThis is verbatim from `ModifiedLSChannel-12.tex`, line 249, in the downloaded v7 source, md5 `bcff52a86196a2e1a0054a34c48a8117`. The channel formula is at line 153; its spectral multipliers and determinant are at lines 237–247. The [literature note](../Library/QuantumContext/roofehkarimipour2024wernerholevo.md) records these locators.\n\nThe module defines, for every real $x$, the complex-linear map on $M_3(\\mathbb C)$\n\n$\\Lambda_x(X)=(1-x)X-\\frac{x}{2}X^{\\mathsf T}+\\frac{x}{2}\\operatorname{tr}(X)I_3.$\n\nThis is `wernerHolevo`; the source uses the same formula with $0\\le x\\le1$. `IsGKSLGenerator G` means that there exist $m\\in\\mathbb N$, a Hermitian $H\\in M_3(\\mathbb C)$ and matrices $V_k\\in M_3(\\mathbb C)$ indexed by $k\\in\\mathrm{Fin}\\,m$ such that, for every $X\\in M_3(\\mathbb C)$,\n\n$G(X)=-i(HX-XH)+\\sum_k\\left(V_kXV_k^\\dagger-\\frac12(V_k^\\dagger V_kX+XV_k^\\dagger V_k)\\right).$\n\nThe empty jump family is allowed. `generatorExp G` is the ordinary exponential of the $9\\times9$ complex matrix of $G$ in the matrix-unit basis, transported back to a linear map. Equality is equality on the entire complex matrix space.\n\nIssue [#21274](https://github.com/the-omega-institute/trureturing/issues/21274) fixes the statement: for every $x\\in[0,2/3]$, there exists such a $G$ with $\\operatorname{generatorExp}(G)=\\Lambda_x$ if and only if $x=0$. This statement is proved.","title":"The noisy qutrit Werner–Holevo channel is a GKSL exponential only at zero noise on $[0,2/3]$","url":"https://the-omega-institute.github.io/trureturing-pages/research/roofeh-karimipour-2024-werner-holevo-lindblad-embedding/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2caf4e3b5e26660f6df592622efa1e46aebd3fbe783e0af2cd29ffd22725d1bd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rosenzweig-stanfill-2026-open-problem-1-3-bell-nonvanishing","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rosenzweig and Stanfill state in Open Problem 1.3 (arXiv:2606.04225v1, page 2):\n\n> Show that $p_{2m,S_1}(\\tfrac{1}{2}-m)\\neq0$ for all $m\\in\\mathbb{N}$ where the sequence $S_1$ satisfies \\eqref{Eq:S1}.\n\nTheir Definition 1.1 gives the Bell-polynomial expression\n\n> Given a sequence of numbers, $S$, indexed over a set $J\\supseteq \\mathbb{N}$, we define $p_{j,S}(t):=\\sum_{k=0}^j \\frac{(-1)^k}{k!} \\widehat{B}_{j,k}(s_1,\\dots,s_{j-k+1})t^k=\\frac{(-1)^j}{j!}\\det\\mathcal{N}_{j,S}(t),\\quad j\\in\\mathbb{N}_0,$\n\nand equation (1.8) gives\n\n> $s_k^{(1)}=(1-2^{1-k})\\frac{2 B_{k}}{k}=-\\frac{2}{k}B_k(1/2),\\quad k\\in\\mathbb{N}.$\n\nThe Lean encoding uses `m : ℕ` with `1 ≤ m` for the source's positive naturals and uses the first Bell-polynomial expression.","title":"Rosenzweig–Stanfill Open Problem 1.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/rosenzweig-stanfill-2026-open-problem-1-3-bell-nonvanishing/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb352e1804dbb3d53b99f29816e6a179df0438563365cec5db87c32c0c53ca1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rosenzweig-stanfill-2026-open-problem-1-6-even-residues","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bart Rosenzweig and Jonathan Stanfill, *On the fundamental solutions of two\nnonlocal parabolic equations related to logarithmic Laplacians*,\narXiv:2606.04225v1, Open Problem 1.6(iv), printed page 4:\n\n> Consider the notation of Theorem 1.5. Then the following are conjectured to be true:\n>\n> (iv) For every α ∈ (0, π), (1.13) is equal to zero whenever m ≥ 0 is even.\n\nThe Lean proposition is\n`∀ m : ℕ, Even m → ∀ α : ℝ, 0 < α → α < Real.pi → paperBracket m α = 0`.\nIt includes m = 0 and α = π/2. The bracket uses the literal finite sums\nin (1.13), the primary recursive c-array in (1.14), its separate Bernoulli\nformula at π/2, and the partial ordinary Bell profile sum in (1.20)–(1.21).\nThe convention is B₁ = −1/2. Natural subtraction and integer division\nare used only in indices; coefficient arithmetic takes place in ℂ.\nThe external factor 2 exp(γ_E m)/π is nonzero, so the zero bracket gives\nthe zero value of (1.13).","title":"Rosenzweig–Stanfill even residue vanishing","url":"https://the-omega-institute.github.io/trureturing-pages/research/rosenzweig-stanfill-2026-open-problem-1-6-even-residues/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb777816c673b6f672d8365e6e96691895b518788b095612a92560935f99de7d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rossetti-mancini-winter-schindler-2025-w3-ppt-measurement-not-separable","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rossetti, Mancini, Winter and Schindler, *Observational entropy of quantum\ncorrelations and entanglement*, arXiv:2510.10058v1, Section X.0.4, ask:\n\n> Nevertheless, none of the tests conducted so far have succeeded in detecting entanglement, leaving open the question of whether (M′, 𝟙 − M′) belongs to the SEP class.\n\nThe source and its equations (62), (79), and (83) are identified in the\n[literature note](../Library/QuantumContext/rossetti2025observational.md).\nThe positive membership assertion is refuted under the literal conventions of\n[preregistration #19016](https://github.com/the-omega-institute/trureturing/issues/19016).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Meaning |\n| --- | --- | --- |\n| Three ordered qubits | `Fin 3 → Fin 2` | Eight computational basis configurations |\n| Normalized W and anti-W vectors | `W3`, `W3bar` | The source's three-term vectors with real coefficient $\\sqrt{1/3}$ |\n| Weight-one and weight-two subspaces | `P1`, `P2` | Sums of the three corresponding basis projectors |\n| Six invariant projectors | `Q1` through `Q6` | The six right-hand sides on printed page 22 |\n| Equation (83) | `Mprime` | Coefficients $1,1/2,0,0,2/3,1/12$ |\n| Equation (62): effects proportional to separable states | `IsSEPEffect` | A nonnegative real multiple of `IsFullySeparableLocal`, with three sites and two local basis states |\n| Both effects belong to SEP | `claim` | `IsSEPEffect Mprime` and `IsSEPEffect (1 - Mprime)` |\n| Negative answer | `result` | The negation of the entire conjunction |\n\n`Pi.single` and `Matrix.vecMulVec` supply the basis vectors and rank-one\noperators. The existing `PronkoFredkinNonCyclicAnnihilation.tensor` supplies\nthe tensor product in the proof. These choices preserve the source expressions.\nThe real scalar and density-state representation follow the source's explicit\ngloss of equation (62); they impose no extra hypothesis on the asserted membership.","title":"The optimal W-state PPT measurement is outside SEP","url":"https://the-omega-institute.github.io/trureturing-pages/research/rossetti-mancini-winter-schindler-2025-w3-ppt-measurement-not-separable/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b334096231d984c65af388464c84c339791a85066c880e229c6593e76ac8b8ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rotundo-schwonnek-2023-extended-mub-norm-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Antonio F. Rotundo and René Schwonnek, arXiv:2303.11382v1, Conjecture 1\n(Extended MUB regime), states that a doubly stochastic matrix has mixed operator\nnorm d^(1−lambda−mu) when ((1−mu)/mu)((1−lambda)/lambda) is at least the\nsquare of its second largest singular value. Equation (7) gives the proposed\nvalue for the 1/mu to 1/(1−lambda) norm. Issue #11781 preregistered this\nstatement, its quantifiers, and the refutation route.","title":"The extended MUB mixed-norm equality is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/rotundo-schwonnek-2023-extended-mub-norm-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c1046701c294736b1c51190ae18f6bef15e8993014b622958cd3303c0ad4cd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:rouleux-2019-affine-determinant-antisymmetric-forms","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Rouleux, *Multimode entanglement for fermions*, arXiv:1910.04790v1 (2019), §3, \"Affine antisymmetric forms in higher dimensions or of lower degree\", states in `3fermiProc.tex`, lines 266–269:\n\n> We conjecture that: (1) the affine determinant is\n> the only non degenerate antisymmetric tensor $\\omega$ on ${\\bf C}^d$ of degree $d$,\n> depending on $d+1$ variables, and: (2)\n> there is no non trivial antisymmetric tensor on ${\\bf C}^d$ of degree $d$, depending on more than $d+1$ variables.\n\nThe preceding discussion specifies \"degree being understood as degree of homogeneity\" (line 247) and uniqueness \"up to a factor\" (lines 264–265). Definition 3, lines 251–253, gives the determinant of the differences from the first point. The source and its conventions are described in [the literature note](../Library/Eigenstructure/rouleux2019multimodeentanglement.md).\n\nIssue [#23528](https://github.com/the-omega-institute/trureturing/issues/23528) fixes the polynomial reading. For natural numbers $d,r$, write $X_{i,j}$ for coordinate $j$ of vector variable $i$, with $0\\le i The matrix $U_4^{c}$ given in (21), for which $\\mathcal{B}_1=\\frac{1}{9}$, leads to the most spin-coherent (classical) basis in $\\mathcal{H}_4$.\n\nHere $\\mathcal{B}_1(U)$ is the mean, over the four basis vectors given by the columns of $U$, of the one-qubit anticoherence $\\mathcal{A}_1(\\psi) = 2[1 - \\operatorname{tr}\\rho_1(F\\psi)^2]$, where $F$ sends a spin-3/2 state to the symmetric state of three qubits and $\\rho_1$ is its one-qubit reduced state (Eqs. (8) and (12)–(15)). The verbatim statements are in [the literature note](../Library/QuantumChannels/rudzinski2024extremequantumness.md).\n\nIssue [#18510](https://github.com/the-omega-institute/trureturing/issues/18510) fixes the reading: 1/9 ≤ ℬ₁(U) for every U ∈ U(4), and the printed $U_4^c$ is unitary with ℬ₁ = 1/9. This is proved.","title":"The most classical orthonormal basis of spin 3/2","url":"https://the-omega-institute.github.io/trureturing-pages/research/rudzinski-burchardt-zyczkowski-2024-spin-three-halves-classical-basis/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6764cf1bd96efdecc0b1bcd8b07523678a48f13d41601bc6c3d0e8f8d35b6001.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sa-ribeiro-prosen-bernard-2024-steady-state-block-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Sá, P. Ribeiro, T. Prosen and D. Bernard, *Symmetry Classes of Classical Stochastic Processes*, J. Stat. Phys. 192, 41 (2025), arXiv:2406.17955, classify the symmetries of Markov generators L, real matrices with nonnegative off-diagonal rates and zero column sums (Eqs. (2)–(4)).\n\nA commuting symmetry U splits L into blocks, and the classification applies to each irreducible block that is itself a Markov generator. The authors write after Eq. (5): \"It is, however, natural to conjecture that the block containing the invariant measure of the Markov process (henceforth referred to as the steady-state block) can still be recast into Markov form.\" Appendix A proves this under the extra assumption (A3).\n\nThe verbatim statements are in [the literature note](../Library/StatisticalMechanics/saribeiroprosenbernard2024symmetryclasses.md).\n\nIssue [#17999](https://github.com/the-omega-institute/trureturing/issues/17999) fixes the reading. Take a Markov generator L, a commuting real orthogonal U, and a stationary probability p. Let the steady-state block be a whole eigenspace of U that contains p, with no further unitary symmetry. Then the restriction of L to it is similar to a Markov generator. This conjecture is refuted.","title":"The steady-state block of a Markov generator with a commuting symmetry need not admit a Markov form","url":"https://the-omega-institute.github.io/trureturing-pages/research/sa-ribeiro-prosen-bernard-2024-steady-state-block-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6cdfb34fc7c58e91d06a07217c65cfac5005b561a5126e2f2fcd584cf9e2b96a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:saaltink-2025-q-transversal-representability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Saaltink, *A theory of q-transversals*, arXiv:2503.12201v1, Section 4.2, page 14, states:\n\n> Conjecture 3. All q-transversal matroids are representable.\n\nThe [source note](../Library/FiniteGeometry/saaltink2025qtransversals.md) records Definition 14, Theorem 15 and the representation convention of Section 3.2. The preregistration is [#19965](https://github.com/the-omega-institute/trureturing/issues/19965). The settling declaration is `D5/S3/Combinatorics/Matroids/QTransversalRepresentability.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite field $\\mathbb F_q$ | `F`, `Field F`, `Fintype F` | Arbitrary finite field |\n| Finite family of subspaces of a finite-dimensional vector space | `m`, `k`, `X : Fin k → Submodule F (Fin m → F)` | Coordinates identify the ambient space with $F^m$ |\n| Every basis of a partial q-transversal admits a partial avoiding transversal | `IsPartialQTransversal` | Every independent finite spanning set admits an injective assignment to indices, with each vector outside its assigned subspace |\n| Rank of the q-transversal matroid | `qTransversalRank` | Supremum of dimensions of partial q-transversals contained in the argument subspace |\n| Section 3.2 matrix rank | `representedRank` | Rank of the representing matrix times the transpose of a basis-coordinate matrix |\n| One representing matrix for the whole q-matroid | `claim`, `result` | One extension and one matrix give equality of the two ranks for every subspace |\n| A field extending $F$ | `Algebra F K`, `FiniteDimensional F K` | A finite-dimensional extension strengthens the source requirement |\n\nThe source's phrase “a dimension n subspace” in the partial definition is read with Theorems 15 and 18, which treat partial q-transversals in every dimension. The public definitions and claim retain the literal conventions of #19965. No assumption that the family is coordinate or nested is imposed.","title":"Every q-transversal matroid is representable over a finite extension","url":"https://the-omega-institute.github.io/trureturing-pages/research/saaltink-2025-q-transversal-representability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22682afef43bf57b9b01631a4dc03de61d44133396c76e28fff7aae4ae02a9bd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sabba-2023-a362534-polarization-transfer-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A362534 (Mohamed Sabba, 2023) concerns upper bounds on the transfer of\nz-magnetization from a group `X_n` of spin-1/2 nuclei to a single spin-1/2\nnucleus `A` in NMR. Its COMMENTS define\n\n> The symmetry-constrained upper bounds are given by the function f(n): (1) for\n> even n, f(n) = (2^(1-n))*n*binomial(n-1, n/2) (2) for odd n, f(n) =\n> (2^(1-n))*n*binomial(n-1, (n-1)/2). The adiabatic bounds are given by the\n> function g(n): (3) for even n, g(n) = 2*(1-(2^(-n))*binomial(n, n/2)) (4) for\n> odd n, g(n) = 2*(1-(2^(-n))*binomial(n, (n-1)/2)).\n\nand close with\n\n> Conjecture: the numerator of g(n) is the denominator of f(n)/g(n).\n\nIssue #10062 fixes the readings: `f(n)` and `g(n)` are the rational numbers\ngiven by these formulas for `n ≥ 1` (the entry's offset); numerator and\ndenominator are those of the fraction in lowest terms, as in the entry's own\nlists `g = 1, 1, 5/4, 5/4, 11/8, 11/8, 93/64, …` and\n`f/g = 1, 1, 6/5, 6/5, 15/11, 15/11, 140/93, …`; the conjecture is asserted for\nevery `n ≥ 1`.","title":"Refutation of the numerator conjecture of OEIS A362534","url":"https://the-omega-institute.github.io/trureturing-pages/research/sabba-2023-a362534-polarization-transfer-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d52ddef6829e9553281520ce7d2794519c0d072bf8e1af801c020fc2d947c409.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sahbi-2026-star-forming-bipartite-all-k-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rafik Sahbi, *Upper k-Star-Forming Sets, k-Independence, and Upper\nDomination*, arXiv:2610.03785v2, Conjecture 4.1:\n\n> For every bipartite graph G and every integer k ≥ 1, βk(G) = SFk(G).\n\nA set I is k-independent when every vertex of the induced graph G[I]\nhas degree less than k. The maximum such cardinality is βₖ(G).\nA set S is k-star-forming when, for every outside vertex v,\nG[S ∪ {v}] contains a copy of K₁,ₖ containing v. The copy need not\nbe induced. SFₖ(G) is the maximum cardinality of an inclusion-minimal\nk-star-forming set.\n\nThe Lean `claim` quantifies over every n and every simple graph G on\nFin n with decidable adjacency, and every natural k, assumes `1 ≤ k` and `G.Colorable 2`,\nand asserts `beta G k = SF G k`.\nEvery finite simple graph can be relabelled onto Fin n; all defining\nadjacency, subset and cardinality conditions are invariant under this\nbijection. Positive integers k correspond exactly to naturals with 1 ≤ k.\nThe theorem `result : ¬ claim` refutes the quoted statement.","title":"Sahbi's Conjecture 4.1: bipartite equality for all positive k","url":"https://the-omega-institute.github.io/trureturing-pages/research/sahbi-2026-star-forming-bipartite-all-k-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f3eb37084178978e2437ab01e9e94ed736ff1fecc4be92ed1dd98d0fb631db4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sahbi-2026-star-forming-bipartite-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rafik Sahbi, *Upper k-Star-Forming Sets, k-Independence, and Upper\nDomination*, arXiv:2610.03785v2, Conjecture 4.2:\n\n> For every bipartite graph G, β₂(G) = SF₂(G).\n\nFor a finite simple graph, a set I is k-independent when every vertex of\nG[I] has degree less than k; βₖ(G) is the maximum such cardinality.\nDefinition 2.1 calls S k-star-forming if, for each v outside S,\nG[S ∪ {v}] contains a copy of K₁,ₖ containing v. The copy need not be\ninduced. A star-forming set is minimal if no proper subset is\nstar-forming; SFₖ(G) is the maximum cardinality among these minimal sets.\n\nThe Lean `claim` quantifies over all n and all decidable simple graphs on\nFin n, assumes `Colorable 2`, and asserts `beta G 2 = SF G 2`.\nThe theorem `result : ¬ claim` refutes this statement. Fin n represents\nevery finite vertex type up to bijection, and all the defining adjacency,\nsubset and cardinality conditions are invariant under such a relabelling.","title":"Sahbi's Conjecture 4.2: bipartite two-independence and upper star formation","url":"https://the-omega-institute.github.io/trureturing-pages/research/sahbi-2026-star-forming-bipartite-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4706ae23da3f018be1d81e1e9b518083215a2a8cc5a178edbc940fe72545692.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sahbi-2026-star-forming-triangle-free-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rafik Sahbi, *Upper k-Star-Forming Sets, k-Independence, and Upper\nDomination*, arXiv:2610.03785v2, Conjecture 7.2:\n\n> For every triangle-free graph G, β2(G) = SF2(G).\n\nA set I is k-independent when every vertex of the induced graph G[I]\nhas degree less than k. The maximum such cardinality is βₖ(G).\nA set S is k-star-forming when, for every outside vertex v,\nG[S ∪ {v}] contains a copy of K₁,ₖ containing v. The copy need not\nbe induced. SFₖ(G) is the maximum cardinality of an inclusion-minimal\nk-star-forming set.\n\nThe Lean `claim` quantifies over every n and every simple graph G on\nFin n with decidable adjacency, assumes `G.CliqueFree 3`,\nand asserts `beta G 2 = SF G 2`.\nEvery finite simple graph can be relabelled onto Fin n; all defining\nadjacency, subset and cardinality conditions are invariant under this\nbijection. Positive integers k correspond exactly to naturals with 1 ≤ k.\nThe theorem `result : ¬ claim` refutes the quoted statement.","title":"Sahbi's Conjecture 7.2: triangle-free equality at two","url":"https://the-omega-institute.github.io/trureturing-pages/research/sahbi-2026-star-forming-triangle-free-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eed6fa014e491c7de50754dbfce2de2bb52978444fed2bbb3d76c7eeab20d508.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sahbi-hypercube-subquorum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every integer `n>=2`, prove that the sub-quorum chromatic number of the\nBoolean cube is\n\n psi_sq(Q_n)=2^(n-1).\n\nDefinition 2.2 permits a partial coloring: it is an onto map from a colored\nsupport `S` to a positive set of `k` colors. At every `v` in `S`, the center\nand the same-color colored neighbors must comprise at least half of the\ncenter and all colored neighbors. The center is counted once and uncolored\nneighbors are excluded.","title":"Sahbi Conjecture 6.4","url":"https://the-omega-institute.github.io/trureturing-pages/research/sahbi-hypercube-subquorum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d2f203a6e599c9543ac02d343f18b1ebc4b48326a64c4b3cc54b5af97a05c9f2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:saied-marshall-anand-rieffel-2025-heralding-rate-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Saied, J. Marshall, N. Anand and E. G. Rieffel, *General protocols for the efficient distillation of indistinguishable photons*, Phys. Rev. Applied 23, 034079 (2025), arXiv:2404.14217v4, distill indistinguishable photons with n-photon protocols built from an n × n unitary whose first row is constant (for example the discrete Fourier transform).\n- **The ideal heralding rate.** For such a unitary, h_n(0) = (−1/n)^{n−1}(n − 1)! Σ_{t=0}^{n−1} (n − t)(−n)^t/t! for every n > 2 (Theorem III.5, Eq. (III.9)), and lim h_n(0) = 1/4 (Theorem III.5(2)).\n- **The conjecture.** After a numerical evaluation for every n from 5 to 10⁵ and selected larger n up to 10⁷ (Figure 4): \"Conjecture III.7. For n ≥ 5, the sequence h_n(0) monotonically decreases to 1/4.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/saied2025photondistillation.md).\n\nIssue [#16743](https://github.com/the-omega-institute/trureturing/issues/16743) fixes the reading: the conjecture is the statement about the sequence of Eq. (III.9), strict decrease for n ≥ 5 together with the limit 1/4. Its correction comment fixes the transcription of Eq. (III.9) ((n − 1)! multiplies the sum). The conjecture is proved.","title":"The ideal heralding rate of Fourier photon distillation decreases strictly to 1/4 from n = 5","url":"https://the-omega-institute.github.io/trureturing-pages/research/saied-marshall-anand-rieffel-2025-heralding-rate-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4bc8f43bbfd9924eb18bb0fec39ee81e1d5eef042baa706d3d16d87718bb2122.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sain-manna-paul-2024-linf-k-number","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Sain, J. Manna, K. Paul, *On local preservation of orthogonality and its application to isometries*, arXiv:2407.08900v1, *Linear Algebra and its Applications* (2024), states:\n\n> Definition 1. Let X be a Banach space. A set A ⊂ S_X is said to be a Koldobsky-set, for the sake of brevity, a K-set, if any operator T ∈ L(X) that preserves Birkhoff-James orthogonality at each point of A, is necessarily a scalar multiple of an isometry.\n\n> Definition 3. Let X be a Banach space having a finite K-set. Then the K-number of X is defined by min{|A| : A is a K-set of X}. We denote the K-number of X by κ(X).\n\nThe Open Question at the end of §2 is:\n\n> Open Question: Characterize the minimal K-sets in ℓ^n_∞ and hence find κ(ℓ^n_∞).\n\nThe spaces are real, and minimality is under inclusion. The result determines the K-number for $n\\ge2$ and characterizes the minimal K-sets consisting of extreme points; it leaves the classification involving non-extreme points open.","title":"The K-number and extreme-point K-sets of real finite-dimensional sup-norm spaces","url":"https://the-omega-institute.github.io/trureturing-pages/research/sain-manna-paul-2024-linf-k-number/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c66519e0fef30bc7331ebd8ca368e63f82d31d04d69fd659b1bad3f4eb02f71.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sajjad-kaur-goodenough-towsley-guha-2026-coherent-information-ordering-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Sajjad, E. Kaur, K. Goodenough, D. Towsley and S. Guha, *Lower bounds on bipartite entanglement in noisy graph states*, arXiv:2404.09014v2; Quantum 10, 2111 (2026), §5.4, “The subsystem with more qubits has higher coherent information?” (`main.tex`, line 951):\n\n> In all of the examples that we have considered, we have observed that the subsystem with the most qubits has the largest coherent information regardless of the types of the two subsystems and their distributions. Therefore, we conjecture that this is generally true but are unable to make a definitive statement.\n\nThe quotation is from the downloaded v2 TeX, `main.tex`, md5 `230ee16b992b9567441d518d7471253e`; the trailing TeX comment is omitted. The source's noise model is in §2.2, lines 186–194, and its entropy and coherent-information definitions are in §2.3, lines 201–216. See also [the literature note](../Library/QuantumContext/sajjadkaurgoodenoughtowsleyguha2026noisygraph.md).\n\nIssue [#21154](https://github.com/the-omega-institute/trureturing/issues/21154) fixes the following non-strict reading. For natural numbers $n_A,n_B$ with $n_B For the case with decay also there are more general examples we can consider but in all cases we conjecture that CoE ≤ QFI with equality at specific points.\n\n[The source note](../Library/QuantumBounds/saleem2025curvature.md) locates Section II,\nEq. (7), the flip-flop Hamiltonian (21), the open evolution (41), the particular\nexamples (42)–(43), and Appendix B. [Preregistration #17681](https://github.com/the-omega-institute/trureturing/issues/17681)\nfixes the literal conventions and the unrestricted-initial-density-matrix reading.\nSection III explicitly introduces losses and mixed probe states after treating pure\nlossless evolution. The resolution is **Refuted**.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Two qubits in the computational basis | Matrices indexed by $\\mathrm{Fin}\\,2\\times\\mathrm{Fin}\\,2$ over $\\mathbb C$ | The matrix type is written out, with no forwarding abbreviation. |\n| Appendix B: $\\sigma_-=(\\sigma_x-i\\sigma_y)/2=|1\\rangle\\langle0|$ | Reused `PronkoFredkinNonCyclicAnnihilation.sigmaMinus`; `jump1`, `jump2` | The two jumps are $\\sigma_-\\otimes I$ and $I\\otimes\\sigma_-$. |\n| Eq. (21), $\\eta_{xy}=1$ | `hamiltonian` | Only the $01,10$ and $10,01$ entries equal $g$. |\n| Eq. (41), dissipator and commutator | `dissipator`, `generator` | Both jumps have the same positive rate $\\kappa$. |\n| Parameter-independent initial density matrix | `claim` quantifies over `rhoZero`; reused `StructuredNegativityCoincidenceRefutation.IsDensity` | Positive semidefinite, trace one. |\n| Evolution with that initial state | `IsTrajectory` | Initial equality for every real $g$ and entrywise time derivatives for every real $g,t$. |\n| Section II, Wootters concurrence | `sigmaYY`, `concurrence` | Characteristic-polynomial roots with multiplicity, real square roots in decreasing order, and $\\max(0,\\lambda_1-\\lambda_2-\\lambda_3-\\lambda_4)$. |\n| Eq. (7), $\\mathrm{CoE}=-\\partial_g^2C$ | `derivG` and the negative iterated scalar derivative in `claim` | Fixed observation time. |\n| Hermitian SLD, $\\partial_g\\rho=(\\rho L+L\\rho)/2$ | `IsSLD`; every such matrix $L$ is quantified in `claim` | QFI is $\\operatorname{Re}\\operatorname{Tr}(\\rho L^2)$. |\n| Positive damping and observation time | `claim` hypotheses $0<\\kappa$, $0 Note that a lower bound is the Chernoff bound of the two pure output states $|0\\rangle\\langle0| = \\mathcal{M}(|00\\rangle\\langle00|)$ and $|+\\rangle\\langle+| = \\overline{\\mathcal{M}}(|00\\rangle\\langle00|)$, which is $\\log 2 = 1$, so $C^{\\mathbb{P}}(\\mathcal{M},\\overline{\\mathcal{M}}) \\geq 1$. It seems reasonable to conjecture that this is optimal, but we do not have at present a proof of it.\n\nThe formal proposition is `claim`, namely $\\operatorname{Tendsto}\\;\\operatorname{exponent}\\;\\operatorname{atTop}\\;(\\operatorname{nhds}(1:\\mathrm{EReal}))$.","title":"The Harrow channel pair's parallel Chernoff exponent is not one","url":"https://the-omega-institute.github.io/trureturing-pages/research/salek-hayashi-winter-2020-harrow-parallel-chernoff-exponent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cdbb93033e1271d5fc2d495f480d485768b383772f82d423376f7681a2537049.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:samadzadeh-mojdeh-nadimi-2024-paired-coalition-binary-tree","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Samadzadeh, Mojdeh and Nadimi, *Paired coalition in graphs*,\narXiv:2402.10842v4, Section 4.2, p. 12, Conjecture 1:\n\n> If T(h) is a perfect binary tree with height h, then PC(T(h)) = 5 if h = 2; 3 if h is odd; 0 otherwise.\n\nThe domain is every positive natural height. The root has depth zero and\nall leaves have depth $h$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Perfect binary tree of height $h$ | `PairedCoalitionBinaryTree.perfectBinaryTree h` | Heap labels $1,\\ldots,2^{h+1}-1$, children $2r,2r+1$. |\n| Dominating set whose induced subgraph contains a perfect matching | `IsPairedDominating` | Reused `SimpleGraph.IsDominating` and a matching subgraph with exactly the selected vertices. |\n| Disjoint sets, neither paired dominating, with paired dominating union | `IsPairedCoalition` | All four requirements are conjuncts. |\n| Vertex partition; each part has another coalition partner | `IsPCPartition` | Nonempty, pairwise disjoint, covering parts; distinct partners in the same finset. |\n| Maximum partition order; zero if none exists | `pcNumber` | Natural supremum of attained orders; finite vertex sets bound all orders. |\n| Every positive height | `claim` | $h:\\mathbb N$ with $1\\le h$. |\n| Height two first, then odd heights, then otherwise | `claim` | Values $5,3,0$ in exactly that order. |\n| Conjecture is false | `result` | Negation of the complete positive-height claim. |\n\nAll declaration names in this dossier belong to\n`D5.S3.Combinatorics.Graph.Domination.SamadzadehMojdehNadimiRefutation.PairedCoalitionBinaryTree`\nunless another owner is explicit. No additional hypothesis restricts the conjecture.","title":"Conjecture 1 on paired coalition numbers of perfect binary trees is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/samadzadeh-mojdeh-nadimi-2024-paired-coalition-binary-tree/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/048a1a03ed214ab9ca38fc7264aa03f363334bddf1e075207cd466ca17bd6d12.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:samanta-mondal-sen-2025-diagonal-unitary-entangling-power-appendix-b-3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Samanta, S. Mondal and A. Sen (De), *Disparity between multipartite entangling and disentangling powers of unitaries: Even vs Odd*, arXiv:2505.18539v1, Appendix B.3, page 8, states:\n\n> It also indicates that such an equivalence may hold for arbitrary diagonal unitary operators acted on arbitrary number of qubits.\n\nThe equivalence is $E_N(U)=D_N(U)$, where Eq. (1) maximizes the generalized geometric measure over fully separable inputs, Eq. (2) defines $D_N(U)=E_N(U^\\dagger)$, and Appendix A.1 defines the generalized geometric measure by overlap with non-genuinely multipartite entangled pure states.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Any number of qubits | `claim` quantifies over every natural `N` | The physical multipartite interpretation concerns $N\\ge2$; the same formal definitions include $N=0,1$. |\n| Every diagonal unitary | Real phase function `θ` on `Fin N → Fin 2`, with diagonal entries $e^{i\\theta_x}$ | Every computational-basis phase is independent. |\n| Fully separable normalized inputs | `IsFullySeparable` | Squared-modulus sum one and a product of local amplitudes. |\n| Non-genuinely multipartite entangled pure states | `IsBiseparable` | Normalization and factorization across a bipartition with both parts nonempty. |\n| GGM | `G` | One minus the supremum of squared overlap over the biseparable subtype. |\n| Entangling and disentangling powers | `E`, `D` | Suprema over fully separable inputs, with the adjoint in `D`. |\n| Equality | `result : claim` | No additional hypotheses. |\n\nThe normalization expression is inlined; the overlap is Mathlib's `dotProduct` applied to the conjugated first argument. Maxima are represented by suprema. The argument identifies the entire value families, so it needs no attainment assertion or restriction on input local phases.","title":"Equal multipartite powers of diagonal phase gates","url":"https://the-omega-institute.github.io/trureturing-pages/research/samanta-mondal-sen-2025-diagonal-unitary-entangling-power-appendix-b-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e7da3cea043660264c552214b43bc8e8f47bdd811a2615a7dadd382bf27fbe3f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:saniga-et-al-2025-hexagon-contextuality-generalization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Saniga, F. Holweck, C. Kelleher, A. Muller, A. Giorgetti and H. de Boutray, *Hexagons govern three-qubit contextuality*, Quantum 9 (2025) 1601, DOI 10.22331/q-2025-01-20-1601; arXiv:2312.07738v3, Section 4, `main.tex`, lines 736–740:\n\n> The message from the above-given observations is more than obvious: the *degree of contextuality* of a contextual quantum geometry of the three-qubit $\\mathcal{W}(5,2)$ whose contexts are i) all the 315 lines of the whole space $\\mathcal{W}(5,2)$, ii) all the 45 lines of an elliptic quadric, iii) all the 105 lines of a hyperbolic quadric or iv) all the 15 lines of a doily is equal to the *number of lines* each of these geometries shares with a classically-embedded hexagon, with the understanding that each shared line corresponds to an unsatisfiable context. A natural generalization of these findings is that this should hold for *any* contextual sub-geometry of $\\mathcal{W}(5,2)$ whose contexts are *lines*.\n\nThe next paragraph begins, “As a particular stance of this conjecture, let us consider the complement of a skew-embedded hexagon” (`main.tex`, lines 742–744), so the proposed generalization extends beyond the four named families. Definitions and source locators are also in [the literature note](../Library/FiniteGeometry/saniga2025hexagons.md).\n\nFor the formal reading, put $V=(\\mathbb F_2^2)^3$. Its nonzero vectors label the canonical three-qubit Pauli observables $P_v$. Write $B$ for the binary symplectic form. Lines are the triples $\\{u,v,u+v\\}$ with $u,v$ nonzero, distinct and $B(u,v)=0$. Set $s(\\ell)=1$ when $\\prod_{q\\in\\ell}P_q=-I$, and $s(\\ell)=0$ otherwise. For a finite family $C$ of these lines, define $d(C)=\\min_{f:V\\to\\mathbb F_2}|\\{\\ell\\in C:\\sum_{q\\in\\ell}f(q)\\ne s(\\ell)\\}|$.\n\nIssue [#24997](https://github.com/the-omega-institute/trureturing/issues/24997) fixes `claim`: for every such $C$ with $d(C)>0$, there exists $H$ satisfying `IsClassicalHexagon` with $d(C)=|C\\cap H|$. The predicate, specified below, includes all the source's classical embeddings. This claim is refuted.","title":"The perp hyperplane refutes the three-qubit hexagon contextuality generalization","url":"https://the-omega-institute.github.io/trureturing-pages/research/saniga-et-al-2025-hexagon-contextuality-generalization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c179944432617917f4e5ba045872e0fb3b494b0c9653c87d0ae97c40144c75ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:santalla-singha-roy-sierra-rodriguez-laguna-2026-even-legged-reconstruction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. N. Santalla, S. Singha Roy, G. Sierra and J. Rodríguez-Laguna, *The hyperlink representation of entanglement and the inclusion-exclusion principle*, arXiv:2601.17926v2, Phys. Rev. A **114**, 022431 (2026), define the entanglement hyperlinks J_I = Σ_{A⊆I} (−1)^{|I|−|A|} S_A of the entanglement entropies S_A of a pure state on N parties (Eq. (5)).\n\nAppendix B proposes the even-legged reconstruction S_A = Σ_{|I| even} Λ_{|I|,|I∩A|} J_I (B2), with Λ_{2l,0} = Λ_{2l,2l} = 0 and Λ_{2l,p} = Λ_{2l,2l−p}. It lists the leg-factors for 2l ≤ 8 (B5) and states: \"It is still to be proved that the leg-factors Λ_{2l,p} exist and do not depend on N\" and \"the validity of Eq. (B2), with leg-factors that do not depend on N, has only been checked up to N = 9\". The verbatim statements are in [the literature note](../Library/Entropy/santalla2026hyperlink.md).\n\nIssue [#16381](https://github.com/the-omega-institute/trureturing/issues/16381) fixes the reading: there are leg-factors, independent of N, with the zero and symmetry laws, such that (B2) holds for every N. It is proved.","title":"Even-legged hyperlinks reconstruct the entanglement entropies with N-independent leg-factors","url":"https://the-omega-institute.github.io/trureturing-pages/research/santalla-singha-roy-sierra-rodriguez-laguna-2026-even-legged-reconstruction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c5e55ec218970af0c6b6559d0f15a8ab7a30d70ea1fa1b9f4315aed4eec585b8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sanz-wolf-perez-garcia-cirac-2009-spin-three-halves-u1-injective-mps","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Sanz, M. M. Wolf, D. Pérez-García and J. I. Cirac, *Matrix product states: Symmetries and two-body Hamiltonians*, Phys. Rev. A 79, 042308 (2009), arXiv:0901.2223v1, Section IV, `Symmetries191208.tex`, lines 748–751:\n\n> Let us remark that this counter-example is applicable to spin $\\ge\n> \\frac{5}{2}$. Indeed, one can prove Theorem \\ref{} for $U(1)$ and\n> spin $\\frac{1}{2}$, which is the content of the following\n> proposition. The case of spin $3/2$ remains an open question.\n\nThe preceding example, lines 726–746, uses general generator eigenvalues, all nonzero, at physical dimension $d=D^2-D$ with bond dimension $D\\ge3$. The question concerns the remaining four-level case. Issue [#23698](https://github.com/the-omega-institute/trureturing/issues/23698) fixes its general-generator reading as follows.\n\nFor $D\\in\\mathbb N$, let $A:\\mathrm{Fin}\\,4\\to\\mathcal M_D(\\mathbb C)$, let $W$ be a unitary $4\\times4$ matrix, and let $\\lambda:\\mathrm{Fin}\\,4\\to\\mathbb R$. Define\n\n$$u(\\varphi)=W\\operatorname{diag}(e^{i\\varphi\\lambda_k})W^\\dagger,\\qquad\n\\Psi_N(w)=\\operatorname{tr}(A_{w_0}\\cdots A_{w_{N-1}}).$$\n\n`IsSymmetric` means that for every real $\\varphi$ and every positive $N$ there is a real $\\theta$, allowed to depend on both, with $u(\\varphi)^{\\otimes N}\\Psi_N=e^{i\\theta}\\Psi_N$. `IsInjective` means that for some natural block length $n$ the map\n\n$$\\Gamma_n:X\\longmapsto\\bigl(w\\longmapsto\\operatorname{tr}(XA_{w_0}\\cdots A_{w_{n-1}})\\bigr)$$\n\nis injective on $\\mathcal M_D(\\mathbb C)$. These are the trace-amplitude, symmetry and injectivity conventions of the source at lines 222–237, 134–140 and 267–276, respectively; [the literature note](../Library/StatisticalMechanics/sanzwolfperezgarciacirac2009mps.md) records the locators.\n\n`claim` asserts universally in $D,A,W,\\lambda$ that if every $A_k\\ne0$, $\\sum_kA_kA_k^\\dagger=I_D$, every $\\lambda_k\\ne0$, at least two charges differ, and `IsSymmetric` holds, then `IsInjective` does not hold. This assertion is refuted. The specified reading concerns four physical levels with general charges; it does not identify their generator with the usual spin-$3/2$ matrix $J_z$.","title":"Four physical levels admit a $U(1)$-symmetric injective matrix-product family","url":"https://the-omega-institute.github.io/trureturing-pages/research/sanz-wolf-perez-garcia-cirac-2009-spin-three-halves-u1-injective-mps/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6497afde9ee18c360e64c4f7eabca6e63c7d2a9d423575a1353fc6284260359.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sarkis-prosniak-nigro-tkatchenko-2026-uniform-overlap-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Sarkis, O. A. Prośniak, S. Nigro and A. Tkatchenko, *Star Topology Optimizes the Charging Power of Quantum Batteries*, arXiv:2603.10853v1, study a battery of N fermionic cells coupled along a graph G.\n- **Overlaps and mode energies.** The adjacency matrix A has orthonormal eigenvectors u_k with eigenvalues ε_k and uniform overlaps w_k = (1ᵀu_k)² (Eq. (11)). With a local field h, the mode energies are E_k(h) = (N/M)ε_k + h (Eq. (14)).\n- **The charging power.** It is a weighted mean of the w_k, and the authors single out w_min = Σ_{k′} w_{k′} over the modes of least energy.\n- **The star.** It gives w_min(S) = N/2 − √(N − 1) (Eq. (23)).\n- **The conjecture.** \"Conjecture 1 (Star optimality for the uniform overlap). Let G be a non-empty graph on N vertices. Its minimal energy uniform overlap w_min(G) is upper bounded by w_min(G) ≤ w_min(S) = N/2 − √(N − 1).\" The authors base it on an exhaustive check for N ≤ 7 and random samples.\n\nThe verbatim statements are in [the literature note](../Library/GraphInvariants/sarkis2026starbattery.md).\n\nIssue [#16578](https://github.com/the-omega-institute/trureturing/issues/16578) fixes the reading.\n- Because E_k(h) is increasing in ε_k, w_min(G) is the squared length of the projection of 1 onto the lowest adjacency eigenspace.\n- \"Non-empty\" means at least one edge.\n- The conjecture is refuted.","title":"A 1225-vertex graph refutes the star bound on the minimal-energy uniform overlap of quantum batteries","url":"https://the-omega-institute.github.io/trureturing-pages/research/sarkis-prosniak-nigro-tkatchenko-2026-uniform-overlap-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/403835bc1eb606ebf4b3a1f28c6031c8843e2097b2868fa4073bd0ec5ea0ce87.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:schnell-ketzmerick-eckardt-2018-bose-selection-cubic-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Schnell, R. Ketzmerick and A. Eckardt, *On the number of Bose-selected modes in driven-dissipative ideal Bose gases*, Phys. Rev. E 97, 032136 (2018), DOI 10.1103/PhysRevE.97.032136, arXiv:1802.04088v2, section “Rates with product structure”, `main.tex` lines 1173–1178 (TeX macros expanded and mathematical delimiters normalized):\n\n> We expect that there is a generalization of the above arguement also to non-ohmic systems described by arbitrary spectral densities. We have checked functions of the form (remember that $J$ must be odd) $J(\\varepsilon)\\propto\\vert\\varepsilon\\vert^d\\mathrm{sgn}(\\varepsilon)$ with some power $d$ (not necessarily integer) and similarly observe $M_S\\leq2M_B-1$.\n\nHere $M_B$ is the number of baths and $M_S=|S|$ the number of selected modes. The two-bath matrix in `main.tex` lines 1062–1083 uses opposite bath spectra and projector couplings, with weights $f_i=|\\langle i|f\\rangle|^2$ and $g_i=|\\langle i|g\\rangle|^2$. The selection condition is Eq. `MS-problem`, lines 374–382. These definitions are also recorded in [the literature note](../Library/StatisticalMechanics/schnellketzmerickeckardt2018bose.md).\n\nIssue [#25984](https://github.com/the-omega-institute/trureturing/issues/25984) fixes a generic reading for $M_B=2$. For $m\\in\\mathbb N$, take unrestricted real energies $\\varepsilon$ and coordinatewise positive real vectors $F,G$, and put $f_i=F_i/\\sum_jF_j$, $g_i=G_i/\\sum_jG_j$. The module's `asym` is\n$A_{ij}=J_d(\\varepsilon_j-\\varepsilon_i)(f_if_j-g_ig_j)$,\nwhere `spow` defines $J_d(0)=0$ and $J_d(x)=x|x|^{d-1}$ for $x\\ne0$.\n\n`IsSelected A S w` means that, for every index $i$, membership in $S$ implies $w_i>0$ and $(Aw)_i=0$, while nonmembership implies $w_i=0$ and $(Aw)_i<0$. Write $V_{d,m}$ for `violating d m`: the parameter triples $(\\varepsilon,F,G)$ with $F,G>0$ admitting such $S,w$ with $|S|\\ge4$. The proposition `claim` says that for every real $d>0$ and every natural $m$, $\\operatorname{int}(V_{d,m})=\\varnothing$ in the full real parameter space $\\mathbb R^{3m}$. This generic reading is weaker than an unconditional three-mode ceiling; it is refuted.","title":"The expected two-bath Bose-selection bound is refuted for cubic spectra","url":"https://the-omega-institute.github.io/trureturing-pages/research/schnell-ketzmerick-eckardt-2018-bose-selection-cubic-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30e07eb41a0d5209ad1b86ff258e423f487a41cec4347aca5772446cf7d07409.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:schoute-grinko-subasi-volkoff-2024-flat-spectrum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Schoute, D. Grinko, Y. Subaşı and T. Volkoff, *Quantum programmable reflections*, arXiv:2411.03648v2, Section 4.2.1, Conjecture 4.6:\n\n> The probabilities $q_j$ for $j\\in\\{(n\\bmod 2)/2,\\ldots,n/2\\}$ in Eq. (181), which define the probe state $\\Phi_n$, could be chosen such that $\\lVert|v_J\\rangle\\rVert^2=1/\\binom{n+2}{2}$ for every $J\\in\\{n\\bmod 2,\\ldots,n-2,n\\}$.\n\nEquation (183) is the equivalent system\n\n$$\n\\frac{2J+1}{\\binom{n+2}{2}}\n =\\sum_{j\\ge J/2}\\frac{q_j}{2j+1}\n \\left|\\sum_{m=-j}^{j}C^{J0}_{jm,j-m}\\right|^2.\n$$\n\nIssue [#17082](https://github.com/the-omega-institute/trureturing/issues/17082) fixes the doubled-integer Racah convention and this system as the binding statement. The settlement is **Proved** by `D5.S3.Quantum.Measurement.SGVFlatSpectrum.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every copy number $n\\ge1$ | `claim` quantifies natural $n$ with $1\\le n$ | Exact |\n| $j=n/2-k$, $0\\le k\\le\\lfloor n/2\\rfloor$ | The finite range in `claim` | Exact reindexing |\n| Probabilities | Nonnegativity and the sum equal to one in `claim` | Exact |\n| Real Condon–Shortley coefficients | `S` uses the frozen doubled-spin `CG` owner; `equal_spin_coefficient` proves its source Racah expression on the admissible summation inputs | Exact binding convention |\n| Coherent sum over magnetic indices | `S` sums $i=0,\\ldots,2j$, with $m=j-i$ | Exact |\n| $J\\le n$ of parity $n$ | The bound and remainder equality in `claim` | Exact |\n| Terms with $2j\\ge J$ | The conditional inside the finite sum | Exact |\n| Flat spectrum (182) | The equality (183) in `claim`; source Eq. (181) divides by $2J+1$ | Source algebraic equivalence |\n\nInteger division inside Racah's formula is Euclidean division; factorials use `Nat.factorial` after `Int.toNat`. Natural subtractions are truncated, and the summation guards preserve their intended ranges. The coefficient owner `DampedSpinKernelPositivityRefutation.CG` and this module have generality I. `S` calls the owner in doubled-integer argument order; the Scribe displays the source Racah expression. Inside `equal_spin_coefficient`, the square-root product and summation-range correspondence establish equality on every admissible summation input.","title":"Explicit flat-spectrum probe probabilities","url":"https://the-omega-institute.github.io/trureturing-pages/research/schoute-grinko-subasi-volkoff-2024-flat-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cdcc00954b66638cdad530bb0343ea016282a9885b1ebd9fbd6fe9de8a814d47.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sedlak-reitzner-chiribella-ziman-2016-ancilla-free-tester-robustness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Sedlák, D. Reitzner, G. Chiribella and M. Ziman, *Incompatible measurements on quantum causal networks*, Phys. Rev. A 93, 052323 (2016), arXiv:1511.00976v2, Section VI.C, ask (verbatim from `ti_pra_07.tex`, line 807, retaining the source's TeX macros):\n\n> It is an open question whether the equality $\\Rt({\\st A}, {\\st B}) = \\Rm( \\st P, \\st Q )$ holds for all ancilla-free testers.\n\nHere $R_{\\rm t}$ and $R_{\\rm m}$ denote tester and measurement robustness. Section VI restricts attention to testers with the same normalization state (`ti_pra_07.tex`, lines 758 and 772); that hypothesis is retained. Proposition 3 characterizes ancilla-free testers as $T_a=M_a\\otimes\\rho$ for an output POVM $M$ and an input density matrix $\\rho$ (lines 333–338). Definitions and locators are in [the literature note](../Library/Observer/sedlakreitznerchiribellaziman2016testers.md).\n\nIssue [#23653](https://github.com/the-omega-institute/trureturing/issues/23653) fixes the following finite-dimensional reading. Let $y,x,j,k$ be finite types with decidable equality, indexing output coordinates, input coordinates and the two outcome sets. Let $\\rho$ be a complex matrix on $x$ with $\\rho\\succeq0$ and $\\operatorname{Tr}\\rho=1$. Let $M=(M_a)_{a\\in j}$ and $N=(N_b)_{b\\in k}$ be complex output POVMs: each effect is positive semidefinite and each family sums to $I_y$. With $r=\\operatorname{rank}\\rho$, the statement is\n\n$$\nR_{\\rm t}\\bigl((M_a\\otimes\\rho)_a,(N_b\\otimes\\rho)_b\\bigr)\n=R_{\\rm m}\\bigl((M_a\\otimes I_r)_a,(N_b\\otimes I_r)_b\\bigr).\n$$\n\nIn the module, `IsTester` means positive semidefinite effects summing to $I_y\\otimes\\sigma$ for some positive semidefinite $\\sigma$ of trace one. `TesterCompatibleAt` at a real $\\lambda$ requires noise testers $E,F$ and a joint tester $C$ with\n\n$$\n\\sum_b C_{ab}=(1-\\lambda)A_a+\\lambda E_a,\n\\qquad\n\\sum_a C_{ab}=(1-\\lambda)B_b+\\lambda F_b.\n$$\n\nThe noises have separate existential witnesses; their normalization states are not prescribed to equal $\\rho$, and their effects need not be products. `POVMCompatibleAt` uses the same marginal equations with noise POVMs and a joint POVM, each positive semidefinite and summing to the identity. `testerRobustness` and `povmRobustness` are the real infima of their respective feasible $\\lambda\\in[0,1]$. The source uses minima; attainment is not part of `claim` or `result`. The displayed equality, with these infimum definitions and $I_r$ indexed by `Fin ρ.rank`, is proved.","title":"Ancilla-free testers with a common normalization state have canonical POVM robustness","url":"https://the-omega-institute.github.io/trureturing-pages/research/sedlak-reitzner-chiribella-ziman-2016-ancilla-free-tester-robustness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/130116b60e888d4a4e7c5b2aae19bacbab6264cabe1b5eeb1f60d942077a3174.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sekatski-guryanova-kothakonda-skotiniotis-2025-amplitude-damping-measure-prepare-cloning","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Sekatski, Y. Guryanova, N. B. T. Kothakonda and M. Skotiniotis, *Cloning Quantum Channels*, arXiv:2509.08059v1 (2025), Section V, amplitude-damping channels, performance comparison, state:\n\n> In particular, it achieves ${\\rm F}_{\\rm CJ}^{\\ADset|\\rm\n> coh}=1$ at $N=M$, which, for amplitude-damping channels, we conjecture to be impossible for any measure-and-prepare process.\n\nThis quotation is from the downloaded v1 source `arxiv_v1.tex`, lines 1684–1685, md5 `abcdd29ea99d1c5adf0206c6b9a9d211`. The source macro `\\ADset` denotes the amplitude-damping family; the coherent process is the subject of “it”. The arXiv abstract page supplies the arXiv-issued DOI in the front matter.\n\nThe source defines measure-and-prepare processes as a measurement consuming $N$ channel queries, followed by an outcome-dependent global $M$-qubit channel preparation (TeX lines 1094–1103). Its preparations need not be tensor powers of an estimated channel; that narrower class is introduced separately at lines 1105–1115.\n\nIssue [#20683](https://github.com/the-omega-institute/trureturing/issues/20683) fixes the target as impossibility of one such process reproducing $\\mathcal E_\\gamma^{\\otimes N}$ for every $\\gamma\\in[0,1]$, at $N=M\\geq1$. This is proved for finite outcome sets with the trace-pairing interpretation specified below. The compiled statement also allows every $N\\geq0$ and every $M\\geq1$ independently.\n\nPrecisely, the module defines the following objects:\n\n- `adChoi` is the unnormalized, reference-first Choi matrix $J_\\gamma$ of the source's Kraus operators $A_0=\\operatorname{diag}(1,\\sqrt{1-\\gamma})$ and $A_1=\\begin{pmatrix}0&\\sqrt\\gamma\\\\0&0\\end{pmatrix}$ (source TeX lines 1538–1553). Its entry at $((r,o),(r',o'))$ is $\\sum_{k=0}^1 A_k(o,r)\\overline{A_k(o',r')}$.\n- `D5.S3.Quantum.Recovery.ProductPrefixRigidity.productMap (H := fun _ => Fin 2 × Fin 2) (fun _ : Fin n => J_γ)` supplies the coordinate tensor power: for assignments $x,y:\\operatorname{Fin}n\\to\\operatorname{Fin}2\\times\\operatorname{Fin}2$, its entry is $\\prod_i J_\\gamma(x_i,y_i)$. At $n=0$ this is the one-dimensional matrix with entry one.\n- `IsChannelChoi` requires a preparation matrix to be positive semidefinite and to have reference marginal equal to the identity: $\\sum_o K((r,o),(r',o))=\\delta_{r,r'}$, where $r,r',o:\\operatorname{Fin}M\\to\\operatorname{Fin}2$ and each pair denotes its sitewise assignment.\n- `claim` says that, for every pair of natural numbers $N,M$ with $M\\geq1$, there are no finite $L$, fixed complex matrices $T_\\ell$ on the $N$-fold Choi space and fixed preparation matrices $K_\\ell$ on the $M$-fold Choi space satisfying `IsChannelChoi`, such that for every $\\gamma\\in[0,1]$ the numbers $p_\\ell(\\gamma)=\\operatorname{tr}(T_\\ell J_\\gamma^{\\otimes N})$ are real and nonnegative, sum to one, and satisfy $\\sum_{\\ell=0}^{L-1}p_\\ell(\\gamma)K_\\ell=J_\\gamma^{\\otimes M}$. The real nonnegative requirement is expressed using the complex order. No further positivity or tester constraints are imposed on $T_\\ell$.\n\nThe definitions and the source quotations are also described in [the literature note](../Library/QuantumContext/sekatskiguryanovakothakondaskotiniotis2025cloning.md).","title":"No finite measure-and-prepare process perfectly clones the amplitude-damping family","url":"https://the-omega-institute.github.io/trureturing-pages/research/sekatski-guryanova-kothakonda-skotiniotis-2025-amplitude-damping-measure-prepare-cloning/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83b2735e8d03a2ed43d801b8408d57720757e8235f0583a1815e8ad8facb6d86.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:seong-bae-2026-optimal-mirrored-witness-sum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Seong, A. Bera, B. C. Hiesmayr, D. Chruściński and J. Bae, *Mirrored Entanglement Witnesses for Multipartite and High-Dimensional Quantum Systems*, arXiv:2510.06863v2, Quantum Sci. Technol. 11, 015037 (2026), ask in “Open problems”, item 2:\n\n> Optimal EWs can be mirrored via another EW, as shown in Eq. (\\ref{eq:gmeew_form}). Since optimal EWs are generally not extremal, yet, it remains unanswered whether a pair of optimal EWs can be mirrored through another optimal EWs.\n\nThis quotation is from the downloaded v2 source `mn_revision_clean.tex`, line 1749, MD5 `08daa608d307e41e61bcd9c0d9f06c31`. The generalized mirroring relation is $W+M=K$ in lines 1688–1692; the witness definition, detection sets, optimality and spanning criterion are in lines 276–281, 388–396 and 485–490. See [the literature note](../Library/QuantumContext/seongberahiesmayrchruscinskibae2026mirrored.md).\n\nIssue [#20970](https://github.com/the-omega-institute/trureturing/issues/20970) fixes a nontrivial reading: for every natural number $d\\ge3$, there are optimal witnesses $W,M$ on $\\mathbb C^d\\otimes\\mathbb C^d$ such that $W+M$ is optimal, each detects a density matrix the other does not, and some density matrix is detected by both. This is proved.\n\nHere a density matrix is positive semidefinite with trace one; a witness is Hermitian, block-positive on every product vector, and has negative real trace pairing with some density matrix. Optimality means that for every real $\\epsilon>0$ and every nonzero positive semidefinite $P$, the operator $A-\\epsilon P$ is not block-positive. Write $D_A=\\{\\rho:\\rho\\text{ is a density matrix},\\ \\operatorname{Re}\\operatorname{tr}(A\\rho)<0\\}$.","title":"Distinct optimal entanglement witnesses can have an optimal sum","url":"https://the-omega-institute.github.io/trureturing-pages/research/seong-bae-2026-optimal-mirrored-witness-sum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/298a00bcb814e92a81d8d541187e3da9f5bf9e32f2621b65675099c2e9bc6626.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:separable-occupied-diagonal-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For each positive length n, let M_n(e) count actual permutations avoiding\n2413 and 3142, with no proper direct cut, and having at least one position\nj with j-sigma(j)=e. Count each permutation once. Put b=sqrt(2)-1,\nrho=b squared, a=1-b and O(e)=(2/a)sum_(n>=1)rho^n M_n(e).\nThe target is a uniform positive constant c_* such that, for every integer e,\n\n$$\nO(e)\\ge\\frac{c_*}{\\sqrt{2+|e|}\\log(2+|e|)}.\n$$\n\nThe empty permutation is excluded and the singleton is retained. The raw\nmass C=(a/2)O is distinct from the normalized occupation. This is an\noccupation obligation motivated by Dougherty-Bliss, Galvan, Polley and\nShuster, *Enumerating separable derangements*, arXiv:2608.27583v1,\nDefinitions 2, 5, 6 and equations (2)-(4), rather than a settlement of the\npaper's full Conjecture 15.\n[Issue 15063](https://github.com/the-omega-institute/trureturing/issues/15063)\nregisters the exact target.","title":"Actual separable occupied-diagonal lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/separable-occupied-diagonal-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/227d35dcb1f3780e4d32f7cbf7ccd560b7c50cf176226550869bf31ae9d4d3ab.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:serrano-ensastiga-2026-purity-time-reversal-overlap-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Serrano-Ensástiga, O. Giraud and J. Martin (arXiv:2507.12680, quant-ph;\nPhys. Rev. A 113 (2026) 012415) bound the sector lengths of multiqubit states\nthrough the reduced states `ρ_A` of a pure state and their time-reversed\npartners `ρ̃ = σ_y^{⊗k} ρ^* σ_y^{⊗k}`, with the overlap `R_ρ = Tr(ρ ρ̃)`.\nTheir Proposition 1 gives `Tr(ρ_A²) + R_{ρ_A} ≥ 2^{−min(k, N−k)}`, and they\nstate:\n\n> Let $A|\\bar{A}$ be a bipartition of an $N$-qubit system into $k=|A|$ and\n> $N-k=|\\bar{A}|$ qubits. Then, the minimum of\n> $\\mathrm{Tr}(\\rho_A^2) + R_{\\rho_A}$ taken over all pure states\n> $|\\psi\\rangle$ of the $N$-qubit system, is given by\n> $\\min_{|\\psi\\rangle} \\big(\\mathrm{Tr}(\\rho_A^2) + R_{\\rho_A}\\big) =\n> 2^{k-N}$ if $2k > N$, and $2^{-k + 1}$ if $2k \\leq N$.\n\nIssue #11730 fixes the reading: `A` is any set of `k` qubits with\n`1 ≤ k ≤ N − 1`, a pure state is a unit vector of amplitudes on the `2^N`\nconfigurations, `ρ_A` is the partial trace over the other qubits, and the\nminimum means a lower bound for every unit vector together with a unit vector\nthat attains it. The result proves the conjecture.","title":"The minimum of purity plus time-reversal overlap","url":"https://the-omega-institute.github.io/trureturing-pages/research/serrano-ensastiga-2026-purity-time-reversal-overlap-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dfd9d709aaba179680f48042755475e9af0078686832f81c07a0672f6b7746bb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:seshadreesan-berta-wilde-2015-renyi-cmi-local-channel","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. P. Seshadreesan, M. Berta, M. M. Wilde, *Rényi squashed entanglement, discord,\nand relative entropy differences*, §3, Conjecture 1:\n\n> Conjecture 1 The Rényi conditional quantum mutual information obeys monotonicity with respect to CPTP maps on system A, i.e., for ρABE ∈ S(HABE) and α ∈ (0, 1) ∪ (1, 2], Iα(A; B|E)ρ ≥ Iα(A′; B|E)τ, where τA′BE ≡ MA→A′(ρABE) and MA→A′ is a CPTP map acting on system A alone.\n\nEquation (3.18) and the source's hypothesis statements appear in\n[the literature note](../Library/QuantumChannels/seshadreesan2015renyi.md).\nIssue [#16068](https://github.com/the-omega-institute/trureturing/issues/16068)\npreregisters the witness.","title":"Local-channel monotonicity of Rényi conditional quantum mutual information","url":"https://the-omega-institute.github.io/trureturing-pages/research/seshadreesan-berta-wilde-2015-renyi-cmi-local-channel/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb92c49af977a9233f246176e512389cfe5cad4927649ebd772402879ccb7a0f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:seshadri-leditzky-siddhu-smith-2023-signalling-game-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Seshadri, F. Leditzky, V. Siddhu and G. Smith, *On the separation of correlation-assisted sum capacities of multiple access channels*, arXiv:2205.13538v3 (2023), Section 3.2.1, define the signalling game using question sets $\\mathcal X_1$ and $\\mathcal X_2$, answers $\\mathcal Y_1=\\mathcal X_2$ and $\\mathcal Y_2=\\mathcal X_1$, and the winning condition\n\n> Formally, we consider question sets $\\mathcal{X}_1$, $\\mathcal{X}_2$ and answer sets $\\mathcal{Y}_1 = \\mathcal{X}_2$ and $\\mathcal{Y}_2 = \\mathcal{X}_1$, and the winning condition is defined as $\\mathcal{W} = \\{(x_1, x_2, y_1, y_2) \\in (\\mathcal{X}_1 \\times \\mathcal{X}_2) \\times (\\mathcal{Y}_1 \\times \\mathcal{Y}_2) \\mid y_1 = x_2,\\ y_2 = x_1\\}$.\n\nThe source proves the classical value $1/\\max(|\\mathcal X_1|,|\\mathcal X_2|)$ and then asks whether the quantum and no-signalling values coincide with it. Its Section 3.2.1, line 880, states:\n\n> For $2 \\leq |\\mathcal{X}_1|, |\\mathcal{X}_2| \\leq 10$, we verify that the numerically computed value for $\\omega^{\\text{NS}}(G_{\\text{s}})$ matches $\\omega^{\\text{cl}}(\\mathcal{N}_{G_{\\text{s}}}) = 1/\\max(|\\mathcal{X}_1|, |\\mathcal{X}_2|)$. Thus, we expect $\\omega^{\\text{cl}}(\\mathcal{N}_{G_{\\text{s}}}) = \\omega^{\\text{Q}}(\\mathcal{N}_{G_{\\text{s}}}) = \\omega^{\\text{NS}}(\\mathcal{N}_{G_{\\text{s}}})$ for the signalling game.\n\nThe preregistration reads this as the finite-game value question for $m=|\\mathcal X_1|\\ge1$ and $n=|\\mathcal X_2|\\ge1$, with uniformly drawn questions.","title":"The signalling game has equal classical, quantum and no-signalling values","url":"https://the-omega-institute.github.io/trureturing-pages/research/seshadri-leditzky-siddhu-smith-2023-signalling-game-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3080bbb50d5d0ee09df287ea6c6176712168d54a91c885a65c21adbc01ea9c99.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shah-kiselev-2026-late-growing-ratio-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Kian C. Shah and Arthemy V. Kiselev, arXiv:2610.08636v1, §3:\n\n> Definition 3.1. For σ ∈ S_N define the partial sums\n> T_k := Σ_{i=N−k+1}^{N} (σ(i) − 1 − p), k = 1, …, N − 1.\n> … We denote by Φ_p the set of all permutations σ ∈ S_N with\n> σ(1) = 1 and T_k ⩾ 0 for all k = 1, …, N − 1.\n\nHere N = 2p. The conjecture states:\n\n> Conjecture 3.1. As p → +∞, the ratio |Φ_p|/(2p − 2)!\n> converges from above to 1 + 0.\n\nThe Lean `claim` uses the existing zero-based `PhiTailEncoding.phi` and\nasserts convergence to 1 together with\n`∀ n : ℕ, 2 ≤ n → (2*n-2).factorial < (phi n).card`.\nAt p = 1 the ratio equals 1. “From above” means the stated strict\ninequality, without an additional monotonicity assertion.","title":"Shah–Kiselev Conjecture 3.1: the late growing ratio","url":"https://the-omega-institute.github.io/trureturing-pages/research/shah-kiselev-2026-late-growing-ratio-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b7f2c93f27285ad68c6e0b446bff58fd7b81b5658c71edbbe3832ca3a83647e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shala-2024-coset-like-lattice-question-7-4","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Shala, *The Probabilistic Zeta Function of a Finite Lattice*, arXiv:2112.13766v6 (published in Rocky Mountain Journal of Mathematics 54 (2024), 1511–1526), asks in Question 7.4: “Does there exist a weakly coset-like atomistic lattice that is not strongly coset-like? In particular, is the d-divisible partition lattice weakly coset-like?”\n\nThe negation of the first question is formalized as the universal implication that every finite atomistic weakly coset-like lattice is strongly coset-like. The settling theorem refutes that implication and answers the existence question affirmatively; Refuted refers to this universal implication.\n\n### Statement echo\n\nEquation (3.1) is represented by `P`, with Mathlib's incidence-algebra Möbius function and the rational ratio of join-irreducible cardinalities. Definitions 5.1 and 5.2 are represented by `StronglyCosetLike` and `WeaklyCosetLike`. The universal first question is `claim`, and the settling node proves `result : ¬ claim`.","title":"Shala Question 7.4: weakly but not strongly coset-like atomistic lattices","url":"https://the-omega-institute.github.io/trureturing-pages/research/shala-2024-coset-like-lattice-question-7-4/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3d35c7e007e009b4373db7f5f52b281dde1b8865858f8f16b45b1b1ce2d359e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shao-zhao-2026-cycle-italian-domination-recurrence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Shao and C. Zhao, *Counting Weight-k Italian Dominating Sets on Trees and Cycles*, arXiv:2610.00108v1, §9 (verbatim):\n\n> Several directions remain open for future work: … • Deriving a complete linear recurrence for the total count Σ_k d_I(C_n, k) using the transfer matrix formulation (the observed limiting ratio ≈ 2.7843 is the dominant eigenvalue of the corresponding transfer matrix).\n\nThe preregistered issue is [#12435](https://github.com/the-omega-institute/trureturing/issues/12435). The settled statement is that, for every natural `n ≥ 3`, the total count `a(n)` satisfies\n`a(n+5) = 2a(n+4) + 2a(n+3) + a(n+2) − a(n+1) − a(n)`, with\n`a(3), …, a(7) = 23, 60, 167, 467, 1297`.","title":"Shao–Zhao cycle Italian-domination recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/shao-zhao-2026-cycle-italian-domination-recurrence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00f956a251bf185e603abcc52f9d4e0901351894280d54be31ec59566c2eb3f5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shap2610","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Boris Shapiro, *Transposition Cayley Spectra and Subgraph Counts*,\narXiv:2610.14373v1, Section 4.1, page 14, Conjecture 1(iii), states:\n\n> For a tree T, a word of length 2|E(T)| in the edges of T using every edge has product 1 if and only if every edge occurs exactly twice and, for every vertex x of T, the subword formed by the edges incident to x is a non-crossing double occurrence word.\n\nEach edge acts by the transposition of its endpoints. Multiplication uses\n$(\\sigma\\rho)(v)=\\sigma(\\rho(v))$. Alternating occurrences of distinct letters\nare crossings; nested pairs are allowed. The one-vertex tree and empty word\nare included.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every finite tree | `n : ℕ`, `T : SimpleGraph (Fin n)`, `T.IsTree` in `claim` | Relabeling of $[n]$; no bound on $n$ |\n| A word in the edges | `w : List (Sym2 (Fin n))`; every member belongs to `T.edgeFinset` | No nonedge letters |\n| Length twice the edge count | `w.length = 2 * T.edgeFinset.card` | Exact equality |\n| Uses every edge | Every member of `T.edgeFinset` belongs to `w` | Onto edge support |\n| Product is identity | `(w.map edgeSwap).prod = 1` | Existing endpoint transposition; literal list product |\n| Every edge occurs twice | `w.count a = 2` for each tree edge | Exact multiplicity |\n| Every vertex's incident subword is noncrossing | `NoncrossingDoubleOccurrence` of the order-preserving membership filter | Existing `Crossing` predicate on the position pairs |\n| If and only if | Both directions of `result : claim` | No dropped clause |\n\n`NoncrossingDoubleOccurrence` displays the source convention literally:\nevery occurring letter has two copies and distinct letters have no crossing\nposition pairs. `claim` retains the preregistered body. The proofs reuse\n`SupportAndEdgeWordReflection.edgeSwap` and\n`NoncrossingNonnestingGraphRecurrence.Crossing`.","title":"Shapiro Conjecture 1(iii): identity words on finite trees","url":"https://the-omega-institute.github.io/trureturing-pages/research/shap2610/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1548c45b9d452b98ac08910dc2518307037731eb503d5e5b0dea92664190e790.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shapiro-2026-unweighted-fourth-moment-inversion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Boris Shapiro, “The (n−2,2)-Spectrum of a Graph”, arXiv:2605.17501v2,\nSection 12, Outlook item 1:\n\n> Compute an explicit closed formula for the unweighted fourth moment and invert it, modulo Laplacian and cubic data, on the finite list of four-edge support forests. The degree-three inversion is complete by the cubic inversion theorem above.\n\nThis settlement refutes the inversion implication SHAP-1 of #14353. It does\nnot assert that an explicit fourth-moment formula is impossible.\nThe source locator is `Library/GraphInvariants/shapiro2026spectrum.md`.","title":"shapiro 2026 unweighted fourth moment inversion","url":"https://the-omega-institute.github.io/trureturing-pages/research/shapiro-2026-unweighted-fourth-moment-inversion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd5e67792de17fe0e242c5f6501b75a7471a99ff97511e16aa4b0e4032cee145.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sharma-primitive-polynomial-conjecture-4-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The target is the literal Conjecture 4.2 in Avnish K. Sharma's\n*Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence\nResults and Conjectures*, arXiv:2608.07262v2. For every odd prime \\(p\\), every\nfinite field \\(K\\) with characteristic \\(p\\) and \\(|K|=p^2\\), and every\n\\(\\lambda\\in K\\) satisfying the source multiplicative-generator condition\n\\(\\operatorname{ord}(\\lambda)=|K|-1\\), the source claim says that\n\\(X^p+X+\\lambda\\) is the monic minimal polynomial of a multiplicative\ngenerator in an extension \\(L/K\\) of degree \\(p\\). “Primitive” here means\nmultiplicative order; it does not mean `Polynomial.IsPrimitive`.\n\nThe [primary statement](https://arxiv.org/html/2608.07262v2), printed page 16,\nreads (mathematical typography transliterated):\n\n> Let p be an odd prime. Then, for every primitive element λ∈F_(p²), the polynomial x^p+x+λ is primitive over F_(p²).\n\nThe negated statement proved here is\n\\[\n\\neg\\bigl[\\forall p\\text{ odd prime},\\ \\forall K\\text{ finite field of\ncharacteristic }p\\text{ with }|K|=p^2,\\ \\forall\\lambda\\in K,\n\\ \\operatorname{ord}(\\lambda)=|K|-1\\ \\Longrightarrow\n\\exists L/K,\\alpha\\in L:\\ [L:K]=p\\ \\land\n\\operatorname{minpoly}_K(\\alpha)=X^p+X+\\lambda\\ \\land\n\\operatorname{ord}(\\alpha)=|L|-1\\bigr].\n\\]\nThe extension is a finite field extension. The minimal polynomial is monic.\nThe existential Conjecture 4.1 is not the target.\n\nThe Lean definition `fullClaim` preserves all of these quantifiers and the\nsource conclusion. The candidate coefficient is the source leading\ncoefficient, so the public `result` is exactly `Not fullClaim` after unfolding\nthe coefficient parameter.","title":"Sharma's primitive-polynomial Conjecture 4.2","url":"https://the-omega-institute.github.io/trureturing-pages/research/sharma-primitive-polynomial-conjecture-4-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a7c68c1b05863eecef17fdd49b9cb30698718f88a48b081c26f1fd6ffe1873bf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shchukin-van-loock-2016-higher-order-epr-tightness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Shchukin and P. van Loock, *Higher-order EPR correlations and inseparability conditions for continuous variables*, arXiv:1511.08444v1 (Phys. Rev. A 93, 032114, 2016), subsection “Approach two”, state the following conjecture in `bip-ent2.tex`, lines 961–963:\n\n> Our conjecture is that the inequality \\eqref{eq:nS:1} cannot be violated by Gaussian states (including all inseparable ones) and the inequality \\eqref{eq:nS:2} is tight. We verified this by numerical computations, but we could not find rigorous proofs.\n\nHere the source's inequalities (nS:1) and (nS:2), at lines 914–924, concern two bosonic modes and every positive integer $n$:\n\n$$\n\\left\\langle(\\hat a^{\\dagger n}\\pm\\hat b^n)(\\hat a^n\\pm\\hat b^{\\dagger n})\\right\\rangle\n\\ge\n\\begin{cases}\nn! & \\text{for separable states},\\\\\n0 & \\text{for all states}.\n\\end{cases}\n$$\n\nIssue [#24246](https://github.com/the-omega-institute/trureturing/issues/24246) fixes the reading to the tightness clause: for every $n\\ge1$, either sign $\\sigma\\in\\{1,-1\\}$ and every $\\varepsilon>0$, there is a normalized finitely supported occupation-basis vector $\\psi$ with $\\|(\\hat a^n+\\sigma\\hat b^{\\dagger n})\\psi\\|^2<\\varepsilon$. This is proved. The Gaussian clause is a separate question.\n\nThe source and its bibliographic locator are recorded in [the literature note](../Library/Observer/shchukinvanloock2016higherorderepr.md).","title":"The zero higher-order EPR bound is tight at every positive order","url":"https://the-omega-institute.github.io/trureturing-pages/research/shchukin-van-loock-2016-higher-order-epr-tightness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a37066d5f61bff32633679d860c8b89c7c6a714aaeec8a74d68900dea3a82ab0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shieh-yang-yu-dotted-machine-fixed-point-convergence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Michael Yang, Hansen Shieh, and Ashley Yu, *Stack-Sorting with\nDotted-Pattern-Avoiding Stacks*, arXiv:2411.11914v2, Section 6, printed\npage 11, state:\n\n> Conjecture 6.2. All permutations in Sₙ for all n ≥ 1 are eventually mapped to a fixed point of the 21̇-machine after a finite number of iterations through the machine.\n\nThe preceding sentence asks whether this machine has any cycle of length\nat least two. The machine applies the dotted-pattern map first and West's\nstack-sorting map second.","title":"Convergence of the Shieh-Yang-Yu dotted-pattern machine","url":"https://the-omega-institute.github.io/trureturing-pages/research/shieh-yang-yu-dotted-machine-fixed-point-convergence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46ab2fe57e8dca2d76923c18d20afdbd47139a0269b4d34faeee8a4fe62ba15b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shieh-yang-yu-twelve-dot-machine-central-binomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Michael Yang, Hansen Shieh and Ashley Yu, *Stack-Sorting with Dotted-Pattern-Avoiding Stacks*, arXiv:2411.11914v2,\nSection 6, Conjecture 6.1 (printed page 10):\n\n> The machine-sortable permutations under the 12̇-machine map in Sₙ are enumerated by (2n−2 choose n−1).\n\nThe 12̇-machine is s ∘ s₁₂̇, with s West's stack-sorting map. By the source's Proposition 3.1, s₁₂̇ reverses each\npeak run, a peak run being a maximal block of consecutive entries that starts at a left-to-right maximum. A\npermutation is machine-sortable when the machine maps it to the identity.","title":"The 12̇-Machine Sorts a Central Binomial Number of Permutations","url":"https://the-omega-institute.github.io/trureturing-pages/research/shieh-yang-yu-twelve-dot-machine-central-binomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ade3f2b2929e594f9889ddafc4905da5ad07bae1a30e5d1e120dab59f9eb96c9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shin-four-point-sumset-compression","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every natural h>=2, put D_h=choose(h+2,2)+1 when h is odd, and\nD_h=choose(h+2,2) otherwise. Must every four-element integer set A have\na four-element integer set B in [0,D_h] with |hB|=|hA|? Here hA is the set\nof sums of exactly h elements of A with repetitions allowed. This is the\nquestion after Corollary 10.5 and equation (193) in arXiv:2609.01690v1,\nequivalently nu(h,4)=D_h for all h>=2. It is restated as open in\narXiv:2609.08915v1, Question 12.5.","title":"Shin's four-point sumset compression question","url":"https://the-omega-institute.github.io/trureturing-pages/research/shin-four-point-sumset-compression/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d524445565b193d51d1c1ede8a1cf0dd44b15cd6db154dab710badfa3bc57e1e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:shirokov-2026-moreau-yosida-eof-selective-locc-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. E. Shirokov, *The Moreau-Yosida approximation of the EoF: basic properties\nand accuracy estimates*, arXiv:2609.30246v1, Section 7, p. 20, asks:\n\n> Open question: can the function $E^{\\lambda}_F$ increase under selective LOCC-operations?\n\nThe conjectural sentence is:\n\n> So, we may conjecture, at the moment, that the function $E^{\\lambda}_F$ does not increase under selective LOCC-operations as well.\n\nIssue [#11557](https://github.com/the-omega-institute/trureturing/issues/11557)\npreregisters the source, quantified statement, Tier 1 classification and literature\ncheck. The selective inequality is the probability-weighted average of normalized\nterminal values being at most the input value, for every positive parameter and\nbipartite state. A finite-dimensional one-round local projective measurement is\nalready sufficient to refute that universal claim.","title":"Selective LOCC can increase Shirokov's Moreau-Yosida entanglement of formation","url":"https://the-omega-institute.github.io/trureturing-pages/research/shirokov-2026-moreau-yosida-eof-selective-locc-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e97fe0850246dd3c9018b99a8783d034aef38d69b61d3d263abe0d189dbf8dd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:siddiqui-wang-2026-concealment-kernel-necessity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. A. Siddiqui and Z. Wang, *Operational Concealment of Measurement Incompatibility by Quantum\nChannels: Rank Loss versus Contraction*, arXiv:2607.11762v2, Section III, prove that two channels with\na common output space and equal adjoint kernels conceal the same POVM pairs, and state:\n\n> Whether equality of adjoint kernels is also necessary for concealment-equivalence remains open.\n\nThe definitions are quoted in [the literature note](../Library/QuantumChannels/siddiquiwang2026concealment.md).\n- A pair of POVMs is *concealed* by a channel $\\mathcal E$ relative to an input set $\\mathcal T$ if\n compatible POVMs reproduce its statistics on every output $\\mathcal E(\\rho)$, $\\rho\\in\\mathcal T$.\n- The adjoint kernel is $\\ker(\\mathcal E^\\dagger)$ on Hermitian operators.\n\nIssue [#14233](https://github.com/the-omega-institute/trureturing/issues/14233) reads the affirmative\nanswer with the hypothesis in its strongest form. For all finite dimensions, all trace-preserving\nKraus channels $\\mathcal E_1,\\mathcal E_2$ with common input and output spaces, and every tomographically\ncomplete $\\mathcal T$: if the two channels conceal the same pairs for every pair of finite outcome\nsets, then their adjoint kernels coincide.","title":"Concealment-equivalent channels with different adjoint kernels","url":"https://the-omega-institute.github.io/trureturing-pages/research/siddiqui-wang-2026-concealment-kernel-necessity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/580778861ede5a0f7b80d8f9609067652760f3cfbc3be018d36375697ef0d91c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:simic-2018-iterated-moment-minor-inequalities","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Simić, *Moment inequalities of the second and third order*, §2:\n\n> Conjecture 1 The inequality φ(r₁,s₁;u₁,v₁)φ(r₂,s₂;u₂,v₂) ≥ [(ξ((r₁+r₂)/2,(s₁+s₂)/2) − ξ((r₁+s₂)/2,(s₁+r₂)/2))(ξ((u₁+u₂)/2,(v₁+v₂)/2) − ξ((u₁+v₂)/2,(v₁+u₂)/2)) − (ξ((r₁+u₂)/2,(s₁+v₂)/2) − ξ((r₁+v₂)/2,(s₁+u₂)/2))(ξ((u₁+r₂)/2,(v₁+s₂)/2) − ξ((u₁+s₂)/2,(v₁+r₂)/2))]² holds for arbitrary r_i, s_i, u_i, v_i ∈ I, i ∈ {1, 2}?\n\nWith σ_a(x) := ξ(x, x+a) and\nθ_a(x,y) := σ_a(x)σ_a(y) − σ_a²((x+y)/2):\n\n> Conjecture 2 Is it true that θ_a(x,y)θ_a(z,v) ≥ [θ_a((x+z)/2,(y+v)/2) − θ_a((x+v)/2,(y+z)/2)]²?\n\nFor a positive random variable X with mean m, the definitions are\n\n$$\\lambda_s=\\begin{cases}\n\\log m-\\mathbb E\\log X,&s=0,\\\\\n\\mathbb E(X\\log X)-m\\log m,&s=1,\\\\\n\\dfrac{\\mathbb E X^s-m^s}{s(s-1)},&s\\ne0,1,\n\\end{cases}\\qquad\n\\xi(s,t)=\\lambda_s\\lambda_t-\\lambda_{(s+t)/2}^2,$$\n\n$$\\varphi(s,t;u,v)=\\xi(s,t)\\xi(u,v)\n-\\left[\\xi\\!\\left(\\frac{s+u}{2},\\frac{t+v}{2}\\right)\n-\\xi\\!\\left(\\frac{s+v}{2},\\frac{t+u}{2}\\right)\\right]^2.$$\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.MomentMinors`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| Positive probability law of X | `Defs.Admissible μ I` | A Borel probability measure on ℝ, positive almost everywhere, with integrable identity. |\n| Moments exist for every s ∈ I | `Defs.Admissible.integrable_rpow` | Integrability of each real power; log X at 0 ∈ I and X log X at 1 ∈ I are separate conditions. |\n| Interval I | `I.OrdConnected` | Any order-connected real set, with all needed midpoints still in I. |\n| λ_s, including s = 0,1 | `Defs.lam`, `Defs.mean` | Exactly the displayed three-case definition, with m the integral of the identity. |\n| ξ, φ | `Defs.xi`, `Defs.phi` | The displayed second and iterated second minors. |\n| σ_a, θ_a | `Defs.sigma`, `Defs.theta` | The displaced pair and its midpoint second minor. |\n| Conjecture 1 | `Defs.claim1` | All eight endpoint parameters independently range over I, in the quoted index order. |\n| Conjecture 2 | `Defs.claim2` | Every real a,x,y,z,v such that x,y,z,v and x+a,y+a,z+a,v+a belong to I. No sign restriction on a. |\n| Both assertions | `Defs.claim`, `Result.result` | The conjunction, for every interval and every admissible law. |","title":"Iterated moment minor inequalities","url":"https://the-omega-institute.github.io/trureturing-pages/research/simic-2018-iterated-moment-minor-inequalities/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17c404636f31b7052cde644bc587a4b385d6a52a018b1f3455654ba42654c886.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:simnacher-2021-two-qubit-separable-numerical-range-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Simnacher, J. Czartowski, K. Szymański, K. Życzkowski, *Confident\nentanglement detection via the separable numerical range*, Phys. Rev. A 104,\n042420 (2021), arXiv:2107.04365v1, Conjecture 8:\n\n> For a single measurement on a two-qubit system, the minimal volume ratio is $\\mu_{2,1} = \\frac{1}{2}$.\n\nHere $\\mu_{2,1}$ is the minimum over Hermitian two-qubit observables $A$ of\n$\\operatorname{vol}L_{\\mathrm{Sep}}(A)/\\operatorname{vol}L(A)$, where\n$L_X(A)=\\{\\operatorname{Tr}\\rho A\\mid\\rho\\in X\\}$ for the set of all states\n($L$) or of separable states ($L_{\\mathrm{Sep}}$), and vol is the Euclidean\nvolume (Definitions 1 and 2). The paper proves $\\mu_{2,1}\\ge\\sqrt2-1$\n(Proposition 7) and $\\mu_{2,1}\\le\\frac12$ with the projector onto\n$|\\phi^+\\rangle=(|00\\rangle+|11\\rangle)/\\sqrt2$.","title":"The minimal separable numerical range ratio of one two-qubit observable","url":"https://the-omega-institute.github.io/trureturing-pages/research/simnacher-2021-two-qubit-separable-numerical-range-ratio/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8b81766701c7090d6e24e5ab214924ddb42804b4c438c08608e622bdc9d9ed97.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:singh-nechita-2022-odd-hadamardness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Singh and I. Nechita, *Diagonal unitary and orthogonal symmetries in quantum theory II: Evolution operators*, J. Phys. A: Math. Theor. 55 (2022) 255302, arXiv:2112.11123v1, Section 6, `main.tex` line 848, immediately following Table 1:\n\n> We note that for the odd values of $d$ considered above, $\\min \\mathfrak h(C) = d^3 + d(d-1)$, obtained for matrices $C$ with the property that all the scalar products between different works have absolute value $1$. We conjecture that this is the case for all odd integers $d$.\n\n“Works” is the source's wording. The source defines its measure in Eq. `eq:Hada_measure`, `main.tex` lines 780–783. For a real sign matrix, write $G=CC^{\\mathsf T}$ and $\\mathfrak h(C)=\\operatorname{Tr}(G^2)=\\sum_{i,j}G_{ij}^2$; conjugate transpose and transpose coincide here. The source and its table are described in [the literature note](../Library/Eigenstructure/singhnechita2022evolution.md).\n\nIssue [#22426](https://github.com/the-omega-institute/trureturing/issues/22426) fixes the reading encoded by `claim`: for every odd natural $d$, there exists an integer $d\\times d$ matrix $C$ with every entry equal to $1$ or $-1$, such that $\\mathfrak h(C)=v_d$ and every integer sign matrix $B$ of that order satisfies $v_d\\leq\\mathfrak h(B)$. Here $v_d=d^3+d(d-1)$ is evaluated in the natural numbers, including subtraction, and then embedded in the integers. Thus attainment and the lower bound are both explicit; no infimum convention is used. The conjecture is refuted, using order nine.","title":"Odd order does not ensure the conjectured minimum of Hadamardness","url":"https://the-omega-institute.github.io/trureturing-pages/research/singh-nechita-2022-odd-hadamardness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1451e005a1fa8f73f3bf053c410e3c7d649a647533e5622bb58c67dc8170ff43.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:singhvi-sabary-barlev-yaakobi-2024-deletion-channel-max-input-entropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Singhvi, Sabary, Bar-Lev and Yaakobi, *Conditional Entropies of k-Deletion/Insertion Channels*, arXiv:2407.10026v2, Section VII-A, ask:\n\n> While we provide a complete characterization of the sequences that minimize input entropy, the precise identification of the sequences that maximize input entropy for the 2-deletion and 2-insertion channels is still unresolved. Determining these sequences and the associated maximal entropy values remains an important open direction.\n\nIssue #19173 fixes the formal statement. For a received binary word, the posterior on uniformly transmitted inputs is proportional to the number of increasing index sets realizing the received word as a subsequence. Entropy is Shannon entropy in bits. The claim gives the two alternating families and their complements, the exact deletion and large insertion maxima, and the finite insertion cases of received lengths two, three and four.\n\n### Statement echo\n\nThe Lean `claim` in `D5/S3/Entropy/SinghviMaxInputEntropy.lean` inlines the two posterior definitions, quantifies over the same finite binary words, uses the frozen embedding count and Shannon entropy, divides by `Real.log 2`, and states all seven clauses of the preregistration. The public `result : claim` is the settling declaration.","title":"Maximal input entropy for binary two-deletion and two-insertion channels","url":"https://the-omega-institute.github.io/trureturing-pages/research/singhvi-sabary-barlev-yaakobi-2024-deletion-channel-max-input-entropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ebb5b7965854454123a931bcf9d83137b8573d05688dc670037158dd5e4cdb4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sitnik-1993-exponential-remainder-differences","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. M. Sitnik, *Inequalities for the exponential remainder of the Taylor\nseries* (*Неравенства для остатка ряда Тейлора экспоненты*),\nVladivostok preprint (1993), https://arxiv.org/abs/1608.08784v2,\nappendix (Дополнение), additional problem list, Problem 11,\nPDF pages 27–28:\n\n> 11. Определим функцию при x > 0. gₙ(x) = Rₙ₋₁(x)/Rₙ(x). Исследовать, при каких k разности этой функции при любом фиксированном x неотрицательны ∆ᵏgₙ(x) > 0? Гипотеза: при любых k. При k = 1 гипотеза сводится к (42), при k = 2 к неравенству задачи 1 на стр. 20.\n\nTranslation: For $x>0$, define $g_n(x)=R_{n-1}(x)/R_n(x)$.\nInvestigate for which $k$ the differences of this function, at every\nfixed $x$, are nonnegative, $\\Delta^k g_n(x)>0$.\nConjecture: for all $k$. For $k=1$ the conjecture reduces to (42),\nand for $k=2$ to the inequality of Problem 1 on page 20.\n\nEquation (1) defines\n\n$$R_n(x)=e^x-\\sum_{j=0}^{n}\\frac{x^j}{j!},\\qquad x\\ge0.$$\n\nThe difference is forward in the natural index:\n\n$$\\Delta^k g_n(x)=\\sum_{j=0}^{k}(-1)^{k-j}\\binom{k}{j}g_{n+j}(x).$$\n\nThe Lean assertion quantifies over every natural $k\\ge1$, every natural\n$n\\ge1$ and every positive real $x$, and asserts $\\Delta^k g_n(x)\\ge0$.\nThis weakens the source's displayed strict inequality. A negative\ninstance refutes both the nonnegative and the strict readings.\n\n| Source object | Lean declaration | Correspondence |\n| --- | --- | --- |\n| $R_n(x)$ | `R` | Exponential minus the finite Taylor sum through degree $n$. |\n| $g_n(x)$ | `g` | $R_{n-1}(x)/R_n(x)$, used only at $n\\ge1$. |\n| $\\Delta^k g_n(x)$ | `(fwdDiff 1)^[k] (fun m => g m x) n` | Iteration of Mathlib's positive unit-step difference. |\n| Universal assertion | `claim` | All $k\\ge1$, $n\\ge1$, $x>0$, with a nonnegative conclusion. |\n| Negative instance | `witness_neg` (private) | $\\Delta^3 g_1(1)<0$. |\n| Refutation | `result` | Exactly $\\neg\\,\\texttt{claim}$. |\n\nAll declaration names have prefix `D5.S3.Analytic.ExpRemainderDifferences`.","title":"A negative third difference of exponential remainder ratios","url":"https://the-omega-institute.github.io/trureturing-pages/research/sitnik-1993-exponential-remainder-differences/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab99d10d5c26302656a0d59d02bc04939111b551279e19f5636a9e3a082b76c8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:siudzinska-2020-generalized-pauli-volume-ratios","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Katarzyna Siudzińska, *Geometry of generalized Pauli channels*, Phys. Rev. A 101, 062323 (2020), arXiv:2002.04657v2, states three conjectures in Section 4.2, “$N=d+1$ and $N=d$ mutually unbiased bases”. The v2 TeX source `GGPC_volume5.tex`, lines 243–246, reads:\n\n> and conjecture that the general formula reads\n> $\\frac{V_{CP}(d)}{V_P(d)}=\\frac{d}{(d+1)!}.$\n\nLines 276–279 read:\n\n> and conjecture that the general formula reads\n> $\\frac{V_{G}(d)}{V_{CP}(d)}=\\frac{d^2-1}{d^2}\\left(\\frac{d-1}{d}\\right)^{d}.$\n\nLines 302–305 read:\n\n> from which we conjecture that\n> $\\frac{V_{EB}(d)}{V_{G}(d)}=\\frac{1}{d+1}.$\n\nFor a natural number $d\\ge2$, put $m=d+1$ and $s(x)=\\sum_{i=1}^{m}x_i$ for $x\\in\\mathbb R^m$. The module defines precisely these four sets:\n\n- $P=\\{x:\\ -1/(d-1)\\le x_i\\le1\\text{ for every }i\\}$ (`boxRegion`).\n- $C=\\{x:\\ -1/(d-1)\\le s(x)\\text{ and }s(x)\\le1+dx_j\\text{ for every }j\\}$ (`cpRegion`).\n- $G=C\\cap\\{x:\\ x_i\\ge0\\text{ for every }i\\}$ (`generatorRegion`).\n- $E=G\\cap\\{x:\\ s(x)\\le1\\}$ (`ebRegion`).\n\nFor these sets write $V_A=\\operatorname{channelVolume}(d,A)=(\\sqrt{d-1}/d)^m\\operatorname{Leb}_m(A)$, for $A=P,C,G,E$. In Lean, `channelVolume` uses `(volume A).toReal`; the proof obtains finite positive Lebesgue measures for all four sets when $d\\ge2$. Thus $V_P,V_C,V_G,V_E$ match the source's $V_P(d),V_{CP}(d),V_G(d),V_{EB}(d)$ as coordinate-region volumes. The source locates the box and its weight at lines 210–228, complete positivity at lines 359–364, nonnegative eigenvalues at lines 256–267, and the nonnegative entanglement-breaking region at lines 282–293; see also [the literature note](../Library/QuadraticForms/siudzinska2020pauligeometry.md).\n\nIssue [#22754](https://github.com/the-omega-institute/trureturing/issues/22754) fixes the statement as the conjunction of these three identities for every natural $d\\ge2$. This is proved for the specified coordinate regions.","title":"Three volume ratios for generalized Pauli eigenvalue regions","url":"https://the-omega-institute.github.io/trureturing-pages/research/siudzinska-2020-generalized-pauli-volume-ratios/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2cda75a0f88ddbd8cf91ac945362a178b16f930083fdf5a46bc9a5a0deee923.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:skalski-stroinski-2025-ising-uniqueness-sets","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Skalski and Stroiński work on `X = {-1,+1}^k` with the Rademacher functions\n`r_j(x) = x_j` and the Walsh functions `w_L(x) = ∏_{j∈L} r_j(x)`; for\n`1 ≤ q ≤ k` they set\n\n> `B^k_q = Lin{w_L : L ⊂ {1,…,k} and |L| ≤ q}`.\n\nFollowing Bogdan, Bosy and Skalski, a subset `U ⊂ X` is a set of uniqueness\nfor `(B^k_q)_+`, the nonnegative functions of `B^k_q`, if `φ = 0` is the only\nsuch function vanishing on `U`, and `u(k,q)` is the size of the smallest one.\nThe section on extremal sizes closes with the conjecture\n\n> `u(k,2) = k+1`, i.e., for every `k` there are no sets of uniqueness having at\n> most `k` elements.\n\nIssue #10022 fixes the readings: points of `X` are functions `Fin k → Bool`\nwith `true` read as `+1`; `B^k_2` is the real span of the `w_L` with\n`|L| ≤ 2`; the equality is read for `k ≥ 3`, and the edge `k = 2` is recorded\nseparately.","title":"Skalski and Stroiński's conjecture on the smallest set of uniqueness for the Ising cone","url":"https://the-omega-institute.github.io/trureturing-pages/research/skalski-stroinski-2025-ising-uniqueness-sets/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d03066804883198541ca6530b67f325ec1de8efa03eca7a9ceaef73d8e08e65e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:slbd2026","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Matéo Spriet, Christopher Langrenez, Raymond Brummelhuis and Stephan De Bièvre, *What Is Special About the Kirkwood–Dirac Distributions? Only They Produce Natural Conditional Expectations*, arXiv:2511.01996v2, §6.3, immediately after (6.23), state:\n\n> In view of what precedes, it is tempting to conjecture that, if $\\hat \\rho$ and $\\hat X$ are both KD real, the optimal observable $\\hat L$ must not be KD-real itself, provided $\\rm I_{QF}$ does not vanish.\n\nThe conjecture is unnumbered; Proposition 6.5 is the published pure-state theorem. The journal article, DOI 10.3390/sym18061008, retains the conjecture after (168), with the pure-state result numbered Proposition 7. The claim is bound to arXiv v2. Source and literature evidence are in [#22488](https://github.com/the-omega-institute/trureturing/issues/22488); the source note is [sprietlangrenez2026kdreal](../Library/HilbertOperators/sprietlangrenez2026kdreal.md).\n\n### Statement echo\n\nAll module-local names below belong to `D5.S3.Quantum.Information.KirkwoodDiracSLDRefutation`; the four public definitions and `result` are in its `SprietKirkwoodDiracSLD` namespace.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite Hilbert space of positive dimension | `d`, the positivity hypothesis, and complex matrices indexed by `Fin d` in `claim` | All positive finite dimensions. |\n| Two orthonormal CSCO eigenbases | `a`, `b` in `Matrix.unitaryGroup` | Their columns are the eigenvectors; eigenvalue labels play no role. |\n| Complementarity | Every entry of the overlap matrix is nonzero | No mutual-unbiasedness hypothesis is added. |\n| Left KD distribution | `qLeft` | Conjugate overlap times the matrix coefficient. |\n| Left observable symbol | `symbolLeft` | Inverse overlap times the matrix coefficient. |\n| KD-real Hermitian sector | `IsKDReal` | Hermitian symmetry and zero imaginary symbol at every label pair. |\n| Density matrix | Reused `SLDSpectralEnergy.IsDensity` | Positive semidefinite with trace one. |\n| KD-real state and generator | The two `IsKDReal` hypotheses on the same bases | No reality or commutativity assumption is added. |\n| SLD tangent equation and Hermitian solution | Reused `FisherDiscord.Defs.IsSLD` | The tangent is $i(\\rho X-X\\rho)=-i[X,\\rho]$. |\n| Existence of the SLD | Existential conclusion in `claim` | Existence is asserted, rather than assumed. |\n| Nonvanishing source QFI | Four times reused `FisherDiscord.Defs.fisherSLD` | The owner is one quarter of $\\operatorname{Tr}(\\rho L^2)$. |\n| Every such SLD fails to be KD-real | Universal conclusion over `L` | `result` negates the whole literal claim. |\n\nThe public definitions retain the literal source expressions. Density, SLD and Fisher information use their existing owners. Each binder or conjunct above either represents the quoted mathematical setting or makes an implicit existence assertion explicit; none narrows the source class. Faithfulness is a property of the counterexample, not a hypothesis restricting `claim`.","title":"A faithful mixed-state counterexample to KD-real SLD exclusion","url":"https://the-omega-institute.github.io/trureturing-pages/research/slbd2026/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81bd89a9636a2f5413b13d494910a150386d0702d077cb831ab3b9976096e499.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sloane-2019-a309221-knight-walk-range-integrality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A309221 (N. J. A. Sloane, 2019) normalizes the expected number of\ndistinct squares visited by a knight's random walk on an infinite chessboard,\nA326954/A326955 (the starting square counted):\n\n> a(0)=1; for n>0, a(n) = (A326954(n)/A326955(n))*2^(3*n-3). (It is only a\n> conjecture that this is always an integer).\n\nIssue #10116 fixes the readings: `E(n)` is the expected number of distinct\nsquares of `ℤ²` visited by a walk from the origin whose `n` steps are chosen\nindependently and uniformly among the eight knight moves, the start included;\nequivalently `E(n) = S(n)/8^n` with `S(n)` the total over all `8^n` step\nsequences; the conjecture asserts that `E(n)·2^(3n−3)` is an integer for every\n`n ≥ 1`.","title":"The normalized expected range of a knight's random walk is an integer","url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6de9f8531cdbb5b6538cdd81ddc8ea81d5d72874b2f8f7455c12ced9edaabe60.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sobolewski-ulas-2022-meta-fibonacci-parity-positions-question-7-5","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Bartosz Sobolewski and Maciej Ulas, *Solutions of certain meta-Fibonacci\nrecurrences*, arXiv:2204.04011v1, Question 7.5, pp. 16–17; journal locator:\nActa Arithmetica 209 (2023), 269–289, DOI 10.4064/aa220613-19-1.\n\n> Let u_i = (u_i(n)) for i = 0, 1 be the increasing sequence of\n> nonnegative integers such that r(u_i(n)) = i. Is the sequence u_i is\n> k-regular for some k?\n\nHere $h(n)=1$ for $n\\leq1$, $h(n)=h(n-h(n-1))+h(n-2)$ for $n>1$,\nand $r(n)=h(n)\\bmod2$. Section 2 requires $k\\geq2$ and defines regularity\nas finite generation of the integer span of the $k$-kernel.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| The recurrence $h=h_{1,1}$ | `h` | Both initial values are one. Truncated negative arguments map to zero, also of value one. The termination minimum is inactive. |\n| $r(n)=h(n)\\bmod2$ | `r` | Natural remainder, values zero and one. |\n| Increasing nonnegative occurrence positions | `u` | `Nat.nth`, with zero-based index. Block balance proves both sets infinite. |\n| Kernel indices $j\\geq0$, $0\\leq a Theorem~\\ref{thm:damping} settles the coherence damping inequality for ground states with real non-positive hopping; whether it extends to systems with complex hopping (e.g., in a magnetic field) or frustrated lattices remains open.\n\nTheorem 1 states |ρ^tw_ij| ≤ ρ^spin_ij = |ρ^spin_ij| for D-parity-changing elements, where ρ^spin is the ordinary partial trace over B and ρ^tw = tr_B[(−1)^{N_B} ρ_ABD]. The verbatim passages are in [the literature note](../Library/QuantumContext/sokolovs2026paritysuperselection.md).\n\nIssue [#19153](https://github.com/the-omega-institute/trureturing/issues/19153) fixes the reading: for number-conserving free-fermion Hamiltonians with complex hopping, the ground state satisfies |ρ^tw_ij| ≤ |ρ^spin_ij| for every D-parity-changing pair and every tripartition into adjacent blocks. It is refuted.","title":"Complex hopping can increase parity-twisted ground-state coherences","url":"https://the-omega-institute.github.io/trureturing-pages/research/sokolovs-2026-complex-hopping-coherence-damping-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b793d8d5c23bb513d58e509acf19afd8f08adf591e33f81ecdbbd8877bb98df2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:solfanelli-buffoni-cuccoli-campisi-2019-metrotropy-half-ergotropy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Solfanelli, L. Buffoni, A. Cuccoli and M. Campisi, *Maximal energy extraction via quantum measurement*, J. Stat. Mech. (2019) 094003, arXiv:1905.10262v2, compare the metrotropy $\\mathcal M$, the maximal energy extractable by a non-selective rank-one projective measurement, with the ergotropy $\\mathcal W$, the maximal energy extractable by a unitary. For stationary states they prove $\\mathcal M\\le\\mathcal W/2$; for non-stationary states they prove $\\mathcal M\\le\\mathcal W$. Section 3.1 of the source TeX `Metrotropy02.tex`, lines 277–308, ends:\n\n> Whether the relation $\\mathcal M \\leq \\mathcal W/2$ holds in the case $[H,\\rho]\\neq 0$ remains an open question.\n\nThe repository proposition `claim` reads the question for finite dimensions. For every $d$ and all $d\\times d$ complex matrices $h,\\rho$ with $h$ Hermitian, $\\rho$ a density matrix and $h\\rho\\ne\\rho h$,\n\\[\n\\operatorname{metrotropy}(h,\\rho)\\le\\operatorname{ergotropy}(h,\\rho)/2 .\n\\]\nHere `measured u ρ` is the non-selective measurement $\\sum_k\\pi_k\\rho\\pi_k$ with $\\pi_k$ the projector onto the $k$th column of a unitary $u$; `metrotropy h ρ` is $\\operatorname{Re}\\operatorname{Tr}(h\\rho)$ minus the infimum of the energies of `measured u ρ` over all unitaries $u$; `ergotropy h ρ` is $\\operatorname{Re}\\operatorname{Tr}(h\\rho)$ minus the infimum of the energies of $v\\rho v^\\dagger$ over all unitaries $v$.\n\nThe Lean declaration is `result : \\lnot claim`.","title":"Measurement can extract more than half the ergotropy of a non-stationary state","url":"https://the-omega-institute.github.io/trureturing-pages/research/solfanelli-buffoni-cuccoli-campisi-2019-metrotropy-half-ergotropy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf7a6ad4bba3930319d9e5d3f4f90e64ea28d25b8988b6cb8923e155f4391dc7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:solid-resolving-tournament","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Hakanen and Pavan, *Solid-Resolving Sets on Directed Graphs*, arXiv:2610.35041v1,\nConjecture 3.4, page 8, asserts that every tournament of orders\n$6,7,8,9,10,11$ has solid-metric dimension at least $4,5,5,5,5,6$, respectively.\nThe source is recorded in `Library/ConceptDynamics/hakanenpavan2026solid.md`.\nPreregistration: [#22881](https://github.com/the-omega-institute/trureturing/issues/22881).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| All tournaments of order $n$ | Every relation on `Fin n` satisfying the reused loopless exclusive-or `IsTournament` | No strong-connectivity or diameter hypothesis |\n| Directed distance, including unreachable vertices | `SolidResolvingTournamentBase.directedDistance`, the extended-natural infimum of directed walk lengths | Deleting repeated-vertex cycles identifies the infimum with shortest-path length |\n| Distance to a nonempty vertex set | `SolidResolvingTournamentBase.distanceToSet`, the infimum over its subtype | Infinity is retained when every member is unreachable |\n| Definition 1.1, every nonempty $X\\ne\\{v\\}$ | `SolidResolvingTournamentBase.IsSolidResolving` | Literal subset quantifier and inequality |\n| Minimum solid-set cardinality | `SolidResolvingTournamentBase.solidMetricDimension` | Natural infimum; full-set solidity gives attainment |\n| Orders $6,7,8,9,10,11$ and bounds $4,5,5,5,5,6$ | The six conjuncts of `SolidResolvingTournament.claim`, proved by `result` | No dropped clause or extra assumption |\n\nThe namespaces in this dossier have prefix `D5.S3.Combinatorics.Graph.Walks`.\nAll auxiliary profile hypotheses are discharged inside the settling proof.","title":"Tournament solid-metric lower bounds","url":"https://the-omega-institute.github.io/trureturing-pages/research/solid-resolving-tournament/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18d47319860bcf02fe36c5a5969b05516dde2036793c82f12229538a9e22035c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:somer-krizek-uniform-subsequence-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Somer and Krizek, Conjecture 4.2, printed page 8, proposes the following.\nFor every `k >= 2`, let an integer sequence satisfy\n\n`w(n+k) = a1*w(n+k-1) + ... + ak*w(n)`\n\nfor every `n >= 0`. Let `p` be prime, `e >= 1`, and `E > 0`. Assume that\n`gcd(ak,p)=1`, that the least positive period modulo `p^e` is exactly\n`p^e E`, and that every residue modulo `p^e` occurs exactly `E` times in\nthat least period. For every positive `g` coprime to `p` and every\nnonnegative `s`, put `r=E/gcd(g,E)`. The conjecture asserts that every\nresidue occurs exactly `r` times among `w(s+ng)` for `0 <= n < p^e r`.","title":"Somer-Krizek uniform-subsequence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/somer-krizek-uniform-subsequence-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/400d38aacde59778ed304b7d04fab4cef006a77b9d0f3b5a6cc2fb066f4a782b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-chen-djokovic-2023-unique-basis-eof","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Song, Chen and Đoković, *The existence of distinguishable bases in three-dimensional subspaces of qutrit-qudit systems under one-way local operations and classical communication*, arXiv:2307.01980v3, p. 3, asks:\n\n> Further, by calculation one can see that every state in the span of the three states has entanglement more than 1.52 ebits. Hence, the entanglement of formation (EOF) of every mixed state whose range is contained in the span is also more than 1.52 ebits [19]. We don't know whether this EOF is exactly equal to that of the pure state |Ψ₁⟩.\n\nThe vectors are those of Eqs. (2)–(3), on $\\mathbb C^3\\otimes\\mathbb C^9$. Write $P_j=|\\Psi_j\\rangle\\langle\\Psi_j|$, $S=\\operatorname{span}_{\\mathbb C}\\{\\Psi_1,\\Psi_2,\\Psi_3\\}$ and $H_0=H(4/9,1/3,2/9)$, with entropy in bits.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Eq. (2), the twelve nonzero entries of $G$ | `UniqueBasisFormationRefutation.G` | Literal real square roots embedded in $\\mathbb C$; indices decreased by one. |\n| Eq. (3), the columns $r_{3a+j}$ | `Psi k (a,b)` | Entry $G(b,3a+k)$, with zero-based `Fin` indices. |\n| The span of the three states | `spanPsi` | Complex linear span of `Set.range Psi`. |\n| Every density operator supported in $S$ | `claim`: positivity, trace one, range inclusion | All matrices on `Fin 3 × Fin 9`, with no added restriction. |\n| EOF equal to that of $\\Psi_1$ | `FormationBounds.E_F`, `PureStateHandshake.rankOneDensity (Psi 0)` | Existing finite-ensemble formation in bits and rank-one density. |\n| A negative answer | `result : ¬ claim` | Refutes the complete universal equality. |","title":"Formation is not constant on the Song–Chen–Đoković subspace","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-chen-djokovic-2023-unique-basis-eof/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/91fcd322f5c4a3a87bbb812622042363783f56ea44d3e9a9bde55699d45c3e20.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-dalfo-fiol-zhang-2026-token-graph-laplacian-radius-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Song, C. Dalfó, M. À. Fiol and S. Zhang, *The Algebraic Connectivity and\nLaplacian Spectral Radius of Token Graphs*, arXiv:2610.00500v1, section 1,\npage 4, state:\n\n> Conjecture 1.1. Let G be a graph of order n(≥ 4), the equality\n> ρ(F_k(G)) = ρ(G) holds for all k with 2 ≤ k ≤ ⌊n/2⌋ if and only if G ≅ S_n.\n\nHere the token vertices are k-subsets, adjacency means that their symmetric\ndifference is an edge of G, and ρ is the largest Laplacian eigenvalue.\nThe literal statement has no connectedness hypothesis. Issue #12457\npreregisters this reading and the counterexample K₂ ⊔ 2K₁.","title":"A disconnected counterexample to the token-radius conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-dalfo-fiol-zhang-2026-token-graph-laplacian-radius-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/54c7e739636b43cf9f8ecf9312ed52cc2cbd29f0c5e84a7d6a8783efe2adcdd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-et-al-2024-steering-temporal-compatibility-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Song, V. Narasimhachar, B. Regula, T. J. Elliott and M. Gu, *Causal classification of spatiotemporal quantum correlations*, arXiv:2306.09336v2 (Phys. Rev. Lett. 133, 110202), Discussion:\n\n> Some entangled states are not atemporal, so is atemporality a strictly stronger notion of quantum correlations? And if so, is it guaranteed by steering or Bell non-locality?\n\nHere:\n- **Temporally compatible from A to B:** R = (id ⊗ Λ)K with K = {ρ_A ⊗ I/2, S} for a quantum channel Λ, the source's Eq. (3). The reverse direction is symmetric.\n- **Atemporal:** temporally compatible in neither direction.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/song2024causalclassification.md).\n\nIssue [#19923](https://github.com/the-omega-institute/trureturing/issues/19923) fixes the reading of the steering clause: every two-qubit density operator that is EPR-steerable in at least one direction would be atemporal. This is refuted. The Bell-nonlocality clause is not addressed.","title":"Steering does not guarantee atemporality","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-et-al-2024-steering-temporal-compatibility-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23d44be857322de8dfa42dc6e5d07d816c650a921dec01b964ebe5f6882b3678.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-kwon-scarani-2025-steady-prior-reversibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Song, H. Kwon and V. Scarani, *Exact and approximate conditions of tabletop reversibility: when is Petz recovery cost-free?*, arXiv:2510.26895v1, Section V, question 2, page 9:\n\n> In all of the examples we considered, a dynamics appears to be tabletop time-reversible whenever a prior state is a steady state. We do not have a proof that this is true in general, and leave it for a future investigation.\n\nThe [source note](../Library/Divergence/song2025tabletop.md) gives the definitions and locators. [Preregistration #20652](https://github.com/the-omega-institute/trureturing/issues/20652) specifies the finite-matrix statement. The universal assertion, including positive definite steady priors, is **Refuted** by `D5.S3.Quantum.Information.SongKwonScaraniSteadyPriorReversibility.result`.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Equation (1), forward dilation | `channel` | The environmental partial trace of $U(X\\otimes\\xi)U^\\dagger$, with system-first product indices |\n| Hilbert–Schmidt adjoint used in Definition 1 | `channelAdjoint` | The environmental partial trace of $U^\\dagger(Y\\otimes I_E)U(I_S\\otimes\\xi)$ |\n| Definition 1, equation (2), Petz recovery | `petz` | $\\sqrt\\gamma\\,N^\\dagger(N(\\gamma)^{-1/2}X N(\\gamma)^{-1/2})\\sqrt\\gamma$; matrix inverse and `CFC.sqrt` |\n| Definition 2, equation (3), same interaction | `tabletopReverse` | $\\operatorname{Tr}_E[U^\\dagger(X\\otimes\\eta)U]$; $U$ is fixed |\n| Definition 2, equation (4), allowed environment | `IsTTR` | A positive semidefinite trace-one $\\eta$ whose reverse equals the Petz map on every complex matrix $X$ |\n| Finite system and environment | The types `S`, `E` and their `Fintype`, `DecidableEq` instances in `claim` | Matrix indices, with no dimension restriction on the universal claim |\n| Physical forward dynamics | Unitarity of `U`, positivity and trace one of `xi` in `claim` | The source's unitary dilation and environmental density state |\n| Full-rank prior | Positive definiteness and trace one of `gamma` in `claim` | A restriction of the source's priors; a refutation in this subclass refutes the general assertion |\n| Steady state | `channel U xi gamma = gamma` in `claim` | Exact stationarity |\n| TTR whenever steady | `IsTTR U xi gamma` conclusion of `claim` | Same interaction, arbitrary reverse environmental density matrix |\n| Negative answer | `result` | Negation of the complete universal claim |\n\nThe public definitions keep the preregistered expressions. The proof reuses `PartialTraceMutualInformation.partialTraceRight`, `NMLWMeasuredFidelitySeparation.sqrt_diagonal`, and the public Fuchs qutrit-state positivity and trace lemmas. The environmental matrix is literal witness data.","title":"A steady prior does not imply tabletop time reversibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-kwon-scarani-2025-steady-prior-reversibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/261e7efe9c3d99c6f033afb9f0ccf5867a178b929f8d79ec5d771c5385da7b4e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-li-2022-dequantumness-power-strict-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Song and N. Li, *Quantumness and Dequantumness Power of Quantum Channels*, Entropy 24(8), 1146 (2022), DOI [10.3390/e24081146](https://doi.org/10.3390/e24081146), Section 3, page 5 of the [publisher PDF](https://mdpi-res.com/d_attachment/entropy/entropy-24-01146/article_deploy/entropy-24-01146.pdf), in the paragraph following the amplitude-damping calculation:\n\n> Though $D_2(\\Lambda)=D_2'(\\Lambda)$ in this case, we conjecture this equality may fail in certain cases. But we have not found the counterexample satisfying $D_2(\\Lambda)>D_2'(\\Lambda)$ yet.\n\nFor a two-state qubit ensemble $\\mathcal E=\\{(p,\\rho),(1-p,\\sigma)\\}$, let $p\\in[0,1]$ and let $\\rho,\\sigma$ be positive semidefinite complex two-by-two matrices of trace one. Equation (1), on page 2, specializes to\n\n$$Q(\\mathcal E)=-\\sum_{i,j=1}^2p_ip_j\\operatorname{tr}[\\rho_i,\\rho_j]^2.$$\n\nIn the module, `quantumness` takes the real part of each trace. With all suprema below over these two-state ensembles, its definitions are\n\n$$Q_{\\max,2}=\\sup_{\\mathcal E}Q(\\mathcal E),\\qquad D_2(\\Lambda)=\\sup_{\\mathcal E}\\bigl(Q(\\mathcal E)-Q(\\Lambda(\\mathcal E))\\bigr),$$\n\n$$D_2'(\\Lambda)=\\sup_{\\mathcal E:\\,Q(\\mathcal E)=Q_{\\max,2}}\\bigl(Q(\\mathcal E)-Q(\\Lambda(\\mathcal E))\\bigr).$$\n\nThis is the two-ingredient reading used in the source's page-5 calculation, where the maximal initial quantumness is $1/4$; it does not optimize over ensembles of arbitrary size. The definitions and source locations are also given in [the literature note](../Library/Entropy/songli2022dequantumness.md).\n\nIssue [#22778](https://github.com/the-omega-institute/trureturing/issues/22778) fixes `claim` as universal equality: every completely positive trace-preserving complex-linear qubit map satisfies $D_2=D_2'$. The resolution `refuted` negates that equality claim and supplies the strict separation sought by the authors; it confirms their quoted conjecture.","title":"Unrestricted two-state dequantumness power can exceed its maximal-quantumness restriction","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-li-2022-dequantumness-power-strict-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbbf44a25594cf34cdc9abe13812c622cba3503b5bedb5c53c713cabc0332fb4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:song-lin-2026-oriented-circulant-zero-transfer-parity-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Song and Lin study the continuous-time quantum walk `U(t) = exp(-i t H)` on a\nmixed graph, where the Hermitian adjacency matrix `H` has entry `i` on an arc\n`u → v`, `-i` on the reversed arc and `0` elsewhere on an oriented graph. A\ngraph has zero transfer from `u` to `v` if `U(t)_{u,v} = 0` for every time\n`t ≥ 0`. For the oriented circulant graphs `G(ℤ_n, C)`, with arcs `a → b` for\n`b - a ∈ C`, `C ⊆ ℤ_n ∖ {0}`, `C ∩ -C = ∅`, and connectedness assumed\nthroughout the section, they state Conjecture 4.1 of arXiv:2608.10643v1:\n\n> Let Γ = G(ℤ_n, C) be an oriented circulant graph with n ≡ 2 (mod 4). If\n> zero transfer occurs between vertex v and 0, then v must be odd.\n\nIssue #10142 fixes the readings: vertices are `ZMod n`; `U(t)` is the matrix\nexponential of `-(i t) H`, taken as the frozen propagator\n`hamiltonianPropagator H t = exp (t • (-i) • H)` of\n`D5/S3/Quantum/Dynamics/ProjectionProbabilityFlow`; zero transfer between `v` and `0` is read in both\ndirections at once, which only strengthens the hypothesis; connectedness is\nthat of the underlying undirected graph; \"v is odd\" means that the\nrepresentative of `v` in `0, …, n − 1` is odd.","title":"Song and Lin's parity conjecture for zero transfer on oriented circulant graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/song-lin-2026-oriented-circulant-zero-transfer-parity-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3bac50cb6382950f4f02c7809a56966edf664187743d893d12b740439190c59.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sp322844-serpentine","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stefano Spezia, OEIS A322844, COMMENTS revision 68, states:\n\n> For n > 1, a(n) is the absolute value of the trace of the 2nd exterior power of an n X n square matrix M(n) defined as M[i,j,n] = j + n*(i-1) if i is odd and M[i,j,n] = n*i - j + 1 if i is even (see A317614). Equivalently, a(n) is the absolute value of the coefficient of the term [x^(n-2)] in the characteristic polynomial of the matrix M(n), or the absolute value of the sum of all principal minors of M(n) of size 2.\n\nThe matrix definition is shared with A317614 revision 194. With one-based indices $1\\leq i,j\\leq n$, its integer entries are $M_{ij}=j+n(i-1)$ for odd $i$ and $M_{ij}=ni-j+1$ for even $i$. Put $\\varepsilon=n\\bmod 2$ and\n\n$$\na(n)=\\frac{n^2}{12}\\bigl(3(1+n^2)-2(2+n^2)\\varepsilon\\bigr).\n$$\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| One-based integer serpentine entries | `SpeziaSerpentineSecondExteriorTrace.M` | `Fin n` indices are increased by one; entry arithmetic is integer arithmetic. |\n| Source rational expression for $a(n)$ | `SpeziaSerpentineSecondExteriorTrace.a` | Rational division and natural remainder; no truncated rational quotient. |\n| Second exterior trace, characteristic coefficient, principal-minor sum | `SpeziaSerpentineSecondExteriorTrace.traceExt` | The binding convention uses $(-1)^k[x^{n-k}]\\chi_M$ for $k\\leq n$ and zero for $k>n$. It does not construct an exterior-power linear map. |\n| First conjecture, every $n>1$ | `SpeziaSerpentineSecondExteriorTrace.claim`, `result` | Integer absolute value, cast to the rationals, equals $a(n)$. |\n| Second conjecture, every $n$ and $k>2$ | `SpeziaSerpentineHigherExteriorTrace.claim`, `result` | Includes $n For k > 2, the trace of the k-th exterior power of the matrix M(n) is equal to zero.\n\nThe matrix definition is shared with A317614 revision 194. With one-based indices $1\\leq i,j\\leq n$, its integer entries are $M_{ij}=j+n(i-1)$ for odd $i$ and $M_{ij}=ni-j+1$ for even $i$. Put $\\varepsilon=n\\bmod 2$ and\n\n$$\na(n)=\\frac{n^2}{12}\\bigl(3(1+n^2)-2(2+n^2)\\varepsilon\\bigr).\n$$\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| One-based integer serpentine entries | `SpeziaSerpentineSecondExteriorTrace.M` | `Fin n` indices are increased by one; entry arithmetic is integer arithmetic. |\n| Source rational expression for $a(n)$ | `SpeziaSerpentineSecondExteriorTrace.a` | Rational division and natural remainder; no truncated rational quotient. |\n| Second exterior trace, characteristic coefficient, principal-minor sum | `SpeziaSerpentineSecondExteriorTrace.traceExt` | The binding convention uses $(-1)^k[x^{n-k}]\\chi_M$ for $k\\leq n$ and zero for $k>n$. It does not construct an exterior-power linear map. |\n| First conjecture, every $n>1$ | `SpeziaSerpentineSecondExteriorTrace.claim`, `result` | Integer absolute value, cast to the rationals, equals $a(n)$. |\n| Second conjecture, every $n$ and $k>2$ | `SpeziaSerpentineHigherExteriorTrace.claim`, `result` | Includes $n For k > 2, the trace of the k-th exterior power of M(n) is equal to zero.\n\nThe same block and OEIS A330287, COMMENTS, revision 31, specify:\n\n> The matrix M(n) is defined as M[i,j,n] = i*j if i < 3 or j < 3 and M[i,j,n] = 2*(i + j) - 4 otherwise.\n\nIndices are one-based. For every positive natural $n$, interpret the integer entries in $\\mathbb C$ and take every natural $k>2$, including $k>n$. The exterior power is represented by the existing `compoundMatrix`; its trace is `Matrix.trace`. The target is exactly this second conjecture, as preregistered in [#21999](https://github.com/the-omega-institute/trureturing/issues/21999).\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| The literal piecewise entry formula | `SpeziaPerimeterHigherExteriorTraces.M` | Source indices are `i.val+1`, `j.val+1`; the comparison is in natural numbers and entries and subtraction are in $\\mathbb C$. |\n| A square matrix of positive order | `claim` quantifies over natural $n>0$ | Positivity is the explicit preregistered scope. |\n| Every exterior order $k>2$ | `claim` quantifies over natural $k>2$ | No upper bound $k\\le n$ is added. |\n| Trace of the exterior power equals zero | `Matrix.trace (compoundMatrix (M n) k)` equals zero | The existing ordered-subset basis also includes the empty-space case. |\n| The universal conjecture holds | `SpeziaPerimeterHigherExteriorTraces.result` | The theorem proves `claim` without factorization or rank hypotheses. |","title":"Higher exterior traces of Spezia's perimeter matrix","url":"https://the-omega-institute.github.io/trureturing-pages/research/sp330700/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d5c08e4eabbdff21cca2c5f7fff64242bcc1ac87863b7e91caaf6be32dd7e45.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sp3921","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Stefano Spezia's OEIS A392131, revision 16, FORMULA states:\n\n> Conjecture: lim_{n->oo} A392132(n)/a(n) = 1.\n\nOEIS A392132, revision 10, FORMULA states the same conjecture as:\n\n> Conjecture: lim_{n->oo} a(n)/A392131(n) = 1.\n\nIn A392131, $a(n)$ is the signed determinant; in A392132, $a(n)$ is the permanent. These are two records of one assertion. The source is OEIS-only, submitted in 2025 and published in 2026, with no source DOI or arXiv identifier. Issue [#21886](https://github.com/the-omega-institute/trureturing/issues/21886) fixes the conventions and the settlement criterion. The source quotations and locators are in [the Library note](../Library/Asymptotics/spezia2026pellcirculant.md).\n\n### Statement echo\n\nLet $P_0=0$, $P_1=1$ and $P_{k+2}=2P_{k+1}+P_k$. With indices in $\\operatorname{Fin}(n)$, the matrix is $C_n(i,j)=P_{n-1-\\operatorname{val}(i-j)}$, with natural subtraction truncated at zero and modular subtraction in $i-j$. Its first row is $(P_{n-1},P_0,\\ldots,P_{n-2})$, the initial right rotation specified by the source COMMENT and four-by-four EXAMPLE. The formalization reuses the existing Pell sequence `D5.S1.Recurrence.PellCompanionGcd.P` and Mathlib's `Matrix.circulant`, `Matrix.det` and `Matrix.permanent`.\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| A392131 COMMENT: first column reversed Pell vector, first row initially rotated right | `C`, with the literal binding vector and Mathlib's modular circulant indexing |\n| A392131 determinant, including the source empty value | `A392131 n = Matrix.det (C n)`; the determinant retains its sign |\n| A392132 permanent of the same matrix | `A392132 n = Matrix.permanent (C n)` |\n| The ratio tends to one | `claim`: every real $\\varepsilon>0$ admits $N\\ge2$ such that every $n\\ge N$ has $|A392132(n)/A392131(n)-1|<\\varepsilon$ |\n| No additional matrix-family hypothesis | `result : claim` has no hypotheses |\n\nThe threshold condition $N\\ge2$ specifies an equivalent tail of the limit; it imposes no extra condition on the sequence. Both empty values are one. The zero determinant at $n=1$ does not affect a tail assertion.","title":"Spezia's Pell circulant permanent-to-determinant ratio","url":"https://the-omega-institute.github.io/trureturing-pages/research/sp3921/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e02f6372e043b036c9d4b9b83335e77a530dcee318ed647b0b71d29ad597eed4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:spitzer-nechita-2026-quantum-graph-component-count","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. L. Spitzer and I. Nechita, *Quantum Graph Theory by Example*, arXiv:2603.23651v3 [math.OA, cross-listed math-ph, quant-ph] (Quantum 10, 2228, 2026), leave the component-count question in the conclusion. The source text states in `content/conclusion.tex`, line 8:\n\n> It would be interesting to determine the exact number of connected components of $X_{A, \\ccdot, C}$ in terms of $\\mathfrak{G}(A, C)$, and in particular to understand the role played by isolated strange edges with phase $\\pi$.\n\nThe paper defines the strange graph $\\mathfrak{G}(A,C)$ from the nonzero two-vertex blocks of a quantum graph of the form $X_{A,B,C}$. The formal module fixes the $B=\\operatorname{diag}A$ reading through the following declarations. For $A,C : \\operatorname{Mat} n$ and $d : \\operatorname{Fin} n \\to \\mathbb C$, `realignedProjector A C d` is the matrix with diagonal-coordinate entries $d$ and off-diagonal blocks determined by $A$ and $C$, and `adjacency A C d` is the linear map obtained from that realigned matrix. `HasComponents G k` means that there are $k$ nonzero star projections summing to the identity whose cross terms $P_s G(P_t Z)$ vanish for every $s \\ne t$ and every $Z$. `underlyingGraph A C` has an edge exactly when the corresponding off-diagonal entry of $A$ or $C$ in either direction is nonzero. `IsLoopFreeStrangeComponent A C d Q` says that a connected component $Q$ has support $\\{i,j\\}$ for distinct vertices, is adjacent, has $A_{ij}=1/2$, and has $d_i=d_j=0$. The two counts are `graphComponentCount A C` and `loopFreeStrangeComponentCount A C d`, the latter counting such components.\n\nThe Lean `claim` quantifies over every natural $n$, every $A,C : \\operatorname{Mat} n$, and every $d : \\operatorname{Fin} n \\to \\mathbb C$. Under $A_{ii}=d_i$, $C_{ii}=d_i$, Hermiticity and idempotence of `realignedProjector A C d`, and Hermiticity of `offDiag A`, it asserts that the set of $k$ for which `HasComponents (adjacency A C d) k` holds has greatest element `graphComponentCount A C + loopFreeStrangeComponentCount A C d`.","title":"The component count is the graph count plus loop-free strange components","url":"https://the-omega-institute.github.io/trureturing-pages/research/spitzer-nechita-2026-quantum-graph-component-count/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd4cea9a99c0ffb29e46f8f025060708202b19d2ea6f595ad0d187cf0fafa93e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:squarefree-mean-partition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A360070, at revision #18 of Jan 29 2023, by Gus Wiseman, Jan 27 2023:\n\n> Numbers for which there exists an integer partition such that the parts have the same mean as the\n> multiplicities.\n\n> Conjecture: No term > 1 is squarefree.\n\nThe entry's worked example:\n\n> A partition of 20 with the same mean as its multiplicities is (5,4,3,2,1,1,1,1,1,1), so 20 is in\n> the sequence.","title":"Squarefree Numbers Have No Partition Whose Parts and Multiplicities Agree in Mean","url":"https://the-omega-institute.github.io/trureturing-pages/research/squarefree-mean-partition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/090273638ddfce560266e12d41aa9d43e493e961298f5e4a98659e41d9a7b28a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sra2609-koteljanskii","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Suvrit Sra, *A Koteljanskii inequality for permanents*, arXiv:2609.39979v1,\nSection 4(d), asks:\n\n> Is $f_A$ Lorentzian for every inverse $M$-matrix $A$?\n\nHere, for $n\\ge1$,\n\n$$f_A(x_0,x)=\\sum_{S\\subseteq[n]}\\operatorname{per}(A_S)^{-1}\nx_0^{n-|S|}\\prod_{i\\in S}x_i.$$\n\nAn inverse $M$-matrix is entrywise nonnegative, invertible, and has inverse\noff-diagonal entries at most zero. The empty principal permanent is one.\nLorentzian means the coefficient-topology closure of the recursively defined\nstrict Lorentzian class of Brändén–Huh, Definition 2.1; it is not defined here\nby only a necessary Hessian test. The complete source expressions are in\n[the source note](../Library/QuadraticForms/sra2026koteljanskii.md) and\n[the Lorentzian note](../Library/Combinatorics/brandenhuh2020lorentzian.md).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Every positive dimension, every real inverse $M$-matrix | `claim`: $n>0$, real `Matrix (Fin n) (Fin n)`, `InverseMMatrix` | No symmetry or strict-entry-positivity assumption is added. |\n| Principal-submatrix permutation permanent | `PSW.principalPermanent`, `Matrix.permanent` | The subtype supplies exactly the selected indices, including the empty case. |\n| Displayed reciprocal polynomial | `reciprocalPolynomial` | `none` is $x_0$ and `some i` is source variable $x_{i+1}$. |\n| Brändén–Huh positive homogeneous class, strict recursion, closure | `positiveHomogeneous`, `quadraticHessian`, `strictlyLorentzian`, `Lorentzian` | All degree-$d$ coefficients are quantified; eigenvalue indices count multiplicity; finite-dimensional coefficientwise epsilon approximation expresses the closure. |\n| Affirmative universal assertion is false | `result` | The theorem proves the literal negation of `claim`. |","title":"Sra's reciprocal-permanent Lorentzian question","url":"https://the-omega-institute.github.io/trureturing-pages/research/sra2609-koteljanskii/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb92ccbd11d35a7e749778b380ecd90145dba24ddc52a9af6a26a7010291f841.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sra2609-symmetric-alpha-permanent","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Suvrit Sra, *A Koteljanskii inequality for permanents*, arXiv:2609.39979v1,\nSection 4(b), asks:\n\n> For symmetrizable inverse $M$-matrices and $\\alpha\\notin\\{\\tfrac12,1\\}$,\n> decide whether $S\\mapsto\\operatorname{per}_\\alpha(A_S)$ is log-supermodular.\n\nThe order is positive as in equation (3.1). The universal affirmative\nassertion quantifies over $n\\ge1$, every real symmetrizable inverse $M$-matrix,\nevery permitted positive order, and every pair of index subsets.\nThe source definitions are quoted in\n[the Library note](../Library/QuadraticForms/sra2026koteljanskii.md).\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Every $n\\ge1$, real inverse $M$-matrix | `claim`, imported `InverseMMatrix` | Exactly the source's nonnegativity, invertibility and inverse off-diagonal sign conditions. |\n| Positive diagonal similarity to symmetry | `Symmetrizable` | The diagonal function is strictly positive; its pointwise reciprocal gives the inverse diagonal. |\n| Equation (3.1), all cycles including fixed points | `alphaPermanent` | `cycleType.card` counts nontrivial cycles, and the fixed-point cardinality adds one-cycles. |\n| Principal restriction, including the empty case | `principalAlphaPermanent` | `Matrix.submatrix` on the index subtype; empty product and permutation sum give one. |\n| $\\alpha>0$, $\\alpha\\ne1/2,1$ and every $S,T$ | `claim` | All three order conditions and both subset binders are retained. |\n| Log-supermodular inequality | `claim` | The $S,T$ product is at most the union-intersection product. |\n| Universal affirmative assertion is false | `result` | Literal negation of `claim`. |","title":"Sra's symmetric alpha-permanent question","url":"https://the-omega-institute.github.io/trureturing-pages/research/sra2609-symmetric-alpha-permanent/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd99ab47111d154b3a372246acdaf54fe338727011a1b452407a2e6fa7ac786a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:stacey-2025-autocorrelation-norm-clifford-invariance-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. C. Stacey, *SICs, the Stabilizer Polytope and Majorization*, arXiv:2509.16819v1, Section III, defines A(ψ)_{mk} = Σ_j p_{m,j} p_{m,j+k}, where p_{m,j} are the outcome probabilities in the Weyl–Heisenberg mutually unbiased bases. It then states:\n\n> Acting upon $\\ket{\\psi}$ with a Clifford unitary can at most permute the rows and columns of the matrix $A$. … Indeed, let $||\\cdot||$ be any unitarily invariant norm; because permutation matrices are a very special subset of the unitary matrices, left- and right-multiplying $A$ by any pair of permutation matrices will leave $||A||$ unchanged. It seems, then, that we have a large family of new candidate monotones that are all extremized by Weyl--Heisenberg SIC fiducials.\n\nThe verbatim passage is in [the literature note](../Library/QuantumContext/stacey2025sicsmajorization.md).\n\nIssue [#18928](https://github.com/the-omega-institute/trureturing/issues/18928) fixes the reading: for d = 5, the Euclidean operator norm of A is invariant under every Clifford unitary, for every pure state. It is refuted.","title":"A Clifford unitary that changes the operator norm of the MUB autocorrelation matrix","url":"https://the-omega-institute.github.io/trureturing-pages/research/stacey-2025-autocorrelation-norm-clifford-invariance-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cca8f6e802ad19d934e7848ea50121ac852566b86eb5da7aa1107fda3a822a6e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:stampach-waclawek-2026-birman-weight-expansion","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Štampach and J. Waclawek, *Optimal discrete p-Hardy–Rellich–Birman\ninequalities*, arXiv:2605.25238v1, Conjecture 5.6(iii), p. 28:\n\n> Let $\\ell\\in\\mathbb N$ and $p>1$, and let $\\tilde{\\rho}^{(\\ell,p)}$ be defined by (2.17) and (2.16).\n\n> For all $n\\geq\\ell$, the terms $\\tilde{\\rho}_{n}^{(\\ell,p)}$ admit a power series expansion in negative powers of $n$ with entirely non-negative coefficients; cf. (2.15).\n\nHere $\\mathbb N$ means positive integers. The convention is\n$\\tilde{\\mathfrak g}_n=n^{1-1/p}\\prod_{j=1}^{\\ell-1}(n-j)$ for $n\\geq0$,\nwith zero extension to negative integers;\n$(\\nabla u)_n=u_n-u_{n-1}$ and $(\\mathrm{div}\\,u)_n=u_{n+1}-u_n$;\n$\\nu^{\\langle a\\rangle}=\\nu|\\nu|^{a-1}$, with\n$0^{\\langle a\\rangle}=0$; and\n$\\tilde\\rho_n=(-1)^\\ell[\\mathrm{div}^{\\ell}(m\\mapsto(\\nabla^{\\ell}\\tilde{\\mathfrak g})_m^{\\langle p-1\\rangle})]_n/\\tilde{\\mathfrak g}_n^{p-1}$,\nwith signed power taken before the divergence iterate.\n\nThe Lean `claim` asks for one non-negative coefficient sequence\n$c:\\mathbb N_0\\to\\mathbb R$ whose series converges to\n$n^{\\ell p}\\tilde\\rho_n$ at every $n\\geq\\ell$. It allows any\nnon-negative constant coefficient. The source's (2.15) normalization implies\nthis claim by $c_0=((1/q)_\\ell)^p$ and $c_k=c_0 A_k$ for $k\\geq1$,\nwhere $1/q=1-1/p>0$ and the rising factorial is positive.\nThus negating the weaker claim refutes part (iii).","title":"Štampach–Waclawek's alternative Birman weight expansion","url":"https://the-omega-institute.github.io/trureturing-pages/research/stampach-waclawek-2026-birman-weight-expansion/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4cd2c02d85a36516d43c712aee442335f0ed6e988cf4a57c0ace6d0f69ee294b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:starke-2025-qudit-swapping-product-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. S. Starke, M. L. W. Basso, L. C. Céleri and J. Maziero,\n“Entanglement swapping for partially entangled qudits and the role of quantum\ncomplementarity”, arXiv:2508.00813v2, p. 8, Eq. (59), conjecture:\n\n> Therefore, we conjecture that the improved upper bound has the form\n> $\\big\\langle E_{l_1}\\big(|\\hat{\\phi}_{pq}^{AB}\\rangle\\big)\\big\\rangle\n> \\le \\frac{E_{l_1}(|\\xi\\rangle_{AC})E_{l_1}(|\\eta\\rangle_{C'B})}{d-1}.$\n\nFor every dimension $d\\ge2$, the inputs are\n$|\\xi\\rangle_{AC}=\\sum_jc_j|jj\\rangle$ and\n$|\\eta\\rangle_{C'B}=\\sum_kb_k|kk\\rangle$, with\n$\\sum_j|c_j|^2=\\sum_k|b_k|^2=1$; the paper's second coefficient $d_k$ is renamed\n$b_k$. The measure is the unscaled ordered sum\n$E_{l_1}(a)=\\sum_{j\\ne k}|a_ja_k|$.\n\nThe Bell outcome $(p,q)$ gives\n$\\phi_{pq}=d^{-1/2}\\sum_kc_{p\\oplus k}b_k\\bar\\omega^{qk}|p\\oplus k,k\\rangle$,\nwhere $\\omega=\\exp(2\\pi i/d)$ and $\\oplus$ is addition modulo $d$.\nIts probability is its squared Hilbert norm; its normalized state is obtained\nby dividing by that norm. The average includes all $d^2$ outcomes, with\nzero-probability outcomes contributing zero. Each conditional state is in\nSchmidt form, since $k\\mapsto p\\oplus k$ is a bijection.","title":"The improved entanglement-swapping product bound fails in dimension four","url":"https://the-omega-institute.github.io/trureturing-pages/research/starke-2025-qudit-swapping-product-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0993fcaa2efd7501fe9fefa899ec633c9363952c02baa9a8749c997b90c9d82.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:stepanov-herrmann-guhr-2021-correlation-top-eigenvector-weight","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Stepanov, H. Herrmann and T. Guhr, *Generic features in the spectral decomposition of correlation matrices*, arXiv:2104.08966v1 (Journal of Mathematical Physics 62 (2021), 083505), leave the domain $A\\setminus(A_1\\cup A_2)$ open. They define\n$A=\\{(c,\\sigma):c>0,\\ \\sigma>0,\\ c^2+\\sigma^2<1\\}$,\n$A_1=\\{(c,\\sigma)\\in A:c\\geq\\frac12\\ \\text{or}\\ \\sigma\\sqrt3c\\ \\text{or}\\ c<\\sigma<1-c\\}$.\nFor an orthonormal eigenbasis, $w_j=\\langle v_j,\\delta_n\\rangle^2$ with $\\delta_n=(1,\\ldots,1)/\\sqrt n$, and the source's definition at TeX lines 877--879 compares $w_1$ with the largest weight among the other eigenvectors. Question III.5 in the source TeX, lines 525--527, asks:\n\n> Given $(c,\\sigma) \\in A \\setminus (A_1 \\cup A_2)$ and $n\\geq 2$ sufficiently large, is there an $n\\times n$ correlation matrix $C$ with mean correlation $c$ and standard deviation $\\sigma$ such that $w_1 i.\" Definition 39\nsets `dis(w) = (d(1), …, d(2n))`, where `d(i)` is the spot car `i` takes minus\nthe spot `w(i)` it prefers, so `d(i) = 0` exactly when car `i` is lucky.\n\nCorollary 41 reads: \"For every w ∈ Q_n, we have 1 ≤ |{i ∈ [2n] : d(i) = 0}| ≤ n\",\nso the number of nonzero entries of `dis(w)` lies in `[n, 2n - 1]`. The article\nthen states:\n\n> Problem 48. Determine if, for i ∈ [n, 2n − 1], there exists w ∈ Q_n such that\n> dis(w) has exactly i nonzero entries. Equivalently, can we always find a\n> Stirling permutation with i unlucky cars?","title":"Stirling Lucky Displacement Spectrum","url":"https://the-omega-institute.github.io/trureturing-pages/research/stirling-lucky-displacement-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1740b111ea41c07530d4c7979c40f7e984718a186c6544e6e60eb369ef7ccc94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:stratton-hsieh-skrzypczyk-2025-bound-coherence-higher-dimensions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Stratton, C.-Y. Hsieh and P. Skrzypczyk, *Unspeakable coherence concentration*, arXiv:2512.04255v1 (2025), published as Phys. Rev. A 113, 062429 (2026), Section IX, Discussion, `main.tex` line 387:\n\n> Finally, a family of states with bound-coherence was shown to exist, i.e., states that are not maximally coherent, nor incoherent, but for which no allowed unitary exists to perform coherence concentration. We leave it open to determine if such states exist in higher dimensions.\n\nIssue [#22599](https://github.com/the-omega-institute/trureturing/issues/22599) fixes the reading for two identical copies with the truncated number observable $L=\\sum_{n=0}^{d-1}n|n\\rangle\\langle n|$. For every integer $d\\ge3$, there is a positive semidefinite matrix $\\rho$ of trace one with $[\\rho,L]\\ne0$, which is not an equal-amplitude pure state, such that for every unitary $V$ commuting with $L\\otimes I+I\\otimes L$,\n\n$M^{(j)}\\bigl(\\operatorname{tr}_B[V(\\rho\\otimes\\rho)V^\\dagger]\\bigr)\\le M^{(j)}(\\rho)\\quad(1\\le j\\le d-1).$\n\nHere $\\rho^{(j)}$ retains the entries with row number minus column number equal to $j$, and $M^{(j)}(\\rho)=\\|\\rho^{(j)}\\|_1$. An equal-amplitude pure state means $|\\psi\\rangle\\langle\\psi|$ with $\\psi_n=e^{i\\theta_n}/\\sqrt d$ for real phases $\\theta_n$. This reading is proved. The source definitions and their locators are in [the literature note](../Library/Eigenstructure/strattonhsiehskrzypczyk2026concentration.md).","title":"Bound-coherent states exist in every dimension at least three","url":"https://the-omega-institute.github.io/trureturing-pages/research/stratton-hsieh-skrzypczyk-2025-bound-coherence-higher-dimensions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71e366bbecea408722a89ffc45bb7c21ed0a03a2dda8c38250d6057cefc047aa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:streif-leib-2020-level-one-qaoa-genuine-superposition","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Streif and M. Leib, *Forbidden subspaces for level-1 QAOA and IQP circuits*, arXiv:2007.13563v1 (Phys. Rev. A 102, 042416 (2020)), study when a level-one QAOA state can have support on a target subspace. In the paper's Section “More than two target states,” `main.tex`, lines 335–341, they write:\n\n> Based on the assumption that the only solution can be found for $\\beta = \\frac{1}{4} \\pi, \\frac{3}{4} \\pi, \\frac{5}{4} \\pi, \\frac{7}{4} \\pi$ we show in the following that level-1 QAOA can not map to a genuine superposition of more than $2$ but less than $2^n$ target states. We consider a genuine superposition of target states a superposition where no amplitude vanishes.\n\nThey then state:\n\n> Based on this observation as well as the calculation for the 2 target state case and numerical investigations we conducted we conjecture that the only possible solutions are $\\beta = \\frac{1}{4} \\pi, \\frac{3}{4} \\pi, \\frac{5}{4} \\pi, \\frac{7}{4} \\pi$.\n\nThe module formalizes the unrestricted exclusion as $\\mathit{claim} := \\forall n,e,\\beta,\\gamma,t,\\quad 2 < t.card \\mathbin{\\to} t.card < 2^n \\mathbin{\\to} \\neg\\mathit{IsGenuineSuperposition}(\\mathit{qaoaState}\\ n\\ e\\ \\beta\\ \\gamma)\\ t$, where `qaoaState` is the one phase-layer and one mixer-layer state and `IsGenuineSuperposition` means nonzero amplitudes exactly on $t$. The settlement is `result : ¬ claim`.","title":"Level-one QAOA reaches a genuine four-state superposition on three qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/streif-leib-2020-level-one-qaoa-genuine-superposition/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/53e88a1858a381b4a9a2274bbaaaf408bc8a8346d4a76e774b58e7221c588d0d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sud-marshall-wang-rieffel-wudarski-2022-maten-worst-case-fidelity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Sud, J. Marshall, Z. Wang, E. Rieffel and F. A. Wudarski, *Dual-map framework for noise characterization of quantum computers*, arXiv:2112.04414v1 (2021); Phys. Rev. A 106, 012606 (2022), Section IV, “Local vs Non-Local Channels”, state:\n\n> We conjecture that the effective channel with the maximally entangled $\\chi$ matrix is the worst case scenario for the proposed MATEN protocol.\n\n> However, this conjecture requires more rigorous treatment, which we leave as an open problem.\n\nThese sentences occur in `main.tex`, lines 289 and 290, respectively, in the downloaded v1 source (709 lines; MD5 `b2275e7db7e3a63cc4c871c172f7943a`). The source specifies $\\chi=|\\Psi\\rangle\\langle\\Psi|$ with $|\\Psi\\rangle=\\tfrac12\\sum_{i=0}^3|ii\\ldots i\\rangle$ on $N$ four-dimensional process factors (lines 286–289), and evaluates $F(\\chi,\\chi_{\\mathrm{MA}})=4^{-N}$ (lines 292–296). Its MATEN definition is at lines 226–232. See also [the literature note](../Library/QuantumContext/sudmarshallwangrieffelwudarski2022dual.md).\n\nIssue [#20689](https://github.com/the-omega-institute/trureturing/issues/20689) fixes the settlement as a universal fidelity lower bound and attainment by cyclic wire permutations. The source's particular GHZ witness and its trace-preservation assertion require the qualification in Triage.\n\nWrite $a\\in\\{0,1,2,3\\}^N$ for a Pauli string and $P_a=\\bigotimes_k\\sigma_{a_k}$, with $\\sigma_0=I$, $\\sigma_1=X$, $\\sigma_2=Y$ and $\\sigma_3=Z$, without Hilbert–Schmidt normalization. The formal definitions are:\n\n- `IsProcessCPTP` means $\\chi\\succeq0$ and $\\sum_{a,b}\\chi_{ab}P_bP_a=I$. This is the process representation of $\\mathcal E_\\chi(\\rho)=\\sum_{a,b}\\chi_{ab}P_a\\rho P_b$, matching the source's CP and TP conditions at lines 93–100.\n- `marginal` is the single-site partial trace: $(\\chi_k)_{ij}=\\sum_{y:y_k=i}\\chi_{y,y[k\\leftarrow j]}$. `productMap` forms $\\chi_{\\mathrm{MA}}=\\bigotimes_k\\chi_k$, entrywise $(\\chi_{\\mathrm{MA}})_{ab}=\\prod_k(\\chi_k)_{a_kb_k}$.\n- The fidelity used is squared Uhlmann fidelity, $F(\\rho,\\sigma)=(\\operatorname{Re}\\operatorname{tr}\\sqrt{\\sqrt\\rho\\,\\sigma\\sqrt\\rho})^2$, exactly `sqFidelity`. This convention agrees with the source's projector calculation and distinguishes squared fidelity from root fidelity.\n- Put $U_N=(\\mathtt{qubitPermutation}(\\mathtt{finRotate}\\,N))^{\\mathsf T}$. Here the untransposed permutation matrix has entries $1$ when $y=x\\circ\\mathtt{finRotate}\\,N$ and $0$ otherwise. `shiftCoeff` is $\\psi_N(a)=2^{-N}\\operatorname{tr}(P_aU_N)$, and `rankOneDensity` gives $\\chi_N=\\psi_N\\psi_N^\\dagger$, with entries $\\psi_N(a)\\overline{\\psi_N(b)}$.","title":"The sharp worst-case MATEN process fidelity is $4^{-N}$ for $N\\ge2$","url":"https://the-omega-institute.github.io/trureturing-pages/research/sud-marshall-wang-rieffel-wudarski-2022-maten-worst-case-fidelity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa2b678dfad277e2833bf871635ea7a311156ad9633e09302afafeda9ff98cd8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sujsuntinukul-chesneau-2026-damascus-violation-set-question-ii","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chanatip Sujsuntinukul and Christophe Chesneau, *Variants of the Damascus inequality*, arXiv:2601.00916v4, Section 5, page 19, ask:\n\n> In addition to the above, it is natural to ask whether analogous structural properties hold for I_n^m when m is arbitrary. Can one prove or disprove that I_n^m (i) stays uniformly away from 1, (ii) is bounded, or (iii) forms an increasing family in n?\n\nThe source concludes: “Thus this question, together with question (iii), remains open.” This dossier settles question (ii).\n\nWrite $f_n(x)=(x^n-1)/(x^{n+1}+1)$, $H_m=\\{x\\in(0,\\infty)^m:\\prod_jx_j=1\\}$ and $I_n^m=\\{x\\in H_m:\\sum_j f_n(x_j)>0\\}$. For all natural numbers $m,n\\ge1$, the result is\n\n$$I_n^m\\text{ is bounded}\\quad\\Longleftrightarrow\\quad m\\le3\\ \\lor\\ (m=4\\land n\\le4)\\ \\lor\\ (m\\in\\{5,6\\}\\land n=1).$$\n\nThe classification is the repository's answer, rather than a proposition quoted from the paper.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Summand of inequality (1), page 2 | `term` | Literal real quotient $(x^n-1)/(x^{n+1}+1)$; no transformed predicate. |\n| $H_m$ consists of positive coordinates with product one | `H` | Functions `Fin m → ℝ`; every coordinate is positive and occurs once in the product. |\n| $I_n^m\\subset H_m$ has strictly positive summand sum | `I` | Membership in `H m` and a strict inequality $0<\\sum_j f_n(x_j)$. |\n| Arbitrary dimension and positive integer exponent | `claim`, binders $m,n$ and guards $1\\le m$, $1\\le n$ | Both parameters vary without upper bounds. Dimensions one and two are included; dimension zero and exponent zero are outside the question. |\n| Boundedness | `Bornology.IsBounded (I n m)` | The usual sup norm on a finite real coordinate space. |\n| Answer to question (ii) | `result : claim` | Exactly the displayed three-branch classification, in both directions. The classification branches express the answer; they are not additional hypotheses. |","title":"Boundedness of the Damascus violation sets","url":"https://the-omega-institute.github.io/trureturing-pages/research/sujsuntinukul-chesneau-2026-damascus-violation-set-question-ii/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fdfd747fb5ce3a1e5118849cad226775c988f0df7869be24c2fbfa46660e05c5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun-2013-circular-primitive-permutation-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhi-Wei Sun, “Some new problems in additive combinatorics”, arXiv:1309.1679,\nConjecture 3.8:\n\n> Let F_q be a finite field with q > 7 elements, and let a_0 be any element\n> of F_q. Then there is a circular permutation (a_1, …, a_{q−1}) of all the\n> nonzero elements of F_q such that all the q − 1 elements a_0 + a_1a_2,\n> a_0 + a_2a_3, …, a_0 + a_{q−2}a_{q−1}, a_0 + a_{q−1}a_1 are primitive\n> elements of the field F_q.\n\nThe Lean definition `claim : Prop` quantifies over every finite field and\nevery offset. An equivalence from `Fin (q - 1)` to the nonzero subtype\nexpresses that every nonzero element appears exactly once. `finRotate`\nadvances cyclically, including the last-to-first edge. `IsPrimitiveRoot z\n(q - 1)` means exact multiplicative order `q - 1`, hence precisely that\n`z` generates the multiplicative group, whose order is `q - 1`.","title":"Sun's Conjecture 3.8: circular permutations with primitive adjacent products","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun-2013-circular-primitive-permutation-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6eead09e31ca9bae31967eb3ddf0160d5d58ae188352563a11c6cb850e312bd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun-2021-conjecture-4-9-weighted-distinct-sums","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhi-Wei Sun, *On permutations of {1,...,n} and related topics*, J. Algebraic Combin. 54 (2021), 893–912, DOI 10.1007/s10801-021-01028-8, arXiv:1811.10503v6, Conjecture 4.9, printed page 18:\n\n> Conjecture 4.9. Let $n\\in\\mathbb Z^+$ and let $F$ be a field with $p(F)>n+1$, where $p(F)=p$ if the characteristic of $F$ is a prime $p$, and $p(F)=+\\infty$ if the characteristic of $F$ is zero. Let $A$ be any finite subset of $F$ with $|A|\\ge n+\\delta_{n,3}$, where $\\delta_{n,3}$ is 1 or 0 according as $n=3$ or not. Then, for the set $S(A):=\\{\\sum_{k=1}^n k a_k:a_1,\\ldots,a_n\\text{ are distinct elements of }A\\}$, we have $|S(A)|\\ge\\min\\{p(F),(|A|-n)n(n+1)/2+n(n^2-1)/6+1\\}$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A field $F$ | `F : Type`, `[Field F]` in `claim` | The preregistered carrier is literal. |\n| Characteristic zero gives infinity; prime characteristic gives that prime | `pF` | `ringChar`, zero test and `ENat` top/cast. |\n| Positive $n$ | `n : Nat`, `1 ≤ n` | Positive integers represented by positive naturals. |\n| $p(F)>n+1$ | Characteristic hypothesis in `claim` | Strict comparison in extended naturals. |\n| Finite $A$ | `A : Finset F`, `[DecidableEq F]` | Decidable equality is a classical representation requirement; it excludes no classical field. |\n| $|A|\\ge n+\\delta_{n,3}$ | Size hypothesis in `claim` | The conditional is exactly 1 at $n=3$ and 0 otherwise. |\n| Distinct selected elements; weights $1,\\ldots,n$ | `weightedDistinctSums` | Embeddings from `Fin n` into $A$, weighted by the natural value plus 1. |\n| Numerical expression and minimum with $p(F)$ | `sunBound` and the conclusion of `claim` | Natural subtraction is exact under the size hypothesis; both divided products are integral, so natural division preserves the source expression. |\n| Refutation | `result : ¬ claim` | The witness is $F=\\mathbb Q$, $n=2$, $A=\\{0,1,2\\}$. |","title":"Sun's weighted distinct-sum bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun-2021-conjecture-4-9-weighted-distinct-sums/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19a51820accfb0a9498206cfdcfd96b9ac110268854fbebcd9a9b3d737fee3b7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun-lowercase-turan-conjecture-52","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sun, arXiv:2608.13192v1, Conjecture 5.2 asks for both assertions, for\nevery integer `n >= 1`:\n\n$$\ng_n(x)^2>g_{n-1}(x)g_{n+1}(x)\\quad\\text{for every real }x\\leq-1,\n$$\n\nand\n\n$$\nv_n(x)^2>v_{n+1}(x)v_{n-1}(x)\\quad\\text{for every real }x\\leq-1/8.\n$$\n\nThe source's literal initial values and multiplied quadratic/cubic\nrecurrences are reproduced in `Library/Recurrence/sun2026generalizations.md`.\nBoth endpoints are included. A proof of only one clause, a finite-index\ncheck, or a non-strict statement does not settle this source question.","title":"Sun Conjecture 5.2: both lowercase strict Turan clauses","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun-lowercase-turan-conjecture-52/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52e5d95c9296f90acbbedc1079d31da10a3617bb5e0c6ac539fdadc00680295f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun-luo-ye-2020-projection-matrix-amgm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Sun, Z.-Q. Luo, and Y. Ye, *On the efficiency of random permutation for ADMM and coordinate descent*, arXiv:1503.06387v3, Section 3.3, Conjecture 3.1, equation (37), PDF p.20, state:\n\n> We observe that we only need to prove a matrix AM-GM inequality for projection matrices. We conjecture that the following matrix AM-GM inequality holds.\n>\n> **Conjecture 3.1** (matrix AM-GM inequality for projection matrices). Suppose $P_i \\in \\mathbb R^{N \\times N}, i=1,\\dots,n$ are projection matrices, then\n> $$\\frac{1}{n!}\\sum_{\\sigma=(\\sigma_1,\\dots,\\sigma_n)\\in\\Gamma} P_{\\sigma_n}P_{\\sigma_{n-1}}\\cdots P_{\\sigma_1}\\preceq\\left(\\frac1n\\sum_i P_i\\right)^n.$$\n\nHere $\\Gamma$ is the set of permutations of $\\{1,\\dots,n\\}$ and $X\\preceq Y$ means that $Y-X$ is positive semidefinite. The source and its coordinate-descent motivation are recorded in [the Library note](../Library/QuadraticForms/sunluoye2020projectionamgm.md). Preregistration: [#15647](https://github.com/the-omega-institute/trureturing/issues/15647).\n\nThe inspected mathematical source is arXiv v3. The matching journal record is\n*Mathematics of Operations Research* 45(1) (2020), 233–271,\nDOI [10.1287/moor.2019.0990](https://doi.org/10.1287/moor.2019.0990);\nthe journal PDF was not independently inspected. The conjecture number above\nis the arXiv v3 number.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Real $N\\times N$ matrices, indexed by $1,\\dots,n$ | `claim`: dimensions `n N`, family on `Fin n`, real matrices on `Fin N` | Finite reindexing |\n| Projection matrices | `claim`: symmetry and idempotence, inlined literally; the private instance proof uses Mathlib `IsStarProjection` | Orthogonal real projections; no rank restriction in the claim |\n| $n$ projection matrices | `claim`: $1\\le n$ | The source's reciprocal requires a positive number of factors |\n| Arithmetic mean, raised to $n$ | `meanProj`, and its $n$th power in `claim` | The inverse of the real cast of $n$ scales the sum |\n| Average over all permutations, with the first entry rightmost | `symProd` | The real inverse factorial scales the sum of reversed permutation words |\n| Positive-semidefinite order | `claim`: the mean power minus the symmetrized product is `Matrix.PosSemidef` | Exact difference direction |\n| Refutation of the universal assertion | `result`: negation of `claim` | Witness $n=N=3$; no assumptions on the result |\n\n### Structured block target and source identities\n\nThe separately registered target [#17674](https://github.com/the-omega-institute/trureturing/issues/17674),\nwith its [pre-probe scope notice](https://github.com/the-omega-institute/trureturing/issues/17674#issuecomment-6287957803),\nrestricts (37) to the following complete domain. For every integer $n\\ge1$\nand positive integers $d_1,\\ldots,d_n$, put $N=\\sum_i d_i$ and take real\n$A_i\\in\\mathbb R^{N\\times d_i}$ with $A_i^\\top A_i=I_{d_i}$ such that\nthe actual concatenation $A=[A_1,\\ldots,A_n]$ is square and nonsingular.\nSet $P_i=I_N-A_iA_i^\\top$, $M=n^{-1}\\sum_iP_i$, and\n$G=(n!)^{-1}\\sum_{\\sigma\\in\\Gamma}P_{\\sigma_n}\\cdots P_{\\sigma_1}$.\nThe target is $G\\preceq M^n$, with all block permutations uniformly\nweighted and the number of blocks as the exponent. The six-dimensional\nwitness below refutes this universal restricted assertion.\n\nThe block dimensions and normalization come from source Sections 2.1.1\nand 2.2, equations (7)–(8) and (23). Claim 4.1, equations (51a)–(51b),\nPDF p.23, assumes this square nonsingular, block-orthonormal structure\nand states identities. In the original block coordinates, let $L_\\sigma$\nhave block $(\\sigma_i,\\sigma_j)$ equal to\n$A_{\\sigma_i}^\\top A_{\\sigma_j}$ for $j\\le i$ and zero otherwise,\nand put $Q_\\sigma=L_\\sigma^{-1}$ and\n$Q=(n!)^{-1}\\sum_\\sigma Q_\\sigma$. Then those identities are\n\n$$\nI-AQ_\\sigma A^\\top=P_{\\sigma_n}\\cdots P_{\\sigma_1},\\qquad\nI-AQA^\\top=G.\n$$\n\nThey hold for the witness below; Claim 4.1 is not refuted. Lemma 3,\nequation (52), and Corollary 4.1, equation (53), prove the weaker bound\n$G\\preceq M$. The target $G\\preceq M^n$ is a strengthening on that\nstructured domain and a restriction of unrestricted Conjecture 3.1.\nThe unrestricted conjecture was already refuted by the historical Lean\nresult above before the block probes. No literal published, still-open\nfull-block assertion matching #17674 was established by the inspected\npre-probe record. The block result is retained here as exposition, with\nno new open-problem-resolution admission or literature-priority claim.","title":"The Sun–Luo–Ye projection-matrix AM-GM conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun-luo-ye-2020-projection-matrix-amgm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57063423c4b7ed7972fd958e38ac0801b3ee6579689c778bcbbfcb0e2b274378.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun210842","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhi-Wei Sun, *Arithmetic properties of some permanents*, arXiv:2108.07723v7,\nConjecture 4.2(i), equations (4.3) and (4.4), asks whether for every odd prime\n$p$ the permanents of the matrices with entries $|j-k|$ and $|j-k+1|$ are\ncongruent to $-1/2$ and $1/2$ modulo $p$, respectively. The integer encoding is\n$A_s(j,k)=|((j.val:\\mathbb Z)+1)-((k.val:\\mathbb Z)+1)+s|$ for\n$j,k : \\operatorname{Fin} p$ and $s=0,1$.\n\n### Statement echo\n\nThe source permanent is the finite sum over permutations of products of\nrow-indexed entries. The Lean claim clears the invertible denominator 2 and\nstates both divisibilities for the same universally quantified odd prime.","title":"Sun Conjecture 4.2(i) absolute-difference permanent congruences","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun210842/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/678fdc72991805cb99df4635fe1664dfa5bb01d025f2c860d8fa329219e19b2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sun228884","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhi-Wei Sun's OEIS A228884, COMMENTS clause (ii), states: “For any composite\nnumber $n$, all prime divisors of $a(n)$ are smaller than $n$.” The source\ndefines $a(n)$ as the determinant of the $n\\times n$ matrix whose $(i,j)$\nentry is $gcd(i-j,n)$ for $1\\leq i,j\\leq n$.\n\nThe formal claim uses the literal integer-entry matrix `C`, its determinant\n`D`, the hypotheses $4\\leq n$ and $\\neg\\operatorname{Nat.Prime}(n)$, and\ninteger divisibility by a natural prime $q$.\n\n### Statement echo\n\nThe settling declaration is\n`D5/S3/Arith/Divisors/SunA228884.result : D5/S3/Arith/Divisors/SunA228884.claim`.\nIts Scribe mirror expands the complete proposition, including every binder and\nthe integer cast in the divisibility clause.","title":"Composite prime divisors of the gcd determinant","url":"https://the-omega-institute.github.io/trureturing-pages/research/sun228884/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c148f6469e6dae4aa9b80105ebd1567ab0d22cb8677ad87c00e4f68f2f951768.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:sutter-popp-hiesmayr-2025-qutrit-weyl-bloch-norm-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. C. Sutter, C. Popp and B. C. Hiesmayr, Phys. Rev. A 113, 032440 (2026); arXiv:2508.18393v2,\nSection VI (p. 10):\n\n> Going forward, it is furthermore an open problem whether for general bipartite qudit states, an\n> entanglement criterion based on the entry-wise 1-norm of the Bloch vector, similar to Thm. IV.1, can\n> be derived. … A quick numerical optimization for $d=3$ yields $\\Vert\\beta\\Vert_1 \\lesssim 25.9735$ for\n> general pure states $|\\psi\\rangle\\in\\mathbb C^3\\otimes\\mathbb C^3$, whereas separable pure states seem\n> to satisfy $\\Vert \\beta\\Vert_1 \\leq 25$.\n\nHere $\\beta_{ij,kl}={\\rm tr}(\\rho\\,(W_{i,j}\\otimes W_{k,l})^\\dagger)$ with the Weyl–Heisenberg operators\n$W_{k,l}=\\sum_j\\omega^{jk}|j\\rangle\\langle j+l|$, $\\omega=e^{2\\pi i/3}$, and $\\Vert\\beta\\Vert_1=\\sum|\\beta_{ij,kl}|$.\nThe verbatim texts are in [the literature note](../Library/QuantumChannels/sutter2026grouptheoretic.md).\nIssue [#14780](https://github.com/the-omega-institute/trureturing/issues/14780) preregisters the reading\nand the proof plan.","title":"A sharp Weyl–Bloch ℓ1 entanglement criterion for two qutrits","url":"https://the-omega-institute.github.io/trureturing-pages/research/sutter-popp-hiesmayr-2025-qutrit-weyl-bloch-norm-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b95b5968512a5aa8df57636fc9a7e7c2d65b4d93158d7eaca4a6f73db96db4d8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:suvagiya-conjecture28-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 28 in section 12.3 of arXiv:2607.17343v2 states that for every\ninteger m >= 4, the minimum spectral radius over all independent edge\nsignings of C_(8m)(1,2) equals the greatest real root of\nx^4−2x^3−6x^2+12x−4. The radius is the attained maximum absolute eigenvalue\nof the real symmetric adjacency matrix. Both Hamilton-cycle sign products\nare included.","title":"Suvagiya Conjecture 28 refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/suvagiya-conjecture28-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/416aa67dbe8385b6e24e750c5bbc19ff3aa43a8cbfe76a2cf700bb962ec17ffb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:svinin-2015-discrete-polynomial-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. K. Svinin, *On integrals for some class of ordinary difference equations\nadmitting a Lax pair representation*, arXiv:1505.06394v2, Conjecture 1,\nEq. (48), asks whether\n\n$$\n\\langle S,T\\rangle_s^r(i)=\\sum_{j=0}^{r-2}(-1)^j\nu(i+r-j-2)u(i+s+j)S_{s+j+2}^{r-j-2}(i)T_{s-r+j}^{j}(i+r-j)\n$$\n\nis an identity for every $r\\ge1$, $s\\ge r-1$ and $i\\in\\mathbb Z$ in the\nspecialization $(h,n)=(1,1)$. The source says: “Unfortunately, by now we can prove this conjecture only for\nr = 1, 2, 3.” The settlement is Proved, under preregistration\n[#19287](https://github.com/the-omega-institute/trureturing/issues/19287).\nThe coefficient field is an arbitrary function $u:\\mathbb Z\\to R$ into any\ncommutative ring $R$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Correspondence |\n| --- | --- | --- |\n| Eq. (4): strictly increasing tuples $0\\le\\lambda_1<\\cdots<\\lambda_r\\le s-1$ | `T` | Maps `Fin r → Fin s.toNat`, filtered by `StrictMono`; factor $a$ has offset $a$. |\n| Section 2.3: weakly increasing tuples with descending offsets | `S` | The `Monotone` filter and integer offset $r-1-a$ reproduce the factor pattern. The printed first-offset $k$ is read as $r$. |\n| Section 3: $(h,n)=(1,1)$ | `T`, `S` | The lattice increments in the two literal sums are one. |\n| Section 3.4: alternating binary operation | `bracket` | The sum has indices $j=0,\\ldots,r$ and exactly the source's degree, subscript and lattice shifts. |\n| Conjecture 1, Eq. (48), all degrees | `claim`, `result` | All rings, functions, degrees $r\\ge1$, subscripts $s\\ge r-1$ and lattice points are quantified. |\n| Degree-zero and nonpositive-subscript conventions | `T`, `S` | Empty products are one; positive-degree maps into `Fin 0` do not exist. Natural subtraction is truncated; lattice arithmetic uses integers. |\n\nNo nonvanishing, field-inverse, characteristic or integral-domain hypothesis is\nadded. For $r=1$ the right-hand side is empty. The Library note\n`Library/RecurrenceComposition/svinin2015discrete.md` gives the verbatim source\nexpressions and locator.","title":"svinin 2015 discrete polynomial identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/svinin-2015-discrete-polynomial-identity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b4cbe06d5c0bde1ae6a8e84f44173e2dd7e79ed68224b6d5c956e20d043b7eea.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:szalay-toth-2025-entanglement-depth-generator-injectivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Sz. Szalay and G. Tóth, *Alternatives of entanglement depth and metrological entanglement criteria*, arXiv:2408.15350v3; Quantum 9, 1718 (2025), Section 4.7, state:\n\n> On the other hand, for $q\\not\\in\\field{N}$ the equation becomes transcendental, and we expect the injectivity of $s_q$.\n\n> For the case $d\\not\\in\\field{N}$, we expect the injectivity of $\\Dim_d$,\n> although the meaning of $\\Dim_d$ is not established for noninteger $d$,\n> which is just a parameter then.\n\nThese are verbatim TeX quotations from `depths.tex`, line 2437 and lines 2445–2447, respectively, in the v3 source archive (MD5 `bbacd570d915c7fdda31d547fe476fec`). The source macros `\\field{N}` and `\\Dim_d` denote $\\mathbb N$ and $\\operatorname{Dim}_d$.\n\nAn integer partition $\\xi$ of $n$ is a multiset of positive integers summing to $n$ (`depths.tex`, lines 462–466). The source defines\n$s_q(\\xi)=\\sum_{x\\in\\xi}x^q$ for real $q$ (Eq. `eq:sq`, lines 1903–1907), and\n$\\operatorname{Dim}_d(\\xi)=\\sum_{x\\in\\xi}d^x$ (Eq. `eq:Dimd`, lines 2306–2312). Repetitions contribute with their multiplicities. For the second expectation, $d\\ge2$ remains a mathematical parameter; a noninteger value is not a Hilbert-space dimension.\n\nIssue [#21841](https://github.com/the-omega-institute/trureturing/issues/21841) fixes `claim` as the disjunction of these two assertions:\n\n- For every $n\\in\\mathbb N$ and every real $q$ unequal to every natural number, $s_q$ is injective on the partitions of $n$.\n- For every $n\\in\\mathbb N$ and every real $d\\ge2$ unequal to every natural number, $\\operatorname{Dim}_d$ is injective on the partitions of $n$.\n\nThe theorem `result : ¬ claim` refutes both universal assertions.","title":"Noninteger parameters do not ensure injectivity of the Szalay–Tóth partition generators","url":"https://the-omega-institute.github.io/trureturing-pages/research/szalay-toth-2025-entanglement-depth-generator-injectivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18e5cf62e072d41ad0877b4dbabd95cbc3b92a3fea65bb5b390a1330dd46106b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ta-2025-problem-11-6-three-invariant-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lực Ta, *Good involutions of conjugation subquandles*, arXiv:2505.08090v5,\nSection 11, Problem 11.6, asks:\n\n> Let G be a finite nonabelian group of order n. Is the number of good\n> involutions of Conj G completely determined by n, the order of the\n> center Z(G), and the class number k(G) of G? If so, is there an exact\n> formula to compute this number?\n\nHere `s_x(y)=xyx^-1`. A good involution is a function rho with rho squared\nequal to the identity, commuting with every s_x, and satisfying\n`s_(rho(x))=s_x^-1`. The count is of all such functions on the whole group,\nwithout quotienting by quandle isomorphism. This is separate from 11.7's\nquestion about a bound for conjugation subquandles.","title":"Ta Problem 11.6: three invariants do not determine the count","url":"https://the-omega-institute.github.io/trureturing-pages/research/ta-2025-problem-11-6-three-invariant-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0511fcbdf02f38f60ae88f7c286189582b206046f8a1713e966e5101ae6a9e07.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ta-2025-problem-11-7-good-involution-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lực Ta, *Good involutions of conjugation subquandles*, arXiv:2505.08090v5,\nProblem 11.7 (printed page 27), asks:\n\n> In the setting of Corollary 6.12, suppose that |Z(H)|, |X/H| ≥ 2 (so, in\n> particular, Conj X is not connected). Is the upper bound on |Good(Conj X)|\n> in Corollary 6.12 always strict?\n\nCorollary 6.12 gives the bound\n`|Good(Conj X)| <= min(|Aut(Conj X)|, |Z(H)|^(|X/H|))`.","title":"Ta Problem 11.7: the good-involution bound need not be strict","url":"https://the-omega-institute.github.io/trureturing-pages/research/ta-2025-problem-11-7-good-involution-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/45db5a7800681d7ce3279de2c66ae30e43583e15c612ef04a9662b8ac712b67a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:takahashi-et-al-2023-weighted-crown-npa1-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J. Takahashi, C. Rayudu, C. Zhou, R. King, K. Thompson and O. Parekh,\n*An SU(2)-symmetric Semidefinite Programming Hierarchy for Quantum Max Cut*,\narXiv:2307.15688v3, Appendix C.1, leaves the interval\n$(n+2)^2/(4(n+1))= 2`, let `fix(sigma)` be the number of fixed points of a\npermutation `sigma` of `Fin n`, and define\n\n```text\nAPD_m(fix) = sum_{sigma in S_n} sign(sigma) * fix(sigma)^m.\n```\n\nTakemura's Conjecture 1 in arXiv:2512.18169v1 asks to prove that the least\npositive `m` for which this sum is nonzero is `n - 1`:\n\n```lean\ntheorem resultDegree (n : Nat) (hn : 2 <= n) :\n firstAppearanceDegree n fix = n - 1\n```\n\nThe source reports numerical verification through `n = 10` and leaves the\nstatement unproved.","title":"Takemura's fixed-point first appearance degree","url":"https://the-omega-institute.github.io/trureturing-pages/research/takemura-fixed-point-apd-conjecture-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f6bc4a428e623927f381e7a8cdbbc51e06b0cd49b512a7e1793e36527eecdbe.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:takemura-fixed-point-apd-conjecture-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For `n >= 2`, let `fix(sigma)` be the number of fixed points of a\npermutation `sigma` of `Fin n`, and define\n\n```text\nAPD_m(fix) = sum_{sigma in S_n} sign(sigma) * fix(sigma)^m.\n```\n\nTakemura's Conjecture 2 in arXiv:2512.18169v1 asks to prove the value at\ndegree `n - 1`:\n\n```lean\ntheorem resultValue (n : Nat) (hn : 2 <= n) :\n apd n fix (n - 1) = (n.factorial : Int)\n```\n\nThe source reports numerical verification through `n = 10` and leaves the\nstatement unproved.","title":"Takemura's fixed-point first appearance value","url":"https://the-omega-institute.github.io/trureturing-pages/research/takemura-fixed-point-apd-conjecture-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a70b353e54da4a88219c3dc242b58081500776cef4b21ed95229cb35bfc7df11.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tamer-et-al-2026-eight-vertex-qutrit-dimension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"U. Tamer, B. Ünal, A. Dizdar, Z. Gedik and Ö. E. Müstecaplıoğlu, *The Quad-C₅ Graph: Maximum Contextuality Gap on Eight Vertices*, arXiv:2605.12828v3, rank eight-vertex exclusivity graphs by their contextuality gap.\n- For a graph G they define the dimension-restricted value η_d(G), the largest λ_max(Σ_i |v_i⟩⟨v_i|) over unit vectors v_i ∈ ℝ^d orthogonal on every edge (Eq. (13)).\n- They define the minimum dimension d*(G) = min{d : η_d(G) > α(G)} (Eq. (25)).\n- For the Wagner graph W and the graph ranked fifth in Table V(a), their non-convex search finds no three-dimensional value above α = 3. They mark d* = 4 for both with \"†Numerically indicated; a rigorous three-dimensional upper bound is not established here.\"\n- §IV: \"A formal demonstration that d*(W) = 4 would require an independent upper bound on η₃(W).\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/tamer2026quadc5.md).\n\nIssue [#16223](https://github.com/the-omega-institute/trureturing/issues/16223) fixes the reading. For G ∈ {W, G₅}, with the edge lists of Table VI:\n- every realization in ℝ^d with d ≤ 3 satisfies Σ_i ⟨ψ, v_i⟩² ≤ 3‖ψ‖², that is λ_max ≤ 3;\n- the value 3 is attained in ℝ³;\n- some realization in ℝ⁴ exceeds 3;\n- α(G) = 3.\n\nHence d*(G) = 4.","title":"The Wagner graph and the rank-5 eight-vertex graph have no qutrit violation, so d* = 4","url":"https://the-omega-institute.github.io/trureturing-pages/research/tamer-et-al-2026-eight-vertex-qutrit-dimension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7acd304b48c794d4744cca94726c19efbcc59233c35288602b37c49170ab3baf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tang-zhang-2025-order-minus-two-schoenberg","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Q. Tang, T. Zhang, \"Sharp Schoenberg type inequalities and the de Bruin–Sharma problem\", arXiv:2508.10341v3 (5 Dec 2025), §6, p. 31.\n\n> The identity (5.2) leads us to the following conjectural refinements of Theorem 6.1 in the case of order −2.\n\n> Conjecture 6.3. Let P(z) = z ∏_{j=1}^n (z − z_j), z_j ∈ C \\ {0}, and let w_1, …, w_n denote the zeros of P′(z). Then Σ_{k=1}^n 1/|w_k|² ≤ 3 Σ_{j=1}^n 1/|z_j|² + |Σ_{j=1}^n 1/z_j|². (6.9) Moreover, equality in (6.9) holds whenever z_1, …, z_n all lie on the same line passing through the origin.\n\n> In the case n = 2 we can verify Conjecture 6.3 by an explicit computation. (Proposition 6.4.)\n\nTheorem 6.1 there proves the weaker right side (n+3) Σ 1/|z_j|².\nThe exact public definitions and result are:\n\n```lean\nnoncomputable def P {n : ℕ} (z : Fin n → ℂ) : Polynomial ℂ :=\n Polynomial.X * ∏ j, (Polynomial.X - Polynomial.C (z j))\n\ndef claim : Prop := ∀ (n : ℕ) (hn : 1 ≤ n) (z : Fin n → ℂ), (∀ j, z j ≠ 0) →\n (((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖⁻¹ ^ 2)).sum ≤\n 3 * ∑ j, ‖z j‖⁻¹ ^ 2 + ‖∑ j, (z j)⁻¹‖ ^ 2) ∧\n ((((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖⁻¹ ^ 2)).sum =\n 3 * ∑ j, ‖z j‖⁻¹ ^ 2 + ‖∑ j, (z j)⁻¹‖ ^ 2) ↔\n ∀ j, ∃ r : ℝ, z j = (r : ℂ) * z ⟨0, hn⟩)\n\ntheorem result : claim\n```\n\nHere n≥1, all z_j are nonzero complex numbers, and roots of the derivative\nare counted with multiplicity. No simplicity, distinctness, reality or\ncommon argument hypothesis is imposed on the nonzero roots. Since\nP′(0)=∏_j(−z_j)≠0, no critical point is zero. The final equivalence says\nthat all z_j are real multiples of the nonzero first root, which is exactly\ncollinearity on a line through the origin.","title":"Tang–Zhang: the sharp order −2 Schoenberg inequality","url":"https://the-omega-institute.github.io/trureturing-pages/research/tang-zhang-2025-order-minus-two-schoenberg/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0d688548bda9b177d93becac42dfacad61bfcac519279d91f3014ded526b44a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tang-zhang-2025-real-order-schoenberg","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Q. Tang and T. Zhang, *Sharp Schoenberg type inequalities and the de Bruin–Sharma problem*, arXiv:2508.10341v3, §1.3, Conjecture 1.13. The source is [tang2025schoenberg](../Library/Zeros/tang2025schoenberg.md).\n\n> Conjecture 1.13. Let P(z) = z ∏_{j=1}^n (z − z_j), z_j ∈ C \\ {0}, and let w_1, …, w_n be the zeros of P′(z). Then for every real number p ≥ 1 we have Σ_{k=1}^n 1/|w_k|^p ≤ ((n+1)^p + n − 1)/n · Σ_{j=1}^n 1/|z_j|^p. Moreover, equality holds if and only if z_1 = ⋯ = z_n.\n\nFor every natural $n\\ge1$ and every family $z_1,\\ldots,z_n\\in\\mathbb C\\setminus\\{0\\}$, put\n\n$$\nP(Z)=Z\\prod_{j=1}^n(Z-z_j),\\qquad\nc(n,p)=\\frac{(n+1)^p+n-1}{n}.\n$$\n\nLet $w_1,\\ldots,w_n$ be all roots of $P'$, counted with algebraic multiplicity. No distinctness, simplicity, reality or common argument hypothesis is imposed on the nonzero roots. The result proves the following three clauses:\n\n1. For every real $p\\ge1$,\n $$\\sum_{k=1}^n|w_k|^{-p}\\le c(n,p)\\sum_{j=1}^n|z_j|^{-p}.$$\n2. For every real $p>1$, equality holds if and only if $z_i=z_j$ for every $i,j$.\n3. Conjecture 1.13 as stated is false: at $p=1$, $n=2$ and $(z_1,z_2)=(1,2)$ give equality although $z_1\\ne z_2$.\n\nThe formal `claim` is the conjunction of (i), (ii), and `¬ conjecture`, where the original conjecture has its full dimension, root and order quantifiers:\n\n```lean\ndef conjecture : Prop :=\n ∀ (n : ℕ) (hn : 1 ≤ n) (z : Fin n → ℂ), (∀ j, z j ≠ 0) → ∀ p : ℝ, 1 ≤ p →\n (((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖ ^ (-p))).sum ≤\n c n p * ∑ j, ‖z j‖ ^ (-p)) ∧\n ((((Polynomial.derivative (P z)).roots.map (fun w => ‖w‖ ^ (-p))).sum =\n c n p * ∑ j, ‖z j‖ ^ (-p)) ↔ ∀ i j, z i = z j)\n```\n\nEvery norm power is a real `Real.rpow`, including nonintegral orders. The named conjecture is refuted by its endpoint equality clause. The sharp inequality for all $p≥1$ and the strict-order equality case are proved in the same result.","title":"Tang–Zhang: real-order reciprocal critical-point moments","url":"https://the-omega-institute.github.io/trureturing-pages/research/tang-zhang-2025-real-order-schoenberg/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/174e09dee911ee2f2f09bdf7150f6303975bb112e002e11875a01e14e392271c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tarabunga-frau-haug-tirrito-piroli-2025-shannon-branch-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. S. Tarabunga, M. Frau, T. Haug, E. Tirrito and L. Piroli, *A nonstabilizerness monotone from stabilizerness asymmetry*, arXiv:2411.05766v1 (2024; Quantum Sci. Technol. 10 (2025) 045026), Section V.C, conjecture following Eq. (69). The source member `draft.tex`, line 782, states verbatim:\n\n> Note that, while we are unable to prove Eq. \\eqref{eq:bound_r} for the case $\\alpha=1$, we conjecture that the lower bound remains valid, as we have not found any counter-examples.\n\nThe label `eq:bound_r` is the inequality at `draft.tex` lines 776–779: for fixed rank-$k$ binary $R$, $\\min_{Q',c}S_\\alpha(p_{Q',c,R})\\ge S_\\alpha(q_R)$, stated there for integer $\\alpha>1$. The states indexed by the diagonal part of $Q$ and the translation $t$ give the squared-overlap law $p$; $q_R$ uses zero off-diagonal matrix and zero phase vector after replacing the amplitudes of $\\psi$ by their absolute values. The definitions and source locators are in [the literature note](../Library/Entropy/tarabunga2025asymmetry.md).\n\nIssue [#24993](https://github.com/the-omega-institute/trureturing/issues/24993) fixes the reading at $k=N=n\\ge1$ and $R=I_n$, where the translation quotient has one element. Write $B_n=\\mathbb F_2^n$. For every unit vector $\\psi:B_n\\to\\mathbb C$, strictly upper-triangular $Q'\\in\\mathbb F_2^{n\\times n}$ and $c\\in B_n$, define\n\n$\\phi_d(x)=2^{-n/2}(-1)^{x^\\top(\\operatorname{diag}d+Q')x}i^{\\sum_j\\widetilde c_j\\widetilde x_j}$,\n\n$p(d)=\\left|\\sum_{x\\in B_n}\\overline{\\phi_d(x)}\\psi(x)\\right|^2$, and $q(d)=\\left|\\sum_{x\\in B_n}2^{-n/2}(-1)^{d\\cdot x}|\\psi(x)|\\right|^2$.\n\nHere the quadratic and dot-product exponents of $-1$ are computed in $\\mathbb F_2$ and represented by zero or one, whereas each tilde is the natural representative zero or one and the exponent of $i$ is summed in $\\mathbb N$. Put $S_1(r)=-\\sum_d r(d)\\ln r(d)$, with $0\\ln0=0$. The module's `claim` asserts $S_1(q)\\le S_1(p)$ for every such $n,\\psi,Q',c$, with the unit condition $\\sum_x|\\psi(x)|^2=1$. Its `result` proves the negation. This refutes a subcase of the source's Shannon conjecture.","title":"A three-qubit counterexample to the Shannon branch-and-bound lower bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/tarabunga-frau-haug-tirrito-piroli-2025-shannon-branch-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4b542eb1bcdd98c4f82d2c43de5d9c0481f5b6dcd2d838af042b1346ac555cf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tarnauceanu-2024-reciprocal-subgroup-orders","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Marius Tărnăuceanu, *On the sum of inverses of subgroup orders in finite groups*,\nScientific Studies and Research, Series Mathematics and Informatics 34 (2024),\nno. 1, 115–120, printed p. 119, Conjecture 3.1:\n\n> There is no constant c > 1 such that f(G) > c, for all finite non-cyclic groups G.\n\nOn printed p. 116 the source defines:\n\n> In the current note, we consider the function f(G) = Σ_{H∈L(G)} 1/|H|.\n\nHere $L(G)$ is the whole subgroup lattice, including the trivial and full subgroups.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Sum of reciprocal subgroup orders | `f` | `finsum` over `Subgroup G`, with real reciprocal of `Nat.card H`; for finite groups this is a finite sum. |\n| No constant $c>1$ | `claim` | Negated existential over real $c$, retaining the strict inequality. |\n| All finite non-cyclic groups | `claim` | Every `G : Type`, its group and finite instances, and the hypothesis `¬ IsCyclic G`. Every finite group has such a carrier up to isomorphism. |\n| $f(G)>c$ | `claim` | Exactly the strict inequality `c < f G`. |\n| Conjecture holds | `result` | The complete proposition `claim`, with no extra hypothesis. |\n\nThe definitions are the literal conventions in preregistration\n[#18701](https://github.com/the-omega-institute/trureturing/issues/18701).","title":"Tărnăuceanu's reciprocal subgroup-order conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/tarnauceanu-2024-reciprocal-subgroup-orders/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72b850df5d499ac60ea422d1e6ad50503654f17d252592cc2149abdd3799a26c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-abbott-renou-gisin-brunner-2018-multipartite-partial-tightness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli, A. A. Abbott, M.-O. Renou, N. Gisin and N. Brunner, *Semi-device-independent characterisation of multipartite entangled states and measurements*, arXiv:1805.00377v1 (Phys. Rev. A 98, 052333, 2018), Appendix C, `SDI_manuscript.tex`, line 497:\n\n> We leave it as an open question whether a similar partial tightness result holds in the more general $n$ partite case.\n\nThe source text at lines 474–496 concerns attainment of Eq. (10) for two parties with exactly $k=md$ fully separable measurement operators, for $m=0,\\ldots,d$. Result 2, lines 265–272, bounds the average winning probability by $\\mathcal A_{n,d}\\le(d^n-k+k/d)/d^n$ whenever at least $k$ of the $d^n$ measurement operators are fully separable, for any shared state and local transformations in the source's scenario.\n\nThe game has uniformly distributed inputs $\\mathbf x,\\mathbf y\\in(\\mathbb Z_d)^n$ and winning output $C_1=\\sum_jx_j$, $C_j=y_j-y_1$ for $j\\ge2$, all modulo $d$; $\\mathcal A_{n,d}=d^{-2n}\\sum_{\\mathbf x,\\mathbf y}P(C(\\mathbf x,\\mathbf y)\\mid\\mathbf x,\\mathbf y)$ (lines 137–159). The definitions and source locator are also in [the literature note](../Library/Observer/tavakoliabbottrenougisinbrunner2018sdi.md).\n\nIssue [#23843](https://github.com/the-omega-institute/trureturing/issues/23843) fixes the affirmative reading: for every natural $n,d\\ge2$ and $0\\le m\\le d^{n-1}$, there exist a density matrix on $(\\mathbb C^d)^{\\otimes n}$, local unitary encodings and a projective measurement with exactly $dm$ fully separable outcome operators and score $(d^n-dm+m)/d^n$. This is proved.","title":"Multipartite partial tightness of the entangled-measurement bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-abbott-renou-gisin-brunner-2018-multipartite-partial-tightness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8f3bd5ad26f99565bb78a432e2821702c3cd657e259e283d623b0168d74563e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-bengtsson-gisin-renes-2020-hoggar-sic-compound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli, I. Bengtsson, N. Gisin and J. M. Renes, *Compounds of symmetric informationally complete measurements and their application in quantum key distribution* (Phys. Rev. Research 2, 043122 (2020); arXiv:2007.01007), define a SIC-compound: d³ unit vectors ψ_{jk} in ℂ^d, j ∈ [d²], k ∈ [d], such that\n- for every k the vectors ψ_{jk} form a SIC, |⟨ψ_{jk}, ψ_{lk}⟩|² = (dδ_{jl} + 1)/(d + 1);\n- for every j the vectors ψ_{jk} form an orthonormal basis.\n\nThey construct compounds for d = 2 and d = 4 and prove that none exists for d = 3.\n- For d = 5, 6, 7, 8 they enumerate the Weyl–Heisenberg fiducials and find no compound. The introduction states: \"for d = 5, 6, 7, 8 we give evidence in support of the same conclusion.\"\n- For d = 8 they add: \"dimension 8 also houses SICs that are not based on the WH group … Whether a SIC-compound can be formed from these exceptional SICs is left as an open question.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumContext/tavakolibengtssongisinrenes2020siccompounds.md).\n\nIssue [#18207](https://github.com/the-omega-institute/trureturing/issues/18207) reads the exceptional SICs as the Hoggar SICs: the orbits of a fiducial vector under the three-qubit Pauli group. Its claim is the negative answer to the question. The claim is refuted, which answers the question in the positive: a SIC-compound in dimension 8 is formed from eight Hoggar SICs.","title":"A SIC-compound in dimension 8 from Hoggar SICs","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-bengtsson-gisin-renes-2020-hoggar-sic-compound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/205ee70348af0428b5ede9d1c8b3072230d90c3078411581d90836d36cf551a7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-morelli-2024-conjecture-1-sic-mub-trace-norm-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli and S. Morelli, *Enhanced Schmidt number criteria based on correlation trace norms*, arXiv:2402.09972v2, Phys. Rev. A 110, 062417 (2024), Conjecture 1:\n\n> Provided that both a SIC and a complete set of MUBs exist in dimension $d_A$ and $d_B$, the criteria in Result 1 and 2 are equivalent. It then holds that $\\left\\|\\mathbf{Q}\\right\\|_\\text{tr}=K\\left\\|\\mathbf{P}\\right \\|_\\text{tr}$.\n\nThe SIC and MUB definitions, the correlation matrices P and Q and the constant K = √(d_A(d_A+1)) √(d_B(d_B+1)) are quoted in [the literature note](../Library/QuantumBounds/tavakolimorelli2024schmidttracenorm.md).\n\nIssue [#19503](https://github.com/the-omega-institute/trureturing/issues/19503) fixes the reading. For all d_A, d_B ≥ 2, every density matrix ρ, every pair of local SICs and every pair of complete local MUB families, ‖Q‖_tr = K ‖P‖_tr. The conjecture is proved.","title":"The MUB and SIC correlation trace norms agree up to the factor K","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-morelli-2024-conjecture-1-sic-mub-trace-norm-equality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24f1b62f6c5b47fe019a65c752457d27905efdcc8615e61ad29f718bb1199ce3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-morelli-2024-conjecture-2-incomplete-eam-schmidt-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli and S. Morelli, *Enhanced Schmidt number criteria based on correlation trace norms*, arXiv:2402.09972v2, Phys. Rev. A 110, 062417 (2024), bound the trace norm of the table of joint outcome probabilities of local measurements for states of Schmidt number at most r.\n- **Equiangular measurement (EAM).** n > d unit vectors ψ_a in C^d with Σ_a |ψ_a⟩⟨ψ_a| = (n/d)𝟙 (Eq. (49)) and |⟨ψ_a|ψ_a′⟩|² = (n − d)/(d(n − 1)) for a ≠ a′ (Eq. (50)); its effects are E_a = (d/n)|ψ_a⟩⟨ψ_a|.\n- **The table.** For Alice's EAM {E^A_a} and Bob's {E^B_b}, (P_n)_ab = tr[ρ (E^A_a ⊗ E^B_b)].\n- **Conjecture 2** (§V A, p. 7): \"For any state of Schmidt number r it holds that ‖P_n‖_tr ≤ [d(d − 1) + r(n − d)]/[n(n − 1)]. (52)\" The introduction leaves deciding the conjectures as an open problem.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/tavakolimorelli2024schmidttracenorm.md). Issue [#17745](https://github.com/the-omega-institute/trureturing/issues/17745) fixes the reading:\n- for all d ≥ 1, 1 ≤ r ≤ d, n > d, any two n-element EAMs on C^d chosen independently, and every state of Schmidt number at most r, the bound (52) holds;\n- a state of Schmidt number at most r is a finite convex combination of pure states whose coefficient matrices have rank at most r.\n\nThis conjecture is proved.","title":"Conjecture 2 of Tavakoli–Morelli: the Schmidt-number bound for equiangular measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-morelli-2024-conjecture-2-incomplete-eam-schmidt-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5fb8d844132491ddb5e7c2d15a17f26c072f144a57632af9c0a0b1ddcd6affa6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-morelli-2024-conjecture-3-incomplete-mub-schmidt-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli and S. Morelli, *Enhanced Schmidt number criteria based on correlation trace norms*, arXiv:2402.09972v2, Phys. Rev. A 110, 062417 (2024), bound the trace norm of the table of joint outcome probabilities of local measurements for states of Schmidt number at most r.\n- **Incomplete MUBs.** Each side measures in m mutually unbiased bases of C^d, with m < d + 1; Q_m is the md × md table of the joint probabilities tr[ρ (|g^l_a⟩⟨g^l_a| ⊗ |h^k_b⟩⟨h^k_b|)] of the md local projectors.\n- **Conjecture 3** (§V B, p. 7): \"For any state of Schmidt number r it holds that ‖Q_m‖_tr ≤ 1 + (m − 1)r/d. (54)\" The introduction leaves deciding the conjectures as an open problem.\n\nThe verbatim statements are in [the literature note](../Library/QuantumBounds/tavakolimorelli2024schmidttracenorm.md). Issue [#17745](https://github.com/the-omega-institute/trureturing/issues/17745) fixes the reading:\n- for all d ≥ 1, 1 ≤ m ≤ d, 1 ≤ r ≤ d, any two families of m mutually unbiased bases of C^d chosen independently, and every state of Schmidt number at most r, the bound (54) holds;\n- a state of Schmidt number at most r is a finite convex combination of pure states whose coefficient matrices have rank at most r.\n\nThis conjecture is proved.","title":"Conjecture 3 of Tavakoli–Morelli: the Schmidt-number bound for incomplete sets of MUBs","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-morelli-2024-conjecture-3-incomplete-mub-schmidt-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d38c582999f15950fb77c987da5170223ce1965b77e07d825ff48b777c8eb0d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tavakoli-zambrini-cruzeiro-uola-abbott-2021-pure-state-htmb","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Tavakoli, E. Zambrini Cruzeiro, R. Uola and A. A. Abbott, *Bounding and simulating contextual correlations in quantum theory*, PRX Quantum 2, 020334 (2021), arXiv:2010.04751v3, subsection “Mixed states as resources for quantum contextuality”, state:\n\n> This gives us strong evidence (i.e., up to numerical precision) that pure states cannot violate the HTMB inequality~\\eqref{eq:HTMB}, and we conjecture this to indeed be the case exactly.\n\nThe quotation is from line 506 of the e-print member `Manuscript_infocontext_reresubmission.tex`, md5 `f767464510150f9e02bc504393cf40e8`. The scenario and Eq. (33) are at lines 492–500; the numerical pure-state bound is at lines 502–507. See also [the literature note](../Library/Observer/tavakolizambrinicruzeirouolaabbott2021contextual.md).\n\nAlice receives two trits $x=(x_1,x_2)\\in\\mathbb Z_3^2$, and Bob guesses either coordinate using one of two three-outcome measurements. Write $P_x=\\operatorname{projector}(\\psi_x)=|\\psi_x\\rangle\\langle\\psi_x|$. Issue [#25326](https://github.com/the-omega-institute/trureturing/issues/25326) fixes the finite-dimensional reading: for every $d\\in\\mathbb N$, every family of unit vectors $\\psi_x\\in\\mathbb C^d$ and every pair of POVMs $(M_b)_{b\\in\\mathbb Z_3}$ and $(N_b)_{b\\in\\mathbb Z_3}$, assume\n\n$$\n\\sum_{x:\\,x_1+a x_2=k}P_x=\\sum_{x:\\,x_1+a x_2=0}P_x\n\\qquad(a\\in\\{1,2\\},\\ k\\in\\mathbb Z_3).\n$$\n\nEach POVM has positive semidefinite effects summing to the identity. Then the module's `htmbValue` satisfies\n\n$$\n\\operatorname{htmbValue}(\\psi,M,N)\n=\\frac1{18}\\sum_{x\\in\\mathbb Z_3^2}\n\\left(\\operatorname{Re}\\operatorname{tr}(P_xM_{x_1})+\n\\operatorname{Re}\\operatorname{tr}(P_xN_{x_2})\\right)\n\\le\\frac23.\n$$\n\nThis is `claim`, and `result : claim` proves it.","title":"Pure preparations cannot violate the HTMB inequality in finite dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/tavakoli-zambrini-cruzeiro-uola-abbott-2021-pure-state-htmb/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9989bf21757734e3b38f1322ee4fa8de01b304e2384d9ebf86f30ddb822860a0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tebni341724-row-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The Comments block of [OEIS A341724](https://oeis.org/A341724), revision 26,\nattributes these conjectures to Mélika Tebni in 2023:\n\n> For 0 < k < p and p prime, T(p,k) == 0 (mod p).\n>\n> For 0 <= k < n and n = 2^m (m natural number), T(n,k) == 0 (mod n).\n\nThe Formula block specifies the column EGF as\n$x^k/(k!(1-2\\sinh(-x)))$. The conjectures concern\n$T(n,k)=n![x^n]x^k/(k!(1-2\\sinh(-x)))$ over the rationals.\nThe 2021 Shannon–Ollerton paper describes the array, rather than stating\nthese 2023 conjectures.","title":"Tebni's prime and dyadic row-divisibility conjectures for A341724","url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni341724-row-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dced1903c5f62e01287ac981bc24626a0af4c20ebb64f0231546cdb6e5d8e66e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tebni350759","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Mélika Tebni's OEIS A350759 revision 17 lists these COMMENTS conjectures:\n\n> For all p prime, (a(p) + a(p+1) - 2) == 0 (mod p), \n> a(p+1) == 1 (mod ((p+1)*p)).\n\nIts FORMULA line is the exponential generating function\n`exp(-1 + (1 + x)*(1 - log(1 + x)))`. The resolved Lean statement is\n`D5/S3/Arith/Congruence/TebniA350759PrimeCongruences.claim`: an integer sequence with that exact EGF exists, and every such sequence satisfies both congruences. The settling declaration is\n`D5/S3/Arith/Congruence/TebniA350759PrimeCongruences.result`, which proves `claim`.\n\n### Statement echo\n\nThe Lean `sourceSeries` is the published formal EGF, with `PowerSeries.log ℚ` interpreted as the formal `log(1+X)` and `PowerSeries.subst` taking the inner series first. `IsSourceSequence a` states that the EGF of the integer sequence `a : ℕ → ℤ`, with coefficients cast to `ℚ`, equals `sourceSeries`. The first conjunct of `claim` is existence of such an `a`; its second conjunct quantifies every such `a`, every natural `p`, and `Nat.Prime p`, then asserts\n`Int.ModEq (p : ℤ) (a p + a (p + 1) - 2) 0`\nand\n`Int.ModEq (((p + 1) * p : ℕ) : ℤ) (a (p + 1)) 1`.\nThe natural product is formed before its cast to `ℤ`, and the prime p=2 is included.","title":"Tebni's OEIS A350759 prime congruences","url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni350759/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59bfcd2536e391f77a0e5384b2db54106f6cde96f56f47b361c55971d687155b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tebni364822-cong","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Tebni's OEIS [A364822](https://oeis.org/A364822), revision 25, names the sequence:\n\n> Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).\n\nThe conjecture resolved here is:\n\n> Conjectures: For p prime (p > 2), a(p) == 2 (mod p).\n\nPreregistration: [#18643](https://github.com/the-omega-institute/trureturing/issues/18643).\nThe companion conjecture is recorded in [#18644](https://github.com/the-omega-institute/trureturing/issues/18644).\n\n### Statement echo\n\nThe natural binder $p$ has hypotheses $p$ prime and $2 Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x)).\n\nThe conjecture resolved here is:\n\n> For n = 2^m (m natural number), a(n) == 1 (mod n).\n\nPreregistration: [#18644](https://github.com/the-omega-institute/trureturing/issues/18644).\nThe companion conjecture is recorded in [#18643](https://github.com/the-omega-institute/trureturing/issues/18643).\n\n### Statement echo\n\nThe natural binder $m$ ranges over all natural numbers including zero. The conclusion is an integer $z=a(2^m)$ with $z\\equiv1\\pmod{2^m}$.\nThe rational formal series have coefficients $1/n!$ in even indices for cosh\nand in odd indices for sinh, and zero in their other indices. The coefficient\n$a(n)$ is $n!$ times the ordinary coefficient of their formal quotient.\nThe existential integer equals the rational coefficient after casting;\nthus integrality and the congruence are both conclusions. No finite index bound\nor extra hypothesis is imposed.","title":"Tebni's power-of-two congruence for OEIS A364822","url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni364822-power-of-two-cong/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/208818c05da8d929a929eac30ac4dbda0b07d29b1129d5e557914528a6443a75.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:teimoori-faal-2021-flat-graph-clique-roots-conjecture-1","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Teimoori Faal, *On Clique Roots of Flat Graphs*, arXiv:2112.09721v1,\nConjecture 1, page 6, states:\n\n> The class of 2-connected K₅-free flat graphs without isolated triangles has only clique roots. In particular, this class has always the clique root r = −1 of multiplicity 2.\n\nThe [source note](../Library/GraphInvariants/teimoorifaal2021flat.md)\ncontains the definitions and locators. The preregistration is\n[#19709](https://github.com/the-omega-institute/trureturing/issues/19709).\nThe quoted conjecture is **Refuted** by a twelve-vertex graph.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite simple graph | `V`, `Fintype V`, `SimpleGraph V` in `claim` | Arbitrary finite vertex type; undirected and loopless |\n| $C(G,x)=\\sum_j c_j(G)x^j$, with $c_0=1$ | `cliquePoly` | Integer clique counts from `SimpleGraph.cliqueFinset`; sum through the vertex count, with higher terms zero |\n| Every edge is in at most two triangles (Definition 2.2) | `IsFlat` | Counts the common neighbours of each adjacent pair |\n| No isolated triangle (Definition 4.1) | `NoIsolatedTriangle` | Every triangle has a common neighbour of all three vertices |\n| 2-connected | Existing `D5.S3.StatisticalMechanics.DimerVirialFourthDifferenceRefutation.Biconnected` and `3 ≤ Fintype.card V` | Connected before and after deletion of any vertex; more than two vertices |\n| $K_5$-free | `SimpleGraph.CliqueFree 5` | No five-vertex clique |\n| Has only clique roots | First conjunct of `claim` | Every complex zero has imaginary part zero |\n| $-1$ has multiplicity exactly two | Second conjunct of `claim` | `Polynomial.rootMultiplicity (-1) = 2` |\n| Implementation instances | `DecidableEq V`, `DecidableRel G.Adj` | Finite graph operations; no additional graph restriction |\n| Negative answer | `result : ¬ claim` | The first conjunct fails on one graph satisfying every hypothesis |","title":"A ring of four complete blocks refutes the flat-graph clique-root conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/teimoori-faal-2021-flat-graph-clique-roots-conjecture-1/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6691c5f7f0e215ec197c98db735126ac9a5508703aa6ae831f09ed3aa324eddb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:teimoori-faal-2026-b-restricted-clique-real-stability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Teimoori Faal, *A Bivariate B-Restricted Clique Polynomial: From Local Neighborhoods to Global Expansion*, arXiv:2602.24151v1, Section 7, Open Problem 1, asks:\n\n> Necessity of Conditions: Are the conditions of r-connectivity and chordality also necessary? Is there an r-connected K_{r+3}-free non-chordal graph for which C_B(G;x,y) fails to be real-stable?\n\nThe second question has a positive answer for every integer $r\\geq1$. The polynomial is the literal clique sum\n\n$$C_B(G;x,y)=\\sum_{K\\subseteq V(G),\\ K\\text{ clique}}x^{|K|}y^{|K\\cap B|},$$\n\nincluding the empty clique. Real stability means that the polynomial is not identically zero and has no zero with $\\operatorname{Im}(x)>0$ and $\\operatorname{Im}(y)>0$.","title":"Faal's Open Problem 1: non-chordal graphs with unstable restricted clique polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/research/teimoori-faal-2026-b-restricted-clique-real-stability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4af0b8621c6395d8a4aad42b9332d05d3f8642a1b357f32ef09271414f95d34.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:teretenkov-2024-interaction-picture-asymptotics","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. Madou, I. Remizov, M. Vafadar (eds.), *Open problems in one-parameter operator semigroups theory*, arXiv:2410.00416v2, p. 21, Problem 29 (A. E. Teretenkov), state verbatim:\n\n> Problem 29 (A.E. Teretenkov). Let B be a Banach space. Let P be a projection on a finite-dimensional Banach subspace of B. Let L0 + λL be a generator of C0-semigroup U^λ_t on B for all λ ∈ [0, λsup). Let U^0_t leave both PB and its complement (I − P)B invariant, let PLP = 0. Denote L_t ≡ (U^0_t)^{−1} L U^0_t. Let the integrals ∫_{−∞}^{t} dt_1 … ∫_{−∞}^{t_{k−1}} dt_k P L_{t_1} … L_{t_k} P, k = 1, …, n + 1 be finite for all t ⩾ 0. If is it possible to find such a λ-dependent operator r^{n,λ} on PB, which is polynomial in λ, and λ-dependent semigroup u^{n,λ}_t on PB, whose generator is polynomial in λ, such that P(U^0_{t/λ²})^{−1} U^λ_{t/λ²} P = u^{n,λ}_t r^{n,λ} + O(λ^{2n+2}), λ → +0 for all t > 0 ? Let us emphasize that we do not assume here asymptotic behaviour to be uniform in t, it is just assumed for each fixed t > 0. If there are counterexamples, what further restrictions should be assumed to obtain such an asymptotic estimate?","title":"Interaction-picture asymptotics","url":"https://the-omega-institute.github.io/trureturing-pages/research/teretenkov-2024-interaction-picture-asymptotics/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/787059df105886b806b5343d1ed645693ceadbeaf175c30689dc8a08e00c3adb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:testart-2026-question-one-type-c-tree-bijection","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Benjamin Testart, *On minimal pattern-containing inversion sequences*,\narXiv:2602.12130v2, Section 7, Question 1, printed page 17, asks:\n\n> Find bijections between the families $\\mathrm{IP}_{n,k}$ or $\\mathcal T_{n,k}$ of Section 6 and other families of combinatorial objects known to be counted by the poly-Bernoulli numbers of type C.\n\nThe question offers an IP-side or a T-side answer. The target here is the\nentire T-side family, rather than the source's already published IP/T\ncorrespondence. Section 6 defines unordered rooted increasing trees.\nFor every pair of integers $a,b\\ge1$, write $N=a+b-1$. The source $T(a,b)$\nconsists of trees on distinct labels $0,\\ldots,N$, rooted at zero, whose\ninternal labels are exactly $0,\\ldots,a-1$ and whose leaves are exactly\n$a,\\ldots,N$. Every parent label is smaller than its child label.\nThe target $P(a,b)$ consists of permutations of $[N]$ whose strict\nexcedance-position set is exactly $[b-1]$, allowing fixed points.\nHere $[0]=\\varnothing$.\n\nThe fixed forward map deletes zero, reads each component root-first by DFS\nwith decreasing child-root labels, and makes each word a directed cycle,\nsending each letter to the next and the last to the first. If the resulting\npermutation is $\\sigma$, the output is $F(t)=D_N(\\sigma)$, where\n\n$$\nD_N(\\sigma)(i)=N+1-\\sigma(N-i)\\quad(1\\le i For any integers d, k ≥ 0 and r ≥ 1, there exist integers $a^{(r)}_{d,k,j}$ such that $\\binom{n+dk}{k}^r = \\sum_j a^{(r)}_{d,k,j}\\binom{n+dj}{j}$ for all n = 0, 1, 2, .... Moreover $a^{(r)}_{d,k,j}$ can be defined as following: $a^{(1)}_{d,k,k} = 1$, $a^{(1)}_{d,k,j} = 0$ for j ≠ k and $a^{(r+1)}_{d,k,j} = \\sum_i S_d(k, j, i)a^{(r)}_{d,k,i}$, where, $S_d(k, j, i)$ are integers, independent of r, for all d, k, j, i.\n\nThe formal statement uses natural numbers for every nonnegative source\nparameter and casts binomial values to integers before taking powers. It\nincludes both sentences and proves finite support, at j≤rk for the power\ncoefficients and at j≤i+k for the structure constants.","title":"Thanatipanonda's shifted binomial power linearization","url":"https://the-omega-institute.github.io/trureturing-pages/research/thanatipanonda-2014-shifted-binomial-power-linearization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf762cb7015326f33c1a332b394e32cf2637d2ea53cb14e1d4a866512274a23d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:thankachy-et-al-2024-vertex-position-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Problem 2.4 of M. Thankachy, E. J. Thomas, U. Chandran, J. Tuite,\nG. Di Stefano and G. Erskine, “On the Vertex Position Number of Graphs”,\nasks: “Is the ratio $\\mathrm{vp}(G)/\\mathrm{vp}^-(G)$ bounded for connected\ngraphs?” The source is arXiv:2209.00359v2, page 5, published in Discussiones\nMathematicae Graph Theory 44 (2024), 1169–1188, DOI `10.7151/dmgt.2491`.\n\nFor a fixed vertex $x$, an $x$-position set excludes every other selected\nvertex from every shortest path from $x$ to a selected vertex. The maximum\nsize is $p_x(G)$; its maximum and minimum over vertices are\n$\\mathrm{vp}(G)$ and $\\mathrm{vp}^-(G)$. Endpoints belong to geodesics,\nso $\\{x\\}$ is permitted but a set containing $x$ and another vertex is not.\nThe Lean definitions use the full finite carrier and all its subsets.","title":"The vertex position ratio is unbounded on connected graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/thankachy-et-al-2024-vertex-position-ratio/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1abe830586354e3df5e94cfc412e00f5e50af27bfb4d7868caee0be8a9ad3d03.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:thue-morse-reduced-abelian-odd","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"This dossier deliberately anchors only the equality\n`rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1)` for every nonnegative integer\n`n`, proposed in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1 and\nprinted page 14 of the verified journal version. The shared Library note\nrecords the journal citation. The source sentence is:\n\n> Although it appears that rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for\n> nonnegative integers n, the problem of determining a full recursion for\n> rho^{ab,red}_t(n) seems to be challenging.\n\n`red(w)` collapses every maximal constant run; reduced abelian complexity\ncounts all length-`n` factors up to rearrangement of their reduced words\nwith equal reduced length. The full recursion and four further questions\nin the source paragraph, namely the nonzero sign of `rho(4n+2)-rho(4n)`, a\nrecursion for `rho(4n)`, equation (11), and non-k-automaticity of sequence\n(10), are deliberately out of scope.","title":"Odd-index reduced abelian complexity of the Thue-Morse word","url":"https://the-omega-institute.github.io/trureturing-pages/research/thue-morse-reduced-abelian-odd/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/427dcf659f6df0018d8e2d583d81bf688593c9681d08de5580222bce105065ad.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tian-dokyeesun-klavzar-conjecture34-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 3.4 in section 3.2 of arXiv:2510.01294v2 states:\n\n> If x is not a cut vertex of a graph G, then\n> gp_o(G-x) <= gp_o(G) + deg_G(x).\n\nThe formal claim ranges over all finite simple connected graphs and every\nnon-cut vertex. Deletion is the actual induced graph on vertices other\nthan x. Non-cut means that deletion does not increase the number of\nconnected components. This retains K1, with empty deletion, zero empty\nouter number, and singleton outer number one.\n\nA pair is Z-positionable when every shortest path between its endpoints\nhas no internal vertex in Z. An outer general position set requires this\nfor every pair with at least one endpoint in Z. The outer number is the\nmaximum cardinality among all such subsets.","title":"Tian–Dokyeesun–Klavžar Conjecture 3.4 refutation","url":"https://the-omega-institute.github.io/trureturing-pages/research/tian-dokyeesun-klavzar-conjecture34-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef0502b0da164cba3c248009f8a4f5e054e70fca48a4ff9221c59aa346197060.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tierz-2026-traceless-bps-freeness-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Tierz, *BPS spectra of Tr[Ψ^p] matrix models for odd p*, arXiv:2604.27164 (v1 April 2026, v2 August 2026), studies the cohomology of wedge multiplication by Q_p = Tr(Ψ^p) on the exterior algebra of the N² fermionic creators Ψ_ij.\n- The trace fermion χ splits off (v2 Theorem 2), leaving the cohomology B•_0 of Q_p = Tr(A^p) on the traceless creators A = Ψ − (χ/√N)·1.\n- The traceless diagonal creators anticommute with Q_p, so B•_0 is a graded module over their exterior algebra Λ•(C^{N−1}).\n\nThe source states freeness in two places, verbatim in [the literature note](../Library/HomologicalAlgebra/tierz2026bpsoddp.md):\n- v1 §5 marks \"(conjectural) H•_red is free over Λ•(C^{N−1})\" as \"the content of the open problem\";\n- v2 §6.3 calls an isomorphism of graded modules B•_0 ≅ Λ•(C^{N−1}) ⊗ M^{(p,N)} \"a sufficient explanation for the remaining factors of 1+x\" and says that \"such freeness … is not proved here\".\n\nIssue [#16298](https://github.com/the-omega-institute/trureturing/issues/16298) fixes the reading: for odd p ≥ 5 with p ≤ 2N − 1, B•_0 is a free Λ•(C^{N−1})-module. The admissible case (p, N) = (5, 4) is refuted.","title":"The traceless BPS cohomology at (p, N) = (5, 4) is not free over the diagonal creators","url":"https://the-omega-institute.github.io/trureturing-pages/research/tierz-2026-traceless-bps-freeness-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3cdf1f9080c5dc16963da88ce06a4ce1ec4592221f97b128602ed8c68748242.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ting-fullwood-wu-2026-three-weight-bayesian-inverse","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"O. Ting, J. Fullwood and Z. Wu, *Operational time-reversal symmetry for unital\nqubit channels*, arXiv:2605.10375v1, call a channel $\\mathcal F$ a Bayesian\ninverse of $\\mathcal E$ with respect to $\\rho$ when\n$\\{\\mathcal E(\\rho)\\otimes\\mathbb 1,\\mathscr J[\\mathcal F]\\}=\\{\\mathbb 1\\otimes\\rho,\\mathscr J[\\mathcal E^\\dagger]\\}$,\n$\\mathscr J[\\mathcal N]=(\\mathrm{id}\\otimes\\mathcal N)(\\mathtt{SWAP})$. For Pauli\nchannels $\\mathcal P(A)=\\sum_\\mu p_\\mu\\sigma_\\mu A\\sigma_\\mu$ they settle the case of\ntwo non-zero weights and characterize the general case by three inequalities, and\nin Section V state that for exactly three non-zero weights they found no Bayesian\ninverse away from the maximally mixed state, which is \"probably due to the fact\nthat either the maximally mixed state is the only state for which a Bayesian\ninverse exists in such a case, or that such states are restricted to a\nmeasure-zero subset of the Bloch ball\". The verbatim statements are in\n[the literature note](../Library/QuantumChannels/ting2026operational.md).\n\nIssue [#13585](https://github.com/the-omega-institute/trureturing/issues/13585)\nfixes the reading: for every probability vector with exactly three non-zero\nentries and every qubit density matrix $\\rho$ (pure states included), a CPTP\nBayesian inverse of $\\mathcal P$ with respect to $\\rho$ exists if and only if\n$\\rho=\\mathbb 1/2$; $\\mathcal P^\\dagger=\\mathcal P$ is part of the statement.","title":"Three-weight Pauli channels have Bayesian inverses only at the maximally mixed state","url":"https://the-omega-institute.github.io/trureturing-pages/research/ting-fullwood-wu-2026-three-weight-bayesian-inverse/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2e4448bad003ff633c1dd6757a92917524022c750efe363287b042a5023b91d4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tolias-dornheim-vorberger-2025-kinetic-odd-moments","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"P. Tolias, T. Dornheim and J. Vorberger, *Kinetic contribution to the arbitrary\norder odd frequency moments of the dynamic structure factor*, arXiv:2508.17810v1,\nSec. 2.3, Eq. (16), state: “Therefore, our conjecture states that the following\nresult holds for the interacting uniform electron gas”, followed by\n\n$$\nM_{\\mathrm{S,KIN}}^{(2k+1)}(q)=\n\\left(\\frac{\\hbar q^2}{2m}\\right)^{2k+1}\\frac{1}{2k+2}\n\\sum_{i=0}^{k}\\binom{2k+2}{2i+1}\n\\left(\\frac{2}{\\hbar q}\\right)^{2i}\\langle p^{2i}\\rangle_0,\n$$\n\n“where $k$ is an arbitrary non-negative integer.” The paper explicitly checks\n$k=0,1,2,3$. Its summary also asks whether similar expressions hold for other\nspin statistics. [Issue #13703](https://github.com/the-omega-institute/trureturing/issues/13703)\nfixes the operator representation and averaging conventions. The quotations and\nlocators are in the [literature note](../Library/QuantumStates/tolias2025kineticmoments.md).","title":"All odd kinetic frequency moments","url":"https://the-omega-institute.github.io/trureturing-pages/research/tolias-dornheim-vorberger-2025-kinetic-odd-moments/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f70c38a989609e0e35987798edde09ff4707bf9e18ff62affc43aa0ef821d1a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tononi-lewenstein-2025-temporal-bell-contraction-identity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Andrea Tononi and Maciej Lewenstein, *Temporal Bell inequalities in non-relativistic many-body physics*, Quantum Science and Technology 10 (2025), arXiv:2409.17290v4, Methods, “Derivation of Eq. `ICHexact`”, conjecture in `arxiv.tex`, lines 343–353 (md5 `225051e786fe4712ac07f5226a0a99ac`):\n\n> However, we conjecture that\n>\n> $\\text{Re}\\big[ \\bra{0} f_N f_1 \\sigma^x_{N}(t) f_1^\\dagger \\ket{0} + \\bra{0} f_N f_1 \\sigma^x_{N}(t) f_N^\\dagger \\ket{0} \\big] = \\text{Re}\\big[G_{NN}(t) - G_{1N}(t) \\big]$,\n>\n> and\n> we verified \\cite{sneg} that the exact analytical form of the left- and right-hand sides coincides for $2\\leq N \\leq 5${, and with exact numerics} up to $N=8$ \\cite{mullerrigat2025}.\n> In the absence of a general proof, we assume the relation to hold for any value of $N$.\n\nThe quotation preserves the TeX wording and citation keys, with display delimiters and the equation label omitted. The source's references identify `sneg` as R. Žitko's SNEG package (2011) and `mullerrigat2025` as preliminary results of G. Müller-Rigat, M. Płodzień, D. Farina, M. Lewenstein and A. Tononi (2025).\n\nIssue [#21106](https://github.com/the-omega-institute/trureturing/issues/21106) fixes the reading: for every integer $N\\ge2$ and every real $J,\\mu,t$, use the Jordan–Wigner annihilators $f_j$ on the full occupation space $(\\mathbb C^2)^{\\otimes N}$, its empty vacuum $|0\\rangle$, and\n\n$H=-J\\sum_{j=1}^{N-1}(f_j^\\dagger f_{j+1}+f_{j+1}^\\dagger f_j)-\\mu\\sum_{j=1}^N f_j^\\dagger f_j$,\n\n$h_{jj}=-\\mu$, $h_{j,j+1}=h_{j+1,j}=-J$, with all other entries zero, and $G(t)=e^{ith}$.\n\nThe observable is $\\sigma_N^x=\\big[\\prod_{j Future work will also address the generalization of Proposition 7, as it remains an open problem whether\n> the result holds without the constraints of |a₁|=|a₂| and |b₁|=|b₂|.\n\nTwo pairs of unbiased qubit observables are operationally equivalent when, for every set $S$ of states, one\npair is $S$-compatible exactly when the other is; $S$-compatibility means that some four-outcome POVM has\nboth margins with the same Born probabilities as the pair on every state of $S$. Proposition 7 proves that,\nfor $|a_1|=|a_2|$ and $|b_1|=|b_2|$ in $(1/\\sqrt2,1]$, equivalence forces equality up to signs and exchange.\nThe verbatim texts are in [the literature note](../Library/QuantumContext/torii2025comparing.md). Issue\n[#15522](https://github.com/the-omega-institute/trureturing/issues/15522) preregisters the general statement\nand the route.","title":"Operational comparison of qubit incompatibility without equal lengths","url":"https://the-omega-institute.github.io/trureturing-pages/research/torii-takakura-imamura-2025-proposition-7-generalization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/760a50c64a8eb71255cf8a74b8b9e0bb06c1a8b9101eb81096928e996001cabf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:toth-pitrik-2026-separable-wasserstein-comparison-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"G. Tóth and J. Pitrik, *Quantum Wasserstein distance and its relation to several types of fidelities*, arXiv:2506.14523v6, Conjecture 1:\n\n> {\\bf Conjecture 1.} The following inequality seems to hold $\\tilde D_{\\rm DPT, sep}(\\varrho,\\sigma)^2 \\le \\tilde D_{\\rm DPT}(\\varrho,\\sigma)^2$ if $\\varrho$ and $\\sigma$ are arbitrary mixed states.\n\nThe two squared distances and their modified forms are quoted in [the literature note](../Library/QuantumContext/toth2026wassersteinfidelities.md).\n\nIssue [#19312](https://github.com/the-omega-institute/trureturing/issues/19312) fixes the reading: the inequality for every d, every family of Hermitian observables and all density matrices ρ, σ. It is refuted.","title":"The separable modified Wasserstein distance can exceed the De Palma–Trevisan one","url":"https://the-omega-institute.github.io/trureturing-pages/research/toth-pitrik-2026-separable-wasserstein-comparison-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c538340e58f82428f20fa31757b108c4e3d5ff98e0cf20fb95bf772d96ccd4e8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:tyagi-2026-divisor-nim-sharper-sg-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Satyam Tyagi, “Removing m from m-pile Divisor Nim”, arXiv:2610.06925v1,\n§5, printed p. 23, states:\n\n> For every nonempty position P, g(P) ≤ 2hmin(P).\n\nA position is a finite multiset of positive natural numbers. A move chooses\none occurrence of a heap h and a positive integer d ≤ h, requires d to divide\nevery other heap, and replaces h by h − d, omitting a zero remainder. The\nremoval need not divide h. The Grundy value is the mex of follower values.\n\nThe Lean proposition `claim` states that every positive nonempty multiset P\nand every member m that is at most every member of P satisfy g(P) ≤ 2m.\nThe theorem `result : claim` preserves these quantifiers and the literal move\nrule, including repeated heaps and singletons.","title":"Tyagi's sharper Sprague–Grundy bound for Divisor Nim","url":"https://the-omega-institute.github.io/trureturing-pages/research/tyagi-2026-divisor-nim-sharper-sg-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f6cc81a2c3c96f4382177673c4a96ac6d5e97a9ad751d534d47c99c17343ce2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:uhlmann-2023-diagonal-majorization-question","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jeffrey Uhlmann, *Partial Proof of a Conjecture with Implications for Spectral Majorization*, arXiv:2309.01302v1, Section 4, asks about removing the dimensional restriction from the diagonal-spectrum majorization relationship:\n\n> However, the fact that such a construction does not generally exist for n > 6 does not necessarily imply that the diagonal-spectrum majorization relationship does not hold in general.\n\n> Empirical investigation may be able to provide sufficient evidence to justify a conjecture or, alternatively, empirical and/or analytical efforts may identify counterexamples that constrain such generalizations\n\nThe assertion is that for every natural dimension $n$, every real symmetric positive definite $P$, and every real diagonal $D$, the diagonal of $PDP^{-1}$ majorizes its spectrum. The spectrum is the vector of diagonal entries of $D$. Section 3 defines $x\\prec y$ by $x=Sy$ for a doubly stochastic matrix $S$. Preregistration: [#18572](https://github.com/the-omega-institute/trureturing/issues/18572).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $x\\prec y$ iff $x=Sy$, with $S$ doubly stochastic | `Majorizes y x` | Literal existential definition using Mathlib's `doublyStochastic`. |\n| Every dimension | `n : ℕ` in `claim` | No upper bound is imposed. |\n| Real symmetric positive definite $P$ | `P : Matrix (Fin n) (Fin n) ℝ` and `P.PosDef` | Over the reals, Mathlib's Hermitian condition is symmetry. |\n| Every real diagonal $D$ | `d : Fin n → ℝ` and `Matrix.diagonal d` | All real entries are allowed. |\n| Spectrum of $PDP^{-1}$ is majorized by its diagonal | Final `Majorizes` expression in `claim` | The diagonal is the majorizing vector and `d` the majorized vector. |\n| The unrestricted assertion is false | `result : ¬ claim` | One seven-dimensional instance refutes the universal statement. |\n\nNo positivity requirement on $D$ is added; the chosen witness also has positive diagonal entries. The source's small-dimension theorem retains its dimensional hypothesis.","title":"Uhlmann's unrestricted diagonal-majorization question","url":"https://the-omega-institute.github.io/trureturing-pages/research/uhlmann-2023-diagonal-majorization-question/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1cbedf397e88cd2c3ed6c8aa974e42ac5f27e64b65290f7662b8622ae1248c02.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ulfarsson-2024-alternating-mesh-basis","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Ulfarsson, *BiSC: An algorithm for discovering generalized permutation patterns*, arXiv:2411.17778v1, Conjecture 2.4.2, page 8; DOI 10.48550/arXiv.2411.17778. The source is recorded in [ulfarsson2024bisc](../Library/PermutationPatterns/ulfarsson2024bisc.md). The external named problem is preregistered in [#19120](https://github.com/the-omega-institute/trureturing/issues/19120).\n\n> Let A be the union of the alternating subgroups, and let P be a basis of patterns such that A = Av(P). Then P = {(p, {0, …, n} × {0, …, n}) : p ∈ S_n ∖ A for some n}.\n\nThe settlement is **Proved** under the literal mesh-pattern conventions in #19120. Every mesh-pattern basis of the even permutations consists exactly of fully shaded odd permutations.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A mesh pattern is a permutation together with shaded cells | `MeshPattern`, with length `k`, permutation `p`, and shading `R` | Positions are zero-based; cells range from $0$ through $k$ on each axis |\n| An occurrence has order-isomorphic selected points and no extra point in a shaded region | `Occurs` | Increasing selected positions, relative value order, and both cell cardinalities are retained |\n| $S$ contains permutations of every length | The dependent sum in `Av` and `Alt` | Length zero is included |\n| $\\operatorname{Av}(P)$ avoids all members of $P$ | `Av` | Universal quantification over the forbidden set |\n| $P$ is a basis of $A$ | `IsBasis` | $\\operatorname{Av}(P)=A$ and every subset with that avoidance set equals $P$; equality orientation is immaterial |\n| $A$ is the union of the alternating subgroups | `Alt` | The sign is $1$ |\n| The forbidden patterns are precisely fully shaded odd permutations | `claim` and `result` | $R=\\mathrm{univ}$ and sign different from $1$, quantified over every basis $P$ |\n\nEvery public definition has the literal body of its binding convention in #19120. The result adds no binder or hypothesis beyond `claim`.","title":"Ulfarsson's alternating mesh-pattern basis conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/ulfarsson-2024-alternating-mesh-basis/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/162f0cdce009c1ba43c7487ec8aa7e35a4845dd2d54e39750c5bdddccf3d9712.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:ulg398064","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A398064, revision 39, FORMULA:\n\n> Conjecture: H(x+1) = lim_{n->oo} n^{x+1} * Sum_{k=0..n} (-1)^k/(n+k) * binomial(x, k) such that H(1/4) = 0.79564036... . Changing the upper bound of the sum to oo yields Euler's product representation of the Gamma function.\n\nThe binding interpretation in [#22706](https://github.com/the-omega-institute/trureturing/issues/22706) is over every $x\\in\\mathbb C$. Put\n\n$$S_n(x)=\\exp((x+1)\\log n)\\sum_{k=0}^n\\frac{(-1)^k}{n+k}\\binom{x}{k}.$$\n\nThe logarithm is real for positive $n$. The entire source function satisfies $H(m+1)=m!$ and, away from positive integers,\n\n$$H(y)=\\frac{\\psi(1-y/2)-\\psi((1-y)/2)}{2\\Gamma(1-y)}.$$\n\nThe assertion includes existence of such an entire function and convergence $S_n(x)\\to H(x+1)$ for every such function and every complex parameter. No existence assumption is substituted for this conclusion.\n\n### Statement echo\n\n| Source or binding clause | Lean counterpart |\n| --- | --- |\n| The inclusive finite sum, both endpoints retained | `truncation` |\n| Generalized binomial coefficients | `Ring.choose` in `truncation` |\n| Positive-integer power with real logarithm | `Complex.exp` and `Real.log` in `truncation` |\n| Entire factorial interpolation and the specified quotient | `IsHadamard` |\n| Existence and convergence for every admissible function, every complex parameter | `claim`, proved by `result` |\n| Quoted decimal example | Not an accuracy assertion of `claim`; the exact quarter-value identity is computed below |\n| Separate infinite-sum sentence | Outside `claim`; its omitted-tail limit remains open as a separate Lean result |\n\nAll local names below belong to `D5.S3.Weil.ZetaGamma.HadamardTruncatedBinomialLimit`; intermediate lemmas are private. The definitions retain the preregistered expressions. The all-complex domain and existence conjunct specify the source's otherwise unstated conventions without restricting the parameter domain.","title":"Ulgenes's truncated-binomial Hadamard limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/ulg398064/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b25190afd327b838e83c211fc30b8f4d29db7b41dda1798cbb8a05efef174edb.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:uola-et-al-2021-heisenberg-channel-compression-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"R. Uola, T. Kraft, S. Designolle, N. Miklin, A. Tavakoli, J.-P. Pellonpää, O. Gühne and N. Brunner, *Quantum measurement incompatibility in subspaces*, arXiv:2010.04048v2 (Phys. Rev. A 103, 022203, 2021), section on incompressible incompatibility:\n\n> It is worth mentioning that one can modify the concept of incompressible incompatibility by demanding that a measurement assemblage becomes compatible under every Heisenberg channel (a channel preserving identity, but not necessarily trace preserving) to a smaller dimensional system. Although we will leave open the question of whether this provides a strict subset of measurement assemblages that are compatible in every subspace, …\n\nHere:\n- **Compatible in every subspace of dimension $n$:** every compression $\\{V^\\dagger M_{a|x}V\\}$ by an isometry $V:\\mathbb C^n\\to\\mathbb C^d$ has a parent POVM.\n- **Heisenberg channel:** a completely positive unital map $\\Phi:M_d\\to M_n$, applied to each effect.\n\nThe definitions are quoted in [the literature note](../Library/QuantumContext/uola2021subspaces.md).\n\nIssue [#20047](https://github.com/the-omega-institute/trureturing/issues/20047) fixes the reading: for each $n\\le d$ the two conditions are equivalent, so the subset is not strict. This is proved.","title":"Compatibility under every Heisenberg channel equals compatibility in every subspace","url":"https://the-omega-institute.github.io/trureturing-pages/research/uola-et-al-2021-heisenberg-channel-compression-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bdd757616cf254d4d9cffd6bc1bee59235b66b8645de3986b25bf907276b79e2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:user3-2026-orthogonal-absolute-product-minimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"User3 (MathOverflow user 600125), *A nontrivial lower bound for the\northogonally transformed absolute product*, question 515549, revision 1,\nGUID `CB719581-BCE3-4573-931E-15327002F861`, asks:\n\n> Let $P\\in\\mathbb{R}^{K\\times K}$ be nonsingular and define\n> $$M(Q)=|PQ|\\,|Q^\\top P^{-1}|,\\qquad Q\\in O(K).$$\n> $|M∣$ means the matrix obtained by taking the absolute value of each entry.\n> Since\n> $$|PQ(Q^\\top P^{-1})|=I\\le |PQ|\\,|Q^\\top P^{-1}|=M(Q),$$\n> one obtains the trivial bound $\\rho(M(Q))\\ge 1$.\n> $\\rho(M(Q))$ is the spectral radius of the matrix $M$.\n> The question is whether there exists a nontrivial quantity $\\beta(P)$,\n> depending only on $P$, such that\n> $$\\rho(M(Q))\\ge\\beta(P)>1\\qquad\\text{for all }Q\\in O(K),$$\n> for an appropriate class of nonsingular matrices $P$.\n> Equivalently, can the optimization problem\n> $$\\inf_{Q\\in O(K)}\\rho\\!\\left(|PQ|\\,|Q^\\top P^{-1}|\\right)$$\n> be bounded below, or even characterized explicitly, in terms of intrinsic\n> properties of $P$?\n\nThe quotation is attributed to User3, MathOverflow, under\n[CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0/).\nIts source identity and license are recorded in\n[the Library note](../Library/Eigenstructure/userthree2026orthogonalabsolute.md).\n\nThe single resolution anchor is the exact optimization characterization:\nfor every natural $K>0$ and every $P\\in\\mathbb R^{K\\times K}$ with\n$\\det P\\ne0$,\n\n$$\n\\left(\\forall Q\\in O(K),\\quad\n1\\le\\rho\\!\\left(|PQ|\\,|Q^\\top P^{-1}|\\right)\\right)\n\\quad\\land\\quad\n\\left(\\exists Q\\in O(K),\\quad\n\\rho\\!\\left(|PQ|\\,|Q^\\top P^{-1}|\\right)=1\\right).\n$$\n\nThus the minimum and infimum are both one, with actual attainment.\nIn particular, no $\\beta(P)>1$ can bound the objective below for every\northogonal $Q$, for any nonsingular $P$ in positive dimension. This answers\nthe characterization question and negates the proposed strict improvement.","title":"The orthogonal absolute-product minimum is attained at one","url":"https://the-omega-institute.github.io/trureturing-pages/research/user3-2026-orthogonal-absolute-product-minimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f412d760192cdb53f7d2e29cee63db99a6f222cfaa17d76a8d6c29b5741ce8cc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:utagi-banerjee-srikanth-2024-purely-nonunital-eternal-nonmarkovianity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Utagi, S. Banerjee and R. Srikanth, *On the eternal non-Markovianity of non-unital quantum channels*, arXiv:2203.10625v2, International Journal of Quantum Information 22(1), 2350039 (2024), DOI 10.1142/S0219749923500399, Section IV, conjecture:\n\n> These diverse examples examined in the Section suggest that a purely non-unital channel (i.e., one lacking any unital component) is impossible. Although we have only studied the qubit case, we conjecture that it is true for non-unital channels in arbitrary finite dimensions.\n\nSection V restates:\n\n> However, we conjecture that for a non-unital channel with no such unital component (in the above sense) to be ENM, is impossible.\n\nThe source's Definition 1 attributes purely non-unital non-Markovianity to the translation in the Bloch action. [The source note](../Library/QuantumChannels/utagi2024eternal.md) gives the locators and quotations. [#16599](https://github.com/the-omega-institute/trureturing/issues/16599) fixes the complete qubit conventions and the all-positive-times criterion, without a nonzero negative asymptotic bound.\n\n### Statement echo\n\n| Source clause or explicit convention | Lean counterpart | Scope |\n| --- | --- | --- |\n| Qubit dynamics on all complex matrices | Real-time `PhyslibLeaf.MatrixMap (Fin 2) (Fin 2) ℂ` in `claim` | The qubit case suffices to refute an assertion for arbitrary finite dimensions. |\n| CPTP at every nonnegative time | Existing `CoPRelativeQuantumnessRefutation.IsCPTP` in `claim` | Complete positivity and trace preservation are reused literally. |\n| Dynamics starts at the identity | Identity hypothesis in `claim` | Required on every matrix, not only states. |\n| Eqs. (9)–(11): delete the translation | `unitalPart` | Its complete body is the preregistered subtraction $\\Phi_t(X)-\\operatorname{Tr}(X)(\\Phi_t(I)-I)/2$. |\n| Definition 1: translation alone supplies non-Markovianity | `PurelyNonUnital` | Non-unital at every positive time; the unital part has a CPTP intermediate map for every $0\\le s\\le t$. |\n| Time-local generator | `IsGeneratorAt` | Entrywise real-time derivative on every complex matrix. |\n| Conditional complete positivity | `CondCP` | The existing output-first `MatrixMap.choi_matrix` is tested on every trace-free vector. |\n| Negative generator direction at every $t>0$ | `EternallyNonMarkovian` | No negative asymptotic bound is asserted. |\n| Impossibility conjecture | `claim`, negated by `result` | Every quantifier and hypothesis above is retained. |\n\nThe entrywise derivative and conditional-Choi criterion are the explicit mathematical conventions of #16599. They do not restrict inputs to density matrices or replace the source's unital part by a certificate predicate.","title":"A rotating replacement state refutes purely non-unital eternal non-Markovianity impossibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/utagi-banerjee-srikanth-2024-purely-nonunital-eternal-nonmarkovianity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/542d85b6c1476bf080d8500b9ffad5503e4f9400b41935d6adc4cfb10c439d3b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:van-dam-gill-grunwald-2005-two-singlet-strength","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"W. van Dam, R. D. Gill and P. D. Grünwald, “The Statistical Strength of\nNonlocality Proofs”, IEEE Transactions on Information Theory 51(8),\n2812–2835 (2005), arXiv:quant-ph/0307125v2, Section VI.D, p. 11,\nConjecture 5 states:\n\n> There is an experiment on pairs of Bell singlets, of the $2\\times4\\times4$ type, more than twice as strong as CHSH, and involving joint measurements on the pairs.\n\nThe counts are two parties, four settings per party and four outcomes\nper setting. The source measures strength by KL divergence in bits,\nminimized over every local response mixture, with uniform, product or\narbitrary correlated setting laws (Section IV.B, Definitions 1–3).","title":"Two singlets give more than twice the statistical strength of CHSH","url":"https://the-omega-institute.github.io/trureturing-pages/research/van-dam-gill-grunwald-2005-two-singlet-strength/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a012c0583543973d788bdc666b2b037defeb33182c010fb3e953723b731cd1e6.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:van-herstraeten-2024-photon-addition-coefficients","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z. Van Herstraeten, N. J. Cerf, S. Guha and C. N. Gagatsos (arXiv:2312.02066,\nquant-ph; Phys. Rev. A 110 (2024) 042430) compare a two-mode squeezed vacuum\nwith `k` photons added to each mode against the single-photon-added state. The\nSchmidt coefficients are `q_n^{(kk)} = λⁿ C(n+k,k)² / N_kk`, and the paper\nwrites `q^{(kk)} = D q^{(11)}` with a lower-triangular Toeplitz matrix `D`\nwhose entries are `λN_11/N_kk · c_m^{(kk)} λ^m`, where\n\n> \\binom{n+k+1}{k}^2 = \\sum_{i=0}^{n+1} c_{n-i}^{(kk)} (i+1)^2\n\nand `c_{-1}^{(kk)} = 1`. The columns of `D` sum to `1`. The appendix ends:\n\n> At this point, we conjecture the non-negativity of Eq.\n> \\eqref{eqapp:Coeffcients}, \\textit{i.e.}, we conjecture that the matrix\n> $\\mathbf{D}$ of Eq. \\eqref{eqapp:Dvector} is column stochastic for all\n> $k\\geq 2$, implying the majorization relation $\\hat{\\sigma}^{k,k} \\prec\n> \\hat{\\sigma}^{1,1}$.\n\nIssue #11700 fixes the reading: `c_n^{(kk)}` (`n ≥ 0`) are the numbers with\n`c_{-1}^{(kk)} = 1` that satisfy the expansion for every `n ≥ 0`, and the\nconjecture is `c_n^{(kk)} ≥ 0` for all `k ≥ 2` and `n ≥ 0`. The formal `claim`\nstates that a solution exists and that every solution is non-negative; the\nresult proves it.","title":"Non-negativity of the photon-addition coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/research/van-herstraeten-2024-photon-addition-coefficients/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3da1437a5048d0f4dedfa8855d5b664d917ed88e47f246020afb8303b12d5be7.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:van-herstraeten-et-al-2025-beam-splitter-state-extremality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Z. Van Herstraeten, J. Davis, N. C. Dias, J. N. Prata, N. J. Cerf and\nU. Chabaud, *Extreme non-negative Wigner functions*, arXiv:2512.14831v3,\nSection 6, printed page 15, ask:\n\n> This leads us to formulate the following open problem: for any two Fock-bounded pure states $\\ket\\psi,\\ket\\phi$ is the beamsplitter state $\\hat\\sigma(\\psi,\\phi)$ an extreme WPS?\n\nThe source is [the version-3 paper](https://arxiv.org/abs/2512.14831v3).\nThe Tier-1 statement, finite-matrix conventions and literature reading are\nin [preregistration #14447](https://github.com/the-omega-institute/trureturing/issues/14447).\nThe answer is **Refuted**.","title":"Finite-Fock beam-splitter states need not be extreme Wigner-positive states","url":"https://the-omega-institute.github.io/trureturing-pages/research/van-herstraeten-et-al-2025-beam-splitter-state-extremality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b5b061d90ece4a6af2fe561d542d87b585afaeab458c4672f251a7aa5def4a04.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:van-herstraeten-guha-cerf-2024-phase-covariant-attenuator-coherent-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Conjecture 2 (Phase-covariant attenuator), section IV.B, arXiv:2312.15623v2, PDF p. 12, states: “The minimum output entropy of a phase-covariant attenuator channel ℳη,𝐩 is achieved by coherent states, ∀η,𝐩.” The domain is every transmissivity η in [0,1] and every probability vector p on the natural occupations. The assertion is that some complex coherent amplitude α minimizes the output entropy over all density operators on the full one-mode Fock space. Preregistration #11681 gives this quantifier order and the counterexample.","title":"Van Herstraeten–Guha–Cerf Conjecture 2: coherent inputs for phase-covariant attenuators","url":"https://the-omega-institute.github.io/trureturing-pages/research/van-herstraeten-guha-cerf-2024-phase-covariant-attenuator-coherent-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d756a304f1b1114859a301e58eb400e596841ddb43ddea209fb707e55b74726.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vandre-2024-graph-condensation-lc-equivalence-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vandré, de Jong, Hahn, Burchardt, Gühne and Pappa, *Distinguishing Graph\nStates by the Properties of Their Marginals*, arXiv:2406.09956v2,\nSection V, Conjecture 16, p. 13:\n\n> Given two graphs G and G′ and a condensation set C such that each node in C is connected to at most one node in the neighborhood in V \\ C. If G and G′ are LC-equivalent, it follows that G_c and G′_c are LC-equivalent.\n\nThe paper considers simple connected graphs. Definition 13 (p. 12) replaces\nC by one fresh vertex c, retains edges between vertices outside C, and joins\nc to an outside vertex i exactly when some vertex of C is adjacent to i.\nLocal complementation (Definition 4, pp. 4–5) complements adjacency among\nthe selected vertex's neighbours. The correspondence with local Clifford\noperations of the associated graph states is stated on p. 5 and attributed\nto Van den Nest, Dehaene and De Moor (2004), DOI\n10.1103/PhysRevA.69.022316. The formal statement uses finite labelled graphs\nand finite sequences of these local complementations.\n\nThe complete quantified claim is:\n\n```lean\n∀ (V : Type) [Fintype V] [DecidableEq V] (G H : SimpleGraph V) (C : Finset V),\n G.Connected → H.Connected →\n (∀ s ∈ C, (Finset.univ.filter fun i => i ∉ C ∧ G.Adj s i).card ≤ 1) →\n (∀ s ∈ C, (Finset.univ.filter fun i => i ∉ C ∧ H.Adj s i).card ≤ 1) →\n LCEquivalent G H → LCEquivalent (condense G C) (condense H C)\n```\n\nBoth graphs must satisfy the outside-neighbour bound. This is the more\nrestrictive reading of the premise, so a counterexample satisfying both\nbounds also refutes a reading requiring the bound on only one graph.","title":"Condensation need not preserve LC-equivalence","url":"https://the-omega-institute.github.io/trureturing-pages/research/vandre-2024-graph-condensation-lc-equivalence-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07ad2135836fb1b8bbc1e386e1ec6369c514eeeb05ecc71e6a4aab6bbc91ccbc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vatter-largest-derangement-limit-below-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vincent Vatter, \"An Assortment of Problems in Permutation Patterns:\nUnimodality, Equivalence, Derangements, and Sorting\", arXiv:2602.16355v2,\nSection 4, unnumbered paragraph immediately after Question 4.3:\n\n> Is there a largest possible limit strictly less than 1?\n\nLocator: https://arxiv.org/html/2602.16355v2#S4. This dossier concerns only\nthat sentence. It is distinct from the universal convergence question and\nhas no problem number of its own.\n\nA permutation class is a downset of actual permutations under subsequence\norder-isomorphism. Write `C_n` for its length-`n` slice, `C_n°` for the\nfixed-point-free members, and `r_C(n) = |C_n°| / |C_n|`. Let `Attained(l)`\nmean that some such class has nonempty slices at every length and\n`r_C(n) -> l` as `n -> infinity`. The affirmative largest-attained-limit\nproposition is\n\n```text\nexists L in R, L < 1 and Attained(L) and\n forall l in R, (l < 1 and Attained(l)) implies l <= L.\n```\n\nThe source imposes no growth or finite-basis condition. Nonempty slices are\nprovided by the constructed witnesses, so their ratios do not rely on an\nempty-denominator convention. Even if a broader convention admits additional\nlimits, any proposed maximum `L < 1` is contradicted by the same witness\nconstruction with `q = L`.","title":"Vatter's largest derangement limit below one","url":"https://the-omega-institute.github.io/trureturing-pages/research/vatter-largest-derangement-limit-below-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c689300e1154bc1ed67c105f4ba66009bd50d046bbf382a7ccd3df606d816bd1.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vatter-question-4-3-derangement-ratio","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Vincent Vatter, \"An Assortment of Problems in Permutation Patterns: Unimodality,\nEquivalence, Derangements, and Sorting\", arXiv:2602.16355v2, Section 4:\n\n> A permutation class is a downset (or order ideal) in this poset: a set 𝒞 of permutations such that if π∈𝒞 and σ≤π, then σ∈𝒞.\n\n> For a permutation class 𝒞, let 𝒞° denote the set of derangements in 𝒞.\n\n> Question 4.3. Does the ratio |𝒞°_n|/|𝒞_n| converge for every permutation class 𝒞?","title":"A negative answer to Vatter's Question 4.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/vatter-question-4-3-derangement-ratio/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5df1343a6b5bc4ca0528901ff32d09413bf7596d5642c3ff339dfe233a0ceb90.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:verhagen-tomamichel-haapasalo-2025-flat-pair-endpoint-homomorphisms-remark-15","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Verhagen, Tomamichel and Haapasalo, *Conditions for Large-Sample Majorization of\nPairs of Flat States in Terms of α-z Relative Entropies*, arXiv:2507.07520v3,\nRemark 15, p. 21, ask:\n\n> Namely, it is still an open question whether the homomorphisms for 𝕂 = ℝ₊ or ℝ₊ᵒᵖ that have the same form as (84), but with parameters α = 0, z < 1, or α = 1, z < 1, are monotone or not. The same question of monotonicity can also be asked for the homomorphisms for 𝕂 = 𝕋ℝ₊ that are similar to (85), but with z < 0.\n\n> We suspect that these additional homomorphisms are in fact not monotone either, since they correspond to parameters α, z for which the α-z relative entropy is known not to satisfy DPI for general pairs of quantum states (with some conditions on the support).\n\nThe settled claim excludes $z=0$, which the source treats as the degenerate\ntrace maps. It asserts failure of both real target orders for every\n$\\alpha\\in\\{0,1\\}$ and every real $z<1$, $z\\ne0$, and failure of the ordinary\ntropical order for every real $z<0$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Eq. (31): finite direct sums of two rank-one block matrices; vectors may vanish | `FlatPair`, `FlatPair.A`, `FlatPair.B` | Arbitrary finite block counts and dimensions; sigma indices are written directly |\n| Eqs. (33)–(34): nonzero overlap or both block vectors zero | `FlatPair.IsEverywhereOverlapping` | Every block, including zero blocks |\n| Eq. (37): weights and normalized overlaps | `FlatPair.p`, `FlatPair.q`, `FlatPair.overlapSq` | Unnormalized vectors, real self-dot-product weights |\n| One channel acts on both matrices | `Majorizes` | Existing completely positive trace-preserving `QuantumChannel` |\n| Eqs. (84)–(85) | `Phi`, `PhiT` | Zero blocks contribute zero; the maximum starts at zero |\n| Both real target orders | `IsMonotone`, `IsMonotoneOp` | The same channel preorder and everywhere-overlapping domain |\n| Endpoint suspicion, excluding trace maps | First conjunct of `claim`, proved by `result` | All real parameters $\\alpha\\in\\{0,1\\}$, $z<1$, $z\\ne0$ |\n| Negative tropical suspicion | Second conjunct of `claim`, proved by `result` | Every real $z<0$ in the ordinary tropical order |","title":"Endpoint homomorphisms of everywhere overlapping flat pairs","url":"https://the-omega-institute.github.io/trureturing-pages/research/verhagen-tomamichel-haapasalo-2025-flat-pair-endpoint-homomorphisms-remark-15/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bbdfbe5552f0f5cace5dc45e13b61bb9f601e03dbe51e25a16fa505688258e50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vesperini-bel-hadj-aissa-franzosi-2023-bell-diagonal-entanglement-distance-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"A. Vesperini, G. Bel-Hadj-Aissa and R. Franzosi, *Entanglement and Quantum Correlation Measures from a Minimum Distance Principle*, arXiv:2205.07143v2; published as *Entanglement and quantum correlation measures for quantum multipartite mixed states*, Sci. Rep. 13, 2852 (2023), DOI 10.1038/s41598-023-29438-7. In Applications, “Bell diagonal states”, the v2 source `QCDED-V4.tex`, line 368, states:\n\n> We were not able to find a simple analytic solution of the minimization procedure for the most general case of BD states. However, numerical minimization provided us with empirical evidence that the procedure (22) also leads for these states to the squared concurrence, as shown in figure \\ref{figBD_ED}, which represent a face of the tetrahedral domain of BD states.\n\nThis is a reported numerical belief, rather than a proved identity. The source read is the arXiv v2 TeX, MD5 `232776c9cef202cc624a80958cb4c818`. Its figure caption at line 372 plots $E[\\rho_{BD}]/2$; the Werner-state formula at lines 396–400 is described as “twice the square of the concurrence”. These fix the factor-two reading registered in issue [#20089](https://github.com/the-omega-institute/trureturing/issues/20089).\n\nLet $B_0,B_1,B_2,B_3$ be the projectors onto $(|00\\rangle+|11\\rangle)/\\sqrt2$, $(|00\\rangle-|11\\rangle)/\\sqrt2$, $(|01\\rangle+|10\\rangle)/\\sqrt2$ and $(|01\\rangle-|10\\rangle)/\\sqrt2$, respectively. For $p_\\alpha\\ge0$ and $\\sum_{\\alpha=0}^3p_\\alpha=1$, put $\\rho_{BD}(p)=\\sum_\\alpha p_\\alpha B_\\alpha$ and $c(p)=\\max(0,2\\max_\\alpha p_\\alpha-1)$. Here $c$ denotes Bell-diagonal concurrence; $C_\\mu$ below denotes site quantum correlation distance (QCD).\n\nThe registered claim is that $E(\\rho_{BD}(p))=2c(p)^2$ for every such probability vector, where $E$ minimizes the site QCD of locally rotated mixtures over all finite density-matrix decompositions and component-dependent local rotations, summing both sites. This claim is refuted.","title":"Bell-diagonal entanglement distance is not twice the squared concurrence","url":"https://the-omega-institute.github.io/trureturing-pages/research/vesperini-bel-hadj-aissa-franzosi-2023-bell-diagonal-entanglement-distance-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5255f13778de9f371d82d1787dd90d89958ff49b94b5b9aabc6f7cb1284c7e4c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:visomirski-griffin-2026-frozen-wave-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Matthew Visomirski and Christopher Griffin, *Coherent Structures and Travelling\nWaves in Spatial Replicators from a Biased Volterra Lattice*, arXiv:2601.06314v1,\nSection 4.2, Conjecture 2 (page 19), asserts:\n\n> Consider the one-dimensional spatial replicator, i.e., Eq. (6) with reaction\n> term generated by a $2n$-cycle with $n\\geq2$. Assuming periodic boundary\n> conditions and initial conditions of the form,\n\n$$\nu_i(x,0)=\\begin{cases}(1+\\sin x)/(2n),&i\\text{ odd},\\\\\n(1-\\sin x)/(2n),&i\\text{ even},\\end{cases}\n$$\n\n> the dynamics will evolve to a frozen wave stationary solution. That is,\n> ecological niches will be maintained.\n\nThe simulated parameters are $a=-1/2$, $D=1/100$, on $[-\\pi,\\pi]$ with periodic\nboundary conditions. Preregistration: [#14205](https://github.com/the-omega-institute/trureturing/issues/14205).","title":"Visomirski–Griffin frozen-wave conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/visomirski-griffin-2026-frozen-wave-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2c859399457daaab0e57b0f7cc6cbffdf501635c432f40ad8cf7c1b5495f8fd9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:volkoff-2016-pair-tunneling-ansatz-ground-state-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. J. Volkoff, arXiv:1610.05807v1, Section V.A, Eq. (20), pp. 6–7, fixes even particle number and writes:\n\n> We conjecture that for each N ≥ 4 there exists a value c_N for which |ω_+(c_N)⟩ or |ω_−(c_N)⟩ is the exact ground state.\n\nThe sector has basis $|N-k,k\\rangle$, indexed by $k\\in\\operatorname{Fin}(N+1)$, and Hamiltonian\n$H=a_0^{\\dagger 2}a_1^2+a_1^{\\dagger 2}a_0^2$.\nWith $M=N/2$ and real $c$, the unnormalized ansatz is the coefficient vector of\n$(x^2+2icxy-y^2)^M\\pm(x^2-2icxy-y^2)^M$, multiplied coordinatewise by\n$\\sqrt{(N-k)!k!}$. Normalization is irrelevant for a nonzero eigenvector.\nAn exact ground state has the least real eigenvalue among all nonzero eigenvectors in the full sector.\nIssue #11667 preregisters this quantified reading and its refutation at $N=8$.","title":"Volkoff's exact-ground-state pair-tunnelling ansatz","url":"https://the-omega-institute.github.io/trureturing-pages/research/volkoff-2016-pair-tunneling-ansatz-ground-state-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3809e7ef33e43ec99671658543dcc375b5e04f577a2fb6006b0cd6316b63640a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vom-ende-2022-d-majorization-extreme-class-infinitude","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Frederik vom Ende, *Strict positivity and D-majorization*, Linear and Multilinear Algebra 70(19) (2022), 4023–4048, arXiv:2004.05613v4, asks in the Conclusion and Outlook, TeX file `strictly_positive_qc_v2.tex`, line 1064:\n\n> Whether one ends up with finitely many extreme points after factoring out the unitary equivalents for now remains open.\n\nThe equivalence relation immediately preceding the question, lines 1060–1064, is\n$X\\sim UXU^*$ for unitary $U$ satisfying $[U,D]=0$, with $D=\\operatorname{diag}(d)$, arbitrary $d\\in\\mathbb R_{++}^n$ and arbitrary $A\\in\\mathbb C^{n\\times n}$. Here $M_D(A)=\\{X:X\\prec_D A\\}$, and $X\\prec_D A$ means that a complex linear, completely positive, trace-preserving map sends $A$ to $X$ and fixes $D$ (Definition `defi_matrix_D_maj`, lines 466–469; equation `eq:def_MD`, lines 926–932). See [the literature note](../Library/Eigenstructure/vomende2022strictpositivity.md).\n\nIssue [#22739](https://github.com/the-omega-institute/trureturing/issues/22739) fixes the affirmative reading: for every positive dimension, strictly positive $d$ and complex matrix $A$, the real extreme points of $M_D(A)$ have finitely many classes under this relation. The module's `claim` quantifies over all natural dimensions, including zero, and asks for a finite set of ambient matrices representing every extreme point. Its negation is established using dimension two, so the positive-dimension reading is also refuted.","title":"Infinitely many extreme classes survive the commuting-unitary quotient in matrix D-majorization","url":"https://the-omega-institute.github.io/trureturing-pages/research/vom-ende-2022-d-majorization-extreme-class-infinitude/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca4c6b9d61e5735781b90350d80e92ba6cfe764c2b6c32441f56e76c6753128e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vom-ende-2024-gibbs-stochastic-simple-spectrum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. vom Ende, *Almost All Quantum Channels Are Diagonalizable*, arXiv:2403.19643v3 [quant-ph], Open Systems & Information Dynamics 31(3), 2450012 (2024), DOI 10.1142/S1230161224500124, leaves a construction question in the Outlook.\n\nThe v3 TeX source `channel_diag_approx.tex`, line 511, states:\n\n> for the case of thermal operations, for example, one would have to first show that almost all Gibbs-stochastic matrices have only simple eigenvalues---regardless of the exact Gibbs state\n\nIts footnote, lines 512–513, reads:\n\n> For those interested: The last point is what makes this non-trivial. While the $\\beta$-permutations \\cite{Alhambra19,Mazurek19,PolytopeDegen22} are candidates for Gibbs-stochastic matrices with distinct eigenvalues many of them are only valid in certain parameter regimes, i.e.~only if the eigenvalues of the Gibbs state are interrelated in a specific way.\n> We leave a general construction that is valid for all Gibbs states as an open problem.\n\nThese quotations are from the downloaded v3 source, md5 `18bc93c300903edbf34bf142ad00519d`; the source's definition of a simple eigenvalue as one of algebraic multiplicity one is at lines 123–125. See also [the literature note](../Library/QuantumContext/vomende2024diagonalizable.md).\n\nIssue [#21152](https://github.com/the-omega-institute/trureturing/issues/21152) fixes the positive-Gibbs-vector reading: for every $n\\in\\mathbb N$ and every $d:\\mathrm{Fin}\\,n\\to\\mathbb R$ with $d_i>0$ for all $i$, there exists a real $n\\times n$ matrix $P$ with nonnegative entries, every column summing to one, $Pd=d$, and no repeated roots in its complex characteristic polynomial. This existence statement is proved. Normalization of $d$ is not required; $n=0$ is included. The source's density consequence is separate from the formal claim.","title":"Simple-spectrum Gibbs-stochastic matrices for every positive Gibbs vector","url":"https://the-omega-institute.github.io/trureturing-pages/research/vom-ende-2024-gibbs-stochastic-simple-spectrum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/639c3effd7ed2af892c516d6a043702f6662c8cb9d68e873b09814d0e03b6d80.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vom-ende-chruscinski-kimura-muratore-ginanneschi-2025-conditional-two-positive-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. vom Ende, D. Chruściński, G. Kimura and P. Muratore-Ginanneschi,\n*Universal Bound on the Eigenvalues of 2-Positive Trace-Preserving Maps*,\narXiv:2506.02145v1, §3 “Outlook”, page 11, question (10), asks:\n\n> Hence one may wonder whether all (generators of) completely positive maps, resp. 2-positive maps satisfy\n\n$$\n\\mathrm{tr}(L)\\le d\\min\\Re\\sigma(L)+(d^2-d)\\max\\Re\\sigma(L).\n$$\n\n> A map $L$ is called conditionally 2-positive if $e^{tL}$ is 2-positive for all $t\\ge0$.\n\n> With this, (10) becomes really a conjecture about the spectrum of arbitrary conditionally 2-positive maps.\n\nVCKM-10 quantifies every integer $d\\ge1$ and every complex-linear\nendomorphism $L$ of $\\mathbb C^{d\\times d}$ satisfying this definition.\nThere is no trace-preservation assumption. The trace is the superoperator\ntrace; the spectrum is the set of roots of the characteristic polynomial.\nThe Lean claim also asserts that the superoperator trace is real.","title":"Spectral bound for conditionally 2-positive maps","url":"https://the-omega-institute.github.io/trureturing-pages/research/vom-ende-chruscinski-kimura-muratore-ginanneschi-2025-conditional-two-positive-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec85ba9cb764e3826bf1bd0ca760ca092b00887e784b6b73a98aa2ab1e970a98.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vom-ende-cichy-2025-kernel-trace-criteria-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. vom Ende and S. Cichy, *Simple Sufficient Criteria for Optimality of Entanglement Witnesses*,\narXiv:2505.15615v2, Section IV (p. 12):\n\n> Next: is our trace-based criterion (Coro. 2) actually equivalent to our kernel criterion (Thm. 2)?\n> This is the question mark in Fig. 2, and while the kernel criterion seems stronger we were not able\n> to find a witness for which only the kernel criterion holds.\n\nThe kernel criterion asks for a vector of Schmidt rank $\\min\\{m,n\\}$ in the kernel of\n$W+{\\rm tr}_2(W)\\otimes{\\bf1}$ (if $m\\le n$) or of $W+{\\bf1}\\otimes{\\rm tr}_1(W)$ (if $m\\ge n$); the\ntrace criterion asks for a maximally entangled $\\Omega$ with\n$\\langle\\Omega|W|\\Omega\\rangle=-{\\rm tr}(W)/\\min\\{m,n\\}$. The verbatim texts are in\n[the literature note](../Library/QuantumChannels/vomende2025witnessoptimality.md).\nIssue [#14763](https://github.com/the-omega-institute/trureturing/issues/14763) preregisters the\nreading and the refutation.","title":"The kernel criterion for witness optimality does not imply the trace criterion","url":"https://the-omega-institute.github.io/trureturing-pages/research/vom-ende-cichy-2025-kernel-trace-criteria-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02ec05cb9a460fd9419658c5c0aa49fefe3efcb264d98d11d50f2a8092176cfd.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vom-ende-shahbeigi-2024-stochastic-generator-set-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"F. vom Ende and F. Shahbeigi, *Generating Sets of Stochastic Matrices*, arXiv:2411.18946, study sets G of column-stochastic matrices whose finite products give every stochastic matrix. N_G(s(n)) is the number of factors needed in the worst case (Definition 3). These are the classical counterparts of generating sets of quantum channels: a generating set models the operations an experiment can implement, and N_G bounds the circuit length needed to implement every classical channel.\n\nTheorem 3 states that G = {P} ∪ conv{1, V1, V2, V3} of Eq. (10) generates s(3) with G^20 = s(3). The Outlook asks \"is the upper bound N_G(S) ≤ 20 in Theorem 3 optimal? We conjecture that this is true\".\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/vomendeshahbeigi2024stochasticgenerators.md).\n\nIssue [#17954](https://github.com/the-omega-institute/trureturing/issues/17954) fixes the reading: N_G(s(3)) = 20, that is, G^20 = s(3) and G^19 ≠ s(3). This conjecture is refuted, together with the generation claim of Theorem 3.","title":"The Eq. (10) set does not generate the 3 × 3 stochastic matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/vom-ende-shahbeigi-2024-stochastic-generator-set-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bc04c01288a87a7c39e7409cecfe462db012ea4983b09d6f038ca05da5618ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:vomende-2024-elementary-channel-divisibility","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Frederik vom Ende, *A Sufficient Criterion for Divisibility of Quantum Channels*,\narXiv:2407.17103v3, Section V, asks:\n\n> An open question at this point is whether our sufficient condition is also necessary, and whether our necessary condition is also sufficient.\n\nThe source is [arXiv v3](https://arxiv.org/html/2407.17103v3), with publication\nidentifier [DOI 10.1063/5.0231812](https://doi.org/10.1063/5.0231812).\nDefinition 1 and Lemmas 2–3 specify the elementary channel and inverse;\nProposition 1 and Theorem 1 specify the necessary and sufficient conditions.\nThis dossier anchors the combined classification of the two reversals,\nincluding their fixed-pair and existential-pair readings.\n\nLet the complex dimension be any finite n ≥ 2, let Φ be an actual CPTP map,\nand let (Kⱼ) be any finite Kraus representation of Φ. For an ordered pair of\northogonal unit vectors (x,y), E(x,y) means that Φ composed on the right\nwith the inverse of the elementary channel Ψₜ(x,y) is completely positive\nfor some real 0 < t < 1. N(x,y) means\n⟨y,Kⱼ*Gx⟩ = 0 for every index j and every complex matrix G in the\nHilbert–Schmidt orthogonal complement of the whole complex Kraus span.\nS(x,y) means N(x,y) together with the additional diagonal tests\n⟨x,Kⱼ*Gx⟩ = 0 over the same indices and complement.\n\nThe classification established by the designated result has four cells:\n\n| Reading | Answer |\n| --- | --- |\n| Fixed ordered pair: N(x,y) iff E(x,y) | Proved for every map and representation in the stated scope. |\n| Existential ordered pair: some N pair iff some E pair | Proved for the same map and representation. |\n| Fixed ordered pair: E(x,y) implies S(x,y) | Refuted in every dimension n ≥ 2. |\n| Existential ordered pair: some E pair implies some S pair | Refuted in every dimension n ≥ 2; the counterexample has no S pair in any finite Kraus representation. |\n\nThe Scribe claim's `Proved` kind refers to this combined classification,\nwhose statement includes the counterexample. It does not label either\nE ⇒ S assertion as proved.","title":"Exact classification of elementary channel divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/research/vomende-2024-elementary-channel-divisibility/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c579a7cb48e184f38466818bbab241db451b20e566aa9d06fff793344e2f613.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wall-sun-sun-golden-unit-lift","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let `pi(m)` be the Pisano period, the least positive period of the Fibonacci\nrecurrence modulo `m`. The primary problem is whether there exists a prime `p`\nwith `pi(p) = pi(p^2)`. The paper also records the stronger conjecture that\ninfinitely many such primes exist.\n\nQuoted from the introduction of arXiv:2603.25343v1:\n\n> “A natural question was asked by Wall in his paper: Can there be a prime\n> \\(p\\) such that \\(\\pi(p)=\\pi(p^2)\\)?”\n\n> “It is known that up to \\(10^{14}\\), there are no such primes (cf. [16]).\n> Still, using heuristics and probabilistic arguments, some authors conjecture\n> the existence of infinitely many primes \\(p\\) satisfying\n> \\(\\pi(p)=\\pi(p^2)\\) [7, 11].”\n\nThe same paper identifies the classical case with `d = 5` and says there are no\nknown `WSS(5)` primes.\n\nCandidate formal statement, after defining `pisanoPeriod`:\n\n```text\nExistence: ∃ p : Nat, Nat.Prime p ∧ pisanoPeriod p = pisanoPeriod (p^2)\nStronger: Set.Infinite {p | Nat.Prime p ∧ pisanoPeriod p = pisanoPeriod (p^2)}\n```\n\nThe paper states the difficulty:\n\n> “The question of Wall for these sequences is related to certain deep\n> arithmetic properties of real quadratic fields.”\n\nIt makes this precise for its generalized recurrence: equality of the periods\nmodulo `p` and `p^2` corresponds, subject to stated hypotheses, to failure of\n`p`-rationality of the associated real quadratic field. This is why the global\nexistence question is not a routine finite-period exercise.","title":"Wall-Sun-Sun primes as a golden-unit lift problem","url":"https://the-omega-institute.github.io/trureturing-pages/research/wall-sun-sun-golden-unit-lift/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1b7d18bf57ac83ca1e469cff8fe6a894b7efa721833add364be56087415a39d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wallner-krupke-schmidt-fekete-2026-token-spectral-contraction","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"T. Wallner, D. Krupke, A. Schmidt and S. P. Fekete, *Pass the Bucket: Efficient, Robust, Local Load Balancing for Teams of Heterogeneous Robots*, arXiv:2608.27085v1, Section IV.B.1, page 4:\n\n> Conjecture 1 (Spectral contraction): The damped transfer map Mα is repeatedly applied and its spectral radius satisfies ρ(Mα) < 1, such that u → 0 for 0 < α < 1.\n\nThe statement concerns the paper's centered linear map in its rotational regime. There are $n=m+1$ robots, $m\\ge1$ pair coordinates and positive velocities $v_1,\\ldots,v_n$. The same damping parameter $\\alpha\\in(0,1)$ is used at every step. The delivered `claim` contains both the strict bound for every complex spectral value and convergence $M_\\alpha^k u\\to0$ for every real initial vector.\n\nThe binding source identifiers $B_\\alpha$ and $M_\\alpha$ are spelled `Balpha` and `Malpha` in Lean, as authorized by [#14768](https://github.com/the-omega-institute/trureturing/issues/14768#issuecomment-6082771002).","title":"Token-1 spectral contraction for every finite positive-velocity chain","url":"https://the-omega-institute.github.io/trureturing-pages/research/wallner-krupke-schmidt-fekete-2026-token-spectral-contraction/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/035d0a87b4a279ebafa8a1e56fc839041c111853e85d8307f18efd672d138722.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-2026-hypercube-disorder-free-localization","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wang, arXiv:2609.07267v1, Conjecture 4.1 asks whether every even $d\\ge4$ has disorder-free localization for the spin-coupled Grover walk with $\\phi_\\sigma=|\\sigma|\\pi/d$ and all spins $+1$. Its equivalent clause asks for a nonzero fixed point with nonuniform position distribution. The source identity and locators are in `Library/QuantumChannels/wang2026hypercubelocalization.md`.","title":"Disorder-free localization on even hypercubes","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-2026-hypercube-disorder-free-localization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f60d08b55dbce7a8c22fed1ac82cb45bb111f594f4e21f693bac1cc4a7db5310.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-2026-hypercube-rendezvous-quantum-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Wang, *Quantum-assisted rendezvous on hypercubes: a quantum Bernoulli noise approach*, arXiv:2610.12949v1 (9 October 2026), studies one-step rendezvous on the hypercube {0,1}^d.\n- Inputs: Alice and Bob receive vertices σ, τ with |σ △ τ| = 2t, uniformly.\n- Outputs: each moves t coordinates, Alice A and Bob B, both t-subsets.\n- Winning: they meet, σ △ A = τ △ B, that is A ∩ B = ∅ and A ∪ B = σ △ τ.\n\nLemma 5.1 gives the no-signalling value ω_NS(d,t) = C(2t,t)/C(d,t). Theorems 5.5 and 6.4 show that for t ≥ 2 and d > 2t no finite-dimensional quantum strategy attains it. §7 asks: \"It remains open whether ω_Q(d,t) is then strictly below ω_NS(d,t)\", where ω_Q is the supremum over finite-dimensional quantum strategies (Eq. (2.8)). The verbatim statements are in [the literature note](../Library/QuantumBounds/wang2026rendezvous.md).\n\nIssue [#15814](https://github.com/the-omega-institute/trureturing/issues/15814) fixes the reading:\n- arbitrary finite-dimensional systems, density matrices and POVMs;\n- no-signalling correlations with input-independent marginals;\n- both values as suprema;\n- the claim ω_Q(d,t) < ω_NS(d,t) for all t ≥ 2 and d > 2t.","title":"The quantum value of balanced hypercube rendezvous is strictly below its no-signalling value","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-2026-hypercube-rendezvous-quantum-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d39b4fce04326855703f012b8d1f0e976a097f37854950d64a987a05029cb643.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-chen-song-2026-appt-extreme-six-levels","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"N. Wang, L. Chen and Z. Song, *Extreme points of absolutely PPT states with exactly three distinct eigenvalues*, arXiv:2603.20717v1, Physica Scripta 101 (2026), 315210, DOI 10.1088/1402-4896/ae9148, Section VII, Conclusions, page 29, asks:\n\n> Fourth, what is the achievable upper bound of distinct eigenvalues of extreme two-qutrit AP states?\n\nThe answer is **six**, including attainment.\n\n### Statement echo\n\nA state is a positive semidefinite trace-one matrix on $\\mathbb C^3\\otimes\\mathbb C^3$. The public definition `apStates` is exactly the set of such matrices satisfying the frozen `QutritPerturbationAttainment.APPT`: every global unitary conjugate has positive semidefinite partial transpose. Extremality is `Set.extremePoints ℝ apStates`; the number of distinct eigenvalues is `(spectrum ℂ rho).ncard`, without multiplicity. `claim` is the conjunction of the universal bound $\\le 6$ and existence of an extreme state with exactly six eigenvalues. `result : claim` has no extra hypotheses. The real scalar field is the field of convex coefficients; the complex spectrum belongs to the original complex matrix, rather than a real surrogate. Finite-dimensional Hermitian spectra are finite.","title":"The sharp six-level bound for extreme two-qutrit absolutely PPT states","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-chen-song-2026-appt-extreme-six-levels/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d917fe6c59f96b00fb0249538a1ee43f90bfbc4beb9b78c8682b14a463dd362c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-conjecture326","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Guoqing Wang, *Semigroup rings and algebraically independent sequences with\nrespect to idempotents in commutative semigroups*, arXiv:2509.18724v1,\nConjecture 3.26, states:\n\n> Let S be a commutative periodic semigroup, and let R be a commutative\n> unitary ring. If d(S,R) < infinity then\n> sup {D_bar(a)(S/H): a in S and is not a group} < infinity.\n\nConjecture 5.3 is a separate finite-semigroup assertion and is not a second\nsettlement of this question.\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| every commutative semigroup, including infinite and nonunital ones | `S : Type`, `[AddCommSemigroup S]` |\n| periodic | `Periodic S`: each native `AddSubsemigroup.closure {a}` is finite |\n| positive cyclic generation | `cyclic a`, without adjoining a zeroth multiple |\n| actual cyclic idempotent | `IsCyclicIdempotent` and `cyclicIdempotent`; pointwise existence uses finite cyclic closure |\n| every commutative unitary ring | `R : Type`, `[CommRing R]`; no field hypothesis |\n| ordinary semigroup algebra | upstream `AddMonoidAlgebra R S`, retaining all labels |\n| all independently nonzero positional coefficients | `GoodSequence`: every `c : N -> R`, with `c i != 0` for each occupied position |\n| finite algebraic constant | `algebraicD S R < top`, supremum of finite good lengths |\n| actual principal ideals | `principalIdeal a = {a} union {a+c : c in S}` |\n| Green quotient | native quotient of `greenCon S`, whose relation is equality of those ideals |\n| cyclic nongroup, with its own possible identity | negation of `CyclicIsGroup a`, quantifying identity and inverses inside `cyclic a` |\n| relative constant at the canonical image | `relativeD ((greenCon S).toQuotient a)` |\n| relative irreducibility | no shorter sublist, including empty, has the same sum |\n| one uniform bound over all filtered elements | `nongroupSup S < top` |\n| whole implication and exact negation | closed `claim : Prop`; designated `result : Not claim` |\n\nAll supremums are in ENat. Infinity means unbounded finite lengths; no\ninfinite product or finite truncation substitutes for the source assertion.\nNonempty products require no algebra unit. Length zero contributes only\nzero to the length supremum. The empty sequence sums only to an existing\ntwo-sided identity. In a monoid this is exactly the usual list sum, so the\nempty proper subsequence is included in the counterexample's irreducibility.\nLists enumerate the source's unordered sequences and their submultisets;\ncommutativity makes enumeration irrelevant.","title":"Wang Conjecture 3.26: finite algebraic constant does not bound the relative constants","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-conjecture326/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f608540a0dcd81d30b61ea1df068bf4e0659aaa7e5852e9ebde79154202d660.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-conjecture53","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wang, arXiv:2509.18724v1, Conjecture 5.3, asks whether every finite\ncommutative semigroup S and field K satisfying either the splitting\ncondition or the characteristic-p / p-power-exponent condition obey\n\n$$d(S,K)=\\max_{a\\in S}\\{\\epsilon_a D_{\\bar a}(S/\\mathcal H)\n+d(\\Gamma(H_a),K)\\}.$$\n\nHere epsilon is zero exactly when the positive cyclic subsemigroup is a\ngroup with its own identity. The algebra is the ordinary semigroup algebra,\nthe scalar quantifier is universal over individually nonzero scalars,\nD is the relative irreducible-length constant at the canonical quotient\nimage, and Gamma consists of the actual stabilizer translation maps.\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| every finite commutative semigroup | `S : Type`, `AddCommSemigroup S`, `Finite S` in `claim` |\n| arbitrary field and both alternatives | `Field K`, `Admissible S K` |\n| ordinary basis and convolution | upstream `AddMonoidAlgebra K S` and its `single` |\n| positive cyclic semigroup and idempotent | `positiveMultiple`, `cyclic`, `IsCyclicIdempotent`, `cyclicIdempotent` |\n| period, exponent, splitting | `period`, `semigroupExponent`, `Splitting` |\n| group property in epsilon | `CyclicIsGroup`, `epsilon` |\n| every independently nonzero coefficient | `GoodSequence` quantifies all nonzero scalar functions of position |\n| supremum, including unboundedness | `algebraicD : ENat`, supremum over every finite list |\n| actual principal ideals and congruence quotient | `Below`, `Green`, `greenCon` and its upstream canonical quotient |\n| relative Davenport constant and empty subsequences | `SequenceSum`, `IrreducibleAt`, `relativeD` |\n| actual stabilizer translations, equality of maps | `HClass`, `Gamma`, composition addition |\n| maximum of the exact summands | `rightSide`, an ENat supremum over the finite carrier |\n| whole universal equality and its negation | `claim : Prop`, `result : Not claim` |\n\nThe source's infinite-length notation is interpreted as unbounded admissible\nfinite lengths, without inventing an infinite product. For a general\nnonunital semigroup algebra only positive products are used; a bottom length\nzero does not affect the supremum. The empty sequence sums to an identity\nonly if the semigroup already has one. The model lies in a monoid over Q,\nwhere the usual nonzero empty product is available. No zero-ring claim is\nneeded for the field-valued conjecture.","title":"Wang's Conjecture 5.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-conjecture53/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3eced8bc0b354240b1fd77e7cb2bfbcda08445138a6406f2e484afc08c3783a0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-tian-2026-strong-saturation-exact-formula","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Qinglin Wang and Yingzhi Tian, *Extremal Problems for the Family of k-Strongly\nConnected Digraphs*, arXiv:2605.01269v1, Section 4, Conjecture 1, printed PDF\npage 13, state: “For sufficiently large $n$,\n$\\operatorname{ex}(n,\\mathcal D_k)=\\frac32(k-\\frac43)(n-k+1)+\\binom n2$.”\nThe source is [Conjecture 1](https://arxiv.org/html/2605.01269v1#Thmconj1).\n\n### Statement echo\n\nThe closed proposition `claim` quantifies every $k\\in\\mathbb N$ with\n$1\\le k$, then a threshold $N\\in\\mathbb N$ depending on $k$, then every\n$n\\in\\mathbb N$ with $N\\le n$. Its equality is\n\n$$\n(\\operatorname{ex}(n,\\mathcal D_k):\\mathbb Q)\n=\\frac32\\left((k:\\mathbb Q)-\\frac43\\right)\n\\left((n:\\mathbb Q)-(k:\\mathbb Q)+1\\right)\n+(\\operatorname{Nat.choose}(n,2):\\mathbb Q).\n$$\n\nEvery division and subtraction is rational. There is no rounding, parity,\ndivisibility restriction or error term. The local Section 4 reading $k\\ge2$\nis also contradicted by $k=3$.\n\nStrict digraphs have ordered distinct-endpoint arcs, permit reciprocal arcs,\nand have no parallel arcs. A subdigraph chooses arbitrary vertex and arc\nsubsets, retaining isolated vertices. Dipaths are vertex-simple directed\npaths. A separator leaves a nonstrong or singleton remainder. For nonempty\nrecords the connectivity is the attained minimum separator size. Positive\nstrong connectivity excludes empty records; the source leaves null\nconnectivity unspecified. This explicit totalization retains the source's\nLemma 2.2 consequence that a $k$-strong graph has at least $k+1$ vertices.\nSaturation means forbidden-free and that every missing ordered arc creates\na $k$-strong arbitrary subdigraph. The extremum is the actual attained\nmaximum arc count over a proved nonempty finite family of saturated hosts.\nThe Blueprint explains the invertible relabelling to `Fin n`; it does not\nclaim a separate compiled transport theorem.","title":"The unrestricted exact Wang–Tian saturation formula is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-tian-2026-strong-saturation-exact-formula/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd184d5ed416065b9be1d755ba8901ba03961951aa1370fb916d4a5267ee6d04.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-wang-tang-ren-2018-bernoulli-noise-walk-gaussian-limit","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Wang, C. Wang, Y. Tang and S. Ren, *Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise*, Advances in Mathematical Physics 2018, Article 2507265, DOI 10.1155/2018/2507265, Section 6, “Conclusion Remarks,” published PDF p. 8:\n\n> It is still unclear whether or not the walk has a limit probability distribution for a general localized initial state.\n\n> Thus we conjecture that, for a general localized initial state, the QBN-based walk might still have the same limit probability distribution as the classical random walk.\n\nThe source's coin space $h$ has orthonormal basis $\\{Z_\\sigma:\\sigma\\in\\Gamma\\}$, where $\\Gamma$ consists of the finite subsets of $\\mathbb N$ (Section 2, Eq. (7), p. 2). Its annihilation and creation operators remove or add a coordinate when permitted, and otherwise give zero (Lemmas 1–2, Eqs. (8)–(9)). Lemma 5 and Definition 6, p. 3, specify\n\n$$\nR_k=\\tfrac12(\\partial_k^*+\\partial_k+I),\\qquad\nL_k=\\tfrac12(\\partial_k^*+\\partial_k-I),\\qquad\n\\Phi_{n+1}(x)=R_n\\Phi_n(x-1)+L_n\\Phi_n(x+1).\n$$\n\nA localized initial state is $\\Phi_0(0)=\\varphi$, $\\Phi_0(x)=0$ for $x\\ne0$, with $\\|\\varphi\\|=1$ (Eq. (53), p. 6); the position probabilities are $\\Pr(X_n=x)=\\|\\Phi_n(x)\\|^2$ (Section 3, p. 3).\n\nIssue [#22700](https://github.com/the-omega-institute/trureturing/issues/22700) fixes the reading: for every unit vector $\\varphi\\in h$, the law of $X_n/\\sqrt n$ converges weakly to $N(0,1)$. This is proved in the coefficient-space model below. The source definitions and their locators are also described in [the literature note](../Library/Asymptotics/wangwangtangren2018bernoulliwalk.md).","title":"Every localized quantum Bernoulli noise walk has the standard Gaussian limit","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-wang-tang-ren-2018-bernoulli-noise-walk-gaussian-limit/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6569764e73bff3e7ffaf15a619bf56da11b7050402fe8ad7a2080b5e43cc1eb4.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wang-zhou-chen-fei-2026-conical-design-concurrence-comparability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H.-F. Wang, W. Zhou, L. Chen and S.-M. Fei, *Estimating the concurrence\nfor quantum states via symmetric measurements*, arXiv:2606.31010v2,\nSection IV, after Theorem 2:\n\n> The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.\n\nA finite positive semidefinite family satisfies\n$\\sum_i E_i\\otimes E_i=\\alpha_E I+\\beta_E F$, where\n$\\alpha_E\\geq\\beta_E>0$ and $F$ is the tensor swap. For a positive\nsemidefinite trace-one bipartite state $\\rho$, put\n$P_E(\\rho)_{ij}=\\operatorname{tr}[\\rho(E_i\\otimes E_j)]$ and\n$B_E(\\rho)=c_d(\\|P_E(\\rho)\\|_1-\\alpha_E-\\beta_E)/\\beta_E$,\nwhere $c_d=\\sqrt{2/[d(d-1)]}$ and $d\\geq2$.\nThe bound is the untruncated two-constant expression in Siudzińska,\nJ. Phys. A 58 (2025) 375302, Section 7, Theorem 6, Eqs. (73)–(74),\nwith $S=\\beta_E$ and $\\mathcal C_{\\max}=\\alpha_E+\\beta_E$.","title":"Comparability of conical-design concurrence bounds","url":"https://the-omega-institute.github.io/trureturing-pages/research/wang-zhou-chen-fei-2026-conical-design-concurrence-comparability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c51db2e77ea4cef3e7bd5cab0ab7d13bf0974b5053538a2359ce2f3e681b2721.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wei-guo-2014-exponential-derivative-ratio-complete-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Section 5 of the source identified by\n`D5/L/Asymptotics/weiguo2014completemonotonicity` states:\n\n> Conjecture 12. For given i ∈ {0} ∪ ℕ, the functions ℱ_i(t) and 𝒢_i(t) defined in (10) or (27) are completely monotonic on (0, ∞).\n\nEquation (6) defines\n\n$$F_i(t)=(-1)^i\\left(\\frac1{e^t-1}\\right)^{(i)},\\qquad\nG_i(t)=(-1)^i\\left(\\frac1{1-e^{-t}}\\right)^{(i)},$$\n\nand equation (10) defines\n\n$$\\mathcal F_i(t)=\\frac{F_{i+1}(t)}{F_i(t)},\\qquad\n\\mathcal G_i(t)=\\frac{G_{i+1}(t)}{G_i(t)}.$$\n\nThe source verifies $i=0,1$ in (30). Complete monotonicity means\nsmoothness on $(0,\\infty)$ and $(-1)^kh^{(k)}(t)\\ge0$ for every\nnonnegative integer $k$ and every $t>0$.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.ExpRatioCM`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $F_i(t)$ in (6) | `Defs.F` | Exactly the globally defined signed iterated derivative of $1/(e^t-1)$. |\n| $G_i(t)$ in (6) | `Defs.G` | Exactly the globally defined signed iterated derivative of $1/(1-e^{-t})$. |\n| $\\mathcal F_i$ in (10) | `Defs.ratioF` | Exactly $F_{i+1}/F_i$. The denominator is positive for $t>0$. |\n| $\\mathcal G_i$ in (10) | `Defs.ratioG` | Exactly $G_{i+1}/G_i$. The denominator is positive for $t>0$. |\n| Complete monotonicity | `Defs.CompletelyMonotoneOn` | `ContDiffOn` with smoothness index `⊤` on `Set.Ioi 0`, and the sign condition for every natural derivative order at every positive argument. |\n| Conjecture 12 | `Defs.claim` | Both complete-monotonicity assertions for every natural index, including zero. |\n| Exact index sets | `Positive.cm_of_le_two`, `Result.failure_on_every_tail` | The positive theorem covers $i\\le2$, while the failure theorem excludes every $i\\ge3$. |\n| Failure on every right half-line | `Result.failure_on_every_tail` | For every $i\\ge3$ and real $T$, there are a natural $k$ and $t>\\max\\{T,0\\}$ at which both alternating derivatives are negative. |\n| Universal refutation | `Result.result` | Exactly $\\neg\\,\\texttt{Defs.claim}$. |\n\nThe globally defined derivatives agree with derivatives of the smooth\nrestrictions on the open positive half-line by neighborhood equality.\nNo derivative value at zero is used.","title":"Refutation and exact classification of exponential derivative ratios","url":"https://the-omega-institute.github.io/trureturing-pages/research/wei-guo-2014-exponential-derivative-ratio-complete-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/840a3e8c75eead2f347f05e0635f28cfb1ffb9720edb2f2a4c2b6753a5673bb2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wei-liu-2024-ccz-magic-capacity-threshold","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Fuchuan Wei and Zi-Wen Liu, *Noise robustness and threshold of many-body quantum magic*, arXiv:2410.21215v1, Section VII:\n\n> We conjecture that the magic capacity threshold for $\\mathrm{CC}Z$ under local depolarizing noise is $1/3$.\n\nThe capacity maximizes robustness of magic over pure six-qubit stabilizer inputs, with $\\mathrm{CC}Z$ and independent single-qubit depolarizing noise acting on the three system qubits $A$ and the three reference qubits $B$ untouched. Robustness is faithful: it equals one exactly on the real convex hull $\\mathrm{STAB}_6$ of pure stabilizer density matrices. The single-qubit channel is $\\rho\\mapsto(1-\\lambda)\\rho+\\lambda\\operatorname{Tr}(\\rho)I_2/2$.\n\nThe literal `claim` has two conjuncts: every pure stabilizer output belongs to `STAB 6` for $\\lambda\\in[1/3,1]$, and a pure stabilizer input with output outside `STAB 6` exists for every $\\lambda\\in[0,1/3)$. Noise follows the gate. The result contradicts the first conjunct at $\\lambda=1/2$.","title":"Wei–Liu CCZ magic-capacity threshold","url":"https://the-omega-institute.github.io/trureturing-pages/research/wei-liu-2024-ccz-magic-capacity-threshold/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd04966aee02b3843a1c9678764adafdb45034d917f57542f82c4b09d54398f0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wei-yang-2024-associated-mersenne-cube-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wei and Yang, *Associated Mersenne graphs*, arXiv:2407.08237v1, Section 6,\nQuestion 6.3 asks:\n\n> What is cube polynomial for Associated Mersenne graph $\\mathcal{M}_n$?\n\nThe associated Mersenne graph has the circular run constrained Boolean words as\nvertices, with Hamming distance one as adjacency. Its cube polynomial is\n$C_n(x)=\\sum_k c_n(k)x^k$, where $c_n(k)$ counts induced copies of $Q_k$.","title":"Cube polynomial of the associated Mersenne graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/wei-yang-2024-associated-mersenne-cube-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4cbe37fb5121b78408c95bfb679f0e91ce8bddcfeab1c401cb0ce40e2ae7ee94.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wei-yang-2024-associated-mersenne-degree-series","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wei and Yang, *Associated Mersenne graphs*, arXiv:2407.08237v1,\nQuestion 6.2: “For given n and k, how many vertices of Associated Mersenne\ngraph ℳₙ have degree k?” The vertices are labelled binary words for which\nevery circular run of ones is followed by a strictly longer run of zeros.\nAdjacent vertices differ at one position. The conventions are\n$\\mathbf M_0=\\{\\lambda\\}$ and $\\mathbf M_1=\\{0\\}$.","title":"Degree generating function of Associated Mersenne graphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/wei-yang-2024-associated-mersenne-degree-series/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8321f62a4dfe4f1b79967a397baa5df0db0da4c84e8bd2b0995cc8c8c3076275.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wierichs-kottmann-killoran-2025-critical-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"D. Wierichs, K. Kottmann and N. Killoran, *Unitary synthesis with optimal brick wall circuits*,\narXiv:2511.16736v1, §V (page 9):\n\n> Conjecture 3. The critical values F(C) of a PQC that has full rank almost everywhere in T^d form an at\n> most (d−2)-dimensional subset of G.\n\nA PQC (Appendix F, Definition 1) consists of Clifford gates and $d$ rotations from $\\{R_X,R_Y,R_Z\\}$, each\ninput feeding one rotation without preprocessing, viewed as $F:T^d\\to G\\subset\\mathbb C^{2^n\\times 2^n}$\nwith $d=\\dim G$; $C$ is the set of inputs at which the Jacobian has rank below $d$. The verbatim texts are\nin [the literature note](../Library/QuantumChannels/wierichs2025brickwall.md). Issue\n[#15435](https://github.com/the-omega-institute/trureturing/issues/15435) preregisters the two-qubit,\n$G=SU(4)$, $d=15$ specialization with Hausdorff dimension and the refutation.","title":"Critical values of parametrized quantum circuits","url":"https://the-omega-institute.github.io/trureturing-pages/research/wierichs-kottmann-killoran-2025-critical-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e96043179e4711df0f4d20a105267e8d9cb220a8a45b28a5bd2398ccd60b865.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wilde-2020-coherent-discrimination-triangle-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. M. Wilde, *Coherent Quantum Channel Discrimination* (IEEE ISIT 2020; arXiv:2001.02668), introduces a coherent version of channel discrimination.\n- The verifier controls the two channels by a qubit R₁ in |+⟩.\n- The prover supplies an input ψ_RA and processes the channel output with a unitary V_RBE′F, returning B and a flag qubit F.\n- The verifier uncomputes the controlled isometric extensions and projects R₁F onto a Bell state.\n\nThe success probability p^coh_s(N⁰, N¹) is the supremum of Eq. (17) over the prover's strategies (Eq. (18)), and 1/2 ≤ p^coh_s ≤ 1 (Proposition 2). Section VI asks: \"An intriguing open question is to determine if p^coh_s − 1/2 is a metric on quantum channels.\" The incoherent analogue p^inc_s − 1/2 is a metric: by Eq. (8) it is half the normalized diamond distance (1/2)‖N⁰ − N¹‖_⋄.\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/wilde2020coherentdiscrimination.md).\n\nIssue [#18289](https://github.com/the-omega-institute/trureturing/issues/18289) fixes the reading: the triangle inequality for d = p^coh_s − 1/2 on qubit channels, which a metric would satisfy. The question is answered in the negative.","title":"The coherent discrimination bias is not a metric on quantum channels","url":"https://the-omega-institute.github.io/trureturing-pages/research/wilde-2020-coherent-discrimination-triangle-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9e63cb888b583ea3feaa53efb36f0b3b0ec0d974ffb819b7b6e8b0647e40365.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wiseman-a344611","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Gus Wiseman's OEIS A344611 revision 27, FORMULA, states: “Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...”. Its NAME is “Number of integer partitions of 2*n with reverse-alternating sum >= 0.” Its COMMENTS define “The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.” The OFFSET is “0,2”, including the empty partition. A160786 revision 25 gives “a(n) = A027193(2n+1).” A027193 revision 78 counts partitions into an odd number of parts. The electronic sources have no page numbers, DOI or arXiv identifier.\n\nFor a decreasing positive partition $\\lambda=(\\lambda_1,\\ldots,\\lambda_k)$ of weight $m$, set $R(\\lambda)=\\sum_{i=1}^k(-1)^{k-i}\\lambda_i$, with $R(())=0$. The assertion is\n\n$$\n\\forall n\\in\\mathbb N,\\quad\n\\#\\{\\lambda\\vdash2n:R(\\lambda)\\ge0\\}\n\\le\\#\\{\\mu\\vdash2n+1: \\text{the number of parts of }\\mu\\text{ is odd}\\}.\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive decreasing integer partitions of a specified weight | `Nat.Partition`, with `parts.sort (· ≥ ·)` | Positive multiset parts with the weight constraint; decreasing sorting is inlined. |\n| Reverse signs and the empty sum | `reverseAlternatingSum` | Existing integer recurrence `TrimmedAlternatingPartitions.trimmedSums`, last endpoint with default zero; the update is $z\\mapsto a-z$. |\n| Partitions of $2n$ with $R\\ge0$ | `a344611` | `Nat.card` of the literal nonnegative-endpoint subtype. |\n| Odd number of parts at weight $2n+1$ | `a160786` | `Odd p.parts.card`; no parity restriction on individual parts. |\n| Every $n\\ge0$ and non-strict comparison | `claim`, `result` | The natural-number quantifier includes zero; no added source hypothesis. |","title":"wiseman a344611","url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6925c13e53ff6b6f892bd373a282fa1659b59bea9c5d7bbefa1b55d58deb1818.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wiseman-a387118","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A387118 asks whether the number of integer partitions of $n$ without choosable initial intervals equals the number of non-superdiagonal reversed integer partitions. The conjecture is preregistered in [#18602](https://github.com/the-omega-institute/trureturing/issues/18602).\n\n### Statement echo\n\nThe formal count `a n` ranges over `Nat.Partition n` and filters the descending sort by the negation of `ChoosableInitial`. The comparison count `nonSuperdiagonal n` filters the ascending sort by the negation of `Superdiagonal`. The source predicates and sorting convention are imported from `D5/S1/Words/Compositions/ChoosableInitialIntervals`.","title":"Wiseman A387118 total partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a387118/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7eb8a7b6e89dfae2221673f09c68dd0737a86845543d27d05e53b100571c4c17.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wiseman-a388713","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A388713 asks whether the number of superdiagonal reversed partitions of $n$ with maximum $k$ equals the number of integer partitions of $n$ with maximum $k$ and choosable initial intervals. The conjecture is preregistered in [#18603](https://github.com/the-omega-institute/trureturing/issues/18603).\n\n### Statement echo\n\nThe formal count `T n k` filters `Nat.Partition n` by `p.parts.sup = k` and superdiagonality of the ascending sort. The comparison count `choosable n k` uses the same maximum condition and choosability of the descending sort. The source predicates and sorting convention are imported from `D5/S1/Words/Compositions/ChoosableInitialIntervals`.","title":"Wiseman A388713 maximum-refined partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a388713/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d669c7bac9a3ce3a9bcc0d46e0bc829da07628a24523023d7911a2d91ef8b77d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wiseman-steinberg-hallaji-2023-resolved-count-weak-values","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wiseman, Steinberg and Hallaji, “Obtaining a single-photon weak value from experiments using a strong (many-photon) coherent state”, AVS Quantum Science 5, 024401 (2023), arXiv:2211.15761v2, Section V, page 6:\n\n> This supports earlier approximate results which suggested that if a photon-number resolving detector were used, the weak value post-selected on detecting m photons would exceed the no-click weak value by precisely m times the single-photon number. We leave a rigorous proof of this conjecture for future work.\n\nIssue #17772 fixes the finite-mode Fock conventions. For every $k+1$-mode passive network $A,B$, coherent amplitude $\\alpha\\ne0$, and natural count $m$ of nonzero resolved probability, the conclusion is $W_m-W_0=mw$ when $m>0$. For $m=0$ the difference is zero without requiring a single-photon denominator. The detector acts on mode zero and ignores the remaining modes; the probed mode is also indexed zero, with arbitrary passive evolution between probing and detection.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Arbitrary passive linear network | Every natural $k$ and $A,B:\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin}(k+1))\\simeq_{\\mathbb C}^{\\mathrm{iso}}\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin}(k+1))$ | No fixed dimension, network or intensity |\n| Coherent input in one mode, vacuum in the others | `coherentInput k α`, `EuclideanSpace.single (0 : Fin (k+1)) α` | Literal binding conventions |\n| Weak measurement of number, with other modes ignored | `number`, `numberProjector m`, `generalizedWeakValue` | Unbounded occupation; exact number-domain proofs; projector retains ignored-mode carrier |\n| Detect exactly $m$ photons | Every natural $m$ with `coherentProbability A B α m ≠ 0` | Conditional values retain nonzero-denominator certificates |\n| Compare with no click | Existential nonzero vacuum probability and `coherentWeakValue ... 0` | Vacuum condition is proved, not an added hypothesis |\n| Precisely $m$ times the single-photon value | `claim`, proved by `result` | Same $A,B$ and probed mode; the positive-count branch proves nonzero single-photon probability |\n| Zero count | The explicit `if m = 0` branch of `claim` | Difference zero; no single-photon value assigned when its probability vanishes |\n\nAll public model definitions and the claim retain the mathematical bodies fixed by #17772; its mode-amplitude carrier is written as $\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin} k)$ without a forwarding declaration. `BeamSplitter.exponentialCoeff` and `BeamSplitter.exponentialVector` supply the one-mode coefficients and vectors. The arbitrary finite-mode completion and totalness are supplied by `PassiveFockModel`.","title":"Resolved-count coherent-state weak values","url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-steinberg-hallaji-2023-resolved-count-weak-values/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be740e0ed89ab9c8d295fb94ebe1fe9ffc33ad091584733c6048d4eb0c7af733.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wiseman388-partitions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A388712 states: “Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts without choosable initial intervals.” Its conjecture is: “Also the number of non-superdiagonal reversed partitions of n into k parts, where a sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i = 1..k.” The question is preregistered in [#18601](https://github.com/the-omega-institute/trureturing/issues/18601).\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Partitions of $n$ into $k$ positive parts | `Nat.Partition n`, filtered by `p.parts.card = k` |\n| Initial intervals admit distinct positive representatives | The imported `ChoosableInitial` predicate on descending parts |\n| Without choosable initial intervals | Negation in `WisemanA388712.T` |\n| Reversed partition, weakly increasing | `p.parts.sort (· ≤ ·)` |\n| $y_i\\ge i$ with one-based indices | The imported `Superdiagonal` predicate uses zero-based index plus one |\n| Non-superdiagonal, with the same number of parts | Negation and the same cardinality filter in `WisemanA388712.nonSuperdiagonal` |\n| Equality for every $n,k$ | `WisemanA388712.claim`, proved by `WisemanA388712.result` |\n\nAll declaration names above have namespace prefix `D5.S1.Words.Compositions.`. The source expressions are retained in the two count definitions. The existing predicates are reused without new aliases. Classical finite cardinalities do not restrict the quantified natural numbers.","title":"Wiseman's nonchoosable partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f725d761dab823606f4ad686f3c4fb1b526a7d1224ce6a9269d71bd704119ccc.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wisniewska-2025-gurland-remainder","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Halina Wiśniewska, *Some inequalities for Gurland's ratio of the gamma\nfunctions*, arXiv:2512.07028v1, section 2, Open Problem 1:\n\n> In Corollary 1 we showed that R_n^m(x,y) tends to zero under a sufficient\n> condition. Does R_n^m(x,y) tend to zero hold for every x,y > 0?\n\nFor positive real x,y put a=(x+y)/2 and d=(x-y)/2. In equations (2)–(4),\n\n$$\nG_* = \\frac{\\Gamma(1+x)\\Gamma(1+y)}{\\Gamma(1+a)^2},\\qquad\nH(s,b)=\\sum_{j=0}^{\\infty}\\frac1{(j+b)^s},\n$$\n$$\nS_m=\\sum_{k=1}^{m-1}\\frac{d^{2k}H(2k,1+a)}{k},\\qquad\nR_m=\\log G_*-S_m.\n$$\n\nThe expansion order m tends to infinity through integers, with m >= 2\nin the source. Its n is already summed out. Every positive real pair,\nincluding x=y and positive integer Gamma arguments, is in scope.","title":"Gurland's original logarithmic remainder","url":"https://the-omega-institute.github.io/trureturing-pages/research/wisniewska-2025-gurland-remainder/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa7927900a75bbce2c87065e7c0edc66c5b48397e78fef4d2ad4e878dd496fed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:witula-slota-2007-quasi-fibonacci-order-eleven-degree","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wituła and Słota define the five polynomial sequences by system (3.12) on\nprinted page 5:\n\n> A_{n+1}(δ) = A_n(δ) + 2δB_n(δ) - δE_n(δ),\n>\n> B_{n+1}(δ) = δA_n(δ) + B_n(δ) + δC_n(δ) - δE_n(δ),\n>\n> C_{n+1}(δ) = δB_n(δ) + C_n(δ) + δD_n(δ) - δE_n(δ),\n>\n> D_{n+1}(δ) = δC_n(δ) + D_n(δ),\n>\n> E_{n+1}(δ) = δD_n(δ) + (1 - δ)E_n(δ),\n\nwith the initial values printed immediately below it:\n\n> A_0(δ) = 1, B_0(δ) = C_0(δ) = D_0(δ) = E_0(δ) = 0.\n\nThe sentence on printed page 19 is:\n\n> Problem. Is it true that deg A_n(∆) = deg B_n(∆) = deg C_n(∆) = deg D_n(∆) = deg E_n(∆) = n for every n = 5, 6, . . .?\n\nSection 6 introduces `∆ := 1/δ` and writes the same polynomial family as\n`A_n(∆),...,E_n(∆)`. The formal module names the indeterminate `X`. It reads\n`degree` in `WithBot ℕ`, rather than `natDegree`, takes the printed threshold\nliterally as `5 ≤ n`, and defines the recurrence over the coefficient ring\n`ℤ`.","title":"Degrees of the order-eleven quasi-Fibonacci polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/research/witula-slota-2007-quasi-fibonacci-order-eleven-degree/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a8382a1c23e835158013538f9e3d56e73cd161fe3fd2b2527cc0cfdef7310e2.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wolk-guhne-2016-balanced-cut-gradient-bound-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Wölk and O. Gühne, *Characterizing the width of entanglement*, arXiv:1507.07226v2, New J. Phys. 18 (2016) 123024, Appendix D. The conjecture states that every state of N spins with entanglement only within the subgroups {1, …, N/2} and {N/2+1, …, N} satisfies\n\n> $\\langle B^2 \\rangle \\leq \\langle B^2\\rangle_- + \\frac{\\langle B\\rangle_\\textrm {cl}-\\langle B^2\\rangle_-}{2N}\\,\\langle J^2\\rangle$\n\nwith ⟨B²⟩₋ = 6N. The gradient operator and the reference values are quoted in [the literature note](../Library/QuantumContext/wolk2016entanglementwidth.md). The printed numerator has ⟨B⟩_cl; the preceding sentence and the endpoint values fix it as ⟨B²⟩_cl.\n\nIssue [#19442](https://github.com/the-omega-institute/trureturing/issues/19442) fixes the reading:\n- for every even N = 2m ≥ 4 and every density matrix that is separable across the two halves;\n- the coefficient of ⟨J²⟩ is (⟨B²⟩_cl − 6N)/(2N).\n\nThe conjecture is refuted.","title":"The balanced-cut gradient bound of Wölk and Gühne fails for eight qubits","url":"https://the-omega-institute.github.io/trureturing-pages/research/wolk-guhne-2016-balanced-cut-gradient-bound-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f784fc7ab37b77403c6767443c056f8f8fb7ffc59265d1f488b8efa2758c5a8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wong-bravyi-gosset-liu-2026-conjecture-2-phase-shifted-extension","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wong, Bravyi, Gosset and Liu, *Lee-Yang tensors and Hamiltonian complexity*, arXiv:2602.03605v1, Section 5.5, Conjecture 2:\n\n> Suppose n ≥ 2, s ∈ [0,1), and the interaction graph is connected. Then the second smallest eigenvalue of H_s coincides with the smallest eigenvalue of H_s restricted to the odd-weight subspace.\n\nSection 5.6 asks:\n\n> We leave as an open question whether Conjectures 1, 2, 3 hold for phase-shifted EPR-like Hamiltonians.\n\nThe question settled here is the extension of Conjecture 2. Preregistration: [#18759](https://github.com/the-omega-institute/trureturing/issues/18759). This is an external published question, tier 1.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| $n\\ge2$ | `claim` quantifies over natural `n` with `2 ≤ n`. |\n| Connected interaction graph | `SimpleGraph (Fin n)`, its adjacency decision procedure, and `G.Connected`. The decision procedure is computational infrastructure. |\n| $0\\le s<1$ | The two inequalities in `claim`. |\n| $h(z)=\\lvert00\\rangle\\langle00\\rvert+z\\lvert11\\rangle\\langle00\\rvert+\\overline z\\lvert00\\rangle\\langle11\\rvert+\\lvert z\\rvert^2\\lvert11\\rangle\\langle11\\rvert$ | `hpq` has exactly these four entries, with the identity on all other bits. |\n| $H_s(\\theta)=-\\sum_{\\{p,q\\}\\in E}h_{p,q}(se^{i\\theta_{p,q}})$ | `H` counts each undirected edge once using `p < q`. |\n| Arbitrary edge angles in $[0,2\\pi)$ | Real representatives in the exponential; the counterexample uses only $0$ and $\\pi$, both in the source interval. |\n| Second smallest eigenvalue | `eigAsc`, sorted with multiplicity, and `List.getD` at index $1$. |\n| Smallest eigenvalue on the odd-weight subspace | Restriction by `Matrix.submatrix` to the odd computational basis and index $0$. The odd weight count is inlined literally in `claim`; the private certificate relates it to Mathlib's `hammingNorm`. |\n| Negative settlement | `result : ¬ claim`. |\n\nThe arbitrary default in `List.getD` and the empty non-Hermitian branch of `eigAsc` totalize the definitions. The witness matrices are Hermitian and the relevant lists have the required entries. The computational basis is `Fin n → Fin 2` directly.","title":"The phase-shifted extension of Conjecture 2","url":"https://the-omega-institute.github.io/trureturing-pages/research/wong-bravyi-gosset-liu-2026-conjecture-2-phase-shifted-extension/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3bbe7e727271304780a7e023434080a25dd4d903476fc89edddd191a3358a695.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu-2014-randomized-graph-negativity-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"J.-Y. Wu, M. Rossi, H. Kampermann, S. Severini, L. C. Kwek, C. Macchiavello,\nD. Bruß, *Randomized Graph States and their Entanglement Properties*, Phys.\nRev. A 89, 052335 (2014), arXiv:1403.3828v3, Section V:\n\n> However, even though this conjecture is supported by numerical evidence,\n> it is an open question whether the monotonic behavior of the negativity in\n> terms of the randomness $p$ is a common feature to all RG states.\n\nThe randomized graph state of a graph $G$ is\n$\\rho^p_G=\\sum_{F\\subseteq E(G)}p^{|F|}(1-p)^{|E(G)\\setminus F|}\n|F\\rangle\\langle F|$, where $|F\\rangle=\\prod_{ab\\in F}\\mathrm{CZ}_{ab}\n|+\\rangle^{\\otimes n}$, and the negativity across a bipartition $A|B$ is\n$N(\\rho)=(\\|\\rho^{\\Gamma_A}\\|_1-1)/2$, evaluated \"with respect to all\npossible bipartitions of the qubits\". The conjecture is that $N$ increases\nmonotonically in $p$; the paper reports it numerically for the complete\ngraphs $K_n$ and the star graphs $S_n$ with $n\\le4$.","title":"Monotonicity of the negativity of randomized graph states","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu-2014-randomized-graph-negativity-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb4dd79edd33951e9b63b19f8c2fe0c3ca6e7253caea5a0f5a67c95539f327e3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu-lou-permutation-invariant-niven-digit-sum-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wu and Lou, *Permutation-Invariant Niven Numbers*, arXiv:2508.01611v3,\nsection 8.4, state that the relevant nontrivial distinct-digit permutation-invariant\nNiven numbers have digit sums between 3 and 81 and conjecture that no exceptions\nexist beyond the cases excluded there.","title":"Wu-Lou permutation-invariant Niven digit-sum bound","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu-lou-permutation-invariant-niven-digit-sum-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb265fbfdeee1267890a769fd806d325bf3ecc1eba25e28c639f9c2ec8d412d9.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu-pyramidal-complement-conjecture-one","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chai Wah Wu, *Algorithms for Complementary Sequences*, INTEGERS 25 (2025),\nA95, printed page 12, Conjecture 1, states:\n\n> For k >= 9, (6) holds for all n >= 1.\n\nFor positive `m`, let\n`P_k(m)=m(m+1)(m(k-2)-(k-5))/6`, and let the complementary sequence enumerate\nthe positive integers outside `{P_k(m) | m>=1}` in increasing order. Equation\n(6), on printed page 11, sets `h=floor((6n/(k-2))^(1/3))` and gives `n+h+1`\nwhen `6n` is at least\n`(k-2)h^3+3(k-1)h^2+(2k-1)h+6`; if that test fails, it gives `n+h-1` when\n`6n` is at most `h(h-1)(h(k-2)+k+1)`, and gives `n+h` otherwise.","title":"Wu's k-gonal-pyramidal complement formula","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu-pyramidal-complement-conjecture-one/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c299978e83cc4658e82032d6bde7cbd5bc77a6cd7d53416f0d59bea65e65164e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu-wu-2025-unified-imaginarity-monotone-comparison","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chuanfa Wu and Zhaoqi Wu, *Two imaginarity monotones induced by unified $(\\alpha,\\beta)$-relative entropy* (Commun. Theor. Phys. 77, 095101 (2025), DOI 10.1088/1572-9494/adbdbd; arXiv:2506.09799v1), Section 4, Example 1, state:\n\n> We have not found a proof of $M_{\\alpha,\\beta}^H(\\rho)\\geq M_{\\alpha,\\beta}^E(\\rho)$ for any $\\alpha\\in (0,1)$, $\\beta\\in (0,1]$ and any qubit state $\\rho$, however, numerical experiments shows that it might be true, and it seems difficult to find a counterexample. It is also conjectured that $M_{\\alpha,\\beta}^H(\\rho)\\geq M_{\\alpha,\\beta}^E(\\rho)$ for any $\\alpha\\in (0,1)$, $\\beta\\in (0,1]$ and any quantum state $\\rho$.\n\nThis quotation is from lines 573–579 of the downloaded v1 TeX, `20250301WuCF-imaginarity_monotone_R2.tex` (md5 `eb104c03d3638b12044677758c200bae`).\n\nFix the coordinate basis and let $\\bar\\rho$ mean entrywise complex conjugation, denoted $\\rho^*$ in the paper. For density matrices, write\n\n$D_{\\alpha,\\beta}(\\rho\\Vert\\sigma)=\\big[(\\operatorname{Re}\\operatorname{tr}(\\rho^\\alpha\\sigma^{1-\\alpha}))^\\beta-1\\big]/\\big[(\\alpha-1)\\beta\\big]$,\n\n$M^H_{\\alpha,\\beta}(\\rho)=D_{\\alpha,\\beta}(\\rho\\Vert\\bar\\rho)$, and\n\n$M^E_{\\alpha,\\beta}(\\rho)=\\inf_{\\sigma\\in\\mathcal F}D_{\\alpha,\\beta}(\\rho\\Vert\\sigma)$,\n\nwhere $\\mathcal F$ consists of positive semidefinite matrices of trace one with real entries. Matrix powers are spectral powers from continuous functional calculus. These are the module's definitions; the source's Eqs. (7), (13) and (20) use the trace without an explicit real part, and Eq. (20) uses a minimum. The corresponding source definitions and locators are in [the literature note](../Library/Divergence/wuwu2025unifiedimaginarity.md).\n\nIssue [#20971](https://github.com/the-omega-institute/trureturing/issues/20971) fixes the formal reading: for every natural $d\\ge1$, every density matrix $\\rho\\in M_d(\\mathbb C)$, every $0<\\alpha<1$ and every $0<\\beta\\le1$, $M^E_{\\alpha,\\beta}(\\rho)\\le M^H_{\\alpha,\\beta}(\\rho)$. Singular states are included. This infimum comparison is proved.","title":"Comparing the Wu–Wu unified imaginarity monotones in every finite dimension","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu-wu-2025-unified-imaginarity-monotone-comparison/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c1a392053f1ce292c584561078ed13786cd1861898276e2c78d6e210afe6edf.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu-zhong-wu-2026-ghz-measure-biseparable-half","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Wu, K. Zhong and J. Wu, “A measure for genuine tripartite entanglement”,\narXiv:2605.02876v3, Section V.C, equation (37), state:\n\n> Numerically maximising E_GHZ over all biseparable A|BC states we find the sharp value sup_{ρ ∈ A|BC} E_GHZ(ρ) = ½, (37) attained e.g. by |0⟩_A ⊗ |Φ⁺⟩_BC and coinciding with the product-state value (34); the analytic proof of the exact constant 1/2 remains open. The same 1/2 holds for the B|AC and C|AB partitions by symmetry.\n\nIssue [#12719](https://github.com/the-omega-institute/trureturing/issues/12719)\npreregisters the Tier 1 A|BC statement: every product of arbitrary density\nmatrices has E_GHZ at most one half, and some such product attains one half.\nThe specified class is exactly ρ_A tensor ρ_BC, including mixed factors.","title":"The exact A|BC constant of the GHZ measure","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu-zhong-wu-2026-ghz-measure-biseparable-half/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8190bcf8f414a715670be7ce1bbf278a498e6cc24da142c379b3979f2c17937.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:wu221018-fibdiv","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A221018, revision 35, NAME:\n\n> Number k such that Fibonacci(k) is divisible by k, k + 1 and k - 1.\n\nIts first FORMULA states:\n\n> a(n) % 10 == 0, 2, or 8 (conjectured). - _Chai Wah Wu_, Aug 06 2026\n\nThe source generator starts at two. Write $F_k$ for the standard Fibonacci sequence, with $F_0=0$, $F_1=1$ and $F_{k+2}=F_k+F_{k+1}$. For every natural $k\\ge2$, the three assumptions $(k-1)\\mid F_k$, $k\\mid F_k$ and $(k+1)\\mid F_k$ imply $k\\bmod10\\in\\{0,2,8\\}$. No converse is asserted.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Generator starts at 2 | Natural index $k$ and $2\\le k$ | Covers the source integers and makes natural subtraction ordinary subtraction. |\n| Fibonacci(k) | Existing `Nat.fib k` | Standard initial values and recurrence; no replacement sequence. |\n| Divisible by k, k+1 and k-1 | The three separate divisibility hypotheses of `claim` | Their implication chain is equivalent to the source conjunction. |\n| Final digit 0, 2 or 8 | The three remainder alternatives in `claim` and `result` | Exact necessary condition; no added hypotheses or converse. |","title":"Wu’s adjacent-index Fibonacci divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/wu221018-fibdiv/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8fb2ef2029dcd5ba108478de276f5ccf1a614d8201871bbd06bd07200fcfdb09.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xie-younis-mei-eberly-2024-qudit-tetrahedron-genuine-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Xie, D. Younis, Y. Mei and J. H. Eberly, *Multipartite Entanglement: A Journey Through Geometry*, arXiv:2304.03281v2; Entropy 26(3), 217 (2024), define the genuine requirement in `tetrahedron.tex`, line 57:\n\n> The ``genuine'' requirement requires that a GME measure should only, and always, vanish for biseparable states \\cite{ma2011,xie2021}.\n\nTheir Scalability paragraph, line 220, states:\n\n> Nonetheless, our tetrahedron measure presented in this work maintains its scalability. Specifically, we can consistently partition a large system into four groups and explore genuine quadripartite entanglement among these groups. Essentially, the tetrahedron measure is a genuine four-qudit entanglement measure, where each party possesses more than two dimensions. The extension of entropic fill to qudit systems has been made possible by a recent study \\cite{yang2022}. Additionally, we have conducted numerical verifications, demonstrating the existence of unique nonnegative solutions for Eq.~\\eqref{eqn1} and \\eqref{eqn2} in qudit systems up to $d=50$. Consequently, we conjecture that our tetrahedron measure is also applicable to qudit systems.\n\nThese quotations were checked against the complete v2 TeX source, `tetrahedron.tex` (233 lines, md5 `da0b24bf21c330f90979f63e7679749c`), fetched from https://arxiv.org/src/2304.03281v2. The definitions and further quotations are collected in [the literature note](../Library/QuantumContext/xieyounismeieberly2024journey.md).\n\nIssue [#20861](https://github.com/the-omega-institute/trureturing/issues/20861) fixes the following reading. Each of four parties has Hilbert space $\\mathbb C^d$, with $d\\ge2$, and $\\psi$ is a coefficient function with $\\sum_x\\|\\psi(x)\\|^2=1$. Write $S_X$ for the base-two von Neumann entropy of its reduced density matrix on $X$. Biseparability means factorization across some nonempty proper subset of the four parties.\n\nAn admissible fill consists of six nonnegative real coordinates $\\sigma=(\\sigma_{AB},\\sigma_{AC},\\sigma_{AD},\\sigma_{BC},\\sigma_{BD},\\sigma_{CD})$ and a real parameter $\\lambda$. The face equations are $\\sigma_{ij}+\\sigma_{ik}+\\sigma_{il}=S_i$ for each party $i$. Put $p=\\sqrt{\\sigma_{AB}\\sigma_{CD}}$, $q=\\sqrt{\\sigma_{AC}\\sigma_{BD}}$ and $r=\\sqrt{\\sigma_{AD}\\sigma_{BC}}$. The radical equations are $-p+q+r=\\lambda S_{AB}$, $p-q+r=\\lambda S_{AC}$ and $p+q-r=\\lambda S_{AD}$. These are the source equations labelled `eqn1` and `eqn2`, TeX lines 103–121.\n\nThe source volume, TeX lines 92–101, is $V=(\\sqrt2/3)\\sqrt{2\\sum_{ij}\\sigma_{ij}}\\big((p+q+r)(-p+q+r)(p-q+r)(p+q-r)\\big)^{1/4}$. The formal claim is: for every $d\\ge2$ and every unit $\\psi$, if an admissible fill exists and every admissible fill has $V=0$, then $\\psi$ is biseparable. This implication is refuted.","title":"The entropic tetrahedron is not a genuine four-qudit entanglement measure","url":"https://the-omega-institute.github.io/trureturing-pages/research/xie-younis-mei-eberly-2024-qudit-tetrahedron-genuine-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8aac5757132c0ee8b06a9ebdc5f7d1e6761336ed117d5a3184ce39c21312c446.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xu-2015-oblique-discord-mutual-information-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Jianwei Xu, *Oblique discord*, arXiv:1506.00404v1, p. 3, Conjecture (22):\n\n> Conjecture: I(ρ^{AB}) ≥ I(Φ_A ρ^{AB}) for any Φ_A and any ρ^{AB}, where I(ρ^{AB}) is the mutual information, Φ_A is defined in Eq.(13).\n\nThe normalized basis immediately before Eq. (13) consists of unit vectors\nv_i and their biorthogonal duals w_i, with ⟨v_i,w_j⟩ = δ_ij. The operation is\nT_b(ρ) = ∑_i (|v_i⟩⟨w_i| ⊗ 1) ρ (|w_i⟩⟨v_i| ⊗ 1),\nN_b(ρ) = tr_B(∑_i ⟨w_i|ρ|w_i⟩), Φ_b(ρ) = T_b(ρ)/N_b(ρ).\nThe Lean normalizer uses partialTraceRight over B on the sum of rectangular\nbra-vector sandwiches, extracting the scalar on the Unit factor. The unit\nnorms prove equality with Re(tr(T_b(ρ))) inside phi, and a local have in\nresult checks the same equality at the witness. Mutual information is\nS(ρ_A) + S(ρ_B) − S(ρ). The conjecture quantifies over all finite dimensions,\nall normalized bases and their duals, and all density states on the domain\nof the normalized operation. Its Lean domain explicitly requires a\npositive normalizing trace.","title":"Xu's oblique-discord mutual-information conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/xu-2015-oblique-discord-mutual-information-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/307f35587c6be83413184e87f9f5afbde646790ced164d9a4d336346021a5515.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xu-et-al-2023-noisy-ghz-witness-bound-gap","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X.-Y. Xu, Q. Zhou, S. Zhao, S.-M. Hu, L. Li, N.-L. Liu and K. Chen, *A generic framework for genuine multipartite entanglement detection*, arXiv:2211.05563v1; published as *General framework for genuine multipartite entanglement detection*, Phys. Rev. A 107 (2023) 052405, DOI 10.1103/PhysRevA.107.052405. In the arXiv TeX member `GMEwitness.tex`, subsection “Estimating entanglement measures”, line 325, they ask:\n\n> It remains open whether $\\epsilon_o^{n,d}(p)$ equals $E_G(\\rho_{n,d}(p))$.\n\nThe definitions are at lines 284–311:\n\n- $|GHZ_n^d\\rangle=d^{-1/2}\\sum_{i=0}^{d-1}|i\\rangle^{\\otimes n}$ and $\\rho_{n,d}(p)=pI/d^n+(1-p)|GHZ_n^d\\rangle\\langle GHZ_n^d|$.\n- $E_G(\\psi)=1-\\sup_{\\phi\\text{ biseparable}}|\\langle\\phi,\\psi\\rangle|^2$, where both vectors are unit vectors and biseparability means a product across some nonempty proper bipartition; the mixed-state measure is its convex roof.\n- $\\mathcal W=P/d-|GHZ_n^d\\rangle\\langle GHZ_n^d|$, where $P$ projects onto computational words using at most two distinct symbols. This is the orthogonal sum in lines 297–298, with the distinct arrangements specified by the orthogonal set $S$ at lines 690–692.\n- $w_{n,d}(p)=\\operatorname{Tr}(\\rho_{n,d}(p)\\mathcal W)$, $\\hat E_G(r\\mathcal W)=\\sup_{\\|\\psi\\|=1,\\ \\phi\\text{ unit biseparable}}\\{\\langle\\psi,(r\\mathcal W+|\\phi\\rangle\\langle\\phi|)\\psi\\rangle-1\\}$ over unit vectors, and $\\epsilon_o^{n,d}(p)=\\sup_{r\\in\\mathbb R}\\{r w_{n,d}(p)-\\hat E_G(r\\mathcal W)\\}$.\n\nThe source locators are also collected in [the literature note](../Library/Divergence/xuzhouzhaohu2023gme.md).\n\nIssue [#25044](https://github.com/the-omega-institute/trureturing/issues/25044) fixes the universal reading: for every pair of integers $n,d\\ge3$ and every real $p\\in[0,1]$, $\\epsilon_o^{n,d}(p)=E_G(\\rho_{n,d}(p))$. The formal convex roof takes the real infimum over all finite ensembles with nonnegative weights summing to one, unit pure vectors and exact reconstruction of the density matrix. This universal equality is refuted.","title":"The optimized GHZ witness bound need not equal the geometric GME measure","url":"https://the-omega-institute.github.io/trureturing-pages/research/xu-et-al-2023-noisy-ghz-witness-bound-gap/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e24027d06124294ab51d1b93cb0b55d9d1325b48ca5a535c8bb4ee6acb761d3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xu-imai-guhne-2022-classicalization-simultaneous-unboundedness","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhen-Peng Xu, Satoya Imai and Otfried Gühne, *Fate of multiparticle entanglement when one particle becomes classical*, arXiv:2206.12834v2, Phys. Rev. A 107, L040401, Section V.B, page 5:\n\n> Inspired by those two examples, an interesting question arises whether there exist entanglement measures E and states ρ_ABC such that both Δ_E(ρ_ABC) and Δ̃_E(ρ_ABC) − Δ_E(ρ_ABC) can be arbitrarily large in the sense that they are not limited by the size of C, even if C is only a qubit. We leave this question for further research.\n\nThe answer is **Proved** for the fixed tripartite negativity of Equation (4), with no dimension-dependent rescaling. The source and conventions are in [the Library note](../Library/QuantumContext/xuimaiguhne2023classicalization.md); [#19638](https://github.com/the-omega-institute/trureturing/issues/19638) is the preregistration.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Equation (4), sum of three bipartite negativities | `tripartiteNegativity` | Partial transposes on C, A and B; positive negativity $(\\|\\rho^{T_Y}\\|_1-1)/2$. |\n| Equation (1), $\\sum_i p_i\\sigma_i\\otimes\\tau_i$ | `branch`, `classicalize` | Unnormalized branches $p_i\\sigma_i=\\operatorname{tr}_C(\\rho m_i)$, including zero-probability outcomes. |\n| Equation (2), minimum entanglement loss | `delta` | Infimum over every finite POVM and every finite register dimension; attainment is proved for the constructed family. |\n| Equation (9), discard C and prepare its zero flag | `deltaTilde` | The prepared flag is the zero basis state of a qubit. |\n| Both gaps arbitrarily large on the same state | `claim`, `result` | Every real threshold admits one positive semidefinite trace-one state on $\\operatorname{Fin}(a)\\times\\operatorname{Fin}(b)\\times\\operatorname{Fin}(2)$ exceeding it in both gaps. |\n\nThe density and POVM conditions appear literally in the public proposition and the infimum. The proofs use the frozen `IsDensity` and `IsPOVM` predicates with those same defining conditions. The source prints a sum of negative eigenvalues without an absolute value; its examples use the standard positive negativity adopted here.","title":"Two unbounded classicalization gaps with a qubit","url":"https://the-omega-institute.github.io/trureturing-pages/research/xu-imai-guhne-2022-classicalization-simultaneous-unboundedness/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e8826687e1855087e2bc6d54f983e491187469241de10c24d3fa41ab07a11a58.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xu-yang-2026-mills-remainder-cm","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H.-Z. Xu and Z.-H. Yang, *Higher order monotonicity involving the Mills\nratio with applications*, Journal of Inequalities and Applications\n2026:98, DOI 10.1186/s13660-026-03505-9.\n\nMills-ratio definition, printed p. 2:\n\n> The Mills ratio [3] is defined by\n>\n> $$R\\equiv R(x)=\\frac{1-\\Phi(x)}{\\phi(x)}\n> =e^{x^2/2}\\int_x^\\infty e^{-t^2/2}\\,dt,\\quad x\\ge0,$$\n>\n> which satisfies the differential equation of first order:\n>\n> $$R'=xR-1.\\qquad(1.1)$$\n\nHere $\\phi(x)=e^{-x^2/2}/\\sqrt{2\\pi}$ and\n$\\Phi(x)=\\int_{-\\infty}^x\\phi(t)\\,dt$ are the standard Gaussian density and\ncumulative distribution function used in the preceding source sentences.\n\nThe bound family in (1.5), printed p. 2, is\n\n$$\\varphi_{p,q,r}(x)=\\frac{p}{\\sqrt{x^2+q}+rx},\n\\qquad p,q>0,\\quad r\\ge0.$$\n\nEquation (1.8), printed p. 3, with its complete surrounding statement:\n\n> Note that all bounds of the form $\\varphi_{p,q,r}(x)$ can be derived from the inequality\n>\n> $$Y_{a,b}(x):=(x^2+a)R^2(x)+bxR(x)<(>)c\\qquad(1.8)$$\n>\n> for all $x>0$, $a,b\\in\\mathbb R$, where\n>\n> $$c=\\lim_{x\\to0}Y_{a,b}(x)=\\frac{a\\pi}{2}\n> \\quad\\text{or}\\quad c=\\lim_{x\\to\\infty}Y_{a,b}(x)=b+1.$$\n\nEquation (3.7), printed p. 6, gives the coefficients following the asymptotic\nexpansion in Theorem 3.1:\n\n> where\n>\n> $$u_n\\equiv u_n(a,b)=(b+1)\\frac{2^n\\Gamma(n+1/2)}{\\Gamma(1/2)}\n> +(n-a)(n-1)!\\sum_{k=0}^{n-1}\n> \\frac{2^k\\Gamma(k+1/2)}{\\Gamma(1/2)k!}.\\qquad(3.7)$$\n>\n> In particular, $u_1=b+2-a$, $u_2=3b+7-2a$, $u_3=15b+36-7a$, and $u_4=105b-36a+249$.\n\nConjecture 7.1, printed p. 27, complete statement:\n\n> Moreover, inspired by Theorem 4.2, we propose the following conjecture.\n>\n> Conjecture 7.1 Let\n>\n> $$D_n(x,a)=Y_{a,a-2}(x)-\\left(a-1+\\sum_{k=1}^{n}\n> \\frac{(-1)^ku_k}{x^{2k}}\\right),$$\n>\n> where $u_k$ is defined by (3.7). Then $x\\mapsto(-1)^nD_n(x,-1)$ is CM on $(0,\\infty)$.\n\n\n### Formal reading\n\nThe namespace is `D5.S3.Analytic.XuYangMillsRemainder.Result`.\nThe definitions `millsRatio`, `Y`, `u` and `D` retain the displayed\nGaussian integral, polynomial expression, Gamma coefficients and finite\nremainder. Here $\\Gamma$ is Euler's Gamma function, $k!$ is the factorial of the\nnonnegative integer $k$, and all integrals use Lebesgue measure.\n`CompletelyMonotoneOn f` means $f$ is $C^\\infty$ on $(0,\\infty)$\nand every signed derivative $(-1)^m f^{(m)}(x)$ is nonnegative there.\n`claim` quantifies over every integer $n\\ge2$; `result` states `claim`.\nThe source does not display a lower bound for $n$ in Conjecture 7.1.\nThe restriction $n\\ge2$ is necessary: $n=1$ fails, as explained in Triage.","title":"Complete monotonicity of the Xu–Yang Mills remainders","url":"https://the-omega-institute.github.io/trureturing-pages/research/xu-yang-2026-mills-remainder-cm/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a67b4576728028cba3b55240e5262b7e206dec00e43d44e6e6c36f945e7a645f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:xu-zhao-trivariate-cauchy-coefficient-signs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"For every nonempty positive composition $k=(k_1,\\ldots,k_d)$ and integer\n$\\ell\\ge1$, put $F_k(z)=z^{-d}\\operatorname{Li}_k(z)$ with its removable\nvalue at zero, and\n$C_n^{j;k;\\ell}=[z^n](1-z)^{-j}F_k(z)^{-\\ell}$ as in source equations\n(7)-(8). Conjecture 1.3 asks for $C_n^{0;k;\\ell}<0$ eventually. For every\nfixed integer $j\\ge1$ it also asks for $C_n^{j;k;\\ell}>0$ eventually and,\nif $k_1>1$, for\n$C_n^{j;k;\\ell}>\\binom{n+j-1}{n}/\\zeta(k)^\\ell$ eventually.\nThe threshold may depend on $k,\\ell,j$; the strict inequality and all\nquantifiers are retained.","title":"Xu-Zhao Conjecture 1.3","url":"https://the-omega-institute.github.io/trureturing-pages/research/xu-zhao-trivariate-cauchy-coefficient-signs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3d862d289a83a7ec54121f1c8bb2b026c3da95075f32b061ba2c047487a7225.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yadavalli-andrejic-kunjwal-2024-biased-trine-joint-measurability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. A. Yadavalli, N. Andrejic and R. Kunjwal, *Qualitative equivalence between incompatibility and Bell nonlocality*, arXiv:2008.10100v3; Phys. Rev. A 110, L060201 (2024), Supplemental Material, Section III, Example 1. Its TeX source `qualequiv_supp_arxiv.tex`, lines 151–158, states:\n\n> Based on our numerical evaluation and visual inspection, we conjecture the exact form of $\\eta_1^{(3)}(b)$ to be\n> $$\\eta_1^{(3)}(b)=\\begin{cases}\n> 1+b,\\quad -1\\leq b<-\\frac{1}{3},\\\\\n> \\frac{2}{3},\\quad b\\in\\left[-\\frac{1}{3},\\frac{1}{3}\\right],\\\\\n> 1-b,\\quad \\frac{1}{3}< b\\leq 1.\n> \\end{cases}$$\n\nThe family is defined in lines 112–118, and the threshold is defined in lines 147–150 by triplewise joint measurability if and only if $\\eta\\leq\\eta_1^{(3)}(b)$. For $k\\in\\{1,2,3\\}$ and $x\\in\\{-1,+1\\}$, its effects on $\\mathbb C^2$ are\n\n$$E_k(x)=\\frac12\\big((1+xb)I+\\eta x\\,\\vec n_k\\cdot\\vec\\sigma\\big),\\qquad\n\\vec n_k=\\cos\\frac{2k\\pi}{3}\\vec e_x+\\sin\\frac{2k\\pi}{3}\\vec e_z.$$\n\nThe parameter domain is $-1\\leq b\\leq1$, $0\\leq\\eta\\leq1$ and $|b|\\leq1-\\eta$. Triplewise joint measurability means the existence of eight positive semidefinite complex $2\\times2$ matrices $G_s$, indexed by $s\\in\\{-1,+1\\}^3$, such that $\\sum_sG_s=I$ and $\\sum_{s:s_k=x}G_s=E_k(x)$ for every $k$ and $x$.\n\nIssue [#22692](https://github.com/the-omega-institute/trureturing/issues/22692) fixes the reading: for every real $b,\\eta$ in this domain, these three measurements are triplewise jointly measurable if and only if $\\eta\\leq\\eta_1^{(3)}(b)$ for the displayed piecewise function. This is proved.","title":"The exact triplewise joint-measurability threshold of biased trine qubit measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/yadavalli-andrejic-kunjwal-2024-biased-trine-joint-measurability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/060b9c1290c33dd928ed974d9f485f38a06ad39fb9bc33f2cfacd5985b03af9a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yang-chen-li-fei-wang-2018-coherence-breaking-amendment","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L.-M. Yang, B. Chen, T. Li, S.-M. Fei and Z.-X. Wang, *Amending Coherence-Breaking Channels via Unitary Operations* (Int. J. Theor. Phys. 57, 3776 (2018); arXiv:1810.09642), study incoherent channels.\n- **Definitions.** An incoherent channel is coherence-breaking if it maps every state to an incoherent state. The coherence-breaking index n(Φ) is the least k with Φ^k coherence-breaking.\n- **Results.** For qubit channels with n(Φ) = 1 and 2 they find unitaries Λα that make Λα ◦ Φ and Φ ◦ Λα ◦ Φ not coherence-breaking.\n- **The conjecture.** Section 5 states: \"for a d-dimensional quantum channel Φ with n(Φ) ≥ 2, one may conjecture that the channel Φ can also be amended by Φ ◦ Λα ◦ Φ ◦ · · · ◦ Φ ◦ Λα ◦ Φ [n Φ] with proper unitary operations Λα.\"\n- **The later preprint.** MiS Preprint 7/2019 asserts the qudit statement in its conclusions without a proof.\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/yang2018coherencebreaking.md).\n\nIssue [#18385](https://github.com/the-omega-institute/trureturing/issues/18385) fixes the reading: for every incoherent trace-preserving Kraus channel with n(Φ) = n ≥ 2, some unitary U makes Φ ◦ (Λ_U ◦ Φ)^{n−1} not coherence-breaking, with the same U in every slot. The conjecture is proved.","title":"Every incoherent channel with coherence-breaking index at least two is amended by one unitary","url":"https://the-omega-institute.github.io/trureturing-pages/research/yang-chen-li-fei-wang-2018-coherence-breaking-amendment/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6023b40c057c728b7a171dcd70298f81a8181a561c2513677573464608661b37.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yao-li-sun-2019-coherence-fraction-incoherent-nonmonotone","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Yao, D. Li and C. P. Sun, *Quantum coherence fraction*, Phys. Rev. A 100, 032324 (2019), arXiv:1906.09789v1, Section V, `CF.tex`, lines 422–423:\n\n> First, similar to the behavior of fully entangled fraction \\cite{Badziag2000}, we conjecture that $C_{\\mathcal{F}}(\\rho)$ may not satisfy the monotonicity under general incoherent operations.\n\nThe source fixes the computational basis as the incoherent basis. An incoherent operation has a Kraus representation whose individual branches map diagonal density matrices to diagonal matrices; the normalized nonzero branches are again incoherent states (Section II, lines 122–131). Its quantum coherence fraction is $C_{\\mathcal F}(\\rho)=\\max_{|\\phi\\rangle\\in\\mathcal{MCS}}\\langle\\phi|\\rho|\\phi\\rangle$, where the maximally coherent unit vectors have coordinates of modulus $1/\\sqrt d$ (lines 141–150). The definitions and source locator are in [the literature note](../Library/QuadraticForms/yaolisun2019coherencefraction.md).\n\nIssue [#22563](https://github.com/the-omega-institute/trureturing/issues/22563) fixes `claim` as the universal monotonicity assertion: for every $d,m\\in\\mathbb N$, every complex $d\\times d$ matrix $\\rho$ that is positive semidefinite with trace one, and every family $K:\\mathrm{Fin}\\,m\\to M_d(\\mathbb C)$ satisfying $\\sum_\\alpha K_\\alpha^\\dagger K_\\alpha=I$ and preserving diagonality under each sandwich $X\\mapsto K_\\alpha X K_\\alpha^\\dagger$ for every complex diagonal matrix $X$, one has $C_{\\mathcal F}(\\sum_\\alpha K_\\alpha\\rho K_\\alpha^\\dagger)\\le C_{\\mathcal F}(\\rho)$. Here the module defines $C_{\\mathcal F}(A)=\\sup\\{\\operatorname{Re}(z^\\dagger Az)/d: z\\in\\mathbb C^d,\\ |z_j|=1\\text{ for every }j\\}$ using the real `sSup`.\n\nThe formal resolution is `refuted`, because `result : ¬ claim` negates universal monotonicity. This confirms the source's conjecture of non-monotonicity.","title":"The quantum coherence fraction can increase under incoherent operations","url":"https://the-omega-institute.github.io/trureturing-pages/research/yao-li-sun-2019-coherence-fraction-incoherent-nonmonotone/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd6193e522b181977fc43277300addee80fb144d5c8daba346cc87fe6ebcd063.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yoshida-rapid-decrease","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rintaro Yoshida, *On some questions of Fisk and Brändén*,\n[arXiv:1005.4218v1](https://arxiv.org/abs/1005.4218v1), Open Problem 5,\nprinted page 11, asks:\n\n> Let $f(x)=\\sum_{k=0}^{\\infty}a_kx^k$ where $\\{a_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence. If $S_r[f(x)]=\\sum_{k=0}^{\\infty}(a_k^2-a_{k-r}a_{k+r})x^k:=\\sum_{k=0}^{\\infty}b_kx^k$, is it true that $\\{b_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence?\n\nDefinition 29, printed page 10, requires positive terms and one real $\\alpha$ with\n$\\alpha\\ge\\max\\{2,\\frac{\\sqrt2}{2}(1+\\sqrt{1+a_1})\\}$ such that\n$a_k^2\\ge\\alpha^2a_{k-1}a_{k+1}$ for every $k\\ge1$.\nNegative indices are zero-extended, following Definition 8, printed page 3.\nThe [Library note](../Library/Certificates/yoshida2010fiskbranden.md) carries these locators.\nThe tier-1 preregistration is [#23414](https://github.com/the-omega-institute/trureturing/issues/23414).\n\n### Statement echo\n\nAll declaration names below belong to `D5.S0.Certificates.YoshidaRapidDecreaseRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive real sequence indexed by all nonnegative integers | `RapidlyDecreasing` positivity conjunct | Every natural index, including zero; no finite truncation. |\n| A single threshold parameter and all positive-index inequalities | `RapidlyDecreasing` existential and universal clauses | The source threshold is literal, with real square roots and real division. |\n| $a_j=0$ outside the index domain | Integer conditionals in `RapidlyDecreasing` and `sCoeff` | Zero for negative indices; otherwise `Int.toNat` selects the coefficient. No separate forwarding definition. |\n| $b_k=a_k^2-a_{k-r}a_{k+r}$ | `sCoeff` | Subtraction occurs after casting indices to integers. |\n| The source's coefficient object and closure question | `claim` | An explicit existence conjunct plus closure for every real coefficient sequence and every natural $r\\ge1$, as preregistered. The coefficient representation imposes no convergence condition. |\n| Negative answer | `result` | The literal negation of the complete claim; $r=1$ supplies the failure. |","title":"Yoshida's rapid-decrease closure question","url":"https://the-omega-institute.github.io/trureturing-pages/research/yoshida-rapid-decrease/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2cbdb3ce34547db5709f49d149fe01b948638a75b182247457a5bc25f28d0e39.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yu-2017-skew-coherence-mixed-state-polygamy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Chang-shui Yu, *Quantum coherence via skew information and its polygamy*, Phys. Rev. A 95, 042337 (2017), arXiv:1704.04871v1, Discussion and Conclusions, `coherenceLy.tex`, lines 997–999:\n\n> However, we would like to conjecture that it could hold for the bipartite mixed states with the dimension $N \\geq 6$.\n\nAppendix D, “The conjecture,” lines 1396–1397, repeats:\n\n> In this sense, we would like to conjecture that this relation should be satisfied in $(N\\geq 6)$-dimensional systems.\n\nThe quotations are checked against the v1 TeX member `coherenceLy.tex`, md5 `026b6fb07e6d276a7bd743cc480af58c`. Lemma 1, lines 170–178, defines the reference-basis coherence by $C(\\rho)=1-\\sum_k\\langle k|\\sqrt\\rho|k\\rangle^2$, with the positive square root. Write $Q(\\rho)=1-C(\\rho)$. Theorem 2, lines 543–550, states $Q(|\\Psi\\rangle_{AB})\\le Q(\\rho_A)Q(\\rho_B)$ for every bipartite pure state and its two reduced density matrices. The conjecture asks for the same inequality for arbitrary bipartite mixed states of total dimension $N\\ge6$; the $2\\otimes3$, $3\\otimes3$, $3\\otimes4$ and $4\\otimes4$ figure captions identify $N$ as the total dimension.\n\nIssue [#23835](https://github.com/the-omega-institute/trureturing/issues/23835) fixes the reading: for all natural $m,n$ with $mn\\ge6$ and every positive semidefinite complex matrix $\\rho$ of trace one on $\\mathbb C^m\\otimes\\mathbb C^n$, in the product reference basis,\n\n$Q(\\rho)\\le Q(\\operatorname{Tr}_B\\rho)Q(\\operatorname{Tr}_A\\rho)$.\n\nThis is `claim`; `result : ¬ claim` refutes it.","title":"Yu's mixed-state skew-coherence polygamy conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/yu-2017-skew-coherence-mixed-state-polygamy/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/546d98ac201da455bd04179722ba5b54533292c52f723f638edbabc25126bb39.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yu-duan-xu-2017-schur-channel-diamond-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Nengkun Yu, Runyao Duan and Quanhua Xu, *Bounds on the distance between a\nunital quantum channel and the convex hull of unitary channels*, IEEE\nTransactions on Information Theory 63(2), 1299–1310, Section V, Theorem 3,\nEquation (8), ask:\n\n> It seems quite likely that in Eq. (8) the first inequality should be strict and the second one should be an equality. However, this is still an unsettled issue.\n\nThe equality in the second inequality is\n$D(\\Phi,\\operatorname{Conv}(U_d))=D(\\Phi,\\Lambda_d)$ for every Schur channel\n$\\Phi$, where $\\Lambda_d=S_d\\cap\\operatorname{Conv}(U_d)$.\nThe distance is the unhalved diamond distance. The\n[literature note](../Library/QuantumChannels/yuduanxu2017unitarydistance.md)\ncontains the definitions and source quotations. Preregistration\n[#16745](https://github.com/the-omega-institute/trureturing/issues/16745)\nspecifies the full quantified statement.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Every dimension and every channel | `claim` quantifies natural `d` and `QuantumChannel (Fin d) (Fin d)` | Includes dimension zero; complete positivity and trace preservation are carried by the channel type. |\n| Schur action $\\Phi(X)=C\\circ X$ on every matrix | `IsSchurChannel` | Literal existence of a matrix giving Hadamard action; no replacement by a recovery certificate. |\n| Convex hull of unitary channels | `MixedUnitary` | Finite families, nonnegative real weights of sum one, and unitary conjugation on every complex matrix. |\n| $\\Lambda_d=S_d\\cap\\operatorname{Conv}(U_d)$ | `SchurMixedUnitary` | Literal intersection; diagonal mixtures are obtained in the proof. |\n| Infimum of diamond distances | `distToSet` | Real `sInf` of the image of the unhalved `diamondDistance`; no attained minimum is assumed. |\n| Equality in the second inequality of (8) | `result : claim` | Proved with the Schur hypothesis as the only additional hypothesis. |","title":"Schur-channel diamond-distance equality","url":"https://the-omega-institute.github.io/trureturing-pages/research/yu-duan-xu-2017-schur-channel-diamond-equality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e313702ded5a20f2bc43b11d9969ac1be0e17ee816d316961372b69dc9785cf0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yu-et-al-2021-absolutely-entangled-all-partitions","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Yu, P. Jayachandran, A. Burchardt, Y. Cai, N. Brunner and V. Scarani, *Absolutely entangled sets of pure states for bipartitions and multipartitions*, Physical Review A 104, 032414 (2021), DOI 10.1103/PhysRevA.104.032414; arXiv:2106.16216v3, Section III, immediately after the Corollary (`main.tex`, line 509):\n\n> This result guarantees that, given any $U$ and any partition $P$, there exists at least one entangled state. An interesting open question is whether one can find an AES, such that, for every $U$, there exists a state that is entangled according to all partitions. This would arguably be the strongest form of genuine multipartite entanglement.\n\nThe quotation is from the v3 source member `main.tex`, md5 `e11d4dc540598e9ef271fc11bad389a0`. Definition 3 is at lines 310–317, and the preceding Corollary at lines 499–508. The definitions and source locator are also in [the literature note](../Library/Entropy/yuetal2021absolutelyentangled.md).\n\nIssue [#22238](https://github.com/the-omega-institute/trureturing/issues/22238) fixes the reading. A partition $p=(d_1,\\ldots,d_k)$ is an ordered factorization with $k\\ge2$, every $d_i\\ge2$, and $\\prod_i d_i=d$. Its tensor coordinates are fixed by `partitionEquiv`, the mixed-radix identification of local index tuples with the standard basis of $\\mathbb C^d$. The matrix `productState p τ` has entries equal to the products of the corresponding local entries. `Separable p ρ` means membership in the real convex hull of such products with every local matrix positive semidefinite and of trace one.\n\nThe formal assertion is: for every $d,a,b\\in\\mathbb N$ with $a,b\\ge2$ and $d=ab$, there is a finite set $S$ of positive semidefinite $d\\times d$ complex matrices of trace one such that\n\n$\\forall U\\in SU(d),\\ \\exists\\rho\\in S,\\ \\forall p\\in\\mathrm{Partition}(d),\\ \\neg\\mathrm{Separable}(p,U\\rho U^\\dagger).$\n\nThe member may depend on $U$ and must serve every partition simultaneously. This fixed-coordinate assertion is proved. Purity is used in the construction; it is not an additional predicate in the type of `claim`.","title":"A finite set supplies one state entangled for all fixed tensor partitions in every frame","url":"https://the-omega-institute.github.io/trureturing-pages/research/yu-et-al-2021-absolutely-entangled-all-partitions/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30dbe1760dfea461b47813a7ecc3e5d595cfb3ff203198182472f87243c7bb51.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yu-oh-2013-skew-information-lambda-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Yu and C. H. Oh, *Robertson-Schrödinger uncertainty relation refined by skew information*, arXiv:1303.6404v1, Appendix 2:\n\n> Numerical evidences show that $\\lambda_f=1$ whenever $4f(0)\\le 1$ and we conjecture that $\\lambda_f=\\min\\{1,1/[4f(0)]\\}$ always holds.\n\nHere λ_f is the minimum over x ≥ 0 of F(x) = (1 + x − f(0)(1 − x)²/(2f(x)))/(2f(x)), for f nonnegative on [0, ∞), operator monotone on Hermitian matrices, symmetric (x f(1/x) = f(x)), normalized (f(1) = 1) and regular (f(0) > 0). The definitions are quoted in [the literature note](../Library/QuantumContext/yu2013skewinformation.md).\n\nIssue [#19664](https://github.com/the-omega-institute/trureturing/issues/19664) fixes the reading. For every such f, min{1, 1/(4f(0))} would be the least value of F on [0, ∞). The conjecture is refuted.","title":"λ_f = min{1, 1/[4f(0)]} fails for a regular symmetric operator monotone function","url":"https://the-omega-institute.github.io/trureturing-pages/research/yu-oh-2013-skew-information-lambda-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f74846c27a96c523f742030f92dfb3407ea4826fa9f3e33cfbbe28846cd5ab92.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:yurischev-2017-xstate-unimodality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. A. Yurischev, *Extremal properties of conditional entropy and quantum\ndiscord for XXZ, symmetric quantum states*, Quantum Inf. Process. 16, 249\n(2017), arXiv:1702.03728v3.\n\nIntroduction:\n\n> we suppose the unimodal property for the function $S_{cond}(\\theta)$ (see\n> Appendix). So, if the unimodality hypothesis is valid the only possibility\n> (except the trivial case $S_{cond}(\\theta)=const$) for a single local\n> extremum (minimum or maximum) to appear or disappear inside the open\n> interval by continuous varying the parameters defining the X state is the\n> doubling the extremun at the ends of interval $[0,\\pi/2]$.\n\nThe Appendix writes the conditional entropy of an X state as\n$f_1(x)=-h_2\\bigl(\\frac{1+p_2x}2,\\frac{1-p_2x}2\\bigr)+h_4\\bigl(\\frac{1+p_2x\\pm\\sqrt{r_1}}4,\\frac{1-p_2x\\pm\\sqrt{r_2}}4\\bigr)$\non $x\\in[0,1]$ (Eq. (A1)). Here\n$r_{1,2}=(p_1\\pm p_5x)^2+4w^2(1-x^2)$, $w=(|p_3+p_4|+|p_3-p_4|)/4$, and\n$h_2,h_4$ are Shannon entropies in bits. The Appendix defines weak\nunimodality as follows:\n\n> A function $f(x)$ is a weakly unimodal function in the interval $[a,b]$ if\n> there exists a value $x_m\\in[a,b]$ for which it is weakly monotonically\n> increasing for $x\\le x_m$ and weakly monotonically decreasing for\n> $x\\ge x_m$. […] Analogous definitions are given for the minimum.\n\nIt then states the conjecture:\n\n> The functions $f_1(x)$ and $f_2(x)$ for every choice of parameters\n> $p_1,\\ldots,p_5$ for which all arguments of Shannon functions are\n> non-negative can have at most only one local extremum (minimum or maximum)\n> in the open interval $x\\in(0,1)$. […] It is required to prove or refute\n> this proposition.","title":"Unimodality of the X-state conditional entropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/yurischev-2017-xstate-unimodality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec6018a133a085a57c3ff20265f7bb714d3bab7884827a6eed502fe80513b47e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"OEIS A385672 (Andrei Zabolotskii, 2025) counts the `n`-step walks on the\nsquare lattice by their algebraic area `(Sum_{steps right} y) - (Sum_{steps\nleft} y)`, the integral of `y dx`. Its FORMULA section states\n\n> It appears that T(2*n, n^2 - k) = 2 * A029552(k) for k < n and\n> T(2*n+1, n^2+n - k) = 4 * A098613(k) for k < n.\n\nwhere A029552 and A098613 are the coefficients of `phi(x)/f(-x)` and\n`psi(x^2)/f(-x)`, with generating functions\n`(1 + 2 Σ_{j>0} x^(j^2)) / ∏_{i>0} (1 − x^i)` and\n`Σ_{j>0} x^(j^2 − j) / ∏_{i>0} (1 − x^i)`. Issue #10337 fixes the readings:\nwalks start at the origin, and the coefficients are read through\n`1 / ∏ (1 − x^i) = Σ p(m) x^m` with `p` the partition numbers, so\n`A029552(k) = p(k) + 2 Σ_{j ≥ 1, j² ≤ k} p(k − j²)` and\n`A098613(k) = Σ_{j ≥ 1, j² − j ≤ k} p(k − j² + j)`.","title":"Walks of near-maximal algebraic area","url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebf522f27afc1976aa73c33f725bacce5b760659c1081080c3c3ad99d917fe61.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zeckendorf-polynomial-maximum-order-complexity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Let `s_Z(n)` be the sum of the digits in the Zeckendorf representation, and set\n`S_Z = (s_Z(n) mod 2)` for `n >= 0` and `S_{Z,P} = (s_Z(P(n)) mod 2)` for\n`n >= 0`. For a binary sequence `S`, `M(S,N)` is the least feedback-register\nlength needed to generate its first `N` values, as defined in the paper.\n\nConjecture 3, quoted from arXiv:2106.09959v1:\n\n> “The sequence \\(\\mathcal S_Z\\) along polynomial subsequences, denoted by\n> \\(\\mathcal S_{Z,P}\\) for a polynomial \\(P\\) of degree \\(d\\geq2\\),\n> verifies \\(M(\\mathcal S_{Z,P},N)\\asymp N^{1/(2d)}\\), i.e. there are\n> \\(c,C>0\\) such as for all \\(N\\) large enough we have\n> \\(cN^{1/(2d)}\\leq M(\\mathcal S_{Z,P},N)\\leq CN^{1/(2d)}\\).”\n\nThe paper has already proved the matching lower bound for monic `P` in `Z[X]` of\ndegree `d >= 2` with `P(N_0)` contained in `N_0`; the new mathematical content is\nthe upper bound. A safe first formal target is the concrete case `P(X) = X^2`:\n\n```text\n∃ C > 0, ∃ N0, ∀ N ≥ N0,\n maximumOrderComplexity (fun n => zeckendorfDigitSum (n^2) % 2) N ≤ C * N^(1/4)\n```\n\nThe real-valued bound must be encoded with explicit ceilings and floors.\n\nThe paper states the difficulty:\n\n> “The maximum order complexity of \\(\\mathcal S_\\varphi\\) is algorithmically\n> more difficult to handle.”\n\n> “With our program and our machine, it is not possible to compute the maximum\n> order complexity of a sequence any further than \\(10^9\\) terms.”\n\nThe authors say the square plot supports the conjecture but the cube plot does\nnot clarify it; proving the conjecture would show their lower bound is sharp.","title":"Maximum order complexity along polynomial Zeckendorf subsequences","url":"https://the-omega-institute.github.io/trureturing-pages/research/zeckendorf-polynomial-maximum-order-complexity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/98bd29e89d5f6d4583605cf660cd56d97d15d0940b831a7713e118e9e8206db0.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zelinsky-zhang-conjecture-22-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zelinsky and Zhang, *Kullback-Leibler divergence and primitive non-deficient\nnumbers*, arXiv:2501.04209v2, page 14, Conjecture 22 (verbatim):\n\n> Assume that n is a positive integer and assume that\n> n is not in {1, 12, 24, 30, 36, 48, 60, 72, 120, 180, 240, 360}.\n> Let p be the smallest prime factor of n. Then v(n) >= 1/p^2.\n\nFor a positive integer n, the paper defines\n`v(n) = sum_{d | n, d > 1} (1/d) log((tau(n) - 1)/d)`.\nThe literal claim formalized here quantifies over every positive natural n,\nuses exactly the twelve published exclusions, and takes p to be\n`Nat.minFac n`. No additional restriction on n is imposed.","title":"Refutation of Zelinsky and Zhang's Conjecture 22","url":"https://the-omega-institute.github.io/trureturing-pages/research/zelinsky-zhang-conjecture-22-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7cba6fb6807ba0a58a35ff2605970d615f6eadba3af9aa026fc3f5168282a576.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhan-bie-cyclic-stack-preimages","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Alex Zhan and Stella Bie, *Cyclic-Pattern-Avoiding Stacks*, INTEGERS 26\n(2026), A15, Conjectures 3 and 4 on page 12 ask for the complete fibre of the\nconsecutive cyclic `[123]` stack map. Write the stack top-first. For an\nincoming value `x` and top entries `a,b`, pop `a` exactly when\n`x 4213`.\n\nFor every integer `n >= 4`, let `m=floor(n/2)`, let\n`y_n=(1,...,m,n,...,m+1)`, and let `S_n` contain every permutation of\n`1,...,n`. The two claims are\n\n`|{p in S_n : cyclicStackSort(p)=y_n}| = 1` for every even `n >= 4`,\n\n`|{p in S_n : cyclicStackSort(p)=y_n}| = ceil(n/2)` for every odd `n >= 4`.\n\nEquivalently, for every natural `m >= 2`, the cardinalities at `2*m` and\n`2*m+1` are `1` and `m+1`. There is no additional condition on the input\nbeyond being a permutation, and no finite upper bound on `n`.","title":"Zhan–Bie consecutive cyclic-stack fibres","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhan-bie-cyclic-stack-preimages/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b60a01977d7c854b2aafff1162999235819810a6fe1923b667023e6f6524092.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhan-gupta-hillery-2024-detector-sensor-dicke-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Zhan, H. Gupta and M. Hillery, *Optimizing Initial State of Detector\nSensors in Quantum Sensor Networks*, arXiv:2306.17401v6, ACM Transactions\non Quantum Computing 5(2), Article 10, DOI 10.1145/3655028, Section 5,\nConjecture 1, asks whether a state with uniform coefficient-squares on\na minimizing weight partition is an optimal initial state. The\n[Library note](../Library/QuantumBounds/zhanguptahillery2024detectorsensor.md)\ncontains the source quotation and locators.\n\nThere are $n\\ge3$ qubit sensors, one firing event with equal priors\n$1/n$, and a unitary with eigenvalues $e^{i\\theta}$ and $e^{-i\\theta}$.\nWrite $q=\\lceil n/2\\rceil$ and\n$T=\\tfrac12\\arccos(-(q-1)/q)$. The conjecture concerns\n$\\theta\\in(0,T]\\cup[\\pi-T,\\pi)$.\nFor a tensor eigenbasis word $x$, its weight is the number of plus\neigenvectors. For two distinct sensors, $R_k$ and $L_k$ count weight-$k$\nwords whose selected bits agree or differ. Every minimizing $l$ of\n$(R_k+\\cos(2\\theta)L_k)/(R_k+L_k)$, over $0\\le k\\le n$, is allowed.\nThe proposed state has coefficient-squares $1/\\binom nl$ on weight $l$\nand zero elsewhere, with arbitrary phases.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Qubit tensor eigenbasis | `Fin n → Bool`, true for the plus eigenvector | A coordinate representation of all pure input states |\n| One firing event at sensor $i$ | `fire` | The local eigenphase multiplies each coefficient |\n| Equal priors and global POVM | `isoSuccess`; the frozen `TriangleIndependentSourceNoise.LocalPOVM` | Positive semidefinite effects sum to the identity; no locality restriction |\n| $n\\ge3$ | `n`, `hn` in `claim` | Includes every permitted sensor number |\n| Both angle intervals and threshold endpoints | `T`, the disjunction in `claim` | Radians replace degrees without changing the intervals |\n| Every minimizing partition | `l`, `l ≤ n`, `IsMinOn` on `Set.Iic n` | The selected distinct sensor indices have values zero and one |\n| Uniform coefficient-squares, arbitrary phases | `φ`, the pointwise `Complex.normSq` hypothesis | The weight-cardinality expression is inlined literally |\n| Any normalized competing pure state | `ψ`, the sum of norm-squares equals one | No symmetry assumption on the competitor |\n| Optimal initial state | For every `M`, an attaining `M'` with the success comparison | `result : claim` proves the universal comparison |\n\nThe natural division in `T` gives $\\lceil n/2\\rceil$. Fin-index proof\nfields witness index bounds and impose no additional hypothesis.\nThe existing `LocalPOVM` supplies the stated POVM definition directly.\nThe weight expression is inlined rather than duplicated beside the\nexisting I-level `DynamicCoolingParallelTieUniqueness` definition;\nthe settling module has generality G. No source clause is dropped.","title":"Dicke optimality for detector-sensor identification","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhan-gupta-hillery-2024-detector-sensor-dicke-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4e26c2121fe1a1d8f7d3af6bdd0b3f0702d46da6dd7441211112b970713b24e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhan-gupta-hillery-2024-detector-sensor-strict-averaging-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"C. Zhan, H. Gupta and M. Hillery, *Optimizing Initial State of Detector Sensors in Quantum Sensor Networks*, ACM Transactions on Quantum Computing 5(2), Article 10 (2024), DOI 10.1145/3655028, arXiv:2306.17401v6, Section 5, Conjecture 2 (`optimal.tex`, lines 131–149, label `conj:avg`):\n\n> For a given $U$, consider two initial states $\\ket{\\psi} = \\sum\\limits_j \\psi_j \\ket{j}$ and $\\ket{\\psi'} = \\sum\\limits_j \\psi'_j \\ket{j}$ such that (i) they are two unequal coefficient-squares, i.e., for some $j$, $|\\psi_j|^2 \\neq |\\psi'_j|^2$, and (ii) they have the same objective function value, i.e., $P(\\ket{\\psi}, U) = P(\\ket{\\psi'}, U)$. We claim that the ``average'' state given by $\\ket{\\psi_{avg}} = \\sum_j \\sqrt{\\frac{|\\psi_j|^2 + |\\psi'_j|^2}{2}} \\ket{j}$ has a lower objective function value, i.e., $P(\\ket{\\psi_{avg}}, U) < P(\\ket{\\psi'}, U)$.\n\nThe quotation is from the v6 source TeX. The [Library note](../Library/QuantumBounds/zhanguptahillery2024detectorsensor.md) records the paper and its encoding. In the source's setting (`problem-related.tex`, lines 17–47), exactly one of $n$ qubit sensors fires, with equal priors $1/n$; its unitary has eigenvalues $e^{\\pm i\\theta}$. The objective is the minimum error of a global POVM discriminating the $n$ possible fired states.\n\nWrite $X_n=\\{0,1\\}^{\\mathrm{Fin}\\,n}$, with $1$ representing the plus eigenvector. A probe is an amplitude function $\\psi:X_n\\to\\mathbb C$, normalized by $\\sum_x|\\psi(x)|^2=1$. For $n\\ge1$, the reused `fire` and `isoSuccess` give $\\phi_i(x)=e^{i\\theta(2x_i-1)}\\psi(x)$ and $S_\\theta(\\psi,M)=n^{-1}\\sum_i\\operatorname{Re}\\langle\\phi_i,M_i\\phi_i\\rangle$. The module defines `optSuccess` as the real indexed supremum of $S_\\theta(\\psi,M)$ over `LocalPOVM (Fin n → Bool) (Fin n)`, `errorProb` as $P_\\theta(\\psi)=1-\\operatorname{optSuccess}_\\theta(\\psi)$, and `average` as $a(\\psi,\\psi')(x)=\\sqrt{(|\\psi(x)|^2+|\\psi'(x)|^2)/2}$, included in $\\mathbb C$ with zero imaginary part.\n\nIssue [#26032](https://github.com/the-omega-institute/trureturing/issues/26032) fixes `claim`: for every natural $n$, every $0<\\theta<\\pi$, and every pair of normalized complex probes, unequal coefficient-squares somewhere and equal $P_\\theta$ imply $P_\\theta(a(\\psi,\\psi')) Let l > 1 and define A_l := {n ∈ lZ⁺ : ‖l^m · n‖_l = m + ‖n‖_l for all m ≥ 0}.\n\n> Can one say anything about the set A_l? For example, is it true that for every n ∈ lZ⁺, there exists m₀ ≥ 0 such that l^{m₀} · n ∈ A_l?\n\nThe every-orbit assertion is proved. The broader request to describe $A_l$\nand to bound the first stable index remains open here. Preregistration: #19862.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| $l>1$ | Natural parameter `l` with `1 < l`. |\n| $n\\in l\\mathbb Z^+$ | Natural parameter `n` with `0 < n` and `l ∣ n`. |\n| Minimal number of copies of $l$, with addition, multiplication and parentheses | Public `value` evaluates the existing `KumarPengTranModularComplexity.Expr`; public `complexity` is the natural infimum of attainable `Expr.leaves`. |\n| $A_l$ | Public `A` states positivity, divisibility and the source equality for every natural exponent, including zero. |\n| Some $m_0\\geq0$ with $l^{m_0}n\\in A_l$ | `claim` quantifies over natural `m₀`; public `result : claim` proves it. |\n\nThe natural evaluation and minimum are the literal conventions of #19862.\nThe private `mem_A_iff` identifies them with the frozen evaluator and minimum\nat modulus zero through injectivity of the natural-to-integer cast. The\npublic claim retains the source's natural definitions. No additional premise\nor finite cutoff is present.","title":"Eventual stability of Zhang's l-complexity","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-2024-l-complexity-question-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9c4dbb90fd5759d9d54ed8d6fde328eca2edad7930895643b57897138323177.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-2024-three-operator-clarkson-mccarthy","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Teng Zhang, *Clarkson–McCarthy type inequalities, part I: $\\ell_p$–$\\ell_p$ and $\\ell_q$–$\\ell_p$ Schatten $p$-estimates* (2024 preprint, arXiv:2410.21961v4), Section 2, subsection “Euler's identity”, states Conjecture `conj:sharp-inequality`:\n\n> Let $A,B,C\\in S_p$. Then for $2\\le p<\\infty$,\n>\n> $\\|A+B+C\\|_p^p+\\|A\\|_p^p+\\|B\\|_p^p+\\|C\\|_p^p\\le\\frac{3^{p-1}+1}{2^p}\\left(\\|A+B\\|_p^p+\\|B+C\\|_p^p+\\|C+A\\|_p^p\\right).$\n>\n> For $00$ by taking $A=B=C\\neq0$.\n\nThis quotation is from lines 574–580 of the downloaded TeX member `v1_Clarkson_McCarthy_type_inequalities_part_I.tex`, MD5 `3ba34aa0913b51484593b3594a9f69d6`; only mathematical display layout and whitespace are normalized. The conjecture is introduced by “numerical evidence suggests the following sharp inequality” (line 571). Zhang restates it in *On five questions of Bourin and Lee* (2026), arXiv:2602.01463v2, TeX member `On_five_questions_of_Bourin_and_Lee.tex`, lines 411–419, MD5 `b3e91436b76494a53687b26bf5b2d06e`.\n\nFor a finite complex matrix and $p>0$, write $\\|X\\|_p^p=\\sum_j s_j(X)^p$. Abbreviate the four-term expression on the left by $L_p(A,B,C)$, the sum of the three pairwise terms by $P_p(A,B,C)$, and the proposed coefficient by $c(p)=(3^{p-1}+1)/2^p$.\n\nIssue [#22304](https://github.com/the-omega-institute/trureturing/issues/22304) fixes the reading settled by `claim`: for every real $p\\ge2$, every natural dimension $n$, and every $A,B,C\\in M_n(\\mathbb C)$, $L_p(A,B,C)\\le c(p)P_p(A,B,C)$. This universal statement is refuted. The reverse clause for $0 For A, B ∈ Hₙ and every k ≥ 5, λ(H^k) ≺_w λ(R_k).\n\nHere $H_n$ consists of all complex Hermitian $n\\times n$ matrices,\n$H=A+B$,\n\n$$Q_k=\\sum_{p=0}^k\\binom{k}{p}A^pB^{k-p},\\qquad\nR_k=\\operatorname{Re}Q_k=\\frac{Q_k+Q_k^*}{2}.$$\n\nThe list $\\lambda$ is nonincreasing, and $x\\prec_w y$ means\n$\\sum_{i=1}^r x_i\\le\\sum_{i=1}^r y_i$ for every $1\\le r\\le n$.\nThe paper proves the comparisons for $k=3,4$. Its conjecture imposes\nno positivity assumption on $A$ or $B$.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Quantum.Matrix.TaylorMajorization`;\nthe Pauli matrices are from `D5.S3.Quantum.FiniteDimensional`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $H_n$, $n\\ge1$ | `Matrix (Fin n) (Fin n) ℂ`, `Matrix.IsHermitian` | `Defs.claim` explicitly requires `1 ≤ n` and both Hermitian hypotheses. |\n| $Q_k$ | `Defs.taylorQ` | The ordered binomial sum, with the $A$ powers on the left. |\n| $R_k$ | `Defs.taylorR` | Complex Hermitian symmetrization; `Defs.taylorR_isHermitian` supplies its Hermitian proof. |\n| $\\lambda$, $\\prec_w$ | `Defs.weakMaj` | Mathlib's nonincreasing `eigenvalues₀`, compared by all initial sums up to $n$. The extra $r=0$ sum is automatically zero. |\n| All dimensions, inputs, orders and partial sums | `Defs.claim` | Every natural $n\\ge1$, Hermitian $A,B$, natural $k\\ge5$ and $1\\le r\\le n$. |\n| $\\Delta_k=\\operatorname{Tr}R_k-\\operatorname{Tr}(A+B)^k$ | `Defs.defect` | Real parts of traces; both compared matrices are Hermitian, so their traces are real. |\n| $Z=\\operatorname{diag}(1,-1)$, $X=\\left[\\begin{smallmatrix}0&1\\\\1&0\\end{smallmatrix}\\right]$ | `qubitZ`, `qubitX` | The existing two-dimensional Pauli matrices. |\n| Refutation | `Result.result` | Exactly `¬ Defs.claim`, without additional assumptions. |","title":"Taylor-coefficient weak majorization for Hermitian matrices","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-2026-taylor-coefficient-weak-majorization/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31d39d6ca5ec45959bba81aa47fb40940ce487ced2da63104bc2dc5b46dc2c23.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-2026-threshold-hypergraph-independence-polynomial","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"E. Zhang, *On the Independence Polynomial and Threshold of an Antiregular\nk-Hypergraph*, arXiv:2303.01024v1, Section 5, asks:\n\n> Alternatively, a more general question is, given two T2-threshold\n> k-hypergraphs with the same independence polynomial, are they isomorphic?\n> These are left for future work.\n\nThe universal positive answer is false. The settling declaration is\n`D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation.result`.\nThe preregistration is [#19406](https://github.com/the-omega-institute/trureturing/issues/19406).\nThe formal result uses five vertices and $k=3$; no arbitrary-$N$ theorem is delivered.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| A finite $k$-uniform hypergraph | `KHypergraph k V`, its `edges` and `uniform` fields | Every edge has cardinality $k$. |\n| No hyperedge is contained in an independent subset | `IsIndependent` | Every edge and subset are tested. |\n| Equation (8), the sum of $x^{|S|}$ over independent subsets | `indepPoly` | Integer coefficients; includes the empty set. |\n| Definition 4.1(T2), real labels and a strict threshold on every $k$-subset | `IsT2Threshold` | Literal strict inequality, with no positivity restriction on labels. |\n| Isomorphism | `Isomorphic` | A vertex equivalence preserves membership of every edge. |\n| Any two such hypergraphs with equal polynomials are isomorphic | `claim` | Quantifies over $k$, both finite vertex types and both hypergraphs. |\n| Negative answer | `result` | Negates the complete universal implication using $k=3$ and `Fin 5`. |\n\nThe encoding permits $k=0,1,2$, whereas the source question concerns $k\\ge3$;\nthe counterexample lies within the source range and all its edges are nonempty.\nThe finite-type instances enumerate vertices and decide equality; they add no\nmathematical restriction to the finite source objects. The carrier `Polynomial ℤ`\nis the formal integer-coefficient expression of Equation (8).","title":"Independence polynomials do not determine T2-threshold hypergraphs","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-2026-threshold-hypergraph-independence-polynomial/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a35bdc350c068e4177f221e85eca2e448b214f77c97094124f4e0f341c5ad76.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-character-sum-difference-question-d-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Wenpeng Zhang, *Some interesting number theory problems*, arXiv:2506.17235v1,\nprinted page 3, states problem (B) and question (D) verbatim as:\n\n> (B). Whether there are infinitely many pairs of fundamentally different\n> integer coefficients polynomials f(x) and g(x) (That is, (f(x)/p) ≠ (g(x)/p))\n> such that Σ_{x=1}^{p−1} (f(x)/p) − Σ_{x=1}^{p−1} (g(x)/p) = c, (2) where c is\n> a fixed constant.\n\n> (D). Whether the values of c can only be 0 or 2?\n\nIdentity (2) compares the Legendre character sums of two fundamentally\ndifferent integer-coefficient polynomials. Following the preceding corollary,\nthe formal statement uses one constant for every odd prime.","title":"Refutation of Zhang's character-sum constant restriction","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-character-sum-difference-question-d-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8271dfb5dc9051b830c02aab0fa94dd90adff2922a2b81660df12ba53f48dfed.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-et-al-2019-qubit-incompatibility-probability","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"L. Zhang, H. Xiang, X. Li-Jost and S.-M. Fei, *Incompatibility probability of random quantum measurements*, Phys. Rev. E 100, 062139, arXiv:1912.12321, Section III.C, PDF p. 5, states:\n\n> We *conjecture* that the incompatibility probability of a pair of random qubit measurements $\\mathbf A$ and $\\mathbf B$, generated by $(a_0,\\boldsymbol a)$ and $(b_0,\\boldsymbol b)$ in Eq. (4), is given by $\\mathbf{Pr}[\\Theta^{\\mathrm{NJM}}_{2,2}]=\\frac14.$\n\nIssue [#15842](https://github.com/the-omega-institute/trureturing/issues/15842) fixes the literal statement, the squared-form criterion and the source and literature boundaries.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Uniform random qubit observables in $V=\\{(x_0,\\boldsymbol x): \\lvert x_0\\rvert+\\lVert\\boldsymbol x\\rVert\\le1\\}$ | `OrderedRootCoordinates.V` on $\\mathbb R\\times\\mathbb R^3$ | The entire closed body is retained. |\n| Yu–Liu–Li–Oh square-root function | `OrderedRootCoordinates.hYu` | Both $(1+x_0)^2$ and $(1-x_0)^2$ occur under the square roots. |\n| The set of pairs violating Eq. (4) | `OrderedRootCoordinates.NJM` | Both body memberships and the strict violation of the squared criterion are retained. |\n| Product Lebesgue volume divided by $\\operatorname{vol}(V)^2$ | `OrderedRootCoordinates.incompProb` | The measure is actual product Lebesgue measure. |\n| Probability equals $1/4$ | `OrderedRootCoordinates.claim` | Literal equality, without extra hypotheses. |\n| Negation of the conjecture | `QubitIncompatibilityProbability.result` | Unconditional negation of that claim. |\n\n### Definition conventions\n\nThe main-text display of $h$ omits squares on $1\\pm x_0$. The squared form is the expression in Appendix C and in Yu–Liu–Li–Oh, Eq. (5), arXiv:0805.1538. The formal definitions use that preregistered squared form. The strict-interior coordinate maps are smooth bijections; the closed body is not replaced by an interior in the claim. Body boundaries, zero vectors and cosine endpoints are removed only within the integration proof, where their nullity is proved.","title":"The incompatibility probability of random qubit measurements","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-et-al-2019-qubit-incompatibility-probability/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/62b3fda7b062a192abfd0a8a79c3995b25d24948b3f7de1e0e38f57e26cc5498.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-jing-2025-block-criterion-power-case-theorem-2-3","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Feng Zhang and Hefang Jing, More inequalities for positive semidefinite matrices, Electronic Journal of Linear Algebra 41 (2025), 595–601, state after Remark 2.13:\n\n> “Due to the limitations of our research method, we cannot solve the problem that Theorem 2.3 holds for $1/2 \\le a \\le 1$.”\n\nTheir Theorem 2.3 states:\n\n> “Let $A \\in M_n$. Then $[[|A|^{2a}, A^*], [A, |A|^{2(1-a)}]] \\ge 0$, for $0 \\le a \\le 1/2$ if and only if $A$ is normal.”\n\nThe notation is “$|A| = (A^*A)^{1/2}$ and $A^*$ is the conjugate transpose of $A$” (p. 595). The delivered claim extends the displayed equivalence to every real $a$ with $1/2 \\le a \\le 1$, every natural dimension, and every complex square matrix, using the existing block construction with $|A|=\\operatorname{CFC.sqrt}(A^*A)$ and real powers.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $|A|=(A^*A)^{1/2}$ and $A^*$ is conjugate transpose | The reused ZhangJingBlockNormalityRefutation.block owner | The source expression is preserved through the frozen owner; no duplicate absolute-value definition is introduced. |\n| $1/2\\le a\\le1$ | claim quantifies a, with both inequalities | Both endpoints are included. |\n| Every n and $A\\in M_n$ | claim quantifies n and A : Matrix (Fin n) (Fin n) Complex | The full finite-dimensional family is retained, including n=0. |\n| The block is positive semidefinite iff A is normal | claim uses Matrix.PosSemidef and $A^*A=A A^*$ | Every block and the normality equality are retained. |\n| $1/2\\le a<1$ survives | The owning Scribe displays the full strict-interval quantified application | Both directions, singular matrices, the half endpoint and dimension zero are retained as an application of canonical suppliers. |\n| The closed interval fails | result : \\lnot claim | The conclusion is the literal refutation. |","title":"Zhang–Jing Theorem 2.3: the upper power interval","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-jing-2025-block-criterion-power-case-theorem-2-3/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/894a63943f9ab056b030d9bc6eb886274d1044d5b4f86e83ac00241794e38c50.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-jing-2025-block-normality-question-2-14","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Feng Zhang and Hefang Jing, *More inequalities for positive semidefinite matrices*, Electronic Journal of Linear Algebra 41 (2025), 595–601, Question 2.14, p. 601:\n\n> Let A ∈ M_n. Then [[f(|A|), A*], [A, g(|A|)]] ≥ 0 if and only if A is normal under the conditions that f(x)g(x) = x² for x ≥ 0, f(x) ≥ g(x) (x ≥ 1) and g(x) ≥ f(x) (0 ≤ x ≤ 1).\n\nThe notation on p. 595 is “|A| = (A*A)^{1/2} and A* is the conjugate transpose of A”. The assertion quantifies over every admissible pair of real functions, every natural dimension, and every complex matrix in that dimension. Preregistration: [#18699](https://github.com/the-omega-institute/trureturing/issues/18699).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $|A|=(A^*A)^{1/2}$ | `CFC.sqrt` of the conjugate-transpose product, inlined in `block` | Literal expression; the existing `CFC.abs` construction is not redeclared. |\n| Four displayed blocks, in their given order | `block`, using `Matrix.fromBlocks` and `cfc` | Every block is retained. |\n| $f(x)g(x)=x^2$ for $x\\ge0$ | First quantified premise of `claim` | Same range and equality. |\n| $f(x)\\ge g(x)$ for $x\\ge1$ | Second quantified premise | Same inequality and range. |\n| $g(x)\\ge f(x)$ for $0\\le x\\le1$ | Third quantified premise | Both endpoints included. |\n| Every $A\\in M_n$, block nonnegative iff $A$ normal | Final quantifiers and equivalence in `claim` | `Fin n` labels the indices; positivity is `Matrix.PosSemidef` and normality is $A^*A=AA^*$. |\n| The universal assertion is false | `result` | Its conclusion is the literal negation of `claim`. |\n\nNo continuity, invertibility or spectral injectivity premise is added. The counterexample functions are continuous and satisfy the three required scalar conditions.","title":"Zhang–Jing Question 2.14: block positivity and normality","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-jing-2025-block-normality-question-2-14/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92c5d803a48867f76b8eca04fb27475f5ef12c4b74b4a358e89815b314287b59.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-li-2026-rank-one-hadamard-product-variational","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lei-Hong Zhang and Ren-Cang Li, *On a Conjecture Related to Eigenvalue\nPerturbations*, arXiv:2609.25552v1, §1 Problem 3 and §5:\n\n> Problem 3: Relate min_{W nonsingular} ‖W∘G‖_ui ‖W^{-1}∘G^T‖_ui to min_{P permutation} ‖P∘G‖_ui^2.\n\n> In fact, for rank-one G, Romeo and Tilli [23, Theorem 1.2] proved (1.1) with c1=1 and the unitary restriction (2.3) with c2=1. They explicitly left the corresponding nonsingular version of Problem 3 unresolved; see [23, Remark 3.1].\n\nFor every natural $n$, every complex $n\\times n$ matrix $G$ with\n$\\operatorname{rank}(G)\\le1$, and every unitarily invariant norm $N$, the\nprecise conclusion is\n\n$$\n\\inf_{W\\in\\mathrm{GL}_n(\\mathbb C)}\n N(W\\circ G)N(W^{-1}\\circ G^{\\mathsf T})\n =\\min_{P\\text{ permutation}}N(P\\circ G)^2.\n$$\n\nA permutation minimizing $N(P\\circ G)$ attains the infimum. The transpose\nof $G$ is plain transpose. The zero matrix, zero weights, and $n=0$ are\nincluded.","title":"Zhang–Li rank-one Hadamard product variational problem","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-li-2026-rank-one-hadamard-product-variational/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a343e80b16a2a85bc8857afeb7dc653012a72198fdbc22f2be48ca94b3606a20.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-nechita-2022-projective-test-channel-compatibility-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Qing-Hua Zhang and Ion Nechita, *A Fisher information-based incompatibility criterion for quantum channels*, Entropy 24(6), 805 (2022), DOI 10.3390/e24060805, arXiv:2204.09963v1, Section 4, conjecture immediately following Lemma `lem:channel-POVM-compatible` (`article.tex`, lines 263–265):\n\n> Consider two quantum channels $\\Phi, \\Psi: \\mathcal L(H_d) \\to \\mathcal L(H_d)$ such that, for all orthonormal bases $\\mathbf e = (e_1, \\ldots, e_d)$, $\\mathbf f = (f_1, \\ldots, f_d)$ of $\\mathbb C^d$, the POVMs\n>\n> $\\Big[ \\Phi^*(|e_i\\rangle\\langle e_i|) \\Big]_{i=1}^d \\quad \\text{ and } \\quad \\Big[ \\Psi^*(|f_j\\rangle\\langle f_j|) \\Big]_{j=1}^d$\n>\n> are compatible. Then, $\\Phi$ and $\\Psi$ are compatible channels.\n\nThe source's `\\ketbra` macro is expanded in the displayed quotation. The source archive member `article.tex` has MD5 `614f0961ea15b65fb0b83ac83bd3fffb`.\n\nHere channels are completely positive, trace-preserving complex linear maps. Two POVMs are compatible when a positive normalized family indexed by both outcome sets has them as its marginals. Two channels are compatible when a single channel to the tensor product of their outputs has them as its partial-trace marginals on every input matrix. These definitions are in the source at lines 124–146 and 179–199 and in [the literature note](../Library/QuantumContext/zhangnechita2022fisherchannelincompatibility.md).\n\nIssue [#21608](https://github.com/the-omega-institute/trureturing/issues/21608) fixes the reading: for every positive dimension and every pair of channels on that space, compatibility of the two induced POVMs for every pair of orthonormal bases implies channel compatibility. This is refuted by two copies of the qubit depolarizing channel $\\Phi_{7/10}$.","title":"Compatible projective tests do not imply compatible quantum channels","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-nechita-2022-projective-test-channel-compatibility-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af715fa23754d73deffae1f5e9808c7df60e91f05134692d858e8c386a8cb19c.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-singh-pati-2017-subentropy-strong-subadditivity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"The question attributed to arXiv:1510.08859v2, §7.1 is recorded in\n[zhang2017subentropy](../Library/Entropy/zhang2017subentropy.md).\n\n> A similarly question can be asked for the strong subadditivity of subentropy: Q(ρ ABC) + Q(ρ B) ⩽ Q(ρ AB) + Q(ρ BC).\n\nFor every finite-dimensional tripartite density operator $\\rho_{ABC}$, let\n$\\rho_{AB}=\\operatorname{Tr}_C\\rho_{ABC}$,\n$\\rho_{BC}=\\operatorname{Tr}_A\\rho_{ABC}$ and\n$\\rho_B=\\operatorname{Tr}_{AC}\\rho_{ABC}$. The proposed assertion is\n\n$$\nQ(\\rho_{ABC})+Q(\\rho_B)\n\\le Q(\\rho_{AB})+Q(\\rho_{BC}).\n$$\n\nThe formal statement quantifies over arbitrary finite coordinate\ntypes and the repository's `DensityState`; changing the parentheses of a\nproduct coordinate type only relabels matrix entries. The original assertion\ndoes not impose commutation, full rank, separability or purity.","title":"Zhang–Singh–Pati: strong subadditivity of subentropy","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-singh-pati-2017-subentropy-strong-subadditivity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de86f2fb1dea490ee675ec7c99c402e6837fef24e82d8dc5ee2c956b7eeb62ba.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-tian-2025-steiner-diameter-radius-trees","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Qingnan Zhang and Yingzhi Tian, *On the Steiner k-diameter and Steiner\n(k, k′)-radius of trees*, arXiv:2511.22492v1, p. 14, Conjecture 5.1, states:\n\n> Let n, k, k′ be integers with n ≥ k ≥ 3 and k > k′ ≥ 1 and let T be a tree with order n. Then Sd_k(T) ≤ k/(k − k′) Sr_{k,k′}(T) − k′(k′ − 1)/(k − k′).\n\nHere $d_T(S)$ is the minimum edge count among connected subgraphs containing\n$S$; $\\varepsilon_k(v;T)$ maximizes $d_T(S)$ over $k$-sets containing $v$;\n$Sd_k(T)$ maximizes these vertex eccentricities.\nFor an anchor set $B$, $\\varepsilon_{k,r}(B;T)$ maximizes $d_T(S)$ over\n$k$-sets containing $B$, and $Sr_{k,r}(T)$ minimizes this quantity over\n$r$-sets $B$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A finite simple tree of order $n$ | `V : Type`, `Fintype V`, `DecidableEq V`, `T : SimpleGraph V`, `T.IsTree`; $n$ is `Fintype.card V` | Exact; decidable equality is classical bookkeeping on a finite carrier. |\n| $n\\ge k\\ge3$ | `k ≤ Fintype.card V`, `3 ≤ k` | Exact. |\n| $k>k'\\ge1$ | `r < k`, `1 ≤ r` | Exact; $r$ denotes $k'$. |\n| Minimum connected-subgraph edge count | `steinerDist` | Literal source definition, using a natural infimum. |\n| Vertex eccentricity and diameter | `steinerEcc`, `steinerDiam` | Literal nested maxima, as natural suprema. |\n| Anchored eccentricity and radius | `steinerSetEcc`, `steinerRadius` | Literal extrema over terminal sets of the specified cardinality. |\n| The rational inequality | `claim`, proved by `result` | All distances and parameters are cast into $\\mathbb Q$ before subtraction and division. |\n| Auxiliary cut representation | `Separates`, `cutEdges`, `steinerDist_cut` | Repository-derived objects characterize the source's literal distance; they do not replace its definition. |\n| Arbitrary terminal $k$-set $A$ and anchor $r$-set $B$ | `pointwise_bound` | Strengthening; no disjointness, hull containment or diameter-maximization hypothesis. |","title":"Zhang–Tian Conjecture 5.1: Steiner diameter and anchored radius of trees","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-tian-2025-steiner-diameter-radius-trees/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eef7549d7cb8c4752cb3f7b4f6dee877ff3cacb6728bcb04957aa6402ae95fef.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-wu-fei-2012-relative-entropy-superadditivity-conjecture-4-2","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhang, Wu and Fei, arXiv:1210.4720v4, p. 6, §4, conjecture (4.2), state:\n\n> Based on the above result, we propose the following conjecture: There exist unitary operators U_A ∈ U(H_A), U_B ∈ U(H_B) and U_AB ∈ U(H_A ⊗ H_B) such that S(U_AB ρ_AB U_AB† ‖ σ_AB) ⩾ S(U_A ρ_A U_A† ‖ σ_A) + S(U_B ρ_B U_B† ‖ σ_B), (4.2) where the reference state σ_AB is required to be full-ranked state. Indeed, our numerical calculations show that inequality (4.2) is true.\n\nThe source is recorded in [the Library note](../Library/Divergence/zhangwufei2013superadditivity.md).\nThe assertion ranges over every pair of finite subsystem dimensions and every joint density\n$\\rho_{AB}$ and full-rank joint reference $\\sigma_{AB}$; all four marginals are actual partial traces.\nThe relative entropy is $S(\\rho\\Vert\\sigma)=\\operatorname{Re}\\operatorname{Tr}(\\rho(\\log\\rho-\\log\\sigma))$\nwith natural logarithms. The independent existential joint and local unitaries are unrestricted.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite $H_A,H_B$ | Natural dimensions `a`, `b`, with coordinates `Fin a`, `Fin b` | All finite dimensions; a density state supplies trace one |\n| Joint density matrices $\\rho_{AB},\\sigma_{AB}$ | `DensityState (Fin a × Fin b)` | Positive semidefinite complex matrices of trace one |\n| Full-ranked $\\sigma_{AB}$ | `(CStarMatrix.ofMatrix.symm σ.1).PosDef` | Positive definiteness of the reference matrix |\n| $\\rho_A,\\sigma_A$ | `marginalRight ρ`, `marginalRight σ` | Keeps the first product factor |\n| $\\rho_B,\\sigma_B$ | `marginalLeft ρ`, `marginalLeft σ` | Keeps the second product factor |\n| Three existential unitaries | `UAB`, `UA`, `UB` and `Matrix.unitaryGroup` membership proofs | The membership proofs express unitarity, with no extra hypotheses |\n| Inequality (4.2) | `claim` | Sum of the two marginal divergences is at most the joint divergence |\n| Refutation | `result` | The negation of the complete universally quantified `claim` |","title":"Zhang–Wu–Fei: relative-entropy superadditivity after unitary conjugation","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-wu-fei-2012-relative-entropy-superadditivity-conjecture-4-2/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/685a9bd87b3ad7173a17773026f0fc2a38f80dbe696daf9bda393bfe73d50366.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-xie-2026-bargmann-imaginary-maximizers","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Lin Zhang and Bing Xie, *A Survey of Bargmann Invariants: Geometric Foundations and Applications*, arXiv:2601.01858v2, p. 26, ask:\n\n> The quantity τ_n predicts the upper bound for the imaginary part of any nth-order Bargmann invariant. This immediately raises the question: what is the complete set of n-tuples Ψ = (|ψ_1⟩, …, |ψ_n⟩) in ℂ² that achieve Im Tr(ψ_1 … ψ_n) = τ_n? We leave this characterization, along with the operational interpretation of the condition z ∈ 𝓑_{n+1} ∖ 𝓑_n, as interesting open problems for future work.\n\nThe first question is **Proved** with the exact conventions in [preregistration #19224](https://github.com/the-omega-institute/trureturing/issues/19224). For every $n\\ge3$, a tuple of unit vectors in $\\mathbb C^2$ attains\n\n$$\\tau_n=\\frac{\\cos^n(\\pi/n)}{\\cos^{n-1}(\\pi/[2(n-1)])}$$\n\nif and only if $\\psi_j=\\eta_jUv_j$ for one common $2\\times2$ unitary $U$, independent phases $|\\eta_j|=1$, and\n\n$$v_j=(\\sqrt{1-t_n},\\sqrt{t_n}\\,e^{2\\pi ij/n}),\\qquad t_n=\\frac{1-\\tan(\\pi/[2(n-1)])/\\tan(\\pi/n)}2.$$\n\nThe [Library note](../Library/QuantumContext/zhangxie2026bargmann.md) gives Definition 3.1, Corollary 5.5, Eq. (5.15), and the source question.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every $n\\ge3$ | `claim` universally quantifies `n` with `3 ≤ n` | Exact |\n| Unit vectors in $\\mathbb C^2$ | `Ψ : Fin n → Fin 2 → ℂ`, with `star (Ψ j) ⬝ᵥ Ψ j = 1` for every `j` | Exact; the wave-function condition is inlined |\n| Pure state $|\\psi_j\\rangle\\langle\\psi_j|$ and ordered trace | `bargmann` uses `Matrix.vecMulVec`, `List.ofFn`, `List.prod`, and `Matrix.trace` | Exact; the projector expression is inlined |\n| Maximal imaginary value | `tau` is the source's cosine-power quotient | Exact; natural subtraction occurs in the exponent, real subtraction in the angle |\n| Complete set | `result : claim` proves both directions of the equality/orbit equivalence | Exact |\n| Explicit candidate answer | `maximizer` is the displayed latitude vector; one `U ∈ Matrix.unitaryGroup (Fin 2) ℂ` and phases `η` | The answer proposed in #19224, with no added hypothesis |\n\nNo clause is dropped or weakened. The source's second question is outside this equivalence.","title":"All qubit tuples maximizing the imaginary Bargmann invariant","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-xie-2026-bargmann-imaginary-maximizers/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6afcded004d80b62364534dcead69c69440ead244eb9483716cde15cb9e122e3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-zhao-2024-speyer-g-uniform-normality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Rong Zhang and James Jing Yu Zhao, *Analytic properties of Speyer’s g-polynomial of uniform matroids*, arXiv:2408.15506v1, Conjecture 4.8:\n\n> If $d=d(n)$ is a real function of $n$ such that $d(n)\\to+\\infty$ as $n\\to+\\infty$, then the coefficient of Speyer’s g-polynomial $g_{n,d(n)}(t)$, namely $S_i(n,d(n))$, is asymptotically normal by local and central limit theorems.\n\nThe source defines $g_{n,d}$ for integer ranks and uses the standing restriction $1\\le d\\le\\lfloor n/2\\rfloor$ after its rank-complement symmetry. The formal claim quantifies $d:\\mathbb N\\to\\mathbb N$, imposes that restriction eventually, and assumes $d(n)\\to\\infty$. It does not extend the polynomial to arbitrary real-valued rank functions. Both limits are uniform in every real threshold and use the source's derivative-ratio mean and variance.\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Scope |\n| --- | --- | --- |\n| Proposition 1.1 factorial sum | `SpeyerUniformNormality.g` | Natural subtraction occurs before casting, on the prescribed finite summation range. |\n| Integer matroid rank under the standing half-rank restriction | `d : ℕ → ℕ` and eventual `1 ≤ d n ∧ d n ≤ n / 2` | The source's wording “real function” is interpreted within its integer rank domain; eventual validity allows arbitrary finitely many initial values. |\n| Rank tends to positive infinity | `Tendsto d atTop atTop` | No monotonicity or growth-rate hypothesis is added. |\n| Theorem 4.2 mean and variance | `BernoulliTriangularLimits.mu`, `sigmaSq`, `sigma` | The same derivative ratios and nonnegative square root. |\n| Coefficient probability | `BernoulliTriangularLimits.prob` | Coefficient divided by evaluation at one, and zero at negative integer indices. |\n| Standard normal distribution and density | `StatLean.HypothesisTesting.stdNormalCDF` and `stdNormalPDF` | Reused Gaussian owners inside `CLT` and `LLT`; `claim` is their conjunction for the Speyer polynomial sequence. Their integral and density correspondences are established inside the consuming proofs. |\n| Central limit theorem, uniform over real thresholds | `BernoulliTriangularLimits.CLT` | The epsilon form uses the finite coefficient support. |\n| Local limit theorem, with integer floor | `BernoulliTriangularLimits.LLT` | The epsilon form retains the literal integer floor. |\n| Both limits | `SpeyerUniformNormality.result : claim` | The conclusion is their conjunction for every admissible divergent rank sequence. |","title":"Normal limits for Speyer g-polynomials of growing uniform-matroid rank","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-zhao-2024-speyer-g-uniform-normality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d82cb72fe8d0cd8690a6be3175638a5eb7d2ffc32b151ac5328429029e50be8f.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhang-zhao-2026-domirank-entropy-monotonicity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Yingying Zhang and Chengye Zhao, *Theoretical Analysis of DomiRank\nCentrality: Automorphism, Entropy, and Graph Transformations*,\narXiv:2610.00107v1, Section 4.1, Remark 4.6, identify the unconditional\nassertion\n\n> the DomiRank entropy of a connected non-regular graph decreases\n> monotonically with σ\n\nas an open problem. The claim quantifies over every finite connected\nnonregular unweighted undirected simple graph and every pair\n$0<\\sigma<\\tau<-1/\\lambda_{\\min}(A)$. It asserts\n$H_G(\\tau)\\le H_G(\\sigma)$, with the actual inverse equilibrium\n$\\Gamma(s)=s(I+sA)^{-1}A\\mathbf1$, positive clipping, normalization,\nand Shannon entropy in bits. There is no raw-positivity or extra\nnonzero-normalizer premise. Conjecture 4.7 is excluded.","title":"DomiRank entropy need not decrease with competition","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhang-zhao-2026-domirank-entropy-monotonicity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd7532a0cef2e5077d944e381f222afb9468cfc9eab701a8b51bcb4b328db87b.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhao-pereira-ma-fei-2021-coherence-assistance-roof-equality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M.-J. Zhao, R. Pereira, T. Ma and S.-M. Fei, *Coherence of assistance and assisted maximally coherent states*, Scientific Reports 11, 5935 (2021), arXiv:2103.08818v1, ask in the paragraph “Relation between the convex roof extension and the least concave majorant extension” (`ZhaoMJ-AMC4.tex`, line 303):\n\n> The strict relation between the coherence of convex roof and the coherence of assistance, that is, whether $C_c(\\rho)< C_a(\\rho)$ holds for all mixed quantum states is an interesting topic.\n\nThe source's standing assumption is $f\\in\\mathfrak F\\setminus\\{0\\}$ (lines 189–190): $f$ is real, symmetric and concave on the probability simplex and vanishes at its vertices. The pure-state coherence is $C_f(|\\psi\\rangle)=f(\\mu(|\\psi\\rangle))$, where $\\mu(|\\psi\\rangle)_i=|\\psi_i|^2$ (lines 207–211).\n\nIn the formal reading, for every natural dimension $n$, `IsCoherenceFunction f` requires concavity on $\\Omega_n=\\{p\\in\\mathbb R^n:p_i\\ge0,\\ \\sum_i p_i=1\\}$, invariance under every coordinate permutation on $\\Omega_n$, and $f(e_i)=0$ for every vertex. Nonzeroness means $\\exists p\\in\\Omega_n,\\ f(p)\\ne0$.\n\nA finite pure-state decomposition has nonnegative real weights $p_j$ summing to one, unit vectors $\\psi_j\\in\\mathbb C^n$ for every index, and $\\rho=\\sum_jp_j|\\psi_j\\rangle\\langle\\psi_j|$. Let $D_f(\\rho)$ be the set of all averages $\\sum_jp_jf(\\mu(\\psi_j))$ over these decompositions. The module defines $C_c(\\rho)=\\inf D_f(\\rho)$ and $C_a(\\rho)=\\sup D_f(\\rho)$, using real `sInf` and `sSup`; the source uses minimum and maximum (lines 212–216 and 232–236). `IsMixed ρ` means $\\rho\\succeq0$, $\\operatorname{tr}\\rho=1$ and $\\operatorname{tr}(\\rho^2)\\ne1$.\n\nIssue [#22770](https://github.com/the-omega-institute/trureturing/issues/22770) fixes the claim: for every $n$, every such nonzero $f$, and every mixed $\\rho$, $C_c(\\rho) For n ≥ 2, it holds that max over π in 𝔖ₙ of |SC₁₂̲₃̲⁻¹(π)| equals max over π in 𝔖ₙ of |SC₃₂̲₁̲⁻¹(π)| equals 2ⁿ⁻².\n\nThe formal claim uses the paper's one-based permutation entries and the right-greedy stack convention fixed by its four worked figures. The flag `false` denotes 1-underline(23), and `true` denotes 3-underline(21).","title":"Refutation of Zhao's maximum preimage conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhao-vincular-stack-maximum-preimages-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a699405dc6c063014e2c369bb920216b0741c93f9aadbccf9796216315385f43.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhao-vincular-stack-second-largest-preimages-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhao, arXiv:2410.17057v1, Conjecture 5.2 (Section 5, printed page 20), states:\n\n> The second-largest number of preimages under SC₁₂̲₃̲ that a permutation in 𝔖ₙ can have is 2ⁿ⁻³, for n ≥ 3. Furthermore, the number of permutations π in 𝔖ₙ satisfying |SC₁₂̲₃̲⁻¹(π)| = 2ⁿ⁻³ is 2n − 2.\n\nThe formal claim retains both clauses under the same universal quantifier. The flag `false` denotes 1-underline(23), with one-based permutation entries and the right-greedy convention fixed by the paper's figures.","title":"Refutation of Zhao's second-largest preimage conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhao-vincular-stack-second-largest-preimages-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e2ab7032b75aa4849eadb5917e2a51d88079aef52bfb8d5f54fca2e6acb4953a.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhao-vincular-stack-sorting-schroeder","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhao, arXiv:2410.17057v1, Conjecture 3.30 (Section 3.2, printed page 11):\n\n> The sorting class of SC_{23̲1} is enumerated by |Sort_n(SC_{23̲1})| = S_{n−1}.\n\nSC_{23̲1} is the right-greedy stack map whose stack, read from top to bottom, must avoid the vincular pattern\n23̲-1 (the entries playing 2 and 3 adjacent); Sort_n(SC_{23̲1}) is the set of permutations of length n whose\nimage avoids 231, and S_m is the large Schröder number.","title":"Schröder Enumeration of the Sorting Class of the Stack Avoiding 23-1","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhao-vincular-stack-sorting-schroeder/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd049b10f02e2f23f481a9824f255c64766dceefb15999860c9f036fef4ad74d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhao-vincular-stack-three-sorting-classes","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhao, arXiv:2410.17057v1, Conjecture 5.1 (Section 5, printed page 20):\n\n> The sorting classes of SC_{312} and SC_{31̲2} are identical. That is, Sort_n(SC_{312}) = Sort_n(SC_{31̲2}) =\n> Sort_n(SC_{31̲2̲}). Furthermore, for τ ∈ Sort_n(SC_{312}), we have SC_{312}(τ) = SC_{31̲2}(τ) = SC_{31̲2̲}(τ).\n\nSC_σ is the right-greedy stack map whose stack, read from top to bottom, must avoid σ; σ is the classical\npattern 312, the vincular pattern in which the entries playing 3 and 1 are adjacent, or the vincular pattern in\nwhich the entries playing 1 and 2 are adjacent. Sort_n(SC_σ) is the set of permutations of length n whose image\navoids 231.","title":"Equal Sorting Classes and Outputs of the Stacks Avoiding 312, 31-2 and 3-12","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhao-vincular-stack-three-sorting-classes/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fec5de0c3ab2013b369456a1708d0ce57a1c07552dcfbcc9104294fec7e4e7c8.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zheng-liu-2024-logical-z-support-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Y. Zheng and D. E. Liu, *From Magic State Distillation to Dynamical Systems*, Quantum 9, 1858 (2025), arXiv:2412.04402, model magic state distillation by a stabilizer code as a map of the Bloch ball.\n\nProposition 3 gives a sufficient condition for z = 0 to be a fixed plane of the map: for every logical operator Z̄_i and every stabilizer element s_j, the product s_j Z̄_i contains a Pauli Z tensor factor. The authors write after Eq. (23): \"This condition is satisfied by most known protocols that we have examined and we conjecture it to be generally true for any stabilizer codes by properly choosing the logical Z operators without modifying the stabilizer groups.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/zhengliu2024magicdistillationdynamics.md).\n\nIssue [#17951](https://github.com/the-omega-institute/trureturing/issues/17951) fixes the reading. For every stabilizer code with at least one logical qubit, some nontrivial logical Pauli word L has a literal Z factor in every product s·L with s in the stabilizer group. This conjecture is refuted.","title":"Logical Z operators with Z support on every stabilizer coset need not exist","url":"https://the-omega-institute.github.io/trureturing-pages/research/zheng-liu-2024-logical-z-support-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ded00b16914e6d98671a29b573ced9b83defd3f7e0b3747a7ad1901da4e7e14.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhou-bash-guha-gagatsos-2023-transmissivity-beta-prior-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean\nsquare error for transmissivity sensing”, arXiv:2304.05539v1, Phys. Rev.\nResearch 5, 043033 (2023), Section VI (p. 6):\n\n> The numerical results indicate that the optimal input states with non-integer\n> mean photon number, have the form of Eq. (27), which means that for\n> $\\bar{n} \\in \\mathbb{N}$ the optimal states are Fock states.\n\nand Section VII (p. 7):\n\n> We note that for the beta prior PDF, we were not able to prove analytically\n> that the Fock states or the in-between states of Eq. (27) states are optimal,\n> even though our numerical computations support such conjecture.\n\nThe beta prior is eq. (16), $P(\\tau)=\\tau^{\\alpha-1}(1-\\tau)^{\\beta-1}/\\mathcal B(\\alpha,\\beta)$\nwith $\\alpha,\\beta>0$. The verbatim definitions are in\n[the literature note](../Library/QuantumStates/zhou2023transmissivity.md).\nIssue [#14698](https://github.com/the-omega-institute/trureturing/issues/14698)\npreregisters the reading with the parameters free, the witness and the exact\ncertificates.","title":"In-between-state optimality for transmissivity sensing with a beta prior","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhou-bash-guha-gagatsos-2023-transmissivity-beta-prior-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5992b665b6748d45e07d201081f795e7cef56bf23de7196fb70f50128e19db9e.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhou-bash-guha-gagatsos-2023-transmissivity-in-between-probe-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean\nsquare error for transmissivity sensing”, arXiv:2304.05539v1, Section V,\np. 4, state:\n\n> For the two-point prior PDF (15), we verify that for integer n̄ the optimal\n> state is the Fock state with the same photon number. For real n̄, our numerical\n> results support the conjecture that the optimal state has the form of the\n> state (27), up to a phase, i.e., |Φ′ₙ̄⟩ = e^{iφn̂}|Φₙ̄⟩.\n\nIssue [#11653](https://github.com/the-omega-institute/trureturing/issues/11653)\npreregisters this Tier 1 conjecture, the quantified reading and the witness.\nThe two-point prior is\n$P(\\tau)=q\\delta(\\tau-\\tau_0)+(1-q)\\delta(\\tau-\\tau_1)$,\nwith $q,\\tau_0,\\tau_1\\in[0,1]$.","title":"In-between-state optimality for Bayesian transmissivity sensing","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhou-bash-guha-gagatsos-2023-transmissivity-in-between-probe-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a01682476b434edaa8e87c135c658418a4b5bcb919cb2c2aeaf34e1bf2ae0ae5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhou-wu-chau-2016-error-disturbance-endpoint-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. S. Zhou, S. Wu and H. F. Chau, *State-Independent Error-Disturbance Tradeoff For Measurement Operators*, arXiv:1509.03812v1 (2015), published as *State-independent error-disturbance trade-off for measurement operators*, Physics Letters A 380(22–23), 1918–1924 (2016), DOI 10.1016/j.physleta.2016.03.046, state Conjecture 3 in Section IV. The v1 source `new_912arxiv.tex`, lines 625–638, reads:\n\n> For all $d \\geq 2$, the following trade-off between error and disturbance holds: $\\varepsilon(\\mathcal{A},\\mathcal{A}') + \\eta(\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$ and $\\Delta(\\mathcal{A},\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$, where $f(\\mathcal{A},\\mathcal{B}) = \\min \\{ \\varepsilon(\\mathcal{A},\\mathcal{B}),\\eta(\\mathcal{A},\\mathcal{B}) \\}$. Moreover, the equalities are both attained either when for all $i$, $\\ket{a'_i} = \\ket{a_i}$, or when for all $i$, $\\ket{a'_i} = \\ket{b_i}$.\n\nThe source uses rank-one projective measurements associated with orthonormal bases of $\\mathbb C^d$. Section II defines the outcome-distance by $D(x,y)=\\max_i|x_i-y_i|$ and compares the $\\mathcal B$ statistics before and after the nonselective $\\mathcal A'$ measurement. Its overall quantity is explicitly\n\n> $\\Delta(\\mathcal{A}, \\mathcal{A}', \\mathcal{B}) = \\max_\\rho \\left( \\varepsilon_{\\rho}( \\mathcal{A}, \\mathcal{A}') + \\eta_{\\rho}( \\mathcal{A}', \\mathcal{B})\\right).$\n\nThis is line 221 of the source. Both terms use one common density matrix. Immediately before the conjecture, lines 616–620 relax the endpoint error by\n\n> $\\varepsilon(\\mathcal{A},\\mathcal{B}) = \\min_{\\tilde{\\mathcal B}} \\varepsilon(\\mathcal{A},\\tilde{\\mathcal{B}})$,\n\nwith the explanation\n\n> This relaxed definition of $\\varepsilon$ means that we only distinguish between the classical probability distributions resulting from measurements $\\mathcal{A}$ and $\\mathcal{B}$ with no regard to the labeling of different measurement outcomes.\n\nThe definitions and source locator are collected in [the literature note](../Library/Estimation/zhouwuchau2016errordisturbance.md).\n\nIssue [#23208](https://github.com/the-omega-institute/trureturing/issues/23208) fixes the statement: for every natural $d\\ge2$ and all three unitary measurement bases $A,A',B$, the minimum of the endpoint error, minimized over outcome permutations, and the disturbance caused by $A$ is at most $\\Delta(A,A',B)$. This second inequality is refuted. The first inequality, concerning the sum of two separately maximized quantities, is not settled here.","title":"The Zhou–Wu–Chau endpoint lower bound on overall error is false","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhou-wu-chau-2016-error-disturbance-endpoint-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cac6a9a4550b848f769487afca53cc2e90b63557e946fcb2a494b9eb7829ef2d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhou-yu-arrow-32-13-enumeration","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Robin D.P. Zhou and Xinyang Yu, *Arrow-Wilf equivalences and enumerative results for short arrow\npatterns*, arXiv:2609.29392v1, abstract:\n\n> Together with earlier work of Archer and Laudone, this leaves only (32;1→3) unresolved for |ν|≤2, which we pose as an open problem.\n\nHere `S_n(32; 1 → 3)` consists of permutations of `1, …, n` avoiding the arrow pattern in the\npaper's sense: the selected values have the required order in the one-line word, and the prescribed\narrow holds in `π̂ = θ⁻¹(π)`.","title":"Enumeration of the Arrow Pattern (32; 1 → 3)","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhou-yu-arrow-32-13-enumeration/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af2f70aafc619e730103f033f23619cdf4ded7cba3bd6e0e35714be0c423c427.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhou-yu-arrow-wilf-equivalence","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Robin D.P. Zhou and Xinyang Yu, *Arrow-Wilf equivalences and enumerative results for short arrow\npatterns*, arXiv:2609.29392v1, Section 7:\n\n> We have computed the avoidance counts for the arrow patterns (12; 3 → 3) and (23; 1 → 1) and found that\n> the resulting sequences agree for all values of n tested. This suggests that these two patterns may be\n> arrow-Wilf-equivalent. However, the explicit formulas we obtained for the two sequences are quite\n> different in form, and we have not been able to establish the equivalence by a direct bijection or by\n> any other method. We leave this as an open problem.\n\nArrow-Wilf equivalence means `|S_n(12; 3 → 3)| = |S_n(23; 1 → 1)|` for every `n ≥ 1`, with containment of\nan arrow pattern as defined in Section 1 of the paper through `π̂ = θ⁻¹(π)`.","title":"The Arrow Patterns (12; 3 → 3) and (23; 1 → 1) Are Arrow-Wilf-Equivalent","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhou-yu-arrow-wilf-equivalence/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6cba7aaace951f38d3db439c78fc96ede7649c1ec7477e8dff526739db039ee.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhu-2020-coth-bernoulli-remainder-derivative-signs","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"Zhu, *Completely monotonic integer degrees for a class of special functions*,\nConjecture 6.4:\n\n> Let m ≥ 1, n ≥ 2, and pₙ(t) be defined as (2.2). Then ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−2)} > 0 holds for all t ∈ (0, ∞) and ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−1)} > 0 is not true for all t ∈ (0, ∞).\n\nEquation (2.2) is\n\n$$p_n(t)=\\frac{\\coth t}{t}\n-\\sum_{k=0}^{n}\\frac{2^{2k}B_{2k}}{(2k)!}t^{2k-2}.$$\n\nPut $P(t)=(-1)^nt^mp_n(t)$ and $M=m+2n$. The first clause is (6.12),\nand the failure of universal strict positivity in the second clause is (6.13).\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.CothBernoulliDegree`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $p_n(t)$ in (2.2) | `Defs.p` | The finite Bernoulli sum includes $k=0$ and $k=n$; its powers of $t$ are integer powers. |\n| $P(t)=(-1)^nt^mp_n(t)$ | `Defs.P` | The sign and the natural power $t^m$ are retained. |\n| Integers $m\\ge1$, $n\\ge2$ | Natural `m`, `n` with `1 ≤ m`, `2 ≤ n` | The domains coincide on the stated positive integer ranges. |\n| $P^{(M-2)}(t)>0$ for all $t>0$ | First conjunct of `Defs.claim` | The derivative is `iteratedDeriv (m + 2 * n - 2) (P m n)`. |\n| $P^{(M-1)}(t)>0$ is not true for all $t>0$ | Second conjunct of `Defs.claim` | The universal strict positivity assertion is negated, with the same real argument domain. |\n| Both clauses | `Result.result` | Exactly `Defs.claim`, with no additional hypothesis. |\n\nThe value assigned by real division and integer powers at zero is immaterial:\n`PartialFraction.hasSum_iteratedDeriv_P` uses neighborhood agreement at each\npositive argument to transfer all iterated derivatives.","title":"Derivative signs of the power-weighted hyperbolic cotangent remainder","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhu-2020-coth-bernoulli-remainder-derivative-signs/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b5dd9b8ade6b8b85fe0d7ff639645bba4567eda6cc5ca8e383cae1cf5f95532.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhu-2024-clifford-third-moment-aggregate-lower-bound","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and\n$|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on\n$\\Sigma(d)$ together with the aggregate bounds $6\\le\\kappa(\\Psi,\\Sigma(d))\\le2d+2$,\n$0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})\\le2d-4$ and $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$. Here\n$\\kappa(\\Psi,\\mathscr T)=\\sum_{\\mathcal T\\in\\mathscr T}\\kappa(\\Psi,\\mathcal T)$ and\n$\\mathscr T_{\\rm iso}=\\{\\mathcal T_O : O\\in O_3(d)\\}$. The verbatim statements are in\n[the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\n\nThe pointwise clause was refuted earlier with one qudit at $d=11$\n([dossier](zhu-2024-clifford-third-moment-kappa-negativity.md)); for that state the aggregate bounds\nhold. Issue [#14561](https://github.com/the-omega-institute/trureturing/issues/14561) targets the\nremaining aggregate lower bound in its printed form $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$.","title":"The isotropic aggregate of a two-qudit state in dimension five is below six","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhu-2024-clifford-third-moment-aggregate-lower-bound/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e87bbbfb3929a417eb7954e57be3faebf0731b701feed82a29981a1db9b67fa.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhu-2024-clifford-third-moment-kappa-negativity","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1 (Commun. Math. Phys. 407, 242 (2026)), Conjecture 2: for an odd\nprime $d$ and $|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$, $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ for every\nstochastic Lagrangian subspace $\\mathcal T\\in\\Sigma(d)$, together with three aggregate\ninequalities that the authors derive from it. Here\n$\\kappa(\\Psi,\\mathcal T)=\\operatorname{tr}[R(\\mathcal T)(|\\Psi\\rangle\\langle\\Psi|)^{\\otimes3}]$,\n$R(\\mathcal T)=r(\\mathcal T)^{\\otimes n}$ and $r(\\mathcal T)=\\sum_{(x;y)\\in\\mathcal T}|x\\rangle\\langle y|$.\nThe verbatim statements are in [the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\n\nIssue [#14367](https://github.com/the-omega-institute/trureturing/issues/14367) targets the\npointwise clause for every odd prime $d$, every $n$, every normalized $\\Psi$ and every\n$\\mathcal T\\in\\Sigma(d)$.","title":"A negative Clifford third-moment expectation at dimension eleven","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhu-2024-clifford-third-moment-kappa-negativity/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ebd1d449d947e3469e52b791acd76d5f7bf9f76a3825d8c2a2644143e9198e5.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhu-2024-clifford-third-moment-sigma-lower-bounds","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and\n$|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on\n$\\Sigma(d)$ together with the aggregate bounds $6\\le\\kappa(\\Psi,\\Sigma(d))\\le2d+2$,\n$0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})\\le2d-4$ and $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$. Here\n$\\Sigma(d)$ is the set of stochastic Lagrangian subspaces of $\\mathbb F_d^6$,\n$\\mathscr T_{\\rm sym}=\\{\\mathcal T_O\\mid O\\in S_3\\}$, $\\mathscr T_{\\rm ns}=\\Sigma(d)\\setminus\\mathscr T_{\\rm sym}$\nand $\\kappa(\\Psi,\\mathscr T)=\\sum_{\\mathcal T\\in\\mathscr T}\\kappa(\\Psi,\\mathcal T)$. The verbatim statements are in\n[the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\nThe pointwise clause and the clause $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$ are refuted\n([pointwise](zhu-2024-clifford-third-moment-kappa-negativity.md),\n[isotropic aggregate](zhu-2024-clifford-third-moment-aggregate-lower-bound.md)). Issue\n[#14620](https://github.com/the-omega-institute/trureturing/issues/14620) targets the two remaining lower\nbounds, $6\\le\\kappa(\\Psi,\\Sigma(d))$ and $0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})$.","title":"At dimension five the full and non-symmetric aggregates of a two-qudit state fall below their conjectured lower bounds","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhu-2024-clifford-third-moment-sigma-lower-bounds/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8961e7ce6c00ca08f7bf4e9796a6fea5ba3ab0069c372c61ad8efec0c1175810.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zhu-huang-zhang-nie-zhao-lu-2026-smolin-hilbert-schmidt-distance","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"X. Zhu, K. Huang, C. Zhang, X. Nie, Q. Zhao and D. Lu, *Universal entanglement detection with geometrically optimal witnesses*, arXiv:2610.17592v1 [quant-ph]; Phys. Rev. A 114, 042409 (2026), Appendix, “Four-qubit Smolin state”. The source TeX is `manuscript.tex` (1053 lines, MD5 `db9c376c24127cc118b3e270cf74af42`). Lines 672–687 state:\n\n> A natural conjecture is that\n>\n> $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\|\\mathcal{N}_p(\\rho_S) - \\mathcal{N}_{\\frac{1}{3}}(\\rho_S)\\|, & p \\in (\\frac{1}{3}, 1], \\end{cases}$\n>\n> which simplifies to\n>\n> $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\frac{\\sqrt{3}}{4} \\left( p - \\frac{1}{3} \\right), & p \\in (\\frac{1}{3}, 1]. \\end{cases}$\n\nHere the norm is Hilbert–Schmidt, $\\|X\\|_{\\mathrm{HS}}=\\sqrt{\\operatorname{Re}\\operatorname{tr}(X^\\dagger X)}$. The appendix defines $\\rho_S=(I+\\sigma_x^{\\otimes4}+\\sigma_y^{\\otimes4}+\\sigma_z^{\\otimes4})/16$ at lines 659–661, $D(\\rho)=\\min_{\\sigma\\in\\mathrm{SEP}(\\mathcal P)}\\|\\rho-\\sigma\\|$ at lines 662–665, and $\\mathcal N_p(\\rho_S)=p\\rho_S+(1-p)I/16$ at lines 669–671. Its four representative partitions are $\\{1\\}:\\{2,3,4\\}$, $\\{1\\}:\\{2\\}:\\{3,4\\}$, $\\{1\\}:\\{2\\}:\\{3\\}:\\{4\\}$ and $\\{1,2\\}:\\{3,4\\}$ (line 666).\n\nIssue [#20844](https://github.com/the-omega-institute/trureturing/issues/20844) fixes the reading: for every $p\\in[0,1]$, the distance is the displayed function $f(p)$ for the first three partitions, and zero for the two-pair partition, which the source identifies as separable at line 668. Every minimum is attained. This is proved. The source and definitions are also recorded in [the literature note](../Library/QuantumContext/zhuhuangzhangniezhaolu2026universal.md).","title":"Exact Hilbert–Schmidt distances for noisy four-qubit Smolin states","url":"https://the-omega-institute.github.io/trureturing-pages/research/zhu-huang-zhang-nie-zhao-lu-2026-smolin-hilbert-schmidt-distance/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92e50ce0cc275be1ed6c13db92b78845b40ca4e0e7ef46b5cb8e01c1e9fc76e3.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zietak-2017-strict-spectral-polya-conjecture","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"K. Ziętak, “From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix,” Banach Center Publications 112 (2017), Conjecture 1.1, asks whether Schatten-p approximant residuals converge to the residual of a strict spectral approximant for every complex matrix subspace.\n\nThe source statement is:\n\n> Let M be a linear subspace of Cᵐˣⁿ and let Bₚ ∈ M be the cₚ-approximant to A ∈ Cᵐˣⁿ. As p tends to ∞, the matrix Rₚ = A − Bₚ converges to A − B⁽ˢᵗ⁾, where B⁽ˢᵗ⁾ is the strict spectral approximant to A.\n\n### Statement echo\n\nThe Lean `claim` quantifies over all natural dimensions, complex matrix subspaces, targets, strict spectral approximants, and real-parameter functions of approximants. Its hypotheses require strict spectral minimality and a Schatten-$p$ minimizing approximant for every $p>1$; its conclusion is convergence of residual matrices. The module's `result` is the literal negation `¬ claim`.","title":"Ziętak's strict spectral approximant convergence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/research/zietak-2017-strict-spectral-polya-conjecture/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/205ace44ad35600383ff4f340696b3a44313f6497705e5924d55c740f55d914d.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zjawin-schmid-hoban-sainz-2023-channel-assemblage-losr-refutation","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"B. Zjawin, D. Schmid, M. J. Hoban and A. B. Sainz, *The resource theory of nonclassicality of channel assemblages*, Quantum 7, 1134 (2023), arXiv:2209.10177v3, Conjecture 3:\n\n> Let the two resources $\\In^{\\,\\theta_1}_{\\A|\\X}$ and $\\In^{\\,\\theta_2}_{\\A|\\X}$ belong to the infinite family of assemblages $\\FA$. Then, the conversion $\\In^{\\,\\theta_1}_{\\A|\\X} \\overset{\\text{LOSR}}{\\longrightarrow} \\In^{\\,\\theta_2}_{\\A|\\X}$ is possible if and only if $\\theta_1 \\geq \\theta_2$.\n\nThis is the verbatim text of `channel-quantum.tex`, line 354, in the [v3 source](https://arxiv.org/src/2209.10177v3), md5 `1478712c3cb6058c040039d9ae9978e6`. The source macros `\\FA`, `\\In`, `\\A` and `\\X` denote $\\mathsf R$, $\\mathbf I$, $\\mathbb A$ and $\\mathbb X$. The definitions and locators are also in [the literature note](../Library/QuantumContext/zjawinschmidhobansainz2023channel.md).\n\nThe family has binary outcomes and settings, $a,x\\in\\{0,1\\}$, and angles $\\theta\\in(0,\\pi/2]$. Alice and Bob share $|\\phi\\rangle=(|00\\rangle+|11\\rangle)/\\sqrt2$. Bob applies the controlled gate $|0\\rangle\\langle0|_B\\otimes I+|1\\rangle\\langle1|_B\\otimes U_\\theta$, where $U_\\theta$ has rows $(\\cos(\\theta/2),-\\sin(\\theta/2))$ and $(\\sin(\\theta/2),\\cos(\\theta/2))$. The construction traces out $B$, applies Alice's effect $M_{a|0}=(I+(-1)^a\\sigma_z)/2$ or $M_{a|1}=(I+(-1)^a\\sigma_x)/2$, and traces out $A$ (Eqs. `eq:CR` and `eq:CRfam`, TeX lines 316–333).\n\nIssue [#21490](https://github.com/the-omega-institute/trureturing/issues/21490) fixes the reading: for every pair of angles in $(0,\\pi/2]$, LOSR conversion between these literal assemblages is possible exactly when the source angle is at least the target angle. The conjecture is refuted.","title":"The controlled-rotation channel assemblages do not obey the proposed LOSR angle ordering","url":"https://the-omega-institute.github.io/trureturing-pages/research/zjawin-schmid-hoban-sainz-2023-channel-assemblage-losr-refutation/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4f1a5f8fb988cc7883264f733ca1fa9904c03c23e398f6e149de7b82d707b83.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zozor-bosyk-portesi-2014-qutrit-overlap-bound-non-optimality","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"S. Zozor, G. M. Bosyk and M. Portesi, *General entropy-like uncertainty relations in finite dimensions*, arXiv:1311.5602v3, J. Phys. A 47 (2014) 495302, Section 3, after Corollary 2:\n\n> We suspect that this corollary is also valid when $N=3$, but we have not been\n> able to prove it yet.\n\nCorollary 2 states that for c > 1/√2 and N = 2 or N ≥ 4 the bound (22), B(c) = min over θ ∈ [0, arccos c] of D(θ) + D(arccos c − θ), is c-optimal. That is, it equals the minimum of the entropy sum over all states and all unitary T ∈ U(N) with c(T) = max |T_ij| = c. The entropies, the overlap and the function D are quoted in [the literature note](../Library/QuantumContext/zozor2014entropylikeuncertainty.md).\n\nIssue [#19610](https://github.com/the-omega-institute/trureturing/issues/19610) fixes the reading for the Tsallis pair of order 2, S₂(p) = 1 − Σ p_i², an admissible pair of the corollary. The suspicion then says that, for every c with 1/√2 < c ≤ 1, some T ∈ U(3) with c(T) = c and some state attain B(c). The suspicion is refuted.","title":"The overlap entropy bound is not c-optimal for qutrits","url":"https://the-omega-institute.github.io/trureturing-pages/research/zozor-bosyk-portesi-2014-qutrit-overlap-bound-non-optimality/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a58f03e9f21fdb73757c5c87b29a59137de7ed516d1bd533f0c25ec1b681512.json","evidence":{"truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"},"id":"dossier:zuppardo-ganardi-miller-bandyopadhyay-paterek-2018-nonselective-negativity-optimum","kind":"dossier","scope":"Read the source dossier for its resolution and assumptions.","status":"source-dossier","summary":"M. Zuppardo, R. Ganardi, M. Miller, S. Bandyopadhyay and T. Paterek, *Entanglement gain in measurements with unknown results* (Phys. Rev. A 99, 042319 (2019); arXiv:1806.05465), study how a non-selective measurement in an orthonormal basis changes entanglement. It maps a pure state |φ⟩ to Σ_j |⟨ψ_j|φ⟩|² |ψ_j⟩⟨ψ_j|.\n- **The derived optimum.** For two qubits and the measurement bases (28), they derive the largest final negativity: N^max_f − N_i = 1/4 − N_i/2 (Eq. (33)).\n- **The open point.** \"Although we were not able to provide a proof that this is indeed the optimal choice among all possible measurement bases, we have conducted numerical check which supports optimality.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/zuppardo2019entanglementgain.md).\n\nIssue [#18399](https://github.com/the-omega-institute/trureturing/issues/18399) fixes the reading: for every unit two-qubit ψ and every orthonormal basis, the measured state has negativity at most N(ψψ†)/2 + 1/4, and some basis attains it. This is proved.","title":"The two-qubit non-selective measurement optimum N/2 + 1/4 is global","url":"https://the-omega-institute.github.io/trureturing-pages/research/zuppardo-ganardi-miller-bandyopadhyay-paterek-2018-nonselective-negativity-optimum/"},{"aliases":[],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b02853e80918fee250b7098ca3fdb4695a388b0665cd22fa8ce3c1baf6793d3d.json","evidence":{"source_url":"https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/"},"id":"explanation:thue-morse-reduced-abelian-odd","kind":"explanation","status":"worked-example","summary":"The first seven symbols form 0110100. Compressing constant runs gives 01010, with three zeros and two ones. R counts distinct pairs across all starts, not just this prefix. The displayed reduction is one example, not a finite test of the theorem.","title":"What a reduced class counts","url":"https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/"},{"aliases":["0710.2387","Heinz H. Bauschke, Frank Deutsch, Hein Hundal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b5f728d8259220b6fe2cef8d5353fb4b8acbc12316900db24215c269369023c.json","authors":"Heinz H. Bauschke, Frank Deutsch, Hein Hundal","evidence":{"source_url":"https://arxiv.org/abs/0710.2387"},"id":"literature:arxiv:0710.2387","identifiers":{"arxiv":"0710.2387"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Characterizing arbitrarily slow convergence in the method of alternating projections","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0710.2387","year":2007},{"aliases":["0710.4167","Eric A. Carlen and Elliott H. Lieb"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca45c16ced5aa0f456dcf03ef60604f5f9073d40ee0a715486119632f14e659a.json","authors":"Eric A. Carlen and Elliott H. Lieb","evidence":{"source_url":"https://arxiv.org/abs/0710.4167"},"id":"literature:arxiv:0710.4167","identifiers":{"arxiv":"0710.4167"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Minkowski Type Trace Inequality and Strong Subadditivity of Quantum Entropy II","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0710.4167","year":2008},{"aliases":["0711.2700","Barry Simon"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/688f8dc6054ed702291356cb9df4a765efd6b7da9d72403ea3a798410895c97d.json","authors":"Barry Simon","evidence":{"source_url":"https://arxiv.org/abs/0711.2700"},"id":"literature:arxiv:0711.2700","identifiers":{"arxiv":"0711.2700"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Equilibrium Measures and Capacities in Spectral Theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0711.2700","year":2007},{"aliases":["0712.1208","Paolo Gibilisco, Fumio Hiai, Denes Petz"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/680273869c932136a22d6528ef82b1f44472b0f2d97b39c877cd778324881aa8.json","authors":"Paolo Gibilisco, Fumio Hiai, Denes Petz","evidence":{"source_url":"https://arxiv.org/abs/0712.1208"},"id":"literature:arxiv:0712.1208","identifiers":{"arxiv":"0712.1208"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum covariance, quantum Fisher information and the uncertainty principle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0712.1208","year":2007},{"aliases":["0807.2971","Jerzy Cislo and Marek Wolf"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f634d4cfcf484f313c1e828bc39b9eaf2fa64ee1dac7e6e12032fcc9c6bef4ef.json","authors":"Jerzy Cislo and Marek Wolf","evidence":{"source_url":"https://arxiv.org/abs/0807.2971"},"id":"literature:arxiv:0807.2971","identifiers":{"arxiv":"0807.2971"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Riesz and Baez-Duarte criteria for the Riemann Hypothesis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0807.2971","year":2008},{"aliases":["0808.1850","Steve Fisk"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00bd4eca7dde1812dc695d7a3b9050abe5be194a3906a9ebe6d7c150a9d23d05.json","authors":"Steve Fisk","evidence":{"source_url":"https://arxiv.org/abs/0808.1850"},"id":"literature:arxiv:0808.1850","identifiers":{"arxiv":"0808.1850"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Questions about determinants and polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0808.1850","year":2008},{"aliases":["0808.2544","Yann Bugeaud, Dalia Krieger, and Jeffrey Shallit"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/053da8deffa80bc42cb0186f46c3c4a10aab5ee4a1d5b5baac0570a9cb3faf0a.json","authors":"Yann Bugeaud, Dalia Krieger, and Jeffrey Shallit","evidence":{"source_url":"https://arxiv.org/abs/0808.2544"},"id":"literature:arxiv:0808.2544","identifiers":{"arxiv":"0808.2544"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Morphic and Automatic Words: Maximal Blocks and Diophantine Approximation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0808.2544","year":2009},{"aliases":["0812.0197","Gunnar Carlsson; Vin de Silva"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f82a7f53f35871005cb5f7293123985138d0948755a9aac09cefed0c7881e66.json","authors":"Gunnar Carlsson; Vin de Silva","evidence":{"source_url":"https://arxiv.org/abs/0812.0197"},"id":"literature:arxiv:0812.0197","identifiers":{"arxiv":"0812.0197"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Zigzag Persistence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0812.0197","year":2008},{"aliases":["0901.0827","Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f3c4039fa6a33608b146344b83fff10323e0b835f80b8c5b621b821e05bcc83.json","authors":"Pavel Etingof, Oleg Golberg, Sebastian Hensel, Tiankai Liu, Alex Schwendner, Dmitry Vaintrob, Elena Yudovina","evidence":{"source_url":"https://arxiv.org/abs/0901.0827"},"id":"literature:arxiv:0901.0827","identifiers":{"arxiv":"0901.0827"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Introduction to representation theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0901.0827","year":2009},{"aliases":["0906.2496","10.48550/arxiv.0906.2496","Walter D. Neumann"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/105683d43d6db1575064d9eac3218795352071777f5480d994d8586a53044385.json","authors":"Walter D. Neumann","evidence":{"source_url":"https://arxiv.org/abs/0906.2496"},"id":"literature:arxiv:0906.2496","identifiers":{"arxiv":"0906.2496","doi":"10.48550/arxiv.0906.2496"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On Leighton's graph covering theorem","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0906.2496","year":2009},{"aliases":["0907.3913","Koenraad M. R. Audenaert"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c6862a930678b9888cb8ccb65ca9512bfd35131cae610defcae3eee868841c19.json","authors":"Koenraad M. R. Audenaert","evidence":{"source_url":"https://arxiv.org/abs/0907.3913"},"id":"literature:arxiv:0907.3913","identifiers":{"arxiv":"0907.3913"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Variance bounds, with an application to norm bounds for commutators","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A0907.3913","year":2009},{"aliases":["1004.0864","Haisheng Li"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55f3b7e5f87f82afaac242c7a6a9daa28323dd82a538da7e7330a6454f502cb6.json","authors":"Haisheng Li","evidence":{"source_url":"https://arxiv.org/abs/1004.0864"},"id":"literature:arxiv:1004.0864","identifiers":{"arxiv":"1004.0864"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Twisted modules and pseudo-endomorphisms","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1004.0864","year":2010},{"aliases":["10.48550/arxiv.1004.1740","1004.1740"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aff4035d808ec7a1ee25c53d019fb2b03f56a66c3ceb5c3d260063af6b4d053c.json","evidence":{"source_url":"https://arxiv.org/abs/1004.1740"},"id":"literature:arxiv:1004.1740","identifiers":{"arxiv":"1004.1740","doi":"10.48550/arxiv.1004.1740"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1004.1740","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1004.1740"},{"aliases":["10.48550/arxiv.1005.4218","1005.4218"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07e82d186ac9cd41e00615bc8c3a9c41697de9cb1de937d6d155786594d8ce15.json","evidence":{"source_url":"https://arxiv.org/abs/1005.4218"},"id":"literature:arxiv:1005.4218","identifiers":{"arxiv":"1005.4218","doi":"10.48550/arxiv.1005.4218"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1005.4218","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1005.4218"},{"aliases":["1007.3049","Houmem Belkhechine and Imed Boudabbous"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a9f7671904c0c813713acf11ed65c0e449eb39e5e54816c3a15477a85085d13a.json","authors":"Houmem Belkhechine and Imed Boudabbous","evidence":{"source_url":"https://arxiv.org/abs/1007.3049"},"id":"literature:arxiv:1007.3049","identifiers":{"arxiv":"1007.3049"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Indecomposable tournaments and their indecomposable subtournaments on 5 and 7 vertices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1007.3049","year":2010},{"aliases":["1010.3346","Á. Baricz and S. Ponnusamy"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50ebe96a17a26dcb74d6f380692d60d9a49705b5bd691e1aed04281fcee84881.json","authors":"Á. Baricz and S. Ponnusamy","evidence":{"source_url":"https://arxiv.org/abs/1010.3346"},"id":"literature:arxiv:1010.3346","identifiers":{"arxiv":"1010.3346"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On Turán type inequalities for modified Bessel functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1010.3346","year":2013},{"aliases":["1011.3777","Lasha Ephremidze"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab35d40d8b4956252c678f40ce4744eb72c3785001f091521719bbd96960127f.json","authors":"Lasha Ephremidze","evidence":{"source_url":"https://arxiv.org/abs/1011.3777"},"id":"literature:arxiv:1011.3777","identifiers":{"arxiv":"1011.3777"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"An Elementary Proof of the Polynomial Matrix Spectral Factorization Theorem","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1011.3777","year":2010},{"aliases":["1104.2271","M. Fannes, F. de Melo, W. Roga, K. Życzkowski"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f77ee1d46660f36c2120acb11e04da21fe84fbc97992236383aceca58c1b7bed.json","authors":"M. Fannes, F. de Melo, W. Roga, K. Życzkowski","evidence":{"source_url":"https://arxiv.org/abs/1104.2271"},"id":"literature:arxiv:1104.2271","identifiers":{"arxiv":"1104.2271"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Matrices of fidelities for ensembles of quantum states and the Holevo quantity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1104.2271","year":2011},{"aliases":["1109.5438","Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, Sergei Starchenko"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8633e50ef1270ff652ff94a5990e419b7d54538dbe9836e1f0a0a699fcfe6f89.json","authors":"Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson, Sergei Starchenko","evidence":{"source_url":"https://arxiv.org/abs/1109.5438"},"id":"literature:arxiv:1109.5438","identifiers":{"arxiv":"1109.5438"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Vapnik-Chervonenkis density in some theories without the independence property, I","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1109.5438","year":2011},{"aliases":["1111.6630","F. Alberto Grunbaum, Luis Velazquez"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d03cac6898c505262880985b0f1dc72a1dffda6811dcad99b4c26a452dad4c7f.json","authors":"F. Alberto Grunbaum, Luis Velazquez","evidence":{"source_url":"https://arxiv.org/abs/1111.6630"},"id":"literature:arxiv:1111.6630","identifiers":{"arxiv":"1111.6630"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The quantum walk of F. Riesz","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1111.6630","year":2012},{"aliases":["1112.0749","Helge Glöckner and Lutz G. Lucht"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be38901389af8a7268339b1a84bd6b9b226747e016f32a945b800ab6412cf9a1.json","authors":"Helge Glöckner and Lutz G. Lucht","evidence":{"source_url":"https://arxiv.org/abs/1112.0749"},"id":"literature:arxiv:1112.0749","identifiers":{"arxiv":"1112.0749"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Weighted inversion of general Dirichlet series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1112.0749","year":2011},{"aliases":["1201.1093","M. E. Shirokov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/671e9b9cb3f722a01dd389901bb2fe7daa2b100a22b1e563cdf56122a83a22c3.json","authors":"M. E. Shirokov","evidence":{"source_url":"https://arxiv.org/abs/1201.1093"},"id":"literature:arxiv:1201.1093","identifiers":{"arxiv":"1201.1093"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On a conjectured property of the von Neumann entropy valid in the commutative case","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1201.1093","year":2012},{"aliases":["10.48550/arxiv.1201.5253","1201.5253"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d52c2f419b0a9d36a2af68d9dd159081a970c3f6fa67e36693666b0da1ae0d27.json","evidence":{"source_url":"https://arxiv.org/abs/1201.5253"},"id":"literature:arxiv:1201.5253","identifiers":{"arxiv":"1201.5253","doi":"10.48550/arxiv.1201.5253"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1201.5253","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1201.5253"},{"aliases":["1202.1655","Michal Adamaszek"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e12134d61636d8c0788dccfba1ad9406402d61647fb4bc205c10fb98484b1607.json","authors":"Michal Adamaszek","evidence":{"source_url":"https://arxiv.org/abs/1202.1655"},"id":"literature:arxiv:1202.1655","identifiers":{"arxiv":"1202.1655"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Hard squares on cylinders revisited","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1202.1655","year":2012},{"aliases":["10.48550/arxiv.1204.3829","1204.3829"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb4e6a55ba676fda6607c247108a9a7607fbd9aaf0fef717b0a8b86deeb49d08.json","evidence":{"source_url":"https://arxiv.org/abs/1204.3829"},"id":"literature:arxiv:1204.3829","identifiers":{"arxiv":"1204.3829","doi":"10.48550/arxiv.1204.3829"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1204.3829","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1204.3829"},{"aliases":["1205.4091","Boris Adamczewski and Jason P. Bell"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/67d9ab82b3e4b746a393baa1cab848b3ce719104df9d506bef182db7f7d99e6d.json","authors":"Boris Adamczewski and Jason P. Bell","evidence":{"source_url":"https://arxiv.org/abs/1205.4091"},"id":"literature:arxiv:1205.4091","identifiers":{"arxiv":"1205.4091"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On vanishing coefficients of algebraic power series over fields of positive characteristic","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1205.4091","year":2012},{"aliases":["10.48550/arxiv.1212.6102","1212.6102"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a72f04d25dcc61d844dc072c0c538e2137a8062bca86b068cecbc29760f8dbf9.json","evidence":{"source_url":"https://arxiv.org/abs/1212.6102"},"id":"literature:arxiv:1212.6102","identifiers":{"arxiv":"1212.6102","doi":"10.48550/arxiv.1212.6102"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1212.6102","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1212.6102"},{"aliases":["1302.5296","Sujit K. Choudhary; Sandeep K. Goyal; Thomas Konrad; Sibasish Ghosh"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30a120d3f17602e1c377c5f7f0ce471ff72cdd24dd98fdda553938e22a3c7ad5.json","authors":"Sujit K. Choudhary; Sandeep K. Goyal; Thomas Konrad; Sibasish Ghosh","evidence":{"source_url":"https://arxiv.org/abs/1302.5296"},"id":"literature:arxiv:1302.5296","identifiers":{"arxiv":"1302.5296"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Persistence of Hardy's nonlocality in time","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1302.5296","year":2013},{"aliases":["1303.6404","Sixia Yu; C. H. Oh"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4c5cb4fb79e8077c5d3463df38449070dfa2668693dfcd0fca9a45f12c97847.json","authors":"Sixia Yu; C. H. Oh","evidence":{"source_url":"https://arxiv.org/abs/1303.6404"},"id":"literature:arxiv:1303.6404","identifiers":{"arxiv":"1303.6404"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Robertson-Schrödinger Uncertainty Relation Refined by Skew Information","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1303.6404","year":2013},{"aliases":["1304.1916","Jeremie Lumbroso"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d6cb23b57b828d1f977833150b88216232f3543a8bd765d0794eabb8bb7f713.json","authors":"Jeremie Lumbroso","evidence":{"source_url":"https://arxiv.org/abs/1304.1916"},"id":"literature:arxiv:1304.1916","identifiers":{"arxiv":"1304.1916"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Optimal Discrete Uniform Generation from Coin Flips, and Applications","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1304.1916","year":2013},{"aliases":["1308.2160","Amitai Zernik"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e783ce7a22d6a569b296d8809b472a027d624f70a1b184285451ef8d5dd6cf0b.json","authors":"Amitai Zernik","evidence":{"source_url":"https://arxiv.org/abs/1308.2160"},"id":"literature:arxiv:1308.2160","identifiers":{"arxiv":"1308.2160"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1310.1273","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1310.1273"},{"aliases":["10.48550/arxiv.1310.1312","1310.1312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3913b47ac0e8efcf901c124759c87d3f286c851ddea6ff7ff705b8374adfd8e.json","evidence":{"source_url":"https://arxiv.org/abs/1310.1312"},"id":"literature:arxiv:1310.1312","identifiers":{"arxiv":"1310.1312","doi":"10.48550/arxiv.1310.1312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1310.1312","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1310.1312"},{"aliases":["1310.2416","Andrzej Nowicki"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44ed47c8368fb148858f6e329db295411b0ab5674d6580f34c6ce5d774462e82.json","authors":"Andrzej Nowicki","evidence":{"source_url":"https://arxiv.org/abs/1310.2416"},"id":"literature:arxiv:1310.2416","identifiers":{"arxiv":"1310.2416"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Strong divisibility and lcm-sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1310.2416","year":2013},{"aliases":["1311.0275","T. Baumgratz, M. Cramer, M. B. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantifying Coherence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1311.0275","year":2014},{"aliases":["10.48550/arxiv.1404.0484","1404.0484","Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57c04c8d97af1d6e94ab57834c83fc7feae91056bfe66e9286d2d07dee61c631.json","authors":"Johan Kok; Paul Fisher; Bettina Wilkens; Mokhwetha Mabula; Vivian Mukungunugwa","evidence":{"source_url":"https://arxiv.org/abs/1404.0484"},"id":"literature:arxiv:1404.0484","identifiers":{"arxiv":"1404.0484","doi":"10.48550/arxiv.1404.0484"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Characteristics of Finite Jaco Graphs, J_n(1), n in N","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1404.0484","year":2014},{"aliases":["1404.1714","Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian Mukungunugwa"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da0d98bcec8e21927e051e2057a5c5677ab25fb9bef2b6a62eb0d6f69bc3c689.json","authors":"Johan Kok, Paul Fisher, Bettina Wilkens, Mokhwetha Mabula, Vivian Mukungunugwa","evidence":{"source_url":"https://arxiv.org/abs/1404.1714"},"id":"literature:arxiv:1404.1714","identifiers":{"arxiv":"1404.1714"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Characteristics of Jaco Graphs, J_infinity(a), a in N","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1404.1714","year":2014},{"aliases":["1405.5853","Srinivasan Arunachalam, Nathaniel Johnston and Vincent Russo"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/002f9727e9d506db429eb17f9e1f96fca1fcf8b124128112faff2f1f3a58dbcb.json","authors":"Srinivasan Arunachalam, Nathaniel Johnston and Vincent Russo","evidence":{"source_url":"https://arxiv.org/abs/1405.5853"},"id":"literature:arxiv:1405.5853","identifiers":{"arxiv":"1405.5853"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Is absolute separability determined by the partial transpose?","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1405.5853","year":2015},{"aliases":["10.48550/arxiv.1501.07464","1501.07464"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe9b1b951b466f86a6d963f41b15fea238ce546bba0f9f5caefdd7882c4e9da9.json","evidence":{"source_url":"https://arxiv.org/abs/1501.07464"},"id":"literature:arxiv:1501.07464","identifiers":{"arxiv":"1501.07464","doi":"10.48550/arxiv.1501.07464"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1501.07464","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1501.07464"},{"aliases":["1502.00359","Vladimir Nikiforov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2a9673c40a7a971de06f59fea52b169b1b6b6debb6814af07f1620bcd386e06.json","authors":"Vladimir Nikiforov","evidence":{"source_url":"https://arxiv.org/abs/1502.00359"},"id":"literature:arxiv:1502.00359","identifiers":{"arxiv":"1502.00359"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Extrema of graph eigenvalues","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1502.00359","year":2015},{"aliases":["1505.03589","Jeffrey P. S. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1507.07327","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1507.07327"},{"aliases":["1512.00267","Stefan Mueller and Georg Regensburger"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ccd3c2a97f4391301e9a0ed26732ef19e67bccfc08f2f0f159fa93d7dc7f4e7.json","authors":"Stefan Mueller and Georg Regensburger","evidence":{"source_url":"https://arxiv.org/abs/1512.00267"},"id":"literature:arxiv:1512.00267","identifiers":{"arxiv":"1512.00267"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1607.01215","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1607.01215"},{"aliases":["1607.06585","Lech Jakóbczyk"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbaf0ad770fd8e180adfd1f43cb6ca9ac2bd4a0f2be8a53a858595698f2df897.json","authors":"Lech Jakóbczyk","evidence":{"source_url":"https://arxiv.org/abs/1607.06585"},"id":"literature:arxiv:1607.06585","identifiers":{"arxiv":"1607.06585"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Inequalities for the exponential remainder of the Taylor series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1608.08784","year":1993},{"aliases":["10.48550/arxiv.1609.08231","1609.08231"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/effe21198c88d95001fb6de5a34c5cbf8b03c9f6c8c13023605e70d512d5aaa1.json","evidence":{"source_url":"https://arxiv.org/abs/1609.08231"},"id":"literature:arxiv:1609.08231","identifiers":{"arxiv":"1609.08231","doi":"10.48550/arxiv.1609.08231"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1609.08231","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1609.08231"},{"aliases":["1610.05807","T. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Optimal and near-optimal probe states for quantum metrology of number conserving two-mode bosonic Hamiltonians","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1610.05807","year":2016},{"aliases":["1612.05186","Alexander Hertlein"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e7dbd1c5b595fa9f55d585d62596c77b12060b617c95bd4187b3718de06d532.json","authors":"Alexander Hertlein","evidence":{"source_url":"https://arxiv.org/abs/1612.05186"},"id":"literature:arxiv:1612.05186","identifiers":{"arxiv":"1612.05186"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Robin's inequality for new families of integers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1612.05186","year":2018},{"aliases":["10.48550/arxiv.1701.02724","1701.02724"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f4655f18f92f2251f5fce210629bcd3f63ff2206492eed1e73b3dee67b1fcb2.json","evidence":{"source_url":"https://arxiv.org/abs/1701.02724"},"id":"literature:arxiv:1701.02724","identifiers":{"arxiv":"1701.02724","doi":"10.48550/arxiv.1701.02724"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1701.02724","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1701.02724"},{"aliases":["10.48550/arxiv.1701.04788","1701.04788"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d168e2dd30bcca798f2691abac3d50be355afa95a43665b1dc5c34d73ce557a3.json","evidence":{"source_url":"https://arxiv.org/abs/1701.04788"},"id":"literature:arxiv:1701.04788","identifiers":{"arxiv":"1701.04788","doi":"10.48550/arxiv.1701.04788"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1701.04788","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1701.04788"},{"aliases":["1701.07023","Raluca Dumitru; Jose Franco"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/48d5c9b73db1307b4ece5af2b88cc3431db3ec2b7ca31568dd158418c4f9b033.json","authors":"Raluca Dumitru; Jose Franco","evidence":{"source_url":"https://arxiv.org/abs/1701.07023"},"id":"literature:arxiv:1701.07023","identifiers":{"arxiv":"1701.07023"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the monotonicity of weighted power means of matrices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1701.07023","year":2017},{"aliases":["1702.05705","Tathagata Basak"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/36616c9d437b5ae4a4db8cb597208ad7f9c5b0422e17b85dd5e9f65fe80a12a6.json","authors":"Tathagata Basak","evidence":{"source_url":"https://arxiv.org/abs/1702.05705"},"id":"literature:arxiv:1702.05705","identifiers":{"arxiv":"1702.05705"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The octonions as a twisted group algebra","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1702.05705","year":2017},{"aliases":["1703.08716","Bert L. Hartnell, Douglas F. Rall, Kirsti Wash"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43437f35e402585155c792e3a1f1c1c93f066f94731735ddf070846f03e13d14.json","authors":"Bert L. Hartnell, Douglas F. Rall, Kirsti Wash","evidence":{"source_url":"https://arxiv.org/abs/1703.08716"},"id":"literature:arxiv:1703.08716","identifiers":{"arxiv":"1703.08716"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Theoretical and computational aspects of entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1705.07160","year":2017},{"aliases":["1708.04898","Andreas Bluhm, Lukas Rauber, and Michael M. Wolf"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/692c33bd15d7725abed1721eaf8ca0ab07cb552c934ce2f315fdafdf66cf2434.json","authors":"Andreas Bluhm, Lukas Rauber, and Michael M. Wolf","evidence":{"source_url":"https://arxiv.org/abs/1708.04898"},"id":"literature:arxiv:1708.04898","identifiers":{"arxiv":"1708.04898"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum compression relative to a set of measurements","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1708.04898","year":2018},{"aliases":["1708.05556","Nicolas Gisin"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32c720761c58c4b897c3ecb812abace4e067b3d69ff351276e1ab939c6a7f758.json","authors":"Nicolas Gisin","evidence":{"source_url":"https://arxiv.org/abs/1708.05556"},"id":"literature:arxiv:1708.05556","identifiers":{"arxiv":"1708.05556"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Elegant Joint Quantum Measurement and some conjectures about N-locality in the Triangle and other Configurations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1708.05556","year":2017},{"aliases":["1711.02491","Burton S. Kaliski Jr."],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e22587d0f4d4e73ee0d68fb8ff766efde87aacf657a47a1e17a74df5e4a52c16.json","authors":"Burton S. Kaliski Jr.","evidence":{"source_url":"https://arxiv.org/abs/1711.02491"},"id":"literature:arxiv:1711.02491","identifiers":{"arxiv":"1711.02491"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Strong Quantum Darwinism and Strong Independence is equivalent to Spectrum Broadcast Structure","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1803.08936","year":2019},{"aliases":["1804.00204","Shmuel Friedland and Stephane Gaubert"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f26e86d7c93595005b7bbcfbba2996116b2aff34804101fb713c59d36485b3f.json","authors":"Shmuel Friedland and Stephane Gaubert","evidence":{"source_url":"https://arxiv.org/abs/1804.00204"},"id":"literature:arxiv:1804.00204","identifiers":{"arxiv":"1804.00204"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1807.11609","year":2018},{"aliases":["1811.01205","Haonan Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3fa9bfc01a85237a9d9cede504da09752479ef3f63c032010c10ef1a6026415e.json","authors":"Haonan Zhang","evidence":{"source_url":"https://arxiv.org/abs/1811.01205"},"id":"literature:arxiv:1811.01205","identifiers":{"arxiv":"1811.01205"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1811.12364","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1811.12364"},{"aliases":["1812.03382","C. M. Mynhardt; A. Roux"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1da46379427662ae5173db4c94f45d1b0729115c060c62e2857f6668fb3533f.json","authors":"C. M. Mynhardt; A. Roux","evidence":{"source_url":"https://arxiv.org/abs/1812.03382"},"id":"literature:arxiv:1812.03382","identifiers":{"arxiv":"1812.03382"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Irredundance Graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1812.03382","year":2021},{"aliases":["1812.11378","Yanjun Chu and Zongzhu Lin"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/21b9654e5507db2918f49dfe06b2786317e8d93f4f39d477f387e12f3bab6297.json","authors":"Yanjun Chu and Zongzhu Lin","evidence":{"source_url":"https://arxiv.org/abs/1812.11378"},"id":"literature:arxiv:1812.11378","identifiers":{"arxiv":"1812.11378"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Moduli spaces of conformal structures on Heisenberg vertex algebras","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1812.11378","year":2018},{"aliases":["10.48550/arxiv.1906.05796","1906.05796","Xiaolong Wu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/280b763b627b82503cc81214ba6b0315fb0eaf1fce47a3d467bb0b5f4803ea19.json","authors":"Xiaolong Wu","evidence":{"source_url":"https://arxiv.org/abs/1906.05796"},"id":"literature:arxiv:1906.05796","identifiers":{"arxiv":"1906.05796","doi":"10.48550/arxiv.1906.05796"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A New Type of Abundant Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1906.05796","year":2019},{"aliases":["10.48550/arxiv.1908.02445","1908.02445","Harish Vemuri"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/27183034d414161dd4d1aa6bcbdc54c19c5a12bac7e4fed7dc125a1e0e77af96.json","authors":"Harish Vemuri","evidence":{"source_url":"https://arxiv.org/abs/1908.02445"},"id":"literature:arxiv:1908.02445","identifiers":{"arxiv":"1908.02445","doi":"10.48550/arxiv.1908.02445"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1910.02039","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1910.02039"},{"aliases":["10.48550/arxiv.1910.04790","1910.04790"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8d2e2b4c4f8af5dbac784cd327450771ac6931af5538bb4fc6fabf58dbf9822.json","evidence":{"source_url":"https://arxiv.org/abs/1910.04790"},"id":"literature:arxiv:1910.04790","identifiers":{"arxiv":"1910.04790","doi":"10.48550/arxiv.1910.04790"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1910.04790","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1910.04790"},{"aliases":["1911.00721","Pranab Basu, Navin Kashyap"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0cc791a37a1f33e0fe332ff49abaec053c016f6145cd775df1d11e5577890e1.json","authors":"Pranab Basu, Navin Kashyap","evidence":{"source_url":"https://arxiv.org/abs/1911.00721"},"id":"literature:arxiv:1911.00721","identifiers":{"arxiv":"1911.00721"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Lattice Structure of Linear Subspace Codes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1911.00721","year":2019},{"aliases":["10.48550/arxiv.1912.05110","1912.05110"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f34a9d378b6a7772df4ac83c9c2e47fe8886d5dbbb1967ed3c20eaff53ed632.json","evidence":{"source_url":"https://arxiv.org/abs/1912.05110"},"id":"literature:arxiv:1912.05110","identifiers":{"arxiv":"1912.05110","doi":"10.48550/arxiv.1912.05110"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"1912.05110","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A1912.05110"},{"aliases":["10.48550/arxiv.2003.02112","2003.02112"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa83556436c1d2ed0195e5743fa2375d08e7cc86179d2c328ddf63285cd0e48a.json","evidence":{"source_url":"https://arxiv.org/abs/2003.02112"},"id":"literature:arxiv:2003.02112","identifiers":{"arxiv":"2003.02112","doi":"10.48550/arxiv.2003.02112"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2003.02112","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2003.02112"},{"aliases":["2004.07168","Taha A. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A new phenomenological definition of entropy and application to black holes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2004.07168","year":2020},{"aliases":["10.48550/arxiv.2004.14408","2004.14408"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/17f7562a41b7447a4d1cec5312bda6599ff95d73cd14899b89eaf7f150a51fa9.json","evidence":{"source_url":"https://arxiv.org/abs/2004.14408"},"id":"literature:arxiv:2004.14408","identifiers":{"arxiv":"2004.14408","doi":"10.48550/arxiv.2004.14408"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2004.14408","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2004.14408"},{"aliases":["10.48550/arxiv.2005.13566","2005.13566"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/026f392c164db0485e7af2bbcf52f43e0f3cd11d74219915054e5899407b7bfe.json","evidence":{"source_url":"https://arxiv.org/abs/2005.13566"},"id":"literature:arxiv:2005.13566","identifiers":{"arxiv":"2005.13566","doi":"10.48550/arxiv.2005.13566"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2005.13566","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2005.13566"},{"aliases":["2006.06716","An Huang; Bogdan Stoica; Xuyang Xia; Xiao Zhong"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/088f0a1241203759de61e4f77eeb4a77bab035fd3272737ecfa13746db70995d.json","authors":"An Huang; Bogdan Stoica; Xuyang Xia; Xiao Zhong","evidence":{"source_url":"https://arxiv.org/abs/2006.06716"},"id":"literature:arxiv:2006.06716","identifiers":{"arxiv":"2006.06716"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Palindromic length sequence of the ruler sequence and of the period-doubling sequence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2007.08317","year":2020},{"aliases":["2010.02408","Eric P. Hanson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8a58fc9ca1d2ebc7698c13509ea6a391175196a0023e813e52f8092e5f36fe4f.json","authors":"Eric P. Hanson","evidence":{"source_url":"https://arxiv.org/abs/2010.02408"},"id":"literature:arxiv:2010.02408","identifiers":{"arxiv":"2010.02408"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Entropic Continuity Bounds & Eventually Entanglement-Breaking Channels","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2010.02408","year":2020},{"aliases":["2010.14682","Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22d8b6b58e0269c68cc226734b3015d770a4d64adb6ba6aa226a9cddee7b0dc8.json","authors":"Chi-Fang Chen, Kohtaro Kato and Fernando G. S. L. Brandão","evidence":{"source_url":"https://arxiv.org/abs/2010.14682"},"id":"literature:arxiv:2010.14682","identifiers":{"arxiv":"2010.14682"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Matrix Product Density Operators: when do they have a local parent Hamiltonian?","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2010.14682","year":2020},{"aliases":["2012.01905","Isobel Davies and Orlando Marigliano"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8250e512678322621fe4294bf32cbc87c2d4f54a802daf3d5236fbf2d02ec4d.json","authors":"Isobel Davies and Orlando Marigliano","evidence":{"source_url":"https://arxiv.org/abs/2012.01905"},"id":"literature:arxiv:2012.01905","identifiers":{"arxiv":"2012.01905"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Coloured Graphical Models and their Symmetries","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2012.01905","year":2021},{"aliases":["2012.07453","Hui Li, Jun Wang, Xiao Yao, Zhuan Ye"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/21d229074408f99c0716b1c767bba8ac1862af16beda5545689261c4a8ed8e64.json","authors":"Hui Li, Jun Wang, Xiao Yao, Zhuan Ye","evidence":{"source_url":"https://arxiv.org/abs/2012.07453"},"id":"literature:arxiv:2012.07453","identifiers":{"arxiv":"2012.07453"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Inequalities Concerning Maximum Modulus and Zeros of Random Entire Functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2012.07453","year":2020},{"aliases":["10.48550/arxiv.2012.07872","2012.07872"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2be8d7df85fa7599d4d294afd63d0c02b16fdabc6f59a1a43b40a4fa1a70537d.json","evidence":{"source_url":"https://arxiv.org/abs/2012.07872"},"id":"literature:arxiv:2012.07872","identifiers":{"arxiv":"2012.07872","doi":"10.48550/arxiv.2012.07872"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2012.07872","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2012.07872"},{"aliases":["10.48550/arxiv.2012.13642","2012.13642"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/20b3e89dcc787c4e3775e2e38521dc2a061b7ab34962aa893353740f07b57b71.json","evidence":{"source_url":"https://arxiv.org/abs/2012.13642"},"id":"literature:arxiv:2012.13642","identifiers":{"arxiv":"2012.13642","doi":"10.48550/arxiv.2012.13642"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2012.13642","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2012.13642"},{"aliases":["10.48550/arxiv.2105.10096","2105.10096"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15ddc712d40d14615bee243500d43fb3239c7756288452340ec423d032e5efb5.json","evidence":{"source_url":"https://arxiv.org/abs/2105.10096"},"id":"literature:arxiv:2105.10096","identifiers":{"arxiv":"2105.10096","doi":"10.48550/arxiv.2105.10096"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2105.10096","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2105.10096"},{"aliases":["10.48550/arxiv.2106.09959","2106.09959"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d594435fa875b00ecae818cdb8bfba5324148db6be42e1514bc6588fa5b51070.json","evidence":{"source_url":"https://arxiv.org/abs/2106.09959"},"id":"literature:arxiv:2106.09959","identifiers":{"arxiv":"2106.09959","doi":"10.48550/arxiv.2106.09959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2106.09959","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2106.09959"},{"aliases":["10.48550/arxiv.2108.00943","2108.00943"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30e0d58ee647dbe4e580b1b31aba18343c92e4cfb7420b528aa4f1dc8944f3d2.json","evidence":{"source_url":"https://arxiv.org/abs/2108.00943"},"id":"literature:arxiv:2108.00943","identifiers":{"arxiv":"2108.00943","doi":"10.48550/arxiv.2108.00943"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2108.00943","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2108.00943"},{"aliases":["2108.02729","K. Mahalingam, P. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Counting scattered palindromes in a finite word","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2108.02729","year":2021},{"aliases":["10.48550/arxiv.2108.06662","2108.06662"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/63fa07b9b58b903608a098dcad44c53d046603c1be23c386d0a8741bfaedb7a1.json","evidence":{"source_url":"https://arxiv.org/abs/2108.06662"},"id":"literature:arxiv:2108.06662","identifiers":{"arxiv":"2108.06662","doi":"10.48550/arxiv.2108.06662"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2108.06662","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2108.06662"},{"aliases":["10.48550/arxiv.2108.07723","2108.07723"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cee53622b341890255d6b533ffa6df49702f00f7e694a5589a230e8deab1aaaa.json","evidence":{"source_url":"https://arxiv.org/abs/2108.07723"},"id":"literature:arxiv:2108.07723","identifiers":{"arxiv":"2108.07723","doi":"10.48550/arxiv.2108.07723"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2108.07723","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2108.07723"},{"aliases":["10.48550/arxiv.2109.09629","2109.09629"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3d015b7c91cd68609923a4980815f39caed6bd0e5806f505127ed5323302925.json","evidence":{"source_url":"https://arxiv.org/abs/2109.09629"},"id":"literature:arxiv:2109.09629","identifiers":{"arxiv":"2109.09629","doi":"10.48550/arxiv.2109.09629"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2109.09629","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2109.09629"},{"aliases":["2110.00738","Dinesh Pandey; Kamal Lochan Patra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7593ad69f66fe659264eb4c77cb3931d23479e14ac7e40052945abc69963e9ea.json","authors":"Dinesh Pandey; Kamal Lochan Patra","evidence":{"source_url":"https://arxiv.org/abs/2110.00738"},"id":"literature:arxiv:2110.00738","identifiers":{"arxiv":"2110.00738"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some new central parts of connected graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2110.00738","year":2021},{"aliases":["2110.02098","M. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Thermometry of Gaussian quantum systems using Gaussian measurements","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2110.02098","year":2022},{"aliases":["10.48550/arxiv.2110.03099","2110.03099"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/789f01c93f0dc798e81efa75a541ce6d52fb8885c7c2a977a04a330a86786de2.json","evidence":{"source_url":"https://arxiv.org/abs/2110.03099"},"id":"literature:arxiv:2110.03099","identifiers":{"arxiv":"2110.03099","doi":"10.48550/arxiv.2110.03099"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2110.03099","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2110.03099"},{"aliases":["10.48550/arxiv.2110.07176","2110.07176"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f083ae141b23d409f45896a4a1febf6ba60ab59a9eae1cc7ef8268bf478303af.json","evidence":{"source_url":"https://arxiv.org/abs/2110.07176"},"id":"literature:arxiv:2110.07176","identifiers":{"arxiv":"2110.07176","doi":"10.48550/arxiv.2110.07176"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2110.07176","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2110.07176"},{"aliases":["2112.00444","Michael Geißer, Theresa Körner, Sascha Kurz and Anne Zahn"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/64d95a4ed3c20cc69a9ca379a114d35325bad0835a4c60f30de56247595ef9cd.json","authors":"Michael Geißer, Theresa Körner, Sascha Kurz and Anne Zahn","evidence":{"source_url":"https://arxiv.org/abs/2112.00444"},"id":"literature:arxiv:2112.00444","identifiers":{"arxiv":"2112.00444"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Squares with three digits","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2112.00444","year":2021},{"aliases":["10.48550/arxiv.2112.05281","2112.05281"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e37540507e015e7da04e86eaba1536290f7f681580f99158c4e530b2f63cb77d.json","evidence":{"source_url":"https://arxiv.org/abs/2112.05281"},"id":"literature:arxiv:2112.05281","identifiers":{"arxiv":"2112.05281","doi":"10.48550/arxiv.2112.05281"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2112.05281","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2112.05281"},{"aliases":["10.48550/arxiv.2112.09721","2112.09721"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/470f1cc642d3f29e27641a3396b367355db17e8a0862b4a5764687f444d262dd.json","evidence":{"source_url":"https://arxiv.org/abs/2112.09721"},"id":"literature:arxiv:2112.09721","identifiers":{"arxiv":"2112.09721","doi":"10.48550/arxiv.2112.09721"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2112.09721","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2112.09721"},{"aliases":["2112.13766","Besfort Shala"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/58944250cdc908768340692b5028df9ee604fc0b53f2d52ef9623f56b1ddb329.json","authors":"Besfort Shala","evidence":{"source_url":"https://arxiv.org/abs/2112.13766"},"id":"literature:arxiv:2112.13766","identifiers":{"arxiv":"2112.13766"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Probabilistic Zeta Function of a Finite Lattice","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2112.13766","year":2024},{"aliases":["2201.07156","Matthew A. Graydon and Joshua Skanes-Norman and Joel J. Wallman"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/046fbf5becf152861eecf003278b8e7c0691e9d8ccc296824e9870d1c88d8b99.json","authors":"Matthew A. Graydon and Joshua Skanes-Norman and Joel J. Wallman","evidence":{"source_url":"https://arxiv.org/abs/2201.07156"},"id":"literature:arxiv:2201.07156","identifiers":{"arxiv":"2201.07156"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Designing Stochastic Channels","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2201.07156","year":2022},{"aliases":["2201.13303","B. Braun, K. Bruegge"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/88ac1018df23e232354412c0690b71e1e478adc80224e7318eae32891f312567.json","authors":"B. Braun, K. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2202.11520","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2202.11520"},{"aliases":["10.48550/arxiv.2202.13548","2202.13548"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2b5a86a14f226ebc841e68ac0eeff383f91c0878e9df68e87f775e3d0edb4896.json","evidence":{"source_url":"https://arxiv.org/abs/2202.13548"},"id":"literature:arxiv:2202.13548","identifiers":{"arxiv":"2202.13548","doi":"10.48550/arxiv.2202.13548"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2202.13548","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2202.13548"},{"aliases":["2203.03381","Luca Onnis"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e260f0e9a6bd7512abcdd61641c6fe7649035b4e9b88f63257c758f0fbf7393d.json","authors":"Luca Onnis","evidence":{"source_url":"https://arxiv.org/abs/2203.03381"},"id":"literature:arxiv:2203.03381","identifiers":{"arxiv":"2203.03381"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On a variant of the happy numbers and their generalizations, arXiv:2203.03381v1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2203.03381","year":2022},{"aliases":["2204.01980","Daniel R. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some explicit estimates for the error term in the prime number theorem","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2204.01980","year":2022},{"aliases":["10.48550/arxiv.2204.09910","2204.09910"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15968a73662c18eccd95956e917052c36150cad898f5fc251b049800efb0930d.json","evidence":{"source_url":"https://arxiv.org/abs/2204.09910"},"id":"literature:arxiv:2204.09910","identifiers":{"arxiv":"2204.09910","doi":"10.48550/arxiv.2204.09910"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2204.09910","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2204.09910"},{"aliases":["10.48550/arxiv.2205.00310","2205.00310"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f3f76442bc54e1bb429198b53ff4a2b4603d429bbf0fa0a2d04c0b8601ae29f6.json","evidence":{"source_url":"https://arxiv.org/abs/2205.00310"},"id":"literature:arxiv:2205.00310","identifiers":{"arxiv":"2205.00310","doi":"10.48550/arxiv.2205.00310"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2205.00310","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2205.00310"},{"aliases":["10.48550/arxiv.2206.06653","2206.06653"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8fe87c7d0c720aa8cffab947802fd24ec98f79b645f03625f5f5ebb1119a24bd.json","evidence":{"source_url":"https://arxiv.org/abs/2206.06653"},"id":"literature:arxiv:2206.06653","identifiers":{"arxiv":"2206.06653","doi":"10.48550/arxiv.2206.06653"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2206.06653","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2206.06653"},{"aliases":["10.48550/arxiv.2206.08154","2206.08154"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3ca038301d1ffabf77ab2a72f0d97d157b592781defcf210931c166efcdd4b65.json","evidence":{"source_url":"https://arxiv.org/abs/2206.08154"},"id":"literature:arxiv:2206.08154","identifiers":{"arxiv":"2206.08154","doi":"10.48550/arxiv.2206.08154"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2206.08154","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2206.08154"},{"aliases":["10.48550/arxiv.2206.09879","2206.09879"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0d8074a8f9af4b4b92e8b9391f7f214a31dc761d4fa84facae7dce9b537496e.json","evidence":{"source_url":"https://arxiv.org/abs/2206.09879"},"id":"literature:arxiv:2206.09879","identifiers":{"arxiv":"2206.09879","doi":"10.48550/arxiv.2206.09879"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2206.09879","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2206.09879"},{"aliases":["10.48550/arxiv.2206.14428","2206.14428"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1152a33b2b1e63f146f48cd707490f3a604955d6d926f58c8b0371c6b6307622.json","evidence":{"source_url":"https://arxiv.org/abs/2206.14428"},"id":"literature:arxiv:2206.14428","identifiers":{"arxiv":"2206.14428","doi":"10.48550/arxiv.2206.14428"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2206.14428","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2206.14428"},{"aliases":["10.48550/arxiv.2206.14825","2206.14825"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/015344e8f489891aa4f2a49e8fe5d09a3a74b7f5fc6183ea4661ff759825b66f.json","evidence":{"source_url":"https://arxiv.org/abs/2206.14825"},"id":"literature:arxiv:2206.14825","identifiers":{"arxiv":"2206.14825","doi":"10.48550/arxiv.2206.14825"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2206.14825","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2206.14825"},{"aliases":["10.48550/arxiv.2207.07603","2207.07603"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3dbf47fdd126db9fd13a30d585fba4b62740fa85097fca8b1247ef74ea14001a.json","evidence":{"source_url":"https://arxiv.org/abs/2207.07603"},"id":"literature:arxiv:2207.07603","identifiers":{"arxiv":"2207.07603","doi":"10.48550/arxiv.2207.07603"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2207.07603","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2207.07603"},{"aliases":["10.48550/arxiv.2207.08280","2207.08280"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b36591c72ccfa56a048214880194770bfb5d2363ed24fe93218712db28f7b2d2.json","evidence":{"source_url":"https://arxiv.org/abs/2207.08280"},"id":"literature:arxiv:2207.08280","identifiers":{"arxiv":"2207.08280","doi":"10.48550/arxiv.2207.08280"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2207.08280","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2207.08280"},{"aliases":["10.48550/arxiv.2208.01484","2208.01484"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0fc66eb168b472a6fc19692889a84a58d362015ff5a48c3e0640d4901474bdcb.json","evidence":{"source_url":"https://arxiv.org/abs/2208.01484"},"id":"literature:arxiv:2208.01484","identifiers":{"arxiv":"2208.01484","doi":"10.48550/arxiv.2208.01484"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2208.01484","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2208.01484"},{"aliases":["10.48550/arxiv.2208.05527","2208.05527"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f3a89f64cb5b8d3712889e921717881ecfff26a0434fe2c8dcbb68ee0f2a83f.json","evidence":{"source_url":"https://arxiv.org/abs/2208.05527"},"id":"literature:arxiv:2208.05527","identifiers":{"arxiv":"2208.05527","doi":"10.48550/arxiv.2208.05527"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2208.05527","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2208.05527"},{"aliases":["10.48550/arxiv.2208.06052","2208.06052"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0ea7bc69d65df1206eca8d8237066f45e53f04d8b0b617e875315ef97adce8a.json","evidence":{"source_url":"https://arxiv.org/abs/2208.06052"},"id":"literature:arxiv:2208.06052","identifiers":{"arxiv":"2208.06052","doi":"10.48550/arxiv.2208.06052"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2208.06052","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2208.06052"},{"aliases":["10.48550/arxiv.2209.03909","2209.03909"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c0892d688b10625e69a516ea1a914e615963837254e822868a255889e2e0f33d.json","evidence":{"source_url":"https://arxiv.org/abs/2209.03909"},"id":"literature:arxiv:2209.03909","identifiers":{"arxiv":"2209.03909","doi":"10.48550/arxiv.2209.03909"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2209.03909","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2209.03909"},{"aliases":["10.48550/arxiv.2210.00370","2210.00370"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d7b109780c8f8bbb7fa8061edc6a64913f98f7c2cbd808a8949a561fb4e3823.json","evidence":{"source_url":"https://arxiv.org/abs/2210.00370"},"id":"literature:arxiv:2210.00370","identifiers":{"arxiv":"2210.00370","doi":"10.48550/arxiv.2210.00370"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2210.00370","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2210.00370"},{"aliases":["10.48550/arxiv.2210.11038","2210.11038"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8a5afcfd51e9556c70cac85709037bda378d49514804d4db79e992a7ac87dba.json","evidence":{"source_url":"https://arxiv.org/abs/2210.11038"},"id":"literature:arxiv:2210.11038","identifiers":{"arxiv":"2210.11038","doi":"10.48550/arxiv.2210.11038"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2210.11038","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2210.11038"},{"aliases":["10.48550/arxiv.2211.01495","2211.01495"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd5227092453061beb06eca83cace366d5b0eeb499f7fbcf41a95c5355043cac.json","evidence":{"source_url":"https://arxiv.org/abs/2211.01495"},"id":"literature:arxiv:2211.01495","identifiers":{"arxiv":"2211.01495","doi":"10.48550/arxiv.2211.01495"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2211.01495","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2211.01495"},{"aliases":["10.48550/arxiv.2211.04379","2211.04379"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/972a2b8310ea471c1e646640c5f127ed47b37890c5979c58a7b131feb9699dfe.json","evidence":{"source_url":"https://arxiv.org/abs/2211.04379"},"id":"literature:arxiv:2211.04379","identifiers":{"arxiv":"2211.04379","doi":"10.48550/arxiv.2211.04379"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2211.04379","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2211.04379"},{"aliases":["10.48550/arxiv.2301.05758","2301.05758"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec8b1d0c81e4ba90d7f3488381beb3336b4633d656d9e8a159fa5f76b029bd16.json","evidence":{"source_url":"https://arxiv.org/abs/2301.05758"},"id":"literature:arxiv:2301.05758","identifiers":{"arxiv":"2301.05758","doi":"10.48550/arxiv.2301.05758"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2301.05758","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2301.05758"},{"aliases":["10.48550/arxiv.2301.07934","2301.07934"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8c73d38ce6f2c17e01973046cf3b108e772de432e1b325c10bebb93eac57020.json","evidence":{"source_url":"https://arxiv.org/abs/2301.07934"},"id":"literature:arxiv:2301.07934","identifiers":{"arxiv":"2301.07934","doi":"10.48550/arxiv.2301.07934"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2301.07934","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2301.07934"},{"aliases":["10.48550/arxiv.2302.10491","2302.10491"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8836dabc84f997413b5384796c3117b86aa4909ede9aa1c3e9def47877cdb384.json","evidence":{"source_url":"https://arxiv.org/abs/2302.10491"},"id":"literature:arxiv:2302.10491","identifiers":{"arxiv":"2302.10491","doi":"10.48550/arxiv.2302.10491"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2302.10491","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2302.10491"},{"aliases":["2302.12243","Stanley Gudder"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6983b0992967493aa2888c624a0e7d3f11c8d11072e9620e8b7e4d26e47480c.json","authors":"Stanley Gudder","evidence":{"source_url":"https://arxiv.org/abs/2302.12243"},"id":"literature:arxiv:2302.12243","identifiers":{"arxiv":"2302.12243"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Dual Quantum Instruments and Sub-observables","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2302.12243","year":2023},{"aliases":["2302.12361","Hayato Arai"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b41cf7bba4663b4639bdb9a1397d09c3f0a2eaa311afad08014b12b2f1f9819.json","authors":"Hayato Arai","evidence":{"source_url":"https://arxiv.org/abs/2302.12361"},"id":"literature:arxiv:2302.12361","identifiers":{"arxiv":"2302.12361"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Diversity of Entanglement Structures with Self-Duality and Non-Orthogonal State Discrimination in General Probabilistic Theories","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2302.12361","year":2023},{"aliases":["2302.13647","France Gheeraert, Giuseppe Romana, Manon Stipulanti"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30be9e1383d6bd14f35f3bd17dd80426338c8a6ca170b36e4e6aa617984902df.json","authors":"France Gheeraert, Giuseppe Romana, Manon Stipulanti","evidence":{"source_url":"https://arxiv.org/abs/2302.13647"},"id":"literature:arxiv:2302.13647","identifiers":{"arxiv":"2302.13647"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"String attractors of some simple-Parry automatic sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2302.13647","year":2023},{"aliases":["10.48550/arxiv.2303.01952","2303.01952"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57358f2575c17cf9cb63cd54574246e825c514fe4eae3f4e2b8d576f53814e33.json","evidence":{"source_url":"https://arxiv.org/abs/2303.01952"},"id":"literature:arxiv:2303.01952","identifiers":{"arxiv":"2303.01952","doi":"10.48550/arxiv.2303.01952"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2303.01952","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2303.01952"},{"aliases":["10.48550/arxiv.2303.04771","2303.04771"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/646ce39523df1528dca081f62a3d2a795dbf48f2a83f22c89642d7bbda56dd6f.json","evidence":{"source_url":"https://arxiv.org/abs/2303.04771"},"id":"literature:arxiv:2303.04771","identifiers":{"arxiv":"2303.04771","doi":"10.48550/arxiv.2303.04771"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2303.04771","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2303.04771"},{"aliases":["10.48550/arxiv.2305.08349","2305.08349"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d913e71e4ab50e326961992395f34fe98e46611b0b3ffdecaf00d38a9087359.json","evidence":{"source_url":"https://arxiv.org/abs/2305.08349"},"id":"literature:arxiv:2305.08349","identifiers":{"arxiv":"2305.08349","doi":"10.48550/arxiv.2305.08349"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2305.08349","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2305.08349"},{"aliases":["2306.16172","H. V. Dedania and A. B. Patel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e7f1deea57c96962ba94f989b41914cce2613a3b9bba6f03cc0c88459d81c3a.json","authors":"H. V. Dedania and A. B. Patel","evidence":{"source_url":"https://arxiv.org/abs/2306.16172"},"id":"literature:arxiv:2306.16172","identifiers":{"arxiv":"2306.16172"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Spatial numerical range in non-unital, normed algebras and their unitizations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2306.16172","year":2024},{"aliases":["10.48550/arxiv.2306.16456","2306.16456"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2073dc500d2769eb30127dea02fcb35457f5b1a25380f850e46e42b327b002ad.json","evidence":{"source_url":"https://arxiv.org/abs/2306.16456"},"id":"literature:arxiv:2306.16456","identifiers":{"arxiv":"2306.16456","doi":"10.48550/arxiv.2306.16456"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2306.16456","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2306.16456"},{"aliases":["2307.03977","Jagannath Bhanja"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9a4777abe7d3b107b9e6391a22290d3b7da21f0e8457508ab8c251458baff56.json","authors":"Jagannath Bhanja","evidence":{"source_url":"https://arxiv.org/abs/2307.03977"},"id":"literature:arxiv:2307.03977","identifiers":{"arxiv":"2307.03977"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Sumsets with a minimum number of distinct terms","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2307.03977","year":2025},{"aliases":["10.48550/arxiv.2307.05820","2307.05820"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9cc914fbd513100f06ff55e16a4e39edd2683ca2b36ea978fc1c91db94ab9651.json","evidence":{"source_url":"https://arxiv.org/abs/2307.05820"},"id":"literature:arxiv:2307.05820","identifiers":{"arxiv":"2307.05820","doi":"10.48550/arxiv.2307.05820"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2307.05820","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2307.05820"},{"aliases":["10.48550/arxiv.2307.08520","2307.08520"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a24a14afb3ca462977edf8c31386956f7381436e2b2036acd7c592d0aeaac1a3.json","evidence":{"source_url":"https://arxiv.org/abs/2307.08520"},"id":"literature:arxiv:2307.08520","identifiers":{"arxiv":"2307.08520","doi":"10.48550/arxiv.2307.08520"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2307.08520","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2307.08520"},{"aliases":["10.48550/arxiv.2307.15688","2307.15688"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61066f846e3ce771818f5f00cdd3009c20b459d355fe48b87eb923793382b4d9.json","evidence":{"source_url":"https://arxiv.org/abs/2307.15688"},"id":"literature:arxiv:2307.15688","identifiers":{"arxiv":"2307.15688","doi":"10.48550/arxiv.2307.15688"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2307.15688","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2307.15688"},{"aliases":["10.48550/arxiv.2308.01181","2308.01181"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c586d1fd9307facbd64292b19add9c91ffa42bb577e7ec53841c05b0f4a25232.json","evidence":{"source_url":"https://arxiv.org/abs/2308.01181"},"id":"literature:arxiv:2308.01181","identifiers":{"arxiv":"2308.01181","doi":"10.48550/arxiv.2308.01181"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2308.01181","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2308.01181"},{"aliases":["2308.01894","Ningping Cao; Maxwell Fitzsimmons; Zachary Mann; Rajesh Pereira; Raymond Laflamme"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a407fa63880b13ea7a7bdfeba2203ceb5e359824157761285ca4f7754dfed089.json","authors":"Ningping Cao; Maxwell Fitzsimmons; Zachary Mann; Rajesh Pereira; Raymond Laflamme","evidence":{"source_url":"https://arxiv.org/abs/2308.01894"},"id":"literature:arxiv:2308.01894","identifiers":{"arxiv":"2308.01894"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Maps Between CPTP and HPTP","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2308.01894","year":2023},{"aliases":["10.48550/arxiv.2308.07642","2308.07642"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/14fda44b2fa8bce5e674ef828699a38832cf715758fe9643fb10b255ed11e47f.json","evidence":{"source_url":"https://arxiv.org/abs/2308.07642"},"id":"literature:arxiv:2308.07642","identifiers":{"arxiv":"2308.07642","doi":"10.48550/arxiv.2308.07642"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2308.07642","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2308.07642"},{"aliases":["10.48550/arxiv.2308.10332","2308.10332"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8a1d743f170e27a911b7cd82928cae6613cf1e7d9c3a8138de1d35d538dc57da.json","evidence":{"source_url":"https://arxiv.org/abs/2308.10332"},"id":"literature:arxiv:2308.10332","identifiers":{"arxiv":"2308.10332","doi":"10.48550/arxiv.2308.10332"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2308.10332","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2308.10332"},{"aliases":["10.48550/arxiv.2308.10771","2308.10771"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85a6164e6b2e286669e235b85dadc6cb52d718f9afde59607708570d87fb0577.json","evidence":{"source_url":"https://arxiv.org/abs/2308.10771"},"id":"literature:arxiv:2308.10771","identifiers":{"arxiv":"2308.10771","doi":"10.48550/arxiv.2308.10771"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2308.10771","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2308.10771"},{"aliases":["10.48550/arxiv.2309.09621","2309.09621"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c5dd6ada674d0d6718819958a685ca11643f7850f23a0d23292f2442d372d70.json","evidence":{"source_url":"https://arxiv.org/abs/2309.09621"},"id":"literature:arxiv:2309.09621","identifiers":{"arxiv":"2309.09621","doi":"10.48550/arxiv.2309.09621"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2309.09621","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2309.09621"},{"aliases":["2311.00806","Khoi-Nguyen Huynh-Vu; Lin Htoo Zaw; Valerio Scarani"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ec0363c1f30a2f80ce1bbcd83f3e187060338fef54af88f4df2169b2b4ad239.json","authors":"Khoi-Nguyen Huynh-Vu; Lin Htoo Zaw; Valerio Scarani","evidence":{"source_url":"https://arxiv.org/abs/2311.00806"},"id":"literature:arxiv:2311.00806","identifiers":{"arxiv":"2311.00806"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Certification of genuine multipartite entanglement in spin ensembles with measurements of total angular momentum","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2311.00806","year":2024},{"aliases":["10.48550/arxiv.2311.04689","2311.04689"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83956da881f17664d24ddf4698e6dda08d80a8919035a242c5346f15ae57e87a.json","evidence":{"source_url":"https://arxiv.org/abs/2311.04689"},"id":"literature:arxiv:2311.04689","identifiers":{"arxiv":"2311.04689","doi":"10.48550/arxiv.2311.04689"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2311.04689","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2311.04689"},{"aliases":["2311.13072","Peter Kagey; William Keehn"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d5f849494294e01e355fac4dee8e119d1343ca78e1b38b26409c3a0b035731f.json","authors":"Peter Kagey; William Keehn","evidence":{"source_url":"https://arxiv.org/abs/2311.13072"},"id":"literature:arxiv:2311.13072","identifiers":{"arxiv":"2311.13072"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Counting tilings of the n x m grid, cylinder, and torus","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2311.13072","year":2023},{"aliases":["2311.17448","D. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Open problems on polynomials, their zero distribution and related questions: 2023","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2312.07754","year":2023},{"aliases":["10.48550/arxiv.2312.10855","2312.10855"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d2b9f8247e2f45cd51fa5e5131820f77224d506129825b3ff866aa34486b8243.json","evidence":{"source_url":"https://arxiv.org/abs/2312.10855"},"id":"literature:arxiv:2312.10855","identifiers":{"arxiv":"2312.10855","doi":"10.48550/arxiv.2312.10855"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2312.10855","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2312.10855"},{"aliases":["10.48550/arxiv.2401.07943","2401.07943","Aidan Hennessey"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/100cf3a2eae6434def77e917dcbbeb020596f6c6bf846119ffe3e71bf0fd19cc.json","authors":"Aidan Hennessey","evidence":{"source_url":"https://arxiv.org/abs/2401.07943"},"id":"literature:arxiv:2401.07943","identifiers":{"arxiv":"2401.07943","doi":"10.48550/arxiv.2401.07943"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Tree and Tripod Nim","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2401.07943","year":2024},{"aliases":["10.48550/arxiv.2403.03280","2403.03280"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c556853bb64319511715ff962cdee72917285489607df95737a38b6242917615.json","evidence":{"source_url":"https://arxiv.org/abs/2403.03280"},"id":"literature:arxiv:2403.03280","identifiers":{"arxiv":"2403.03280","doi":"10.48550/arxiv.2403.03280"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2403.03280","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2403.03280"},{"aliases":["10.48550/arxiv.2403.07668","2403.07668"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0b6a424561b51a5f59c150be0f82820cfbbbb13332652021b5a4f1d1c551ab4.json","evidence":{"source_url":"https://arxiv.org/abs/2403.07668"},"id":"literature:arxiv:2403.07668","identifiers":{"arxiv":"2403.07668","doi":"10.48550/arxiv.2403.07668"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2403.07668","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2403.07668"},{"aliases":["10.48550/arxiv.2404.11312","2404.11312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51cb982048df2485f6ce817621ba03d1987653024b5fad2b2a37d8a3ace28b45.json","evidence":{"source_url":"https://arxiv.org/abs/2404.11312"},"id":"literature:arxiv:2404.11312","identifiers":{"arxiv":"2404.11312","doi":"10.48550/arxiv.2404.11312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2404.11312","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2404.11312"},{"aliases":["10.48550/arxiv.2404.18622","2404.18622"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ff2891260316b6bd36f30d2db1ff7414cf1baf5b6f94128baa8538b7a228263.json","evidence":{"source_url":"https://arxiv.org/abs/2404.18622"},"id":"literature:arxiv:2404.18622","identifiers":{"arxiv":"2404.18622","doi":"10.48550/arxiv.2404.18622"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2404.18622","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2404.18622"},{"aliases":["2404.18806","Richard J. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Bivariate Generating Functions Enumerating Non-Bonding Dominoes on Rectangular Boards","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2404.18806","year":2024},{"aliases":["10.48550/arxiv.2405.02727","2405.02727"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32225105cbc43f57f01b1fadcc1c79dae29eab19a77ecf0e7f8f6d15bb5deb73.json","evidence":{"source_url":"https://arxiv.org/abs/2405.02727"},"id":"literature:arxiv:2405.02727","identifiers":{"arxiv":"2405.02727","doi":"10.48550/arxiv.2405.02727"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2405.02727","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2405.02727"},{"aliases":["2405.08678","Robert Fulsche, Franz Luef, Reinhard F. Werner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d811299978feff85366617d7a5486450ef811ce27c5b7017246605e63129f4e.json","authors":"Robert Fulsche, Franz Luef, Reinhard F. Werner","evidence":{"source_url":"https://arxiv.org/abs/2405.08678"},"id":"literature:arxiv:2405.08678","identifiers":{"arxiv":"2405.08678"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Wiener's Tauberian theorem in classical and quantum harmonic analysis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2405.08678","year":2025},{"aliases":["10.48550/arxiv.2405.08939","2405.08939"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5bbbe0605dc3bd4251cdd9833eeff02a561f5ef703c91be811f2787dd68b36c6.json","evidence":{"source_url":"https://arxiv.org/abs/2405.08939"},"id":"literature:arxiv:2405.08939","identifiers":{"arxiv":"2405.08939","doi":"10.48550/arxiv.2405.08939"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2405.08939","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2405.08939"},{"aliases":["2405.09626","Harry Buhrman; Dmitry Grinko; Philip Verduyn Lunel; Jordi Weggemans"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e7c5696b924f0d6185d7a59e4212cbc9a76594d329467c9caeb2592ca7cf21f0.json","authors":"Harry Buhrman; Dmitry Grinko; Philip Verduyn Lunel; Jordi Weggemans","evidence":{"source_url":"https://arxiv.org/abs/2405.09626"},"id":"literature:arxiv:2405.09626","identifiers":{"arxiv":"2405.09626"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Permutation tests for quantum state identity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2405.09626","year":2026},{"aliases":["10.48550/arxiv.2405.19593","2405.19593"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95fbaa3e691044a88a9f4c31f14dd54b8db8a64e6e91b5c6c94c157eb9e3a4f7.json","evidence":{"source_url":"https://arxiv.org/abs/2405.19593"},"id":"literature:arxiv:2405.19593","identifiers":{"arxiv":"2405.19593","doi":"10.48550/arxiv.2405.19593"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2405.19593","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2405.19593"},{"aliases":["2406.04045","B. Bajnok and W. K. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Diameter of Undirected Cayley Graphs of Finite Abelian Groups","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2406.04045","year":2024},{"aliases":["10.48550/arxiv.2406.09956","2406.09956"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a08fa2a8106407181ae9fbd6217082672b213371967ca499a59d1e5c0976ec7.json","evidence":{"source_url":"https://arxiv.org/abs/2406.09956"},"id":"literature:arxiv:2406.09956","identifiers":{"arxiv":"2406.09956","doi":"10.48550/arxiv.2406.09956"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2406.09956","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2406.09956"},{"aliases":["10.48550/arxiv.2406.12083","2406.12083"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/05f65f151bc708636f58af3321dcd1909aed4b341163cce95120a105ad3b8a0f.json","evidence":{"source_url":"https://arxiv.org/abs/2406.12083"},"id":"literature:arxiv:2406.12083","identifiers":{"arxiv":"2406.12083","doi":"10.48550/arxiv.2406.12083"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2406.12083","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2406.12083"},{"aliases":["10.48550/arxiv.2407.01835","2407.01835","Noah Kravitz"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe3be55d6c98539c06c2d2fd0656ba1b5cab122e391490b33cc1c4319efa4a72.json","authors":"Noah Kravitz","evidence":{"source_url":"https://arxiv.org/abs/2407.01835"},"id":"literature:arxiv:2407.01835","identifiers":{"arxiv":"2407.01835","doi":"10.48550/arxiv.2407.01835"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Rearranging small sets for distinct partial sums","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.01835","year":2024},{"aliases":["2407.03278","D.-Ş. Marinescu; C. P. Niculescu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2b7ece9c6c76aa524a1360a334fdd87bea4ff27c26bd95ad616376e49f0555c.json","authors":"D.-Ş. Marinescu; C. P. Niculescu","evidence":{"source_url":"https://arxiv.org/abs/2407.03278"},"id":"literature:arxiv:2407.03278","identifiers":{"arxiv":"2407.03278"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A survey of the Hornich-Hlawka inequality","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.03278","year":2024},{"aliases":["10.48550/arxiv.2407.05768","2407.05768"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52ce029ac50408f309c05e86c675ca60314d023b3ab04bdd30479a3b753bcdeb.json","evidence":{"source_url":"https://arxiv.org/abs/2407.05768"},"id":"literature:arxiv:2407.05768","identifiers":{"arxiv":"2407.05768","doi":"10.48550/arxiv.2407.05768"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2407.05768","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.05768"},{"aliases":["10.48550/arxiv.2407.06338","2407.06338"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba2c2cec09d2329a471e405bb4ee9beeede0f4c759d77c0225bcc10600d6f550.json","evidence":{"source_url":"https://arxiv.org/abs/2407.06338"},"id":"literature:arxiv:2407.06338","identifiers":{"arxiv":"2407.06338","doi":"10.48550/arxiv.2407.06338"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2407.06338","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.06338"},{"aliases":["10.48550/arxiv.2407.08237","2407.08237"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4cdaa31726fe3ae1e44d16269f27701657e88b4f12089dc80fe461f6ca22b713.json","evidence":{"source_url":"https://arxiv.org/abs/2407.08237"},"id":"literature:arxiv:2407.08237","identifiers":{"arxiv":"2407.08237","doi":"10.48550/arxiv.2407.08237"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2407.08237","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.08237"},{"aliases":["2407.08900","D. Sain; J. Manna; K. Paul"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/741f2d02f67893b5a9450f597b08ba385411192f7a49a6262ed306d4d9f4fd96.json","authors":"D. Sain; J. Manna; K. Paul","evidence":{"source_url":"https://arxiv.org/abs/2407.08900"},"id":"literature:arxiv:2407.08900","identifiers":{"arxiv":"2407.08900"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On local preservation of orthogonality and its application to isometries","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.08900","year":2024},{"aliases":["2407.20048","Brennan Benfield, Oliver Lippard"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fac28e1b9320b5a9317899a473a41d87364f12e0102ed05d7f87dc7783b06624.json","authors":"Brennan Benfield, Oliver Lippard","evidence":{"source_url":"https://arxiv.org/abs/2407.20048"},"id":"literature:arxiv:2407.20048","identifiers":{"arxiv":"2407.20048"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Connecting Zeros in Pisano Periods to Prime Factors of K-Fibonacci Numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2407.20048","year":2025},{"aliases":["2408.00487","Riccardo Bonetto; Hildeberto Jardón Kojakhmetov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cde85ad2aac0573f4b32f33e97db698949aae265bef22095acc9ae231684db83.json","authors":"Riccardo Bonetto; Hildeberto Jardón Kojakhmetov","evidence":{"source_url":"https://arxiv.org/abs/2408.00487"},"id":"literature:arxiv:2408.00487","identifiers":{"arxiv":"2408.00487"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Eigenvalues of Graphs with Mixed Algebraic Structure","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2408.00487","year":2024},{"aliases":["2408.07658","R. Grünwald; Zs. Páles"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/245046e07475f5994a5c91f2548d741b9f1a98c595f55e64ee3dff93b7cbdc4b.json","authors":"R. Grünwald; Zs. Páles","evidence":{"source_url":"https://arxiv.org/abs/2408.07658"},"id":"literature:arxiv:2408.07658","identifiers":{"arxiv":"2408.07658"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Comparison of Gini means with fixed number of variables","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2408.07658","year":2024},{"aliases":["2408.14632","Morgan Fiebig, aBa Mbirika, Jürgen Spilker"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1cf157837d29f6323ab90f6a65d8766e7d34f4ffc6c4676f2d279dba5677fb7.json","authors":"Morgan Fiebig, aBa Mbirika, Jürgen Spilker","evidence":{"source_url":"https://arxiv.org/abs/2408.14632"},"id":"literature:arxiv:2408.14632","identifiers":{"arxiv":"2408.14632"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Period patterns, entry points, and orders in the Lucas sequences: theory and applications","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2408.14632","year":2025},{"aliases":["10.48550/arxiv.2408.15000","2408.15000"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/22c210cafd56df6e61f1e25b43426c294ce372b09930058471eb59bad519da0d.json","evidence":{"source_url":"https://arxiv.org/abs/2408.15000"},"id":"literature:arxiv:2408.15000","identifiers":{"arxiv":"2408.15000","doi":"10.48550/arxiv.2408.15000"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2408.15000","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2408.15000"},{"aliases":["10.48550/arxiv.2408.15723","2408.15723"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/224c52db55ae17350a2ad8079a63b708986bba5f8d489d62dc84e8cbf28bb596.json","evidence":{"source_url":"https://arxiv.org/abs/2408.15723"},"id":"literature:arxiv:2408.15723","identifiers":{"arxiv":"2408.15723","doi":"10.48550/arxiv.2408.15723"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2408.15723","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2408.15723"},{"aliases":["2409.02306","William Q. Erickson, Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, Mitchell Minyard, Kyle Rosengartner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bcc487ae8be43658d1adbbe3a89ac4db227f660ddb99f486d93b2b467e7a98c1.json","authors":"William Q. Erickson, Daniel Herden, Jonathan Meddaugh, Mark R. Sepanski, Mitchell Minyard, Kyle Rosengartner","evidence":{"source_url":"https://arxiv.org/abs/2409.02306"},"id":"literature:arxiv:2409.02306","identifiers":{"arxiv":"2409.02306"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Limits and Periodicity of Metamour 2-Distance Graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2409.02306","year":2024},{"aliases":["10.48550/arxiv.2409.11268","2409.11268"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9dcf6d35cb4abe3a5725548cece537847a6a9b337f4bc6717905ced6e355fd48.json","evidence":{"source_url":"https://arxiv.org/abs/2409.11268"},"id":"literature:arxiv:2409.11268","identifiers":{"arxiv":"2409.11268","doi":"10.48550/arxiv.2409.11268"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2409.11268","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2409.11268"},{"aliases":["10.48550/arxiv.2410.00416","2410.00416"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9f18889ae0ecc5ef1171eaf11168909ed2343e0496b198c3fde8a140603754eb.json","evidence":{"source_url":"https://arxiv.org/abs/2410.00416"},"id":"literature:arxiv:2410.00416","identifiers":{"arxiv":"2410.00416","doi":"10.48550/arxiv.2410.00416"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2410.00416","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.00416"},{"aliases":["2410.00529","Pierre Letouzey, Shuo Li, Wolfgang Steiner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/032c4714cc2dc2dc2174a061bd95dc357fb5e827db7602e8ad24b08ab66cca19.json","authors":"Pierre Letouzey, Shuo Li, Wolfgang Steiner","evidence":{"source_url":"https://arxiv.org/abs/2410.00529"},"id":"literature:arxiv:2410.00529","identifiers":{"arxiv":"2410.00529"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Pointwise order of generalized Hofstadter functions G, H and beyond","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.00529","year":2024},{"aliases":["10.48550/arxiv.2410.07467","2410.07467"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6fdf19f2925380e08b19f5691a1331ba293e9893595185c374cacef622f1960f.json","evidence":{"source_url":"https://arxiv.org/abs/2410.07467"},"id":"literature:arxiv:2410.07467","identifiers":{"arxiv":"2410.07467","doi":"10.48550/arxiv.2410.07467"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2410.07467","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.07467"},{"aliases":["2410.11852","Roberto Garrappa, Stefan Gerhold, Marina Popolizio, Thomas Simon"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d2332cbff1cae53f671df4eb44df07417ca95d5605c560e8648bd33b79a68320.json","authors":"Roberto Garrappa, Stefan Gerhold, Marina Popolizio, Thomas Simon","evidence":{"source_url":"https://arxiv.org/abs/2410.11852"},"id":"literature:arxiv:2410.11852","identifiers":{"arxiv":"2410.11852"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On some inequalities for the two-parameter Mittag-Leffler function in the complex plane","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.11852","year":2025},{"aliases":["10.48550/arxiv.2410.13575","2410.13575"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e650e79bb39f0236a4fa428b0036338df5d0454a7a10ea2c31780340c3ad82e3.json","evidence":{"source_url":"https://arxiv.org/abs/2410.13575"},"id":"literature:arxiv:2410.13575","identifiers":{"arxiv":"2410.13575","doi":"10.48550/arxiv.2410.13575"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2410.13575","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.13575"},{"aliases":["10.48550/arxiv.2410.17681","2410.17681"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f9e2d9391319fa0792d2d1cb7845e3c8d499110d1067efa519cc5579d375926.json","evidence":{"source_url":"https://arxiv.org/abs/2410.17681"},"id":"literature:arxiv:2410.17681","identifiers":{"arxiv":"2410.17681","doi":"10.48550/arxiv.2410.17681"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2410.17681","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.17681"},{"aliases":["2410.21215","Fuchuan Wei and Zi-Wen Liu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8258e8328d18c42134f7f340095b3bd2f6ec72dd3559bc40951342f6d126d762.json","authors":"Fuchuan Wei and Zi-Wen Liu","evidence":{"source_url":"https://arxiv.org/abs/2410.21215"},"id":"literature:arxiv:2410.21215","identifiers":{"arxiv":"2410.21215"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Noise robustness and threshold of many-body quantum magic","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.21215","year":2024},{"aliases":["2410.21961","Teng Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/37e14102d30e3a71bc5573a97c758883141ab1ed799b1ffccb431dba7ff0fc5b.json","authors":"Teng Zhang","evidence":{"source_url":"https://arxiv.org/abs/2410.21961"},"id":"literature:arxiv:2410.21961","identifiers":{"arxiv":"2410.21961"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Clarkson–McCarthy type inequalities, part I: ℓ_p–ℓ_p and ℓ_q–ℓ_p Schatten p-estimates","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.21961","year":2024},{"aliases":["10.48550/arxiv.2410.23997","2410.23997"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbd5c9fd5388716445b675fb08f0818325f22309459250ff6cbe9989f9364d55.json","evidence":{"source_url":"https://arxiv.org/abs/2410.23997"},"id":"literature:arxiv:2410.23997","identifiers":{"arxiv":"2410.23997","doi":"10.48550/arxiv.2410.23997"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2410.23997","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2410.23997"},{"aliases":["10.48550/arxiv.2411.03648","2411.03648"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4ab6abe97433b4367b6f17319b9b8eb8de5c51e40b72e52fd0621fce297455a.json","evidence":{"source_url":"https://arxiv.org/abs/2411.03648"},"id":"literature:arxiv:2411.03648","identifiers":{"arxiv":"2411.03648","doi":"10.48550/arxiv.2411.03648"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2411.03648","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.03648"},{"aliases":["10.48550/arxiv.2411.08139","2411.08139"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a182a0bbc656cc37b9b984049ac5308f641e650d47668b56c738fb4be8e57c39.json","evidence":{"source_url":"https://arxiv.org/abs/2411.08139"},"id":"literature:arxiv:2411.08139","identifiers":{"arxiv":"2411.08139","doi":"10.48550/arxiv.2411.08139"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2411.08139","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.08139"},{"aliases":["2411.11914","Michael Yang, Hansen Shieh, Ashley Yu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c2c1a3d2e35d9c355459bcf45c11398b7aaf4707a045dea8ac4e0f2253f83ec.json","authors":"Michael Yang, Hansen Shieh, Ashley Yu","evidence":{"source_url":"https://arxiv.org/abs/2411.11914"},"id":"literature:arxiv:2411.11914","identifiers":{"arxiv":"2411.11914"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Stack-Sorting with Dotted-Pattern-Avoiding Stacks","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.11914","year":2025},{"aliases":["10.48550/arxiv.2411.14488","2411.14488"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/834324f0b8ceafdc65b79f722cea4049563f5edf3a593298c105df6a0c659447.json","evidence":{"source_url":"https://arxiv.org/abs/2411.14488"},"id":"literature:arxiv:2411.14488","identifiers":{"arxiv":"2411.14488","doi":"10.48550/arxiv.2411.14488"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2411.14488","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.14488"},{"aliases":["10.48550/arxiv.2411.17778","2411.17778"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/511c0abe80930306442278fe02b973b4fa005ab056c2d2959106142f7b1f8c01.json","evidence":{"source_url":"https://arxiv.org/abs/2411.17778"},"id":"literature:arxiv:2411.17778","identifiers":{"arxiv":"2411.17778","doi":"10.48550/arxiv.2411.17778"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2411.17778","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.17778"},{"aliases":["2411.18946","Frederik vom Ende; Fereshte Shahbeigi"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af8b1e429909fbcb95bc9f6cbe07c7078becbedf169890307860cb21e0c38d3a.json","authors":"Frederik vom Ende; Fereshte Shahbeigi","evidence":{"source_url":"https://arxiv.org/abs/2411.18946"},"id":"literature:arxiv:2411.18946","identifiers":{"arxiv":"2411.18946"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Generating Sets of Stochastic Matrices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.18946","year":2024},{"aliases":["10.48550/arxiv.2411.19364","2411.19364"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ec1e7524d2669c0ea3824cef8f2a597d5894c850baccb200347c98c7defd23c.json","evidence":{"source_url":"https://arxiv.org/abs/2411.19364"},"id":"literature:arxiv:2411.19364","identifiers":{"arxiv":"2411.19364","doi":"10.48550/arxiv.2411.19364"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2411.19364","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2411.19364"},{"aliases":["10.48550/arxiv.2412.00336","2412.00336"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2771a4d94c0c0b04e99939add9442bd82a351ba9b3892306d08da55848c01a6f.json","evidence":{"source_url":"https://arxiv.org/abs/2412.00336"},"id":"literature:arxiv:2412.00336","identifiers":{"arxiv":"2412.00336","doi":"10.48550/arxiv.2412.00336"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2412.00336","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2412.00336"},{"aliases":["10.48550/arxiv.2412.07667","2412.07667"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d1e51938ea9e25a234f11cd22e97dd404d991c90dbfd6d5463b452e85a235a93.json","evidence":{"source_url":"https://arxiv.org/abs/2412.07667"},"id":"literature:arxiv:2412.07667","identifiers":{"arxiv":"2412.07667","doi":"10.48550/arxiv.2412.07667"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2412.07667","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2412.07667"},{"aliases":["10.48550/arxiv.2412.08022","2412.08022"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b4f101f4d324bcca08a34d9a6e27cca8dfee58299a18c0dd149814a71c87801e.json","evidence":{"source_url":"https://arxiv.org/abs/2412.08022"},"id":"literature:arxiv:2412.08022","identifiers":{"arxiv":"2412.08022","doi":"10.48550/arxiv.2412.08022"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2412.08022","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2412.08022"},{"aliases":["10.48550/arxiv.2501.00357","2501.00357"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/420768246905ec85d172b102bf6a0ee24f5b785815d1fe0482f95336d71711d3.json","evidence":{"source_url":"https://arxiv.org/abs/2501.00357"},"id":"literature:arxiv:2501.00357","identifiers":{"arxiv":"2501.00357","doi":"10.48550/arxiv.2501.00357"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2501.00357","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2501.00357"},{"aliases":["10.48550/arxiv.2501.04209","2501.04209"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1b25cf1c77c0f90d51a6330d11d86eee828b060b02576f388aa781673c549a6.json","evidence":{"source_url":"https://arxiv.org/abs/2501.04209"},"id":"literature:arxiv:2501.04209","identifiers":{"arxiv":"2501.04209","doi":"10.48550/arxiv.2501.04209"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2501.04209","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2501.04209"},{"aliases":["10.48550/arxiv.2501.07269","2501.07269"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a46af6b93dccdafcc0a4a65fcc4eeaa872ccf51a00c6a9e636381ccee4717d0.json","evidence":{"source_url":"https://arxiv.org/abs/2501.07269"},"id":"literature:arxiv:2501.07269","identifiers":{"arxiv":"2501.07269","doi":"10.48550/arxiv.2501.07269"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2501.07269","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2501.07269"},{"aliases":["10.48550/arxiv.2501.14340","2501.14340"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf864970ed55406ed7e583935233dd40514aa0c4c1190379ddfbd1650436bcda.json","evidence":{"source_url":"https://arxiv.org/abs/2501.14340"},"id":"literature:arxiv:2501.14340","identifiers":{"arxiv":"2501.14340","doi":"10.48550/arxiv.2501.14340"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2501.14340","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2501.14340"},{"aliases":["10.48550/arxiv.2501.19272","2501.19272"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e194bf1cc25127637658b3d8333610e1c27878413c3577fe0888c400bfcd22a.json","evidence":{"source_url":"https://arxiv.org/abs/2501.19272"},"id":"literature:arxiv:2501.19272","identifiers":{"arxiv":"2501.19272","doi":"10.48550/arxiv.2501.19272"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2501.19272","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2501.19272"},{"aliases":["10.48550/arxiv.2502.00134","2502.00134"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4e4c554ad3df656098b0449cbeb8d4b3d76cb08b3a3de65eb168826f4292c3d.json","evidence":{"source_url":"https://arxiv.org/abs/2502.00134"},"id":"literature:arxiv:2502.00134","identifiers":{"arxiv":"2502.00134","doi":"10.48550/arxiv.2502.00134"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.00134","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.00134"},{"aliases":["10.48550/arxiv.2502.00777","2502.00777"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b8fe355492b9470375603db36c70828c86e9e4e1299576c6d3261c4db34b6b19.json","evidence":{"source_url":"https://arxiv.org/abs/2502.00777"},"id":"literature:arxiv:2502.00777","identifiers":{"arxiv":"2502.00777","doi":"10.48550/arxiv.2502.00777"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.00777","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.00777"},{"aliases":["10.48550/arxiv.2502.01815","2502.01815"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/80536af187ae03c387ffc860e8984479c905502a63c1826beebaac49b4e42fa0.json","evidence":{"source_url":"https://arxiv.org/abs/2502.01815"},"id":"literature:arxiv:2502.01815","identifiers":{"arxiv":"2502.01815","doi":"10.48550/arxiv.2502.01815"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.01815","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.01815"},{"aliases":["10.48550/arxiv.2502.05939","2502.05939"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1fbf90eea9fa81635470ee5acb0fd8c4e61090acbd26a12bce63fe672b5efc72.json","evidence":{"source_url":"https://arxiv.org/abs/2502.05939"},"id":"literature:arxiv:2502.05939","identifiers":{"arxiv":"2502.05939","doi":"10.48550/arxiv.2502.05939"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.05939","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.05939"},{"aliases":["10.48550/arxiv.2502.05993","2502.05993"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc0858eb5ddf13a2643b85dc26255e25c9f7f50c32140195a4f0e81bbb71c235.json","evidence":{"source_url":"https://arxiv.org/abs/2502.05993"},"id":"literature:arxiv:2502.05993","identifiers":{"arxiv":"2502.05993","doi":"10.48550/arxiv.2502.05993"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.05993","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.05993"},{"aliases":["10.48550/arxiv.2502.10521","2502.10521"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5483cf136737525df145fb0b40bde1659e9adaa0397ecc29c8267b97e1b810a7.json","evidence":{"source_url":"https://arxiv.org/abs/2502.10521"},"id":"literature:arxiv:2502.10521","identifiers":{"arxiv":"2502.10521","doi":"10.48550/arxiv.2502.10521"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.10521","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.10521"},{"aliases":["10.48550/arxiv.2502.10655","2502.10655"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18ea9ff1ba4b870d864d4705ce3586f4a495f1f280db2bf2474729e7cb3529df.json","evidence":{"source_url":"https://arxiv.org/abs/2502.10655"},"id":"literature:arxiv:2502.10655","identifiers":{"arxiv":"2502.10655","doi":"10.48550/arxiv.2502.10655"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.10655","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.10655"},{"aliases":["10.48550/arxiv.2502.11760","2502.11760"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d68cfb45f86318dea9a1c156bb665065cb310ce717ff1db8d30f5d0cc99e6974.json","evidence":{"source_url":"https://arxiv.org/abs/2502.11760"},"id":"literature:arxiv:2502.11760","identifiers":{"arxiv":"2502.11760","doi":"10.48550/arxiv.2502.11760"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.11760","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.11760"},{"aliases":["2502.12240","Alexey Milekhin; Zofia Adamska; John Preskill"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db6449394ec494f78de54a1b13122158543dfff67b91518aec4211d24845d6eb.json","authors":"Alexey Milekhin; Zofia Adamska; John Preskill","evidence":{"source_url":"https://arxiv.org/abs/2502.12240"},"id":"literature:arxiv:2502.12240","identifiers":{"arxiv":"2502.12240"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Observable and computable entanglement in time","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.12240","year":2025},{"aliases":["10.48550/arxiv.2502.14341","2502.14341","Dickson Y. B. Annor"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5a9086a3ec2a3e65df7f2bbbe5f5a2eed0421c454192080a168eb9b001e9cf55.json","authors":"Dickson Y. B. Annor","evidence":{"source_url":"https://arxiv.org/abs/2502.14341"},"id":"literature:arxiv:2502.14341","identifiers":{"arxiv":"2502.14341","doi":"10.48550/arxiv.2502.14341"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Domination Parameters of Graph Covers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.14341","year":2025},{"aliases":["2502.17411"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a7eb612f4c1c0b119f3df273ae11a1159e72ce9560c9fa5f7e0e3c17ace4350d.json","evidence":{"source_url":"https://arxiv.org/abs/2502.17411"},"id":"literature:arxiv:2502.17411","identifiers":{"arxiv":"2502.17411"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2502.17411","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.17411"},{"aliases":["2502.20659","A. Christiana; B. Clingenpeel; H. Guo; J. Oh; J. H. Przytycki; X. Wang; H. Yun"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/092be57550af55248fe6a4ca2b6e7c1bdb398564e8719310ffe875582e13fa5e.json","authors":"A. Christiana; B. Clingenpeel; H. Guo; J. Oh; J. H. Przytycki; X. Wang; H. Yun","evidence":{"source_url":"https://arxiv.org/abs/2502.20659"},"id":"literature:arxiv:2502.20659","identifiers":{"arxiv":"2502.20659"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Low Dimensional Homology of the Yang-Baxter Operators Yielding the HOMFLYPT Polynomial","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2502.20659","year":2025},{"aliases":["10.48550/arxiv.2503.04122","2503.04122"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c9127f982459500a440fe41ba29825c56aeac196d06e7d7a81addd6de17703c5.json","evidence":{"source_url":"https://arxiv.org/abs/2503.04122"},"id":"literature:arxiv:2503.04122","identifiers":{"arxiv":"2503.04122","doi":"10.48550/arxiv.2503.04122"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2503.04122","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2503.04122"},{"aliases":["10.48550/arxiv.2503.12201","2503.12201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b28badfe543eefd4d13bfe101407e0bf9d303a921699122c6174ca8bfde196e9.json","evidence":{"source_url":"https://arxiv.org/abs/2503.12201"},"id":"literature:arxiv:2503.12201","identifiers":{"arxiv":"2503.12201","doi":"10.48550/arxiv.2503.12201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2503.12201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2503.12201"},{"aliases":["10.48550/arxiv.2504.00835","2504.00835"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da46be6539cf4c22b910362cfe99f721ce90340242329c1b48f7b4cd87379b9c.json","evidence":{"source_url":"https://arxiv.org/abs/2504.00835"},"id":"literature:arxiv:2504.00835","identifiers":{"arxiv":"2504.00835","doi":"10.48550/arxiv.2504.00835"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2504.00835","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.00835"},{"aliases":["2504.04879","Véronique Gayrard"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd0d6e8660b22e268ba47b26cdcbeecdb505eaabbb289be6dbf5b63a6deb4778.json","authors":"Véronique Gayrard","evidence":{"source_url":"https://arxiv.org/abs/2504.04879"},"id":"literature:arxiv:2504.04879","identifiers":{"arxiv":"2504.04879"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Mixed memories in Hopfield networks","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.04879","year":2025},{"aliases":["10.48550/arxiv.2504.05123","2504.05123"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83ba7e5ea67a09ca0674cf4d310213812f3c13f0e99e26972cf20ef6ccc9f1c4.json","evidence":{"source_url":"https://arxiv.org/abs/2504.05123"},"id":"literature:arxiv:2504.05123","identifiers":{"arxiv":"2504.05123","doi":"10.48550/arxiv.2504.05123"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2504.05123","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.05123"},{"aliases":["10.48550/arxiv.2504.13108","2504.13108","Owen John Levens and Joel Brewster Lewis and Bridget Eileen Tenner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/020fa87ab9ee9f7714b4a4641626a38040b75881df31f7fbee12b5c509846e7a.json","authors":"Owen John Levens and Joel Brewster Lewis and Bridget Eileen Tenner","evidence":{"source_url":"https://arxiv.org/abs/2504.13108"},"id":"literature:arxiv:2504.13108","identifiers":{"arxiv":"2504.13108","doi":"10.48550/arxiv.2504.13108"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Global patterns in signed permutations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.13108","year":2025},{"aliases":["10.48550/arxiv.2504.13729","2504.13729"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/699856561e06bbf8a2ccd140a0986d09c59064a478b8e97a122261dce90b77bd.json","evidence":{"source_url":"https://arxiv.org/abs/2504.13729"},"id":"literature:arxiv:2504.13729","identifiers":{"arxiv":"2504.13729","doi":"10.48550/arxiv.2504.13729"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2504.13729","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.13729"},{"aliases":["10.48550/arxiv.2504.15401","2504.15401"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d1a713e652d441f2e303ccdf940afeaec18eeb591f03bd0bc8cebfc91b06df6f.json","evidence":{"source_url":"https://arxiv.org/abs/2504.15401"},"id":"literature:arxiv:2504.15401","identifiers":{"arxiv":"2504.15401","doi":"10.48550/arxiv.2504.15401"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2504.15401","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.15401"},{"aliases":["10.48550/arxiv.2504.19661","2504.19661"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/552ede96bb0a5137c7c682f7695685439f055ea20ff9be9afddf28bf0831a7fa.json","evidence":{"source_url":"https://arxiv.org/abs/2504.19661"},"id":"literature:arxiv:2504.19661","identifiers":{"arxiv":"2504.19661","doi":"10.48550/arxiv.2504.19661"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2504.19661","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2504.19661"},{"aliases":["2505.03967","Erwin Miña-Díaz; Olof Rubin"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12f716217b48522d9286b0f541af85ced3ea2213c353a57a7f4d3bc44b988de0.json","authors":"Erwin Miña-Díaz; Olof Rubin","evidence":{"source_url":"https://arxiv.org/abs/2505.03967"},"id":"literature:arxiv:2505.03967","identifiers":{"arxiv":"2505.03967"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Chebyshev polynomials on equipotential curves","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.03967","year":2025},{"aliases":["10.48550/arxiv.2505.04882","2505.04882"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/512dcea3def8675c4f3357d79989ace7e3ac1aabec801c0ccfa27c122bc24721.json","evidence":{"source_url":"https://arxiv.org/abs/2505.04882"},"id":"literature:arxiv:2505.04882","identifiers":{"arxiv":"2505.04882","doi":"10.48550/arxiv.2505.04882"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.04882","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.04882"},{"aliases":["2505.06457","Andrés Carnero Bravo"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/613de7e8205a676de8543e531f8f83798f952ea787deb021055804b49b5c1ba1.json","authors":"Andrés Carnero Bravo","evidence":{"source_url":"https://arxiv.org/abs/2505.06457"},"id":"literature:arxiv:2505.06457","identifiers":{"arxiv":"2505.06457"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On independence complexes of graph products","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.06457","year":2025},{"aliases":["2505.06567","Wanchen Zhang; Zicheng Han; Fei Shi; Xiande Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7cf77b9e275c9d0ebbb2c5d68b1218732820e5946de2f32d487d8a2f3b369375.json","authors":"Wanchen Zhang; Zicheng Han; Fei Shi; Xiande Zhang","evidence":{"source_url":"https://arxiv.org/abs/2505.06567"},"id":"literature:arxiv:2505.06567","identifiers":{"arxiv":"2505.06567"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"New constructions of multipartite entanglement resistant to particle loss","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.06567","year":2025},{"aliases":["10.48550/arxiv.2505.06893","2505.06893"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9bcd180a56536d357897e5c828c4fd7f1e707a96fc27a6947cbdddac555d7393.json","evidence":{"source_url":"https://arxiv.org/abs/2505.06893"},"id":"literature:arxiv:2505.06893","identifiers":{"arxiv":"2505.06893","doi":"10.48550/arxiv.2505.06893"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.06893","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.06893"},{"aliases":["10.48550/arxiv.2505.08090","2505.08090"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61369ffd1865f1efe3a03f902b187beb85625155d727d02eefa98f5a987a9be6.json","evidence":{"source_url":"https://arxiv.org/abs/2505.08090"},"id":"literature:arxiv:2505.08090","identifiers":{"arxiv":"2505.08090","doi":"10.48550/arxiv.2505.08090"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.08090","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.08090"},{"aliases":["10.48550/arxiv.2505.11986","2505.11986"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/366ba4edb3f29a03cf73aec9ef1ee67fcfa9067e4d60b2bbadeaa0c0300b97f5.json","evidence":{"source_url":"https://arxiv.org/abs/2505.11986"},"id":"literature:arxiv:2505.11986","identifiers":{"arxiv":"2505.11986","doi":"10.48550/arxiv.2505.11986"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.11986","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.11986"},{"aliases":["10.48550/arxiv.2505.13796","2505.13796"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c60a9734e97dc81ada77dc7f6bdbad137eb945ecc1a9190e461ac1b5ad71bff.json","evidence":{"source_url":"https://arxiv.org/abs/2505.13796"},"id":"literature:arxiv:2505.13796","identifiers":{"arxiv":"2505.13796","doi":"10.48550/arxiv.2505.13796"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.13796","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.13796"},{"aliases":["2505.15615","Frederik vom Ende and Simon Cichy"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9cdb75adc81652d570a6cc9a61d26b492f64124b77f73988f252909b9e1554d5.json","authors":"Frederik vom Ende and Simon Cichy","evidence":{"source_url":"https://arxiv.org/abs/2505.15615"},"id":"literature:arxiv:2505.15615","identifiers":{"arxiv":"2505.15615"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Simple Sufficient Criteria for Optimality of Entanglement Witnesses","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.15615","year":2025},{"aliases":["10.48550/arxiv.2505.18539","2505.18539"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5356723c5ed1f8d83fd1e8809e71afea7c6ed01ccce27f9e1460ea8667520209.json","evidence":{"source_url":"https://arxiv.org/abs/2505.18539"},"id":"literature:arxiv:2505.18539","identifiers":{"arxiv":"2505.18539","doi":"10.48550/arxiv.2505.18539"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2505.18539","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.18539"},{"aliases":["2505.20811","Hojin Chu, Homoon Ryu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb0c26d00157322ab222bc1734173cb220d640edb6d81ccd892d6d9804306d93.json","authors":"Hojin Chu, Homoon Ryu","evidence":{"source_url":"https://arxiv.org/abs/2505.20811"},"id":"literature:arxiv:2505.20811","identifiers":{"arxiv":"2505.20811"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Linear-Time Computation of the Frobenius Normal Form for Symmetric Toeplitz Matrices via Graph-Theoretic Decomposition","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.20811","year":2025},{"aliases":["2505.21167","Martin Ravn Christiansen"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/80e3b1f0ec521f07927ab4120393b3c33fb97d007630677e92791540a7cf3702.json","authors":"Martin Ravn Christiansen","evidence":{"source_url":"https://arxiv.org/abs/2505.21167"},"id":"literature:arxiv:2505.21167","identifiers":{"arxiv":"2505.21167"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Correlational Bound for Eigenvalues of Fermionic 2-Body Operators","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.21167","year":2025},{"aliases":["2505.23638","A. Benedito; G. Sierra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9580612f29f5ec8a386969c0733bb275c1114b0119713c36dab558ed1ac10ae6.json","authors":"A. Benedito; G. Sierra","evidence":{"source_url":"https://arxiv.org/abs/2505.23638"},"id":"literature:arxiv:2505.23638","identifiers":{"arxiv":"2505.23638"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Visualizing Three-Qubit Entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2505.23638","year":2025},{"aliases":["10.48550/arxiv.2506.01843","2506.01843"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e50f0cf3b961b0573a4c5ba765fa50ecc727139d1ff562a9196e42251f1b793.json","evidence":{"source_url":"https://arxiv.org/abs/2506.01843"},"id":"literature:arxiv:2506.01843","identifiers":{"arxiv":"2506.01843","doi":"10.48550/arxiv.2506.01843"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.01843","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.01843"},{"aliases":["10.48550/arxiv.2506.02455","2506.02455","Jack Allsop and Ian M. Wanless"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33a20a03179c957d42b6d1fa523157cf8b2221a543c153911cc02d24655e0b6b.json","authors":"Jack Allsop and Ian M. Wanless","evidence":{"source_url":"https://arxiv.org/abs/2506.02455"},"id":"literature:arxiv:2506.02455","identifiers":{"arxiv":"2506.02455","doi":"10.48550/arxiv.2506.02455"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Perfect 1-factorisations of K_{11,11}","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.02455","year":2025},{"aliases":["2506.03441","Anuj Apte; Ojas Parekh; James Sud"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b6a70e3064a00c3b481d01037c2d90d711eb87a9d673d0ef8904d87b955f3f4.json","authors":"Anuj Apte; Ojas Parekh; James Sud","evidence":{"source_url":"https://arxiv.org/abs/2506.03441"},"id":"literature:arxiv:2506.03441","identifiers":{"arxiv":"2506.03441"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Conjectured Bounds for 2-Local Hamiltonians via Token Graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.03441","year":2025},{"aliases":["10.48550/arxiv.2506.04002","2506.04002"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30740d614a8b6750a9d01821e39eedaacc9ed01dc32ca6522b8961cfc5e99d1a.json","evidence":{"source_url":"https://arxiv.org/abs/2506.04002"},"id":"literature:arxiv:2506.04002","identifiers":{"arxiv":"2506.04002","doi":"10.48550/arxiv.2506.04002"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.04002","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.04002"},{"aliases":["10.48550/arxiv.2506.04991","2506.04991"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7ea323ce0b6ee4a1a900c1d6bbc9a317eb2bb64edeba099ad56da4481d29ab9.json","evidence":{"source_url":"https://arxiv.org/abs/2506.04991"},"id":"literature:arxiv:2506.04991","identifiers":{"arxiv":"2506.04991","doi":"10.48550/arxiv.2506.04991"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.04991","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.04991"},{"aliases":["2506.09493","Pierre J. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"TRAPs, Generalisations of MZVs, Locality and Resurgence for Quantum Field Theories","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.09493","year":2025},{"aliases":["10.48550/arxiv.2506.09912","2506.09912"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/434d8f58a5347ee5b677fb5d6b1edc6a64875198ce72c1b672086ae46b64dfbf.json","evidence":{"source_url":"https://arxiv.org/abs/2506.09912"},"id":"literature:arxiv:2506.09912","identifiers":{"arxiv":"2506.09912","doi":"10.48550/arxiv.2506.09912"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.09912","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.09912"},{"aliases":["2506.14523","Géza Tóth; József Pitrik"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/80f3a7b4ff39616cc02fe79552b7383da1e375d418be2488f90f6238e4e96eec.json","authors":"Géza Tóth; József Pitrik","evidence":{"source_url":"https://arxiv.org/abs/2506.14523"},"id":"literature:arxiv:2506.14523","identifiers":{"arxiv":"2506.14523"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Wasserstein distance and its relation to several types of fidelities","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.14523","year":2026},{"aliases":["2506.15609","Sophia Denker; Satoya Imai; Otfried Gühne"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73f537f959172fd3d6d486af291a9bdf733bc0929f1545f7bc76eb5ef51f3911.json","authors":"Sophia Denker; Satoya Imai; Otfried Gühne","evidence":{"source_url":"https://arxiv.org/abs/2506.15609"},"id":"literature:arxiv:2506.15609","identifiers":{"arxiv":"2506.15609"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Chiral Symmetries and Multiparticle Entanglement","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.15609","year":2025},{"aliases":["10.48550/arxiv.2506.16432","2506.16432"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e6488ae52081c95ac48b0b86ed6cc455101c7b1e1176795ba8b26a9a5c715cf.json","evidence":{"source_url":"https://arxiv.org/abs/2506.16432"},"id":"literature:arxiv:2506.16432","identifiers":{"arxiv":"2506.16432","doi":"10.48550/arxiv.2506.16432"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.16432","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.16432"},{"aliases":["10.48550/arxiv.2506.17235","2506.17235"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1170cb6de5b56b8b51311d329de874ece97d6177e2ef71e689083e2c54c14fa1.json","evidence":{"source_url":"https://arxiv.org/abs/2506.17235"},"id":"literature:arxiv:2506.17235","identifiers":{"arxiv":"2506.17235","doi":"10.48550/arxiv.2506.17235"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.17235","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.17235"},{"aliases":["10.48550/arxiv.2506.18813","2506.18813"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51c76e54d70a18374c07e19c1ce3d8cf01429e6453a653140ca279249d7b8b00.json","evidence":{"source_url":"https://arxiv.org/abs/2506.18813"},"id":"literature:arxiv:2506.18813","identifiers":{"arxiv":"2506.18813","doi":"10.48550/arxiv.2506.18813"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.18813","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.18813"},{"aliases":["10.48550/arxiv.2506.23936","2506.23936"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f2836a7506f24edff01e347fe20484727433e5d8f1ec758f0cba46089784a02f.json","evidence":{"source_url":"https://arxiv.org/abs/2506.23936"},"id":"literature:arxiv:2506.23936","identifiers":{"arxiv":"2506.23936","doi":"10.48550/arxiv.2506.23936"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2506.23936","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2506.23936"},{"aliases":["10.48550/arxiv.2507.09276","2507.09276"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3040916df867674b37a0853a633a08ad8af466a60e5cb7334e44f48d306debd0.json","evidence":{"source_url":"https://arxiv.org/abs/2507.09276"},"id":"literature:arxiv:2507.09276","identifiers":{"arxiv":"2507.09276","doi":"10.48550/arxiv.2507.09276"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2507.09276","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.09276"},{"aliases":["2507.11596","S. R. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Identically vanishing k-generalized Fibonacci polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.11596","year":2026},{"aliases":["10.48550/arxiv.2507.16500","2507.16500","Johan Kok"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4bbc6228fc9ea10acd4110dff98204800ef0828a4a37947b5a85447e798a4c8d.json","authors":"Johan Kok","evidence":{"source_url":"https://arxiv.org/abs/2507.16500"},"id":"literature:arxiv:2507.16500","identifiers":{"arxiv":"2507.16500","doi":"10.48550/arxiv.2507.16500"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Integer sequences with conjectured relation with certain graph parameters of the family of linear Jaco graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.16500","year":2025},{"aliases":["2507.17370","Sébastien Ferenczi, Luca Q. Zamboni"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/75d33b6227a3d502d310216caa7318228abbeddf059131950d00980299c9cf47.json","authors":"Sébastien Ferenczi, Luca Q. Zamboni","evidence":{"source_url":"https://arxiv.org/abs/2507.17370"},"id":"literature:arxiv:2507.17370","identifiers":{"arxiv":"2507.17370"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Clustering, order conditions, and languages of interval exchanges","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.17370","year":2025},{"aliases":["10.48550/arxiv.2507.18767","2507.18767"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/008cf585fe7e0959a66f6bb527ad318ce5d87b55f20b1eeeae8102f05f21c75d.json","evidence":{"source_url":"https://arxiv.org/abs/2507.18767"},"id":"literature:arxiv:2507.18767","identifiers":{"arxiv":"2507.18767","doi":"10.48550/arxiv.2507.18767"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2507.18767","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.18767"},{"aliases":["10.48550/arxiv.2507.20847","2507.20847"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b52930923bd004f31dbde9ed1555b4eccec2bfc797917474f4702c3d3f13fd5.json","evidence":{"source_url":"https://arxiv.org/abs/2507.20847"},"id":"literature:arxiv:2507.20847","identifiers":{"arxiv":"2507.20847","doi":"10.48550/arxiv.2507.20847"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2507.20847","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.20847"},{"aliases":["10.48550/arxiv.2507.22719","2507.22719"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a984ceaf84c5899013d762df5d3826c1a42401f02974e52ed0174b294091482.json","evidence":{"source_url":"https://arxiv.org/abs/2507.22719"},"id":"literature:arxiv:2507.22719","identifiers":{"arxiv":"2507.22719","doi":"10.48550/arxiv.2507.22719"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2507.22719","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2507.22719"},{"aliases":["10.48550/arxiv.2508.00813","2508.00813"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6d912c430b6f9d74df0129784fc9434a1f99cebda5d735333f366f77c6c77fc.json","evidence":{"source_url":"https://arxiv.org/abs/2508.00813"},"id":"literature:arxiv:2508.00813","identifiers":{"arxiv":"2508.00813","doi":"10.48550/arxiv.2508.00813"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2508.00813","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2508.00813"},{"aliases":["10.48550/arxiv.2508.04054","2508.04054"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1a15a5ac4917d046c3ecca6122feb72e245b91271c3c348848e52b5491c16aa.json","evidence":{"source_url":"https://arxiv.org/abs/2508.04054"},"id":"literature:arxiv:2508.04054","identifiers":{"arxiv":"2508.04054","doi":"10.48550/arxiv.2508.04054"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2508.04054","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2508.04054"},{"aliases":["10.48550/arxiv.2508.06531","2508.06531"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da2942d8ea3e5e917156e4f3d4b8811e77f4abdab33022af22425a3cc37244c5.json","evidence":{"source_url":"https://arxiv.org/abs/2508.06531"},"id":"literature:arxiv:2508.06531","identifiers":{"arxiv":"2508.06531","doi":"10.48550/arxiv.2508.06531"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2508.06531","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2508.06531"},{"aliases":["10.48550/arxiv.2508.08335","2508.08335"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a5a4e9371d15726cd5048d7763e1d5cc2beea2d494508c9f92b2ed03dc4729b.json","evidence":{"source_url":"https://arxiv.org/abs/2508.08335"},"id":"literature:arxiv:2508.08335","identifiers":{"arxiv":"2508.08335","doi":"10.48550/arxiv.2508.08335"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2508.08335","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2508.08335"},{"aliases":["10.48550/arxiv.2508.10341","2508.10341"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c11fa0c1c9f71e60e555c62d1408f03d3c123bf225fe1780fbda6c4b7dd3bc09.json","evidence":{"source_url":"https://arxiv.org/abs/2508.10341"},"id":"literature:arxiv:2508.10341","identifiers":{"arxiv":"2508.10341","doi":"10.48550/arxiv.2508.10341"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2508.10341","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2508.10341"},{"aliases":["10.48550/arxiv.2508.10664","2508.10664"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33b31d0ef7e210eb0e1ab9b2d741a01a1510cdaf7ffb3c1b31bd2eb2c170f22d.json","evidence":{"source_url":"https://arxiv.org/abs/2508.10664"},"id":"literature:arxiv:2508.10664","identifiers":{"arxiv":"2508.10664","doi":"10.48550/arxiv.2508.10664"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2509.02874","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2509.02874"},{"aliases":["2509.04312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31545fee1da071dbfe747f075a26c5da91c7dabecfe1d8c7f7f960c2be39601b.json","evidence":{"source_url":"https://arxiv.org/abs/2509.04312"},"id":"literature:arxiv:2509.04312","identifiers":{"arxiv":"2509.04312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2509.04312","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2509.04312"},{"aliases":["10.48550/arxiv.2509.08059","2509.08059"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a356f1bef4e28100db63bb111d4b8c9d489928596337752fcf33ac5fa9b2105.json","evidence":{"source_url":"https://arxiv.org/abs/2509.08059"},"id":"literature:arxiv:2509.08059","identifiers":{"arxiv":"2509.08059","doi":"10.48550/arxiv.2509.08059"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Reduced complexities for sequences over finite alphabets","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2509.16034","year":2025},{"aliases":["2509.16819","Blake C. Stacey"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a9947ef346866e5d458f4110c650010bca04d7fb490f401ce3cb2de8dde4b862.json","authors":"Blake C. Stacey","evidence":{"source_url":"https://arxiv.org/abs/2509.16819"},"id":"literature:arxiv:2509.16819","identifiers":{"arxiv":"2509.16819"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Log-majorizations between quasi-geometric type means for matrices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2510.04691","year":2026},{"aliases":["10.48550/arxiv.2510.07530","2510.07530"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f238ecadeda62e2104feeda5779dfdfbdf681daa24c96174e3c646b09c93e26.json","evidence":{"source_url":"https://arxiv.org/abs/2510.07530"},"id":"literature:arxiv:2510.07530","identifiers":{"arxiv":"2510.07530","doi":"10.48550/arxiv.2510.07530"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2510.07530","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2510.07530"},{"aliases":["10.48550/arxiv.2510.16561","2510.16561"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d400dbaaae2c6ed39739fda49d1dd2ecde26eeec74c66d4448424ce03fc245ee.json","evidence":{"source_url":"https://arxiv.org/abs/2510.16561"},"id":"literature:arxiv:2510.16561","identifiers":{"arxiv":"2510.16561","doi":"10.48550/arxiv.2510.16561"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2510.16561","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2510.16561"},{"aliases":["2510.23726","Daniel Belkin, James Allen, Bryan K. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Apparent Universal Behavior in 2nd Moments of Random Quantum Circuits","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2510.23726","year":2025},{"aliases":["2510.24280","Anjali Bhagat; Tanmay Kulkarni; Urban Larsson; Divya Murali"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4786b38b1c3ebad0529b8fb0722b9cf8b1f332bfb05b0b30c5ba9605bfc2b5a.json","authors":"Anjali Bhagat; Tanmay Kulkarni; Urban Larsson; Divya Murali","evidence":{"source_url":"https://arxiv.org/abs/2510.24280"},"id":"literature:arxiv:2510.24280","identifiers":{"arxiv":"2510.24280"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Tie-breaking in self interest cumulative subtraction games","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2510.24280","year":2026},{"aliases":["2510.24882","Marc T. Pudelko"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/56e06c55dc90164b5dfad5f196c10701e87d04fa7e63778cf25ea9797d20fa15.json","authors":"Marc T. Pudelko","evidence":{"source_url":"https://arxiv.org/abs/2510.24882"},"id":"literature:arxiv:2510.24882","identifiers":{"arxiv":"2510.24882"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.00314","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.00314"},{"aliases":["10.48550/arxiv.2511.01108","2511.01108"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ddf71b43450cf66d27dc164ac9c19d9be17bd6039bde43cd30ba9abc05f62e65.json","evidence":{"source_url":"https://arxiv.org/abs/2511.01108"},"id":"literature:arxiv:2511.01108","identifiers":{"arxiv":"2511.01108","doi":"10.48550/arxiv.2511.01108"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.01108","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.01108"},{"aliases":["10.48550/arxiv.2511.01719","2511.01719"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e523d1deee8e4ec964aeb0b12f3a923706dabf49a896d551aab4cff5d2f84cd.json","evidence":{"source_url":"https://arxiv.org/abs/2511.01719"},"id":"literature:arxiv:2511.01719","identifiers":{"arxiv":"2511.01719","doi":"10.48550/arxiv.2511.01719"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.01719","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.01719"},{"aliases":["10.48550/arxiv.2511.01996","2511.01996"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d804ab3c6f19da627bb7dca2fc138e9b4a6eed07e614a148eea5bf5a7b6afcf.json","evidence":{"source_url":"https://arxiv.org/abs/2511.01996"},"id":"literature:arxiv:2511.01996","identifiers":{"arxiv":"2511.01996","doi":"10.48550/arxiv.2511.01996"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.01996","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.01996"},{"aliases":["2511.02121","Anastasia Matveeva"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f673333f5aa2f28fbed2b5ecba4b102ea550fd989b084a44a18a7b1f857ac909.json","authors":"Anastasia Matveeva","evidence":{"source_url":"https://arxiv.org/abs/2511.02121"},"id":"literature:arxiv:2511.02121","identifiers":{"arxiv":"2511.02121"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the integrality of some P-recursive sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.02121","year":2025},{"aliases":["10.48550/arxiv.2511.03082","2511.03082"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc016a0ce306881d5e51d40b2b0ac28ac1fb498d3f663cebd7fd28b393519c23.json","evidence":{"source_url":"https://arxiv.org/abs/2511.03082"},"id":"literature:arxiv:2511.03082","identifiers":{"arxiv":"2511.03082","doi":"10.48550/arxiv.2511.03082"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.03082","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.03082"},{"aliases":["10.48550/arxiv.2511.03847","2511.03847"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db414f58dcf13d3e29b7221f2a54dbdfcd4dbb7ed4abf7526315bb1833553b61.json","evidence":{"source_url":"https://arxiv.org/abs/2511.03847"},"id":"literature:arxiv:2511.03847","identifiers":{"arxiv":"2511.03847","doi":"10.48550/arxiv.2511.03847"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.03847","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.03847"},{"aliases":["10.48550/arxiv.2511.09395","2511.09395"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/283a620d50e0fef9fef79432eeb598d9ba8fc00b225e8237fc5641ccb9ed5371.json","evidence":{"source_url":"https://arxiv.org/abs/2511.09395"},"id":"literature:arxiv:2511.09395","identifiers":{"arxiv":"2511.09395","doi":"10.48550/arxiv.2511.09395"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.09395","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.09395"},{"aliases":["2511.15954","Daniel McNulty"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f2445d056660066a43360dfd3417ec5e6f43badb327c6f0d815862779e4a933.json","authors":"Daniel McNulty","evidence":{"source_url":"https://arxiv.org/abs/2511.15954"},"id":"literature:arxiv:2511.15954","identifiers":{"arxiv":"2511.15954"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Graph-Theoretic Approach to Quantum Measurement Incompatibility","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.15954","year":2025},{"aliases":["10.48550/arxiv.2511.16736","2511.16736"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a0d833b34abe6ba7ebf94fcd74522bf9983a0c27e7d2565ecd4df8574d96a09.json","evidence":{"source_url":"https://arxiv.org/abs/2511.16736"},"id":"literature:arxiv:2511.16736","identifiers":{"arxiv":"2511.16736","doi":"10.48550/arxiv.2511.16736"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.16736","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.16736"},{"aliases":["10.48550/arxiv.2511.16959","2511.16959","Saúl A. Blanco, Mikhail P. Golubyatnikov, Elena V. Konstantinova, Natalia V. Maslova, Luka A. Nikiforov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/837b839bb055df1074c9139a5bdf730a833d5076bd827aa67ac12f18417f40eb.json","authors":"Saúl A. Blanco, Mikhail P. Golubyatnikov, Elena V. Konstantinova, Natalia V. Maslova, Luka A. Nikiforov","evidence":{"source_url":"https://arxiv.org/abs/2511.16959"},"id":"literature:arxiv:2511.16959","identifiers":{"arxiv":"2511.16959","doi":"10.48550/arxiv.2511.16959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Generating the symmetric group by three prefix reversals","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.16959","year":2025},{"aliases":["10.48550/arxiv.2511.18330","2511.18330"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc2dc953e6f70998a879b920c993df35e90dd60c89d9041cedb105db4b03deb5.json","evidence":{"source_url":"https://arxiv.org/abs/2511.18330"},"id":"literature:arxiv:2511.18330","identifiers":{"arxiv":"2511.18330","doi":"10.48550/arxiv.2511.18330"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.18330","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.18330"},{"aliases":["10.48550/arxiv.2511.18466","2511.18466"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1167e5bf500a991ad1f0e69b80c5fa4ea7836d918437f031057a0812a208faa.json","evidence":{"source_url":"https://arxiv.org/abs/2511.18466"},"id":"literature:arxiv:2511.18466","identifiers":{"arxiv":"2511.18466","doi":"10.48550/arxiv.2511.18466"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.18466","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.18466"},{"aliases":["10.48550/arxiv.2511.19585","2511.19585"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/54589dab029fa6c364099d91311e9a9d25233be83c042d7d031999e54f2c5502.json","evidence":{"source_url":"https://arxiv.org/abs/2511.19585"},"id":"literature:arxiv:2511.19585","identifiers":{"arxiv":"2511.19585","doi":"10.48550/arxiv.2511.19585"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.19585","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.19585"},{"aliases":["2511.19744","Stephen McKean"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4eb563b095ddd699a1c8199fe464b183cac5e0b63cc0777eb69fbf91252eab6d.json","authors":"Stephen McKean","evidence":{"source_url":"https://arxiv.org/abs/2511.19744"},"id":"literature:arxiv:2511.19744","identifiers":{"arxiv":"2511.19744"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Toda primes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.19744","year":2025},{"aliases":["2511.20345","J. Manna; K. Mandal; K. Paul; D. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"An improvement of the Blanco-Koldobsky-Turnšek characterization of isometries","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.20345","year":2025},{"aliases":["10.48550/arxiv.2511.20925","2511.20925"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0df45fe095fba56939adc7efe8881eb34bbba0f2d7ad0678cdb26332a1848822.json","evidence":{"source_url":"https://arxiv.org/abs/2511.20925"},"id":"literature:arxiv:2511.20925","identifiers":{"arxiv":"2511.20925","doi":"10.48550/arxiv.2511.20925"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.20925","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.20925"},{"aliases":["10.48550/arxiv.2511.22492","2511.22492"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0d601445676f1a2717f7ac3dc0b0a1d3644668d2f8ffaf4c43dd764691535b2a.json","evidence":{"source_url":"https://arxiv.org/abs/2511.22492"},"id":"literature:arxiv:2511.22492","identifiers":{"arxiv":"2511.22492","doi":"10.48550/arxiv.2511.22492"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.22492","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.22492"},{"aliases":["10.48550/arxiv.2511.22755","2511.22755"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4564b6079e5c5a48e172f7556a86799b0895e840dd698a28e17b70cba72e0a2c.json","evidence":{"source_url":"https://arxiv.org/abs/2511.22755"},"id":"literature:arxiv:2511.22755","identifiers":{"arxiv":"2511.22755","doi":"10.48550/arxiv.2511.22755"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2511.22755","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2511.22755"},{"aliases":["2512.00278","O. Lindblad and E. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Simple Eigenvalues and Non-vanishing Eigenvectors of the Anderson Model","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.00278","year":2025},{"aliases":["10.48550/arxiv.2512.02151","2512.02151"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c525b6e7afcaaaa30625a65aa86739b449147614830bbd7795a83e2d7029ac7.json","evidence":{"source_url":"https://arxiv.org/abs/2512.02151"},"id":"literature:arxiv:2512.02151","identifiers":{"arxiv":"2512.02151","doi":"10.48550/arxiv.2512.02151"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2512.02151","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.02151"},{"aliases":["2512.07028","Halina Wiśniewska"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59de1def0a58ac4bec85595402bf062d5a0b5a87415d276b7b428feb99651d66.json","authors":"Halina Wiśniewska","evidence":{"source_url":"https://arxiv.org/abs/2512.07028"},"id":"literature:arxiv:2512.07028","identifiers":{"arxiv":"2512.07028"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some inequalities for Gurland's ratio of the gamma functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.07028","year":2025},{"aliases":["10.48550/arxiv.2512.07127","2512.07127"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a927acc8eb565fece89ba3f59eb9393bdc9c2414b9cf1353f7882859eae69edf.json","evidence":{"source_url":"https://arxiv.org/abs/2512.07127"},"id":"literature:arxiv:2512.07127","identifiers":{"arxiv":"2512.07127","doi":"10.48550/arxiv.2512.07127"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2512.07127","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.07127"},{"aliases":["10.48550/arxiv.2512.08797","2512.08797"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4decdddab27111ef269c3ee91e4e702a6f5df0e4a343b0d163c5889138f1a2b1.json","evidence":{"source_url":"https://arxiv.org/abs/2512.08797"},"id":"literature:arxiv:2512.08797","identifiers":{"arxiv":"2512.08797","doi":"10.48550/arxiv.2512.08797"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2512.08797","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.08797"},{"aliases":["10.48550/arxiv.2512.12681","2512.12681"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c4e6cfacfc43b8a2096b0a6491a312888d7a3c61faf37ce72fc827a751ee38d.json","evidence":{"source_url":"https://arxiv.org/abs/2512.12681"},"id":"literature:arxiv:2512.12681","identifiers":{"arxiv":"2512.12681","doi":"10.48550/arxiv.2512.12681"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2512.12681","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.12681"},{"aliases":["2512.14831","Zacharie Van Herstraeten; Jack Davis; Nuno C. Dias; João N. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Extreme non-negative Wigner functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.14831","year":2025},{"aliases":["10.48550/arxiv.2512.15486","2512.15486"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab9af69cdd2d674d6f04b3c53ba972b84aeddc33790efe7329df9321e5308b54.json","evidence":{"source_url":"https://arxiv.org/abs/2512.15486"},"id":"literature:arxiv:2512.15486","identifiers":{"arxiv":"2512.15486","doi":"10.48550/arxiv.2512.15486"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2512.15486","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2512.15486"},{"aliases":["2512.17285","Szymon Głąb and Franciszek Prus-Wiśniowski"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebfcc17b3c16078d44d9d35eb7f88caff4634034196b4675a34490e3780de7dc.json","authors":"Szymon Głąb and Franciszek Prus-Wiśniowski","evidence":{"source_url":"https://arxiv.org/abs/2512.17285"},"id":"literature:arxiv:2512.17285","identifiers":{"arxiv":"2512.17285"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2601.16161","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2601.16161"},{"aliases":["10.48550/arxiv.2602.03605","2602.03605"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55ee468d11397a62f807bf5239c650a4484b0190c2b0f3428b0d40ab6a9c7b91.json","evidence":{"source_url":"https://arxiv.org/abs/2602.03605"},"id":"literature:arxiv:2602.03605","identifiers":{"arxiv":"2602.03605","doi":"10.48550/arxiv.2602.03605"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2602.03605","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2602.03605"},{"aliases":["10.48550/arxiv.2602.07587","2602.07587"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19b4f6246489cb6a805fe4e54c8abee7cbc187608428c4c73775bba560c26fc2.json","evidence":{"source_url":"https://arxiv.org/abs/2602.07587"},"id":"literature:arxiv:2602.07587","identifiers":{"arxiv":"2602.07587","doi":"10.48550/arxiv.2602.07587"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.06489","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.06489"},{"aliases":["10.48550/arxiv.2603.06930","2603.06930"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15c07627635f0fa819936e52193ac3af244709334876ce0c149b67c2b20e40f0.json","evidence":{"source_url":"https://arxiv.org/abs/2603.06930"},"id":"literature:arxiv:2603.06930","identifiers":{"arxiv":"2603.06930","doi":"10.48550/arxiv.2603.06930"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.06930","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.06930"},{"aliases":["10.48550/arxiv.2603.10414","2603.10414"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce44cbb3124eea136370394d33af64a5118b8220fac37a88f159b4b4de52cd1e.json","evidence":{"source_url":"https://arxiv.org/abs/2603.10414"},"id":"literature:arxiv:2603.10414","identifiers":{"arxiv":"2603.10414","doi":"10.48550/arxiv.2603.10414"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.10414","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.10414"},{"aliases":["10.48550/arxiv.2603.10853","2603.10853"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04403eeac31e49ab3f4a4a05442c0acd322e079a97860a1175c4b1b4e254d425.json","evidence":{"source_url":"https://arxiv.org/abs/2603.10853"},"id":"literature:arxiv:2603.10853","identifiers":{"arxiv":"2603.10853","doi":"10.48550/arxiv.2603.10853"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.10853","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.10853"},{"aliases":["2603.14154","Aleksandrs Sokolovs"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef4d28e78cc7cb52611788fd83d0a5bae2b4b0efd7cd17833a83642ed77d79c9.json","authors":"Aleksandrs Sokolovs","evidence":{"source_url":"https://arxiv.org/abs/2603.14154"},"id":"literature:arxiv:2603.14154","identifiers":{"arxiv":"2603.14154"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Parity superselection obstructs monogamy of mutual information in free fermions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.14154","year":2026},{"aliases":["10.48550/arxiv.2603.14292","2603.14292"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6866b3d4e6f219377bd995eb3659fc629421b6bbc4ec9ae6efc39d470692c51b.json","evidence":{"source_url":"https://arxiv.org/abs/2603.14292"},"id":"literature:arxiv:2603.14292","identifiers":{"arxiv":"2603.14292","doi":"10.48550/arxiv.2603.14292"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.14292","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.14292"},{"aliases":["10.48550/arxiv.2603.17154","2603.17154"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e85fa3ce05917ec7fbe22b87a318c1f1a8c891a1fe8c75dc9995555c83a9e587.json","evidence":{"source_url":"https://arxiv.org/abs/2603.17154"},"id":"literature:arxiv:2603.17154","identifiers":{"arxiv":"2603.17154","doi":"10.48550/arxiv.2603.17154"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.17154","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.17154"},{"aliases":["10.48550/arxiv.2603.20748","2603.20748"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69325a073d4031edc8c2d50ee8891c80334a0047264ed30266cea5c2d0479e2a.json","evidence":{"source_url":"https://arxiv.org/abs/2603.20748"},"id":"literature:arxiv:2603.20748","identifiers":{"arxiv":"2603.20748","doi":"10.48550/arxiv.2603.20748"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.20748","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.20748"},{"aliases":["10.48550/arxiv.2603.20792","2603.20792"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd982b8de5afad97ebd01e0f978474d85b6765ca91fcaa3ee0387a30da3fff16.json","evidence":{"source_url":"https://arxiv.org/abs/2603.20792"},"id":"literature:arxiv:2603.20792","identifiers":{"arxiv":"2603.20792","doi":"10.48550/arxiv.2603.20792"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.20792","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.20792"},{"aliases":["10.48550/arxiv.2603.21645","2603.21645"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15790f89dacdafdf4abb0c2b1240ef072648f7ee5d7f852a601d2b9ec83ec893.json","evidence":{"source_url":"https://arxiv.org/abs/2603.21645"},"id":"literature:arxiv:2603.21645","identifiers":{"arxiv":"2603.21645","doi":"10.48550/arxiv.2603.21645"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.21645","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.21645"},{"aliases":["10.48550/arxiv.2603.22163","2603.22163"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd7848d66206b53d7df1ac3f22f024e0322b50aaedb7132dc1702996d168cf8c.json","evidence":{"source_url":"https://arxiv.org/abs/2603.22163"},"id":"literature:arxiv:2603.22163","identifiers":{"arxiv":"2603.22163","doi":"10.48550/arxiv.2603.22163"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.22163","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.22163"},{"aliases":["10.48550/arxiv.2603.22195","2603.22195"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8231ed9eb2a01bae3cc85c5307342cdfcdfcbf62cbeabffd0c9acadf7b797121.json","evidence":{"source_url":"https://arxiv.org/abs/2603.22195"},"id":"literature:arxiv:2603.22195","identifiers":{"arxiv":"2603.22195","doi":"10.48550/arxiv.2603.22195"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.22195","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.22195"},{"aliases":["10.48550/arxiv.2603.24679","2603.24679"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9346f87632299da4b9904af469dc29c480d8d1cc13677cfb2f71eaa791f2e8de.json","evidence":{"source_url":"https://arxiv.org/abs/2603.24679"},"id":"literature:arxiv:2603.24679","identifiers":{"arxiv":"2603.24679","doi":"10.48550/arxiv.2603.24679"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.24679","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.24679"},{"aliases":["10.48550/arxiv.2603.25113","2603.25113"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ed64f8b9acc70e1a1ffb37c4c6d89c038ca2bb66468b743dbe3205fd16b133d3.json","evidence":{"source_url":"https://arxiv.org/abs/2603.25113"},"id":"literature:arxiv:2603.25113","identifiers":{"arxiv":"2603.25113","doi":"10.48550/arxiv.2603.25113"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.25113","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.25113"},{"aliases":["10.48550/arxiv.2603.25343","2603.25343"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e8b6f478bb1147dfb80ce7fd24a18b4fe7446d37fcc483c7a79b22e82d7b19d.json","evidence":{"source_url":"https://arxiv.org/abs/2603.25343"},"id":"literature:arxiv:2603.25343","identifiers":{"arxiv":"2603.25343","doi":"10.48550/arxiv.2603.25343"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.25343","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.25343"},{"aliases":["10.48550/arxiv.2603.28905","2603.28905"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52669595b180c51621e7e1b95e42ee191059aaa707f6e8997b1d3d55d1697d6f.json","evidence":{"source_url":"https://arxiv.org/abs/2603.28905"},"id":"literature:arxiv:2603.28905","identifiers":{"arxiv":"2603.28905","doi":"10.48550/arxiv.2603.28905"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.28905","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.28905"},{"aliases":["10.48550/arxiv.2603.29571","2603.29571"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa9957e34cdefd4d27cab9fd1021f666f7f16a65d34370f0f9d46a85c3111bba.json","evidence":{"source_url":"https://arxiv.org/abs/2603.29571"},"id":"literature:arxiv:2603.29571","identifiers":{"arxiv":"2603.29571","doi":"10.48550/arxiv.2603.29571"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2603.29571","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2603.29571"},{"aliases":["10.48550/arxiv.2604.00165","2604.00165"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7eeafbb0d57c3154aad8e2e22e1087d595ba310e00ad09c68fccf2650c17f452.json","evidence":{"source_url":"https://arxiv.org/abs/2604.00165"},"id":"literature:arxiv:2604.00165","identifiers":{"arxiv":"2604.00165","doi":"10.48550/arxiv.2604.00165"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.00165","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.00165"},{"aliases":["10.48550/arxiv.2604.01217","2604.01217"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/525465482978ecc6f66c8e88fcdbaa758e0c08e23e8bf1e2b7dbd80e7d7fb3d0.json","evidence":{"source_url":"https://arxiv.org/abs/2604.01217"},"id":"literature:arxiv:2604.01217","identifiers":{"arxiv":"2604.01217","doi":"10.48550/arxiv.2604.01217"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.01217","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.01217"},{"aliases":["2604.02420","Jofre Abellanet-Vidal; Guillem Müller-Rigat; Albert Rico; Anna Sanpera"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2fd34cf963de20d29920911c72113b4949ae3b2ada97412dd298d5ef67b8537f.json","authors":"Jofre Abellanet-Vidal; Guillem Müller-Rigat; Albert Rico; Anna Sanpera","evidence":{"source_url":"https://arxiv.org/abs/2604.02420"},"id":"literature:arxiv:2604.02420","identifiers":{"arxiv":"2604.02420"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Bounding the entanglement of a state from its spectrum","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.02420","year":2026},{"aliases":["10.48550/arxiv.2604.04781","2604.04781"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eefc11ca2ef7497b489d6d9491787e27b2e00ab9ce61be9d935529cb74046f02.json","evidence":{"source_url":"https://arxiv.org/abs/2604.04781"},"id":"literature:arxiv:2604.04781","identifiers":{"arxiv":"2604.04781","doi":"10.48550/arxiv.2604.04781"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.04781","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.04781"},{"aliases":["10.48550/arxiv.2604.08215","2604.08215"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5a579b3ca733f620e394c8e53749eed56e3fce14d70dc6da1120e25062f13c1b.json","evidence":{"source_url":"https://arxiv.org/abs/2604.08215"},"id":"literature:arxiv:2604.08215","identifiers":{"arxiv":"2604.08215","doi":"10.48550/arxiv.2604.08215"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.08215","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.08215"},{"aliases":["2604.08998","Bilal Ahmad Rather"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/928fcf325a7dd91dd80ed83fd2e839e1872ec4a904588f89b5523d2566f66e3f.json","authors":"Bilal Ahmad Rather","evidence":{"source_url":"https://arxiv.org/abs/2604.08998"},"id":"literature:arxiv:2604.08998","identifiers":{"arxiv":"2604.08998"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On roots of domination polynomials for friendship and book graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.08998","year":2026},{"aliases":["10.48550/arxiv.2604.09491","2604.09491"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a024e554d1dfc60277a212d97b09f3a4f8e61d708d0436aa0bfe197dd1bfe73.json","evidence":{"source_url":"https://arxiv.org/abs/2604.09491"},"id":"literature:arxiv:2604.09491","identifiers":{"arxiv":"2604.09491","doi":"10.48550/arxiv.2604.09491"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.09491","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.09491"},{"aliases":["10.48550/arxiv.2604.12557","2604.12557"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32cb7f1c315489c32b36f848e4f2e68cd8a1560e1c64dd350711f66c03a60a28.json","evidence":{"source_url":"https://arxiv.org/abs/2604.12557"},"id":"literature:arxiv:2604.12557","identifiers":{"arxiv":"2604.12557","doi":"10.48550/arxiv.2604.12557"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.12557","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.12557"},{"aliases":["2604.12808","Nathaniel Johnston, Vincent Russo"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/352083245dd3d36c50185218ffaae2139b5c93909acebb758dc8e11b3f9b5a10.json","authors":"Nathaniel Johnston, Vincent Russo","evidence":{"source_url":"https://arxiv.org/abs/2604.12808"},"id":"literature:arxiv:2604.12808","identifiers":{"arxiv":"2604.12808"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Distinguishability of locally diagonal orthogonally invariant quantum states","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.12808","year":2026},{"aliases":["10.48550/arxiv.2604.14391","2604.14391"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/13f55784e446059de88bdc5ad168de800700dc9fb7f534603b929e9838139307.json","evidence":{"source_url":"https://arxiv.org/abs/2604.14391"},"id":"literature:arxiv:2604.14391","identifiers":{"arxiv":"2604.14391","doi":"10.48550/arxiv.2604.14391"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.14391","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.14391"},{"aliases":["2604.14639","Seok Hyun Byun and Svetlana Poznanović"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44c4d46943c192bb2922d03e7c8fc6f2d55eeb1ba2abfd4e8b2263d137b88cf3.json","authors":"Seok Hyun Byun and Svetlana Poznanović","evidence":{"source_url":"https://arxiv.org/abs/2604.14639"},"id":"literature:arxiv:2604.14639","identifiers":{"arxiv":"2604.14639"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Unimodality and log-concavity of generalized Glasby-Paseman sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.14639","year":2026},{"aliases":["2604.15396","B. J. González and E. R. Negrín"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da0db4d2f1098d6f62604a477166d40e9a81d41d11e34d08dd026441e9a7d2a8.json","authors":"B. J. González and E. R. Negrín","evidence":{"source_url":"https://arxiv.org/abs/2604.15396"},"id":"literature:arxiv:2604.15396","identifiers":{"arxiv":"2604.15396"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A new equivalence to the Riemann Hypothesis by means of the Salem integral equation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.15396","year":2026},{"aliases":["2604.17847","Aradhya Goel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0ae5f16b0281b81d38fa8596970d6196d08c74b688b30234dc4a80cde48360cc.json","authors":"Aradhya Goel","evidence":{"source_url":"https://arxiv.org/abs/2604.17847"},"id":"literature:arxiv:2604.17847","identifiers":{"arxiv":"2604.17847"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.23188","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.23188"},{"aliases":["10.48550/arxiv.2604.23959","2604.23959"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c9d4b264ac674836ad7125ee2485639a04695005833d8aafc3ecaa2fe348cff1.json","evidence":{"source_url":"https://arxiv.org/abs/2604.23959"},"id":"literature:arxiv:2604.23959","identifiers":{"arxiv":"2604.23959","doi":"10.48550/arxiv.2604.23959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2604.23959","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2604.23959"},{"aliases":["2604.26927","M. Kvashchuk, P. Chernyshova, L. E. A. Porto, T.-A. Ohst, L. B. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.02876","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.02876"},{"aliases":["10.48550/arxiv.2605.05149","2605.05149"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecf6d2571decfdfd7ef6c093e91f9d2e158fe2be610ad606eb28a06a5b7610fc.json","evidence":{"source_url":"https://arxiv.org/abs/2605.05149"},"id":"literature:arxiv:2605.05149","identifiers":{"arxiv":"2605.05149","doi":"10.48550/arxiv.2605.05149"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.05149","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.05149"},{"aliases":["10.48550/arxiv.2605.10375","2605.10375"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/54efed728abdd96bd6507ab7139ae4cf328f5b175bd48167b61ac6c4c9618ad1.json","evidence":{"source_url":"https://arxiv.org/abs/2605.10375"},"id":"literature:arxiv:2605.10375","identifiers":{"arxiv":"2605.10375","doi":"10.48550/arxiv.2605.10375"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.10375","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.10375"},{"aliases":["2605.12822","Brendan B. Connelly; Ezekiel Ito; Thomas C. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Quad-C5 Graph: Maximum Contextuality Gap on Eight Vertices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.12828","year":2026},{"aliases":["10.48550/arxiv.2605.16151","2605.16151"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f57098d9caf920cedb6da0b3ea9b4a1082aafa88a725c392ae45776e3394493.json","evidence":{"source_url":"https://arxiv.org/abs/2605.16151"},"id":"literature:arxiv:2605.16151","identifiers":{"arxiv":"2605.16151","doi":"10.48550/arxiv.2605.16151"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.16151","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.16151"},{"aliases":["2605.17501","Boris Shapiro"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39e8dbce79be46e3a44482e5fb3dcea53c6945e48463334282bf17e4ac3ca73c.json","authors":"Boris Shapiro","evidence":{"source_url":"https://arxiv.org/abs/2605.17501"},"id":"literature:arxiv:2605.17501","identifiers":{"arxiv":"2605.17501"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.21457","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.21457"},{"aliases":["10.48550/arxiv.2605.22307","2605.22307"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e15100ffd31756022c264885ae7a31f97cdffea948a23d6db95faae26d882a20.json","evidence":{"source_url":"https://arxiv.org/abs/2605.22307"},"id":"literature:arxiv:2605.22307","identifiers":{"arxiv":"2605.22307","doi":"10.48550/arxiv.2605.22307"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.22307","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.22307"},{"aliases":["2605.25238","František Štampach; Jakub Waclawek"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95dcd9b3bbe2dcb64529b1cd2b6594cb6b0f8231e26f99a090ff56c6220dfcf5.json","authors":"František Štampach; Jakub Waclawek","evidence":{"source_url":"https://arxiv.org/abs/2605.25238"},"id":"literature:arxiv:2605.25238","identifiers":{"arxiv":"2605.25238"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2605.29436","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2605.29436"},{"aliases":["2605.30306","Arina Filimonova; Svetlana Puzynina"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/967e574a1006c6b1d60a7262e3e24c447d5edb347813159d3452564ffbde56c6.json","authors":"Arina Filimonova; Svetlana Puzynina","evidence":{"source_url":"https://arxiv.org/abs/2605.30306"},"id":"literature:arxiv:2605.30306","identifiers":{"arxiv":"2605.30306"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.04225","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.04225"},{"aliases":["2606.06655","Miklós Bóna, Balázs Maga, Jacob Richey"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c16b3de5c5860803eec562f45742e5c2e737764ebe464906d88f0fbf49fc281.json","authors":"Miklós Bóna, Balázs Maga, Jacob Richey","evidence":{"source_url":"https://arxiv.org/abs/2606.06655"},"id":"literature:arxiv:2606.06655","identifiers":{"arxiv":"2606.06655"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.22614","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.22614"},{"aliases":["2606.23721","Leila Muney"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/673d154e88a6029d6781fec6986699f845a72d45cbeaca98f6716bf8dd5b6462.json","authors":"Leila Muney","evidence":{"source_url":"https://arxiv.org/abs/2606.23721"},"id":"literature:arxiv:2606.23721","identifiers":{"arxiv":"2606.23721"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.25150","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.25150"},{"aliases":["2606.25493","Gregory Demo, David Lund, Oleksiy Al-saadi"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/beb31d7616be6d78c45711e15b54fcce8e081575005aa538e3f5ac87a23f9146.json","authors":"Gregory Demo, David Lund, Oleksiy Al-saadi","evidence":{"source_url":"https://arxiv.org/abs/2606.25493"},"id":"literature:arxiv:2606.25493","identifiers":{"arxiv":"2606.25493"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.25766","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.25766"},{"aliases":["10.48550/arxiv.2606.25884","2606.25884"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a83b37f7c6d5fb1622bacb65cc3e488119db80dd73d4b2a4e9fef612a84e24f6.json","evidence":{"source_url":"https://arxiv.org/abs/2606.25884"},"id":"literature:arxiv:2606.25884","identifiers":{"arxiv":"2606.25884","doi":"10.48550/arxiv.2606.25884"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.25884","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.25884"},{"aliases":["2606.26786","Henri Riihimäki, Jason P. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Avalanche homology of digraphs via sandpile dynamics","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.26786","year":2026},{"aliases":["2606.27260","Hao Li"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eec93899dbf019bf7ff8fae23a1613d683193ee3e69872998f111de8d9c7c382.json","authors":"Hao Li","evidence":{"source_url":"https://arxiv.org/abs/2606.27260"},"id":"literature:arxiv:2606.27260","identifiers":{"arxiv":"2606.27260"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Conjecture About Arbitrary Even-Order Convexity of w-Optimization of the Split CIF","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.27260","year":2026},{"aliases":["2606.29311","Pramod Kamal Kharel; Manghang Limbu; Nabaraj Khatri; Ashish Khanal; Kiran Adhikari"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8acb869427d858f444c4db9e721f92c52492a3f2c9629ee4f2af508bc4837245.json","authors":"Pramod Kamal Kharel; Manghang Limbu; Nabaraj Khatri; Ashish Khanal; Kiran Adhikari","evidence":{"source_url":"https://arxiv.org/abs/2606.29311"},"id":"literature:arxiv:2606.29311","identifiers":{"arxiv":"2606.29311"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Pseudo entropy and topological phases of matter","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.29311","year":2026},{"aliases":["2606.31010","H.-F. Wang; W. Zhou; L. Chen; S.-M. Fei"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c4951a9f80bc705fa890f83486fa7ca92c452a7e3709a34e7bf633111dc5854.json","authors":"H.-F. Wang; W. Zhou; L. Chen; S.-M. Fei","evidence":{"source_url":"https://arxiv.org/abs/2606.31010"},"id":"literature:arxiv:2606.31010","identifiers":{"arxiv":"2606.31010"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Estimating the concurrence for quantum states via symmetric measurements","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.31010","year":2026},{"aliases":["10.48550/arxiv.2606.31103","2606.31103"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ac6024e10a0fafabe24134b4d521b29fe271ec66d30cabdb909f2ee2bc6e075c.json","evidence":{"source_url":"https://arxiv.org/abs/2606.31103"},"id":"literature:arxiv:2606.31103","identifiers":{"arxiv":"2606.31103","doi":"10.48550/arxiv.2606.31103"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.31103","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.31103"},{"aliases":["10.48550/arxiv.2606.31205","2606.31205"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ed8f4b938aab6e310b494e66efc358649cdff8a6abf03368958d21285af84e13.json","evidence":{"source_url":"https://arxiv.org/abs/2606.31205"},"id":"literature:arxiv:2606.31205","identifiers":{"arxiv":"2606.31205","doi":"10.48550/arxiv.2606.31205"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2606.31205","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2606.31205"},{"aliases":["2607.00404","Ján Mináč, Tung T. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Narayana transformation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.01572","year":2026},{"aliases":["2607.01805","Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0738b1210f0254f11fda9b5ce1b30443bff4a6d6abc6ebaa62396e23c2b28b81.json","authors":"Srikrishna Maity; Shigeru Furuichi; Supriyo Dutta","evidence":{"source_url":"https://arxiv.org/abs/2607.01805"},"id":"literature:arxiv:2607.01805","identifiers":{"arxiv":"2607.01805"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"r-deformed α-z-Rényi relative entropy","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.01805","year":2026},{"aliases":["10.48550/arxiv.2607.02093","2607.02093"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68b339f4f445e8c2df2060ce31b41b15a73e663d89910b86fe24adec1ffc2bca.json","evidence":{"source_url":"https://arxiv.org/abs/2607.02093"},"id":"literature:arxiv:2607.02093","identifiers":{"arxiv":"2607.02093","doi":"10.48550/arxiv.2607.02093"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.02093","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.02093"},{"aliases":["2607.04534","Manuel Israel Cázares, Wenlin Zhang, Haobo Ma"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8083b0ae3f65d5181c7b140330b711d85ca04801cf5b46a0c001131328371bcb.json","authors":"Manuel Israel Cázares, Wenlin Zhang, Haobo Ma","evidence":{"source_url":"https://arxiv.org/abs/2607.04534"},"id":"literature:arxiv:2607.04534","identifiers":{"arxiv":"2607.04534"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.05567","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.05567"},{"aliases":["2607.05847","K. Baek; Y. Jo; H. Nha"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31445eab93820f7a0ff14ba57d1d956f78da11f160fca32e382c254a8465c1ae.json","authors":"K. Baek; Y. Jo; H. Nha","evidence":{"source_url":"https://arxiv.org/abs/2607.05847"},"id":"literature:arxiv:2607.05847","identifiers":{"arxiv":"2607.05847"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.06050","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.06050"},{"aliases":["10.48550/arxiv.2607.06808","2607.06808","Tomasz Kania and Jerzy Kakol"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3b9b1ded5368d5cf6e58d02919a8ea5776bf6c256e6f3c67641911b2807060d.json","authors":"Tomasz Kania and Jerzy Kakol","evidence":{"source_url":"https://arxiv.org/abs/2607.06808"},"id":"literature:arxiv:2607.06808","identifiers":{"arxiv":"2607.06808","doi":"10.48550/arxiv.2607.06808"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.07851","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.07851"},{"aliases":["10.48550/arxiv.2607.09376","2607.09376"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/63282d9567098d0167db96698970c49eaa899b901c6e7ad020d904012fcaa4bd.json","evidence":{"source_url":"https://arxiv.org/abs/2607.09376"},"id":"literature:arxiv:2607.09376","identifiers":{"arxiv":"2607.09376","doi":"10.48550/arxiv.2607.09376"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.09376","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.09376"},{"aliases":["2607.10311","Mohammad Farhan, Sandi Klavžar, Dorota Kuziak, Ismael G. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Multiset resolvability parameters in graphs: A survey with new results and open problems","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.10311","year":2026},{"aliases":["10.48550/arxiv.2607.11194","2607.11194"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6cbba84dea5886fd4bffb85d63c6f5de6d05a18b4ac09fd06d799688a4a6727.json","evidence":{"source_url":"https://arxiv.org/abs/2607.11194"},"id":"literature:arxiv:2607.11194","identifiers":{"arxiv":"2607.11194","doi":"10.48550/arxiv.2607.11194"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.11194","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.11194"},{"aliases":["10.48550/arxiv.2607.11762","2607.11762"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0f6d0fb2b702bb07e86c3cf065deb8d65c3068cd6a8112bce7f3760d61d0a20.json","evidence":{"source_url":"https://arxiv.org/abs/2607.11762"},"id":"literature:arxiv:2607.11762","identifiers":{"arxiv":"2607.11762","doi":"10.48550/arxiv.2607.11762"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.11762","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.11762"},{"aliases":["10.48550/arxiv.2607.12330","2607.12330"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6df3b5caf4d23a7a8d689f74d2fca383424cb5a154f28ec72ab155706f97f1b5.json","evidence":{"source_url":"https://arxiv.org/abs/2607.12330"},"id":"literature:arxiv:2607.12330","identifiers":{"arxiv":"2607.12330","doi":"10.48550/arxiv.2607.12330"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.12330","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.12330"},{"aliases":["2607.14887","Xinru Jiang, Suzhen Wen, Yueming Zhong"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9221a31a16681f47730378f560bc5596308775e062bb3e3d824049db93bc1018.json","authors":"Xinru Jiang, Suzhen Wen, Yueming Zhong","evidence":{"source_url":"https://arxiv.org/abs/2607.14887"},"id":"literature:arxiv:2607.14887","identifiers":{"arxiv":"2607.14887"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Alternating adjacent-sum polytopes: transfer matrices and Ehrhart series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.14887","year":2026},{"aliases":["10.48550/arxiv.2607.15598","2607.15598"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b57dbffe7b973da7f09d45fbd095a2bcbf178167a681b0c9ae18998008b7775d.json","evidence":{"source_url":"https://arxiv.org/abs/2607.15598"},"id":"literature:arxiv:2607.15598","identifiers":{"arxiv":"2607.15598","doi":"10.48550/arxiv.2607.15598"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.15598","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.15598"},{"aliases":["10.48550/arxiv.2607.17343","2607.17343"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/555eb32393103f17bbf973231b8108d9fef2b6b94aae4e2f5786638cef1e0b0d.json","evidence":{"source_url":"https://arxiv.org/abs/2607.17343"},"id":"literature:arxiv:2607.17343","identifiers":{"arxiv":"2607.17343","doi":"10.48550/arxiv.2607.17343"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.17343","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.17343"},{"aliases":["10.48550/arxiv.2607.19412","2607.19412"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/042a1fc7b5e4ca83f50733ff32b5ef9830e418ce423f7c97124f527f51298d33.json","evidence":{"source_url":"https://arxiv.org/abs/2607.19412"},"id":"literature:arxiv:2607.19412","identifiers":{"arxiv":"2607.19412","doi":"10.48550/arxiv.2607.19412"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.19412","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.19412"},{"aliases":["10.48550/arxiv.2607.20750","2607.20750"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46ffb7bd7581e5bd947e4d597a176811fe9140afe14f0170c55beb9e419062d3.json","evidence":{"source_url":"https://arxiv.org/abs/2607.20750"},"id":"literature:arxiv:2607.20750","identifiers":{"arxiv":"2607.20750","doi":"10.48550/arxiv.2607.20750"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.20750","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.20750"},{"aliases":["10.48550/arxiv.2607.21539","2607.21539"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd1c70a4abcb0767549f02c1112797d0098b659c929742e4a67fb04c1789e083.json","evidence":{"source_url":"https://arxiv.org/abs/2607.21539"},"id":"literature:arxiv:2607.21539","identifiers":{"arxiv":"2607.21539","doi":"10.48550/arxiv.2607.21539"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.21539","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.21539"},{"aliases":["10.48550/arxiv.2607.23641","2607.23641"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bb45345c207b2a26a07fc3fb2fb45b91d102cee502c04966bd47180bf05b8fb.json","evidence":{"source_url":"https://arxiv.org/abs/2607.23641"},"id":"literature:arxiv:2607.23641","identifiers":{"arxiv":"2607.23641","doi":"10.48550/arxiv.2607.23641"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.23641","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.23641"},{"aliases":["10.48550/arxiv.2607.24014","2607.24014"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c58fabfa23df6226fc431fd742f42ef20ac8db5310f8ef7c36ebd518d2f90b3b.json","evidence":{"source_url":"https://arxiv.org/abs/2607.24014"},"id":"literature:arxiv:2607.24014","identifiers":{"arxiv":"2607.24014","doi":"10.48550/arxiv.2607.24014"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.24014","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.24014"},{"aliases":["10.48550/arxiv.2607.26214","2607.26214"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a45b1c3918b031a5fc98d72f04f3b66f127e638b91c2562453ac0191c0c68d41.json","evidence":{"source_url":"https://arxiv.org/abs/2607.26214"},"id":"literature:arxiv:2607.26214","identifiers":{"arxiv":"2607.26214","doi":"10.48550/arxiv.2607.26214"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2607.26214","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.26214"},{"aliases":["2607.28345","Alison Gray; Pransu Patel; Isaiah Young"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e594f0e60d39ae937778e43562b9ed14d9d0f9cff94d4bcde1a1d6d500525d14.json","authors":"Alison Gray; Pransu Patel; Isaiah Young","evidence":{"source_url":"https://arxiv.org/abs/2607.28345"},"id":"literature:arxiv:2607.28345","identifiers":{"arxiv":"2607.28345"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The cycle C9 does not admit uniform mixing","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2607.28345","year":2026},{"aliases":["10.48550/arxiv.2608.00109","2608.00109"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1873a83e35720756839ab21e871c6b306a5756277988f4948337e05c05554808.json","evidence":{"source_url":"https://arxiv.org/abs/2608.00109"},"id":"literature:arxiv:2608.00109","identifiers":{"arxiv":"2608.00109","doi":"10.48550/arxiv.2608.00109"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.00109","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.00109"},{"aliases":["10.48550/arxiv.2608.00124","2608.00124"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/261ebbcab1a59b452ba9454da3a7eecdbdf51897af1cddc3e4fc34656a6bd310.json","evidence":{"source_url":"https://arxiv.org/abs/2608.00124"},"id":"literature:arxiv:2608.00124","identifiers":{"arxiv":"2608.00124","doi":"10.48550/arxiv.2608.00124"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.00124","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.00124"},{"aliases":["10.48550/arxiv.2608.00340","2608.00340"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85fec8241e82025f947923b87878cfa7f604c245f73bda298fe13805f40f8aa7.json","evidence":{"source_url":"https://arxiv.org/abs/2608.00340"},"id":"literature:arxiv:2608.00340","identifiers":{"arxiv":"2608.00340","doi":"10.48550/arxiv.2608.00340"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.00340","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.00340"},{"aliases":["10.48550/arxiv.2608.00347","2608.00347"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/62922cfc2c814fde2f5de3f4dc01f6b1a9d860599bba252e6fd566038954222d.json","evidence":{"source_url":"https://arxiv.org/abs/2608.00347"},"id":"literature:arxiv:2608.00347","identifiers":{"arxiv":"2608.00347","doi":"10.48550/arxiv.2608.00347"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.00347","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.00347"},{"aliases":["2608.01167","Hanwool Lee; Teiko Heinosaari"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68c7a9d8e36e77fe83c8fad7497d22231547ce54ea6aa5612782c8fbcdae83b3.json","authors":"Hanwool Lee; Teiko Heinosaari","evidence":{"source_url":"https://arxiv.org/abs/2608.01167"},"id":"literature:arxiv:2608.01167","identifiers":{"arxiv":"2608.01167"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Perfect Discrimination of Non-Orthogonal Quantum States via Adaptive Post-Measurement Queries","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.01167","year":2026},{"aliases":["10.48550/arxiv.2608.03363","2608.03363"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab00ec07ff6b4f1654bc453b27097bc4bf47dea4565d1cde1599c7bf0e799830.json","evidence":{"source_url":"https://arxiv.org/abs/2608.03363"},"id":"literature:arxiv:2608.03363","identifiers":{"arxiv":"2608.03363","doi":"10.48550/arxiv.2608.03363"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.03363","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.03363"},{"aliases":["10.48550/arxiv.2608.03390","2608.03390"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/326fa257c559a51dd6878083ac02a4775a8544d2d9ad0acf8afdede14c8bd1e9.json","evidence":{"source_url":"https://arxiv.org/abs/2608.03390"},"id":"literature:arxiv:2608.03390","identifiers":{"arxiv":"2608.03390","doi":"10.48550/arxiv.2608.03390"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.03390","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.03390"},{"aliases":["10.48550/arxiv.2608.03723","2608.03723","Anna Vishnyakova"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f88e0fe2b69d5c0b41c3158fe70a727348882ec9ce69b4acab1c268f9a04cd3.json","authors":"Anna Vishnyakova","evidence":{"source_url":"https://arxiv.org/abs/2608.03723"},"id":"literature:arxiv:2608.03723","identifiers":{"arxiv":"2608.03723","doi":"10.48550/arxiv.2608.03723"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Polynomially Deformed Normalized Pochhammer Sequences Having Generating Functions With Only Real Non-positive Zeros","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.03723","year":2026},{"aliases":["2608.07262","Avnish K. Sharma"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4dcc4b411b6cbae70770d92c1aaecd6edbf98e155954fd607a14b482461add45.json","authors":"Avnish K. Sharma","evidence":{"source_url":"https://arxiv.org/abs/2608.07262"},"id":"literature:arxiv:2608.07262","identifiers":{"arxiv":"2608.07262"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Primitive Polynomials of the Form g(x)+λ over Finite Fields: Non-Existence Results and Conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.07262","year":2026},{"aliases":["10.48550/arxiv.2608.08692","2608.08692"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72f379007a3a5c43acd35f6841717dd0ae0b4518f536c6d53e421103bdabb37d.json","evidence":{"source_url":"https://arxiv.org/abs/2608.08692"},"id":"literature:arxiv:2608.08692","identifiers":{"arxiv":"2608.08692","doi":"10.48550/arxiv.2608.08692"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.08692","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.08692"},{"aliases":["10.48550/arxiv.2608.08789","2608.08789"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f921f068d208ba52a69aca2fe96ec738bd50c2232a1032a74fe21d1f3c66c12d.json","evidence":{"source_url":"https://arxiv.org/abs/2608.08789"},"id":"literature:arxiv:2608.08789","identifiers":{"arxiv":"2608.08789","doi":"10.48550/arxiv.2608.08789"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.08789","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.08789"},{"aliases":["10.48550/arxiv.2608.10643","2608.10643"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5e00e8abbdc916a401577fbfd7f0effdcc04737dfcf2a9dd1f1ecd473c7a3b4.json","evidence":{"source_url":"https://arxiv.org/abs/2608.10643"},"id":"literature:arxiv:2608.10643","identifiers":{"arxiv":"2608.10643","doi":"10.48550/arxiv.2608.10643"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.10643","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.10643"},{"aliases":["2608.11850","Xiuwu Zhu; Yu Wang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b860781e264d46eb30ff0c412cf44d8df0bd7c22152eb872e34356755bad902e.json","authors":"Xiuwu Zhu; Yu Wang","evidence":{"source_url":"https://arxiv.org/abs/2608.11850"},"id":"literature:arxiv:2608.11850","identifiers":{"arxiv":"2608.11850"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Uniformly Stable Minimal Weyl–Heisenberg Measurements Approaching the SIC Benchmark","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.11850","year":2026},{"aliases":["10.48550/arxiv.2608.13192","2608.13192"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/187bd4d57062292b34b8eead0fa4cda17c8b06198c5e0426c2229790ff9071c0.json","evidence":{"source_url":"https://arxiv.org/abs/2608.13192"},"id":"literature:arxiv:2608.13192","identifiers":{"arxiv":"2608.13192","doi":"10.48550/arxiv.2608.13192"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.13192","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.13192"},{"aliases":["10.48550/arxiv.2608.13320","2608.13320","David V. Feldman and Alexander Wilce"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f2618419af452fac543d5952bfdf27e0470b1140a9d2a6994152ff15ed33a8a.json","authors":"David V. Feldman and Alexander Wilce","evidence":{"source_url":"https://arxiv.org/abs/2608.13320"},"id":"literature:arxiv:2608.13320","identifiers":{"arxiv":"2608.13320","doi":"10.48550/arxiv.2608.13320"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.21936","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.21936"},{"aliases":["10.48550/arxiv.2608.25092","2608.25092"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bdcc16e35cbd420de6074ceead7496ece9a2deab58108d0a45fe2bd8d42666e2.json","evidence":{"source_url":"https://arxiv.org/abs/2608.25092"},"id":"literature:arxiv:2608.25092","identifiers":{"arxiv":"2608.25092","doi":"10.48550/arxiv.2608.25092"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.25092","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.25092"},{"aliases":["10.48550/arxiv.2608.25912","2608.25912"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d032574c05a49ae8393795673ed6d9677b92bfb9c61866f4649228a1b0aa2ef5.json","evidence":{"source_url":"https://arxiv.org/abs/2608.25912"},"id":"literature:arxiv:2608.25912","identifiers":{"arxiv":"2608.25912","doi":"10.48550/arxiv.2608.25912"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.25912","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.25912"},{"aliases":["10.48550/arxiv.2608.27085","2608.27085"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0decc17b72fc1ed92d3a81f1594ffb00e6daee36a56ef630f9fad308683ee3aa.json","evidence":{"source_url":"https://arxiv.org/abs/2608.27085"},"id":"literature:arxiv:2608.27085","identifiers":{"arxiv":"2608.27085","doi":"10.48550/arxiv.2608.27085"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.27085","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.27085"},{"aliases":["2608.27171","Christoph Hirche; Ian George; Theshani Nuradha; Mark M. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Conditional contraction coefficients and their applications to quantum networks","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.27171","year":2026},{"aliases":["2608.27583"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b2861bf923558c3184475c28ef50cc3e7c2b88132dc0612b30761c0152786711.json","evidence":{"source_url":"https://arxiv.org/abs/2608.27583"},"id":"literature:arxiv:2608.27583","identifiers":{"arxiv":"2608.27583"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.27583","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.27583"},{"aliases":["2608.27755","Charles E. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Undecidability of Adjacent Equality for Insertion, Shuffle, and Crossover Language Operations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.27755","year":2026},{"aliases":["10.48550/arxiv.2608.29231","2608.29231"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1112a601941336c2ef05f6047f1a82e482e0f5a57eaea3fe7b501128c60efde.json","evidence":{"source_url":"https://arxiv.org/abs/2608.29231"},"id":"literature:arxiv:2608.29231","identifiers":{"arxiv":"2608.29231","doi":"10.48550/arxiv.2608.29231"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.29231","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.29231"},{"aliases":["10.48550/arxiv.2608.30002","2608.30002","Umesh Shankar"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0f13b26845b370348e0109db1800a518bfeb4250222f030063327d9f27adf54.json","authors":"Umesh Shankar","evidence":{"source_url":"https://arxiv.org/abs/2608.30002"},"id":"literature:arxiv:2608.30002","identifiers":{"arxiv":"2608.30002","doi":"10.48550/arxiv.2608.30002"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Avoiding patterns with three distinct letters in Canon permutations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.30002","year":2026},{"aliases":["2608.30808","Daniele Bartoli; Pantelimon Stănică"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8662afc51485167bb6685d140cea7c0b70ba33d17773653a28767b11952c187f.json","authors":"Daniele Bartoli; Pantelimon Stănică","evidence":{"source_url":"https://arxiv.org/abs/2608.30808"},"id":"literature:arxiv:2608.30808","identifiers":{"arxiv":"2608.30808"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Reduced polynomial lifts of APN permutations over Galois rings and effective non-APN bounds","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.30808","year":2026},{"aliases":["10.48550/arxiv.2608.30851","2608.30851"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39b567c1b238117644ce9663fe64427085924d6ff3c6e8cb8d5123592ddb8aee.json","evidence":{"source_url":"https://arxiv.org/abs/2608.30851"},"id":"literature:arxiv:2608.30851","identifiers":{"arxiv":"2608.30851","doi":"10.48550/arxiv.2608.30851"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2608.30851","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2608.30851"},{"aliases":["2609.00706","Haobo Ma, Wenlin Zhang, Manuel Israel Cázares"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ddfa1374feb7efdd45ce745fa7e65a6255f61a9be63bf5704e266fabdf1534a.json","authors":"Haobo Ma, Wenlin Zhang, Manuel Israel Cázares","evidence":{"source_url":"https://arxiv.org/abs/2609.00706"},"id":"literature:arxiv:2609.00706","identifiers":{"arxiv":"2609.00706"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.00750","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.00750"},{"aliases":["2609.00778","Bikun Li; Liang Jiang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ab0ce16ebf907ade52b1dc174c466a3a6302939e063ac139bf3ccc2025e38d2.json","authors":"Bikun Li; Liang Jiang","evidence":{"source_url":"https://arxiv.org/abs/2609.00778"},"id":"literature:arxiv:2609.00778","identifiers":{"arxiv":"2609.00778"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"High-Rank Encoding Can Improve Approximate Quantum Error Correction","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.00778","year":2026},{"aliases":["10.48550/arxiv.2609.01140","2609.01140"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/564e830a14cb397947d8477567b11c1b5d7a40120a0caac1d77021f173923767.json","evidence":{"source_url":"https://arxiv.org/abs/2609.01140"},"id":"literature:arxiv:2609.01140","identifiers":{"arxiv":"2609.01140","doi":"10.48550/arxiv.2609.01140"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.01140","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.01140"},{"aliases":["10.48550/arxiv.2609.01440","2609.01440"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f13a1ea3f3b22f4d9f310ebbffe657dec1301a8fead15c4c7bfa98d74fdd7be3.json","evidence":{"source_url":"https://arxiv.org/abs/2609.01440"},"id":"literature:arxiv:2609.01440","identifiers":{"arxiv":"2609.01440","doi":"10.48550/arxiv.2609.01440"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.01440","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.01440"},{"aliases":["2609.01562","Juan Gil, Zhenni Liang, Ayodeji Odetola, Michael Weiner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a96d493b1163a89d332455f1ea546c639e56c1028f2c9a44af323bf964c36e6.json","authors":"Juan Gil, Zhenni Liang, Ayodeji Odetola, Michael Weiner","evidence":{"source_url":"https://arxiv.org/abs/2609.01562"},"id":"literature:arxiv:2609.01562","identifiers":{"arxiv":"2609.01562"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Iterated-sumset spectra: The complete exponent law and its rank geometry","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.01690","year":2026},{"aliases":["2609.02559","Kenneth Goodenough, Andrew Landahl, Joon Lee, Antonio Russo and Kevin Thompson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4dced23ddb6d90f7d6975733e08bc20726bc969c349d74d387e6942d2b727aa6.json","authors":"Kenneth Goodenough, Andrew Landahl, Joon Lee, Antonio Russo and Kevin Thompson","evidence":{"source_url":"https://arxiv.org/abs/2609.02559"},"id":"literature:arxiv:2609.02559","identifiers":{"arxiv":"2609.02559"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Optimal Fusion Strategies for Quantum Computation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.02559","year":2026},{"aliases":["10.48550/arxiv.2609.02939","2609.02939","William Kasel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0544f7fa34efffeffac1751cfc4e58effb7f5e69afe0fc361397690d7e55432f.json","authors":"William Kasel","evidence":{"source_url":"https://arxiv.org/abs/2609.02939"},"id":"literature:arxiv:2609.02939","identifiers":{"arxiv":"2609.02939","doi":"10.48550/arxiv.2609.02939"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.03081","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.03081"},{"aliases":["2609.03567","Emmanuel Jeandel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/036ef5fcdfc6475a0121e5ca69676edc125d71f80013e99521dd7671f0dc447e.json","authors":"Emmanuel Jeandel","evidence":{"source_url":"https://arxiv.org/abs/2609.03567"},"id":"literature:arxiv:2609.03567","identifiers":{"arxiv":"2609.03567"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Generalized Fidelity, the Data Processing Inequality, and Convexity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.09753","year":2026},{"aliases":["2609.10078","Jonte R. Hance"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38ded93116994156d9b2b4e7bb9f9be77baaf8b4ac73d2f37b9e4d9d35eaf95a.json","authors":"Jonte R. Hance","evidence":{"source_url":"https://arxiv.org/abs/2609.10078"},"id":"literature:arxiv:2609.10078","identifiers":{"arxiv":"2609.10078"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Why three? A two-level system with four mutually unbiased questions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.10078","year":2026},{"aliases":["10.48550/arxiv.2609.11072","2609.11072"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86a700b9b310a9161d3fc94d5402ed4c71c0e1f02c97deb83ffd3b8045f89d33.json","evidence":{"source_url":"https://arxiv.org/abs/2609.11072"},"id":"literature:arxiv:2609.11072","identifiers":{"arxiv":"2609.11072","doi":"10.48550/arxiv.2609.11072"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.11072","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.11072"},{"aliases":["10.48550/arxiv.2609.11186","2609.11186"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bf31b02b8c5178cbabe9b37d330d15dcb1d26247942939da88cd97056eb0d76.json","evidence":{"source_url":"https://arxiv.org/abs/2609.11186"},"id":"literature:arxiv:2609.11186","identifiers":{"arxiv":"2609.11186","doi":"10.48550/arxiv.2609.11186"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.11186","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.11186"},{"aliases":["2609.12667","A. S. Holevo; M. E. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.17418","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.17418"},{"aliases":["10.48550/arxiv.2609.18476","2609.18476"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4648ae5609bea9578c97b857091f6501efa3b7e5d010674462d8aeb9014f0544.json","evidence":{"source_url":"https://arxiv.org/abs/2609.18476"},"id":"literature:arxiv:2609.18476","identifiers":{"arxiv":"2609.18476","doi":"10.48550/arxiv.2609.18476"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.18476","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.18476"},{"aliases":["10.48550/arxiv.2609.18803","2609.18803"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa6ccc0bc04edcb8a114cf1f1e562cc76cfd6223f9365e64cd121ab0c5b07f3a.json","evidence":{"source_url":"https://arxiv.org/abs/2609.18803"},"id":"literature:arxiv:2609.18803","identifiers":{"arxiv":"2609.18803","doi":"10.48550/arxiv.2609.18803"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.22394","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.22394"},{"aliases":["10.48550/arxiv.2609.22769","2609.22769"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4127d502268cdada002ed35ebe17110dfe9f072a867a259c4483a6752f7cd98c.json","evidence":{"source_url":"https://arxiv.org/abs/2609.22769"},"id":"literature:arxiv:2609.22769","identifiers":{"arxiv":"2609.22769","doi":"10.48550/arxiv.2609.22769"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.22769","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.22769"},{"aliases":["10.48550/arxiv.2609.23106","2609.23106"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f50f952ed6cca56a30547f7995cd0c36cef74a58aac59b0621e9cddace51ba9d.json","evidence":{"source_url":"https://arxiv.org/abs/2609.23106"},"id":"literature:arxiv:2609.23106","identifiers":{"arxiv":"2609.23106","doi":"10.48550/arxiv.2609.23106"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.23106","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.23106"},{"aliases":["10.48550/arxiv.2609.23607","2609.23607"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dab6d4aee2440aaf2a2a23559a53870ab413d992b1e24fb9e45e73e305e820e3.json","evidence":{"source_url":"https://arxiv.org/abs/2609.23607"},"id":"literature:arxiv:2609.23607","identifiers":{"arxiv":"2609.23607","doi":"10.48550/arxiv.2609.23607"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.23607","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.23607"},{"aliases":["2609.24544","Sompong Chuysurichay; Sawian Jaidee; Chatchawan Panraksa; Thomas Ward"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/314d9377e694c1e5b8b3cf9a4c1a61d98f6d183ac7aa0223cc28ee78cb39ab6f.json","authors":"Sompong Chuysurichay; Sawian Jaidee; Chatchawan Panraksa; Thomas Ward","evidence":{"source_url":"https://arxiv.org/abs/2609.24544"},"id":"literature:arxiv:2609.24544","identifiers":{"arxiv":"2609.24544"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.25078","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.25078"},{"aliases":["10.48550/arxiv.2609.25100","2609.25100"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0964b6a5b35c8636f31a4ac29e18e5be8f403b5bf4a1ab87eabfb57752e2a6f2.json","evidence":{"source_url":"https://arxiv.org/abs/2609.25100"},"id":"literature:arxiv:2609.25100","identifiers":{"arxiv":"2609.25100","doi":"10.48550/arxiv.2609.25100"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.25100","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.25100"},{"aliases":["10.48550/arxiv.2609.25128","2609.25128","Haobo Ma, Rafik Sahbi, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e72b926bf0b991fb4705088175a45cc12a4e38d801e312b92900841683ca593.json","authors":"Haobo Ma, Rafik Sahbi, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2609.25128"},"id":"literature:arxiv:2609.25128","identifiers":{"arxiv":"2609.25128","doi":"10.48550/arxiv.2609.25128"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.25552","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.25552"},{"aliases":["2609.25742","Sen-Peng Eu, Zai-Ting Huang, and Louis Kao"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a843c14e095698d64010ea9568a06ff4b59d238ae01903d43d655935e4685170.json","authors":"Sen-Peng Eu, Zai-Ting Huang, and Louis Kao","evidence":{"source_url":"https://arxiv.org/abs/2609.25742"},"id":"literature:arxiv:2609.25742","identifiers":{"arxiv":"2609.25742"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Zero-Run Spectra of the (3,2) Raney numbers Modulo Primes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.25742","year":2026},{"aliases":["2609.26114","David V. Feldman"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e5b5fb88294d190d979da0e5ca2d81358584c20d5a49b5581cb3ff3334788182.json","authors":"David V. Feldman","evidence":{"source_url":"https://arxiv.org/abs/2609.26114"},"id":"literature:arxiv:2609.26114","identifiers":{"arxiv":"2609.26114"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Missing Zigzag: Cycles of Semitone Trichords and a Conservation Law in Equal Temperament","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.26114","year":2026},{"aliases":["2609.26762","Manjul Bhargava"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d90eeaaae704f7f99e75cfc2f2784e5c22bc48f0a00dac176021aba574a58610.json","authors":"Manjul Bhargava","evidence":{"source_url":"https://arxiv.org/abs/2609.26762"},"id":"literature:arxiv:2609.26762","identifiers":{"arxiv":"2609.26762"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The existence and uniqueness of magic-faced hypercubes, and applications to Khajuraho most-perfect magic squares, cubes, and hypercubes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.26762","year":2026},{"aliases":["10.48550/arxiv.2609.27906","2609.27906"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/096bcb130f7d2b686254f0565f22da7745a98251e27a4928cdfb7a3f1169d1ba.json","evidence":{"source_url":"https://arxiv.org/abs/2609.27906"},"id":"literature:arxiv:2609.27906","identifiers":{"arxiv":"2609.27906","doi":"10.48550/arxiv.2609.27906"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.27906","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.27906"},{"aliases":["2609.28511","Jean-Pierre Gazeau"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46c520b94cffff96c85ccd500e7ada28fa0fd71898baf8622069c3985c79a580.json","authors":"Jean-Pierre Gazeau","evidence":{"source_url":"https://arxiv.org/abs/2609.28511"},"id":"literature:arxiv:2609.28511","identifiers":{"arxiv":"2609.28511"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Bi-Oriented Interlaced Matrix Multiplication and an Operator Factorisation of the Quantum Harmonic Oscillator","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.28511","year":2026},{"aliases":["10.48550/arxiv.2609.28594","2609.28594"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1cfa5de736a13c5a84220b7317ad360010f5cd494cd38bed99b261620cb40d02.json","evidence":{"source_url":"https://arxiv.org/abs/2609.28594"},"id":"literature:arxiv:2609.28594","identifiers":{"arxiv":"2609.28594","doi":"10.48550/arxiv.2609.28594"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.28594","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.28594"},{"aliases":["10.48550/arxiv.2609.28808","2609.28808"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6d0bb73f4c5c4aa9c558d6af0d5286d8bc2043bde4dcc236d61dfac8f1fb03a.json","evidence":{"source_url":"https://arxiv.org/abs/2609.28808"},"id":"literature:arxiv:2609.28808","identifiers":{"arxiv":"2609.28808","doi":"10.48550/arxiv.2609.28808"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.28808","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.28808"},{"aliases":["10.48550/arxiv.2609.29392","2609.29392"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fba9d7145affd07cb8df15c1fd67609ee7e1289a6a0151283f5bb9ec13b4c9d9.json","evidence":{"source_url":"https://arxiv.org/abs/2609.29392"},"id":"literature:arxiv:2609.29392","identifiers":{"arxiv":"2609.29392","doi":"10.48550/arxiv.2609.29392"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.29392","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.29392"},{"aliases":["2609.30246","M. E. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Moreau-Yosida approximation of the EoF: basic properties and accuracy estimates","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.30246","year":2026},{"aliases":["10.48550/arxiv.2609.30754","2609.30754"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9abebbffddb02c741f0f0ce9dfb3c002860e7f6ff70a585382a8f1b8f74ea49f.json","evidence":{"source_url":"https://arxiv.org/abs/2609.30754"},"id":"literature:arxiv:2609.30754","identifiers":{"arxiv":"2609.30754","doi":"10.48550/arxiv.2609.30754"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.30754","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.30754"},{"aliases":["10.48550/arxiv.2609.31489","2609.31489"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1303523e5402f83f4f0f97a2266df305a3fd6dd34d904ae43901a544822369b.json","evidence":{"source_url":"https://arxiv.org/abs/2609.31489"},"id":"literature:arxiv:2609.31489","identifiers":{"arxiv":"2609.31489","doi":"10.48550/arxiv.2609.31489"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.31489","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.31489"},{"aliases":["10.48550/arxiv.2609.32077","2609.32077"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/feebee98b61a6fc169017246bdd66446e56945db9074cc16051f7b0ac7bf694a.json","evidence":{"source_url":"https://arxiv.org/abs/2609.32077"},"id":"literature:arxiv:2609.32077","identifiers":{"arxiv":"2609.32077","doi":"10.48550/arxiv.2609.32077"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.32077","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.32077"},{"aliases":["10.48550/arxiv.2609.32161","2609.32161"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b85cfb001850c2b202f18be56013e81e04751367e4af7b5e9599269bc6883beb.json","evidence":{"source_url":"https://arxiv.org/abs/2609.32161"},"id":"literature:arxiv:2609.32161","identifiers":{"arxiv":"2609.32161","doi":"10.48550/arxiv.2609.32161"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.32161","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.32161"},{"aliases":["2609.33421","Benoit Cloitre, Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ace90f93924f57a325aa5896a131e4c006a11bef339b4dead905e87ab353108f.json","authors":"Benoit Cloitre, Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2609.33421"},"id":"literature:arxiv:2609.33421","identifiers":{"arxiv":"2609.33421"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Padovan-automatic description of a nested recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.33421"},{"aliases":["2609.33744","William J. Wesley"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0fdec5b1f598e19670862cd1eda6f1369f52d1c79cdcc137333e3a305e62069e.json","authors":"William J. Wesley","evidence":{"source_url":"https://arxiv.org/abs/2609.33744"},"id":"literature:arxiv:2609.33744","identifiers":{"arxiv":"2609.33744"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Fibonacci numbers are not 3-accessible","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.33744","year":2026},{"aliases":["2609.37973","Paweł Pękała"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a38bb8f9301a175f450f91174c75b747d87aeed785d8238b0f0226a42b24fc63.json","authors":"Paweł Pękała","evidence":{"source_url":"https://arxiv.org/abs/2609.37973"},"id":"literature:arxiv:2609.37973","identifiers":{"arxiv":"2609.37973"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On-line majority edge-colourings of graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.37973","year":2026},{"aliases":["10.48550/arxiv.2609.38243","2609.38243"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd5655f886dac96ffce22ad820468330632fb88845cf1a44a34ae902a3a72642.json","evidence":{"source_url":"https://arxiv.org/abs/2609.38243"},"id":"literature:arxiv:2609.38243","identifiers":{"arxiv":"2609.38243","doi":"10.48550/arxiv.2609.38243"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.38243","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.38243"},{"aliases":["10.48550/arxiv.2609.39979","2609.39979"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18b3d5662f4f3b73646da95557c10815c7c3cebe8aac7965d08efab2137f2532.json","evidence":{"source_url":"https://arxiv.org/abs/2609.39979"},"id":"literature:arxiv:2609.39979","identifiers":{"arxiv":"2609.39979","doi":"10.48550/arxiv.2609.39979"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2609.39979","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.39979"},{"aliases":["10.48550/arxiv.2609.40154","2609.40154","Pui Hang Lee, Pui Yee Lee and Bjørn Kjos-Hanssen"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/540d2b3ea7c4b3e80bf75a60208c5129cb7c2f0b592df2546c794c3133901194.json","authors":"Pui Hang Lee, Pui Yee Lee and Bjørn Kjos-Hanssen","evidence":{"source_url":"https://arxiv.org/abs/2609.40154"},"id":"literature:arxiv:2609.40154","identifiers":{"arxiv":"2609.40154","doi":"10.48550/arxiv.2609.40154"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Černý complexity of binary words","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2609.40154","year":2026},{"aliases":["10.48550/arxiv.2610.00107","2610.00107"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fdf03dbe5ef07ffbaf76718738bafc6cd6757233f7b493b4eb7dd479ad959c1.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00107"},"id":"literature:arxiv:2610.00107","identifiers":{"arxiv":"2610.00107","doi":"10.48550/arxiv.2610.00107"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00107","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00107"},{"aliases":["10.48550/arxiv.2610.00108","2610.00108"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b68212a3980488308451a09957a484513c1770dc01fcdeb2ab896cb31d1385e7.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00108"},"id":"literature:arxiv:2610.00108","identifiers":{"arxiv":"2610.00108","doi":"10.48550/arxiv.2610.00108"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00108","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00108"},{"aliases":["10.48550/arxiv.2610.00293","2610.00293"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4111bc2bbc0f5c3954ed155f1ffed261d94f7c955871463aa157199977ff5b1.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00293"},"id":"literature:arxiv:2610.00293","identifiers":{"arxiv":"2610.00293","doi":"10.48550/arxiv.2610.00293"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00293","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00293"},{"aliases":["10.48550/arxiv.2610.00500","2610.00500"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/017bbb37ad1e8e910925c9202d31e706a0796b160f7167dbf9c1e7d813d3c175.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00500"},"id":"literature:arxiv:2610.00500","identifiers":{"arxiv":"2610.00500","doi":"10.48550/arxiv.2610.00500"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00500","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00500"},{"aliases":["10.48550/arxiv.2610.00579","2610.00579"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a199cdeb7c8cf652e0955d1fd2a81f98018e49d1f233555f735ad825dc88902a.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00579"},"id":"literature:arxiv:2610.00579","identifiers":{"arxiv":"2610.00579","doi":"10.48550/arxiv.2610.00579"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00579","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00579"},{"aliases":["10.48550/arxiv.2610.00765","2610.00765"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0579d350fc1a6a7c959811f5438392401f5259d3b00b9d09b9d29e76a4746ea4.json","evidence":{"source_url":"https://arxiv.org/abs/2610.00765"},"id":"literature:arxiv:2610.00765","identifiers":{"arxiv":"2610.00765","doi":"10.48550/arxiv.2610.00765"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.00765","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.00765"},{"aliases":["2610.01029","Vu Trung Hieu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0838f4f81d338bae68f4a68f404e750e474e2b10ae8ea5ed9764665c6942dff.json","authors":"Vu Trung Hieu","evidence":{"source_url":"https://arxiv.org/abs/2610.01029"},"id":"literature:arxiv:2610.01029","identifiers":{"arxiv":"2610.01029"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A connectedness result for convex polynomial multi-objective optimization without boundedness","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.01029","year":2026},{"aliases":["10.48550/arxiv.2610.01584","2610.01584"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9937a6d1028d18c2f8daece18c2097983405056ec387cf65205dea4c6661bf9.json","evidence":{"source_url":"https://arxiv.org/abs/2610.01584"},"id":"literature:arxiv:2610.01584","identifiers":{"arxiv":"2610.01584","doi":"10.48550/arxiv.2610.01584"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.01584","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.01584"},{"aliases":["10.48550/arxiv.2610.01860","2610.01860"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c25bb24d3eaec71706e62c45ec0d620df5a57853bb8d803ed8801ef58138f60a.json","evidence":{"source_url":"https://arxiv.org/abs/2610.01860"},"id":"literature:arxiv:2610.01860","identifiers":{"arxiv":"2610.01860","doi":"10.48550/arxiv.2610.01860"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.01860","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.01860"},{"aliases":["10.48550/arxiv.2610.02392","2610.02392"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ecc0c8b76b9b6ceae74c0cc094143ce3cdaf401f115fda26186b04e8a0bff9f4.json","evidence":{"source_url":"https://arxiv.org/abs/2610.02392"},"id":"literature:arxiv:2610.02392","identifiers":{"arxiv":"2610.02392","doi":"10.48550/arxiv.2610.02392"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.02392","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.02392"},{"aliases":["10.48550/arxiv.2610.02495","2610.02495"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09d5d14771fc991da4b5fe78fd92d2f48dc275dd6b3b7a61e88e7c88fc12d11f.json","evidence":{"source_url":"https://arxiv.org/abs/2610.02495"},"id":"literature:arxiv:2610.02495","identifiers":{"arxiv":"2610.02495","doi":"10.48550/arxiv.2610.02495"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.02495","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.02495"},{"aliases":["10.48550/arxiv.2610.02833","2610.02833"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5363af503b554fac2354088f746c26278905d0db32c0a9db99b519e14f0cf7f8.json","evidence":{"source_url":"https://arxiv.org/abs/2610.02833"},"id":"literature:arxiv:2610.02833","identifiers":{"arxiv":"2610.02833","doi":"10.48550/arxiv.2610.02833"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.02833","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.02833"},{"aliases":["10.48550/arxiv.2610.03477","2610.03477"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4087a6805033d8856dde2222ad5b0a61afe24246bf8b597ca9f493dcb81e4378.json","evidence":{"source_url":"https://arxiv.org/abs/2610.03477"},"id":"literature:arxiv:2610.03477","identifiers":{"arxiv":"2610.03477","doi":"10.48550/arxiv.2610.03477"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.03477","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.03477"},{"aliases":["10.48550/arxiv.2610.03785","2610.03785"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf9da57d00e4c77fcc30d10d70705e49a6d37c56a4dc4f2795f5f4543fad1d8d.json","evidence":{"source_url":"https://arxiv.org/abs/2610.03785"},"id":"literature:arxiv:2610.03785","identifiers":{"arxiv":"2610.03785","doi":"10.48550/arxiv.2610.03785"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.03785","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.03785"},{"aliases":["10.48550/arxiv.2610.03892","2610.03892"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ba2e1aee7949cfc546a7826f939364721de511a1227c41b437e1c6356cf03db.json","evidence":{"source_url":"https://arxiv.org/abs/2610.03892"},"id":"literature:arxiv:2610.03892","identifiers":{"arxiv":"2610.03892","doi":"10.48550/arxiv.2610.03892"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.03892","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.03892"},{"aliases":["10.48550/arxiv.2610.05302","2610.05302"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0cc87ced23340aede06a95238849b64b7cefc51e925648900a9bfb04c1a285b7.json","evidence":{"source_url":"https://arxiv.org/abs/2610.05302"},"id":"literature:arxiv:2610.05302","identifiers":{"arxiv":"2610.05302","doi":"10.48550/arxiv.2610.05302"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.05302","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.05302"},{"aliases":["10.48550/arxiv.2610.05623","2610.05623","Nicolae Cotfas, Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8def505a648b877ea6d22361647dcf7aed698f15ae4cea9e8601d31baf369e3.json","authors":"Nicolae Cotfas, Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2610.05623"},"id":"literature:arxiv:2610.05623","identifiers":{"arxiv":"2610.05623","doi":"10.48550/arxiv.2610.05623"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Approximation of a continuous-variable Gaussian state by discrete-variable quantum states","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.05623"},{"aliases":["10.48550/arxiv.2610.05776","2610.05776"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e46d443b592c67c927a25091979937e1a2d31c7665546750ba0157eb81067f8.json","evidence":{"source_url":"https://arxiv.org/abs/2610.05776"},"id":"literature:arxiv:2610.05776","identifiers":{"arxiv":"2610.05776","doi":"10.48550/arxiv.2610.05776"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.05776","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.05776"},{"aliases":["10.48550/arxiv.2610.06925","2610.06925"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3231183102f9c6829c9c4d6b6472ed736140e0910a75e93bf4af98c39d3b9623.json","evidence":{"source_url":"https://arxiv.org/abs/2610.06925"},"id":"literature:arxiv:2610.06925","identifiers":{"arxiv":"2610.06925","doi":"10.48550/arxiv.2610.06925"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.06925","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.06925"},{"aliases":["10.48550/arxiv.2610.07999","2610.07999"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/478108ef4afd3e40b101288ca83310f26512a75f7e28683545fbc1f9307073cc.json","evidence":{"source_url":"https://arxiv.org/abs/2610.07999"},"id":"literature:arxiv:2610.07999","identifiers":{"arxiv":"2610.07999","doi":"10.48550/arxiv.2610.07999"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.07999","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.07999"},{"aliases":["10.48550/arxiv.2610.08636","2610.08636"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/53a02c218d55805556b9b49dc534f12e07aeca2cd2eb1bd488b919e01242a5f6.json","evidence":{"source_url":"https://arxiv.org/abs/2610.08636"},"id":"literature:arxiv:2610.08636","identifiers":{"arxiv":"2610.08636","doi":"10.48550/arxiv.2610.08636"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.08636","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.08636"},{"aliases":["10.48550/arxiv.2610.08889","2610.08889"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2cc6f48af57ab236eca09b245bdb49d9a3d9857bd3fc25ca7435490870dd478e.json","evidence":{"source_url":"https://arxiv.org/abs/2610.08889"},"id":"literature:arxiv:2610.08889","identifiers":{"arxiv":"2610.08889","doi":"10.48550/arxiv.2610.08889"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.08889","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.08889"},{"aliases":["10.48550/arxiv.2610.09300","2610.09300","My Hanh Pham, Thanh Vu"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b7b2cdf401457bd5eb1a37e4705babcc8a1d67c24db3e24c27dc43043a01167.json","authors":"My Hanh Pham, Thanh Vu","evidence":{"source_url":"https://arxiv.org/abs/2610.09300"},"id":"literature:arxiv:2610.09300","identifiers":{"arxiv":"2610.09300","doi":"10.48550/arxiv.2610.09300"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On graphs whose powers are well-covered","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.09300","year":2026},{"aliases":["2610.10820","Matthew Chen; Natalie Klco"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf2004e2e5c26b7800681c436f168b14e514eed6eecd56fd740c4a46c28cfa07.json","authors":"Matthew Chen; Natalie Klco","evidence":{"source_url":"https://arxiv.org/abs/2610.10820"},"id":"literature:arxiv:2610.10820","identifiers":{"arxiv":"2610.10820"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Susceptibility of qudit-qudit entanglement to quantum noise: insights from the negativity rank","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.10820","year":2026},{"aliases":["10.48550/arxiv.2610.11446","2610.11446"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a16c204ae93b46c3673439b4b24180ca70c44620eff0e41baffedf1620e011a1.json","evidence":{"source_url":"https://arxiv.org/abs/2610.11446"},"id":"literature:arxiv:2610.11446","identifiers":{"arxiv":"2610.11446","doi":"10.48550/arxiv.2610.11446"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.11446","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.11446"},{"aliases":["10.48550/arxiv.2610.11596","2610.11596"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc8b14297dd5bf7dec8abbc56bf36ed1a6f730760e908db1e741659893179e9d.json","evidence":{"source_url":"https://arxiv.org/abs/2610.11596"},"id":"literature:arxiv:2610.11596","identifiers":{"arxiv":"2610.11596","doi":"10.48550/arxiv.2610.11596"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.11596","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.11596"},{"aliases":["2610.12949","C. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum-assisted rendezvous on hypercubes: a quantum Bernoulli noise approach","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.12949","year":2026},{"aliases":["2610.14373","Boris Shapiro"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90cb3adbd885a37b0378bb21fbe2db08707983acd275107c40dbae74013d7a67.json","authors":"Boris Shapiro","evidence":{"source_url":"https://arxiv.org/abs/2610.14373"},"id":"literature:arxiv:2610.14373","identifiers":{"arxiv":"2610.14373"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Transposition Cayley Spectra and Subgraph Counts","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.14373","year":2026},{"aliases":["2610.15889","Suvrit Sra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08a1dc6ad272a1c714af6580e10af768f0215fb603566a3e9bddd186b528baa6.json","authors":"Suvrit Sra","evidence":{"source_url":"https://arxiv.org/abs/2610.15889"},"id":"literature:arxiv:2610.15889","identifiers":{"arxiv":"2610.15889"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The p-Bures-Wasserstein distance via 3-by-3 block matrices (with application to p-Bures-Wasserstein barycenters)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.15889","year":2026},{"aliases":["10.48550/arxiv.2610.16562","2610.16562"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/227014f8ad4e4f2e5352151d212b68fca18b1d5f451a1d7b8ba0a0b916906de0.json","evidence":{"source_url":"https://arxiv.org/abs/2610.16562"},"id":"literature:arxiv:2610.16562","identifiers":{"arxiv":"2610.16562","doi":"10.48550/arxiv.2610.16562"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.16562","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.16562"},{"aliases":["2610.17267","Grace Natalie Jaffe"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0cdda91df27d6995467208a24017ec855780adba9f9cc85e3bc17b4cc685bf6.json","authors":"Grace Natalie Jaffe","evidence":{"source_url":"https://arxiv.org/abs/2610.17267"},"id":"literature:arxiv:2610.17267","identifiers":{"arxiv":"2610.17267"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Denominators in the Farey Sequence, the Stern Sequence, and the Frobenius Problem with Relatively Prime Multipliers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.17267","year":2026},{"aliases":["10.48550/arxiv.2610.17393","2610.17393"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/847527d45220e50a410ba8dc37bbb3749d37d40dd06e5b6a6f01d46aeee1e1b5.json","evidence":{"source_url":"https://arxiv.org/abs/2610.17393"},"id":"literature:arxiv:2610.17393","identifiers":{"arxiv":"2610.17393","doi":"10.48550/arxiv.2610.17393"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.17393","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.17393"},{"aliases":["2610.18816","Haobo Ma, Maksim E. Shirokov, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5147ddd42ae349e12251a3b9424a83fba862c134a9299e251639da419d3139ac.json","authors":"Haobo Ma, Maksim E. Shirokov, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2610.18816"},"id":"literature:arxiv:2610.18816","identifiers":{"arxiv":"2610.18816"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the coincidence of discrete and continuous definitions of Entanglement of Formation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.18816"},{"aliases":["10.48550/arxiv.2610.19249","2610.19249"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b8d361c3b401b9720fc23f1cc2ed9d45c95a4915ca74986c36b93d990eff324.json","evidence":{"source_url":"https://arxiv.org/abs/2610.19249"},"id":"literature:arxiv:2610.19249","identifiers":{"arxiv":"2610.19249","doi":"10.48550/arxiv.2610.19249"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.19249","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.19249"},{"aliases":["2610.20329","Edgar Alcalá-Arroyo, Alonso Castillo-Ramirez, Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86a7ca53fd5ce0a424bb31c1b5ce6d6ec427bd80b6152c4e28fefde1de75bed5.json","authors":"Edgar Alcalá-Arroyo, Alonso Castillo-Ramirez, Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2610.20329"},"id":"literature:arxiv:2610.20329","identifiers":{"arxiv":"2610.20329"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Generation by lazy and invertible cellular automata over the Klein four-group","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.20329"},{"aliases":["10.48550/arxiv.2610.21593","2610.21593"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19a57d5da37b4db114776d2a20a23911355fb754c702e6f46d49c6d44f65ad87.json","evidence":{"source_url":"https://arxiv.org/abs/2610.21593"},"id":"literature:arxiv:2610.21593","identifiers":{"arxiv":"2610.21593","doi":"10.48550/arxiv.2610.21593"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.21593","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.21593"},{"aliases":["10.48550/arxiv.2610.22248","2610.22248"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/acf8467fcfee5b53613fdfbc5231d888b701f4b0f1179bfac60f7a0e566231a0.json","evidence":{"source_url":"https://arxiv.org/abs/2610.22248"},"id":"literature:arxiv:2610.22248","identifiers":{"arxiv":"2610.22248","doi":"10.48550/arxiv.2610.22248"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.22248","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.22248"},{"aliases":["2610.24059","Arnav Krishnan, Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c08209870521beede1a1b6770f8517440c2709a304e5d9a08f7e7363a560d71.json","authors":"Arnav Krishnan, Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2610.24059"},"id":"literature:arxiv:2610.24059","identifiers":{"arxiv":"2610.24059"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Sharp zero forcing numbers for the generalized Petersen graphs P(n,3) and P(n,4)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.24059"},{"aliases":["10.48550/arxiv.2610.24324","2610.24324"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/89fdf9e637e045b0a98c89b51256cd5d1236b450cb280181ab55f8503965dff6.json","evidence":{"source_url":"https://arxiv.org/abs/2610.24324"},"id":"literature:arxiv:2610.24324","identifiers":{"arxiv":"2610.24324","doi":"10.48550/arxiv.2610.24324"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.24324","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.24324"},{"aliases":["10.48550/arxiv.2610.26875","2610.26875"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc83622d65533183f7b73ea57a3ce5e72970ce84530a1c35778164fb5d16cd48.json","evidence":{"source_url":"https://arxiv.org/abs/2610.26875"},"id":"literature:arxiv:2610.26875","identifiers":{"arxiv":"2610.26875","doi":"10.48550/arxiv.2610.26875"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.26875","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.26875"},{"aliases":["2610.29291","Spencer Daugherty, Michaela Fahnestock, Shawn Huxtable, Michael Larson, Nourah Al Munayes and Storey Peacock"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1e4638ddfbeac5491e806c8b5b024dbd451dcb80a75c0c0cee80318f734e387.json","authors":"Spencer Daugherty, Michaela Fahnestock, Shawn Huxtable, Michael Larson, Nourah Al Munayes and Storey Peacock","evidence":{"source_url":"https://arxiv.org/abs/2610.29291"},"id":"literature:arxiv:2610.29291","identifiers":{"arxiv":"2610.29291"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Counting and Constructing Latin tableaux","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.29291","year":2026},{"aliases":["2610.29608","Haobo Ma, Mircea Merca, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4482837ff4798624b2ed9736a30cbcfca2648c90ffc05b0ee0eca6435e6ef34.json","authors":"Haobo Ma, Mircea Merca, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2610.29608"},"id":"literature:arxiv:2610.29608","identifiers":{"arxiv":"2610.29608"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Residue Sums Modulo Products of Two Distinct Odd Primes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.29608"},{"aliases":["10.48550/arxiv.2610.29846","2610.29846"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1758893d77e2cd5e084efb797ba848dbe8994aa3ee7a9029e4595cb69fa95f5a.json","evidence":{"source_url":"https://arxiv.org/abs/2610.29846"},"id":"literature:arxiv:2610.29846","identifiers":{"arxiv":"2610.29846","doi":"10.48550/arxiv.2610.29846"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"2610.29846","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.29846"},{"aliases":["2610.31475","William F. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Computer-assisted results for n-sided non-transitive dice taking face values in [n]","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2610.40321","year":2026},{"aliases":["2611.01064","Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a825127bd9ce8a62f3d325694f8a852eb9fbec2f63bb0c0a96cc74433b39b77d.json","authors":"Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://arxiv.org/abs/2611.01064"},"id":"literature:arxiv:2611.01064","identifiers":{"arxiv":"2611.01064"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Maximal input entropy for binary two-deletion and two-insertion channels","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2611.01064"},{"aliases":["2611.01243","Max A. Alekseyev"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbce8465fd7ba0b25bc6b1a22630130ea30d468958829507e42b2ae154ad148b.json","authors":"Max A. Alekseyev","evidence":{"source_url":"https://arxiv.org/abs/2611.01243"},"id":"literature:arxiv:2611.01243","identifiers":{"arxiv":"2611.01243"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Halving Partitions and Hyperbinary Expansions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2611.01243","year":2026},{"aliases":["2611.04568","Wenlin Zhang, Mazhar Ali, Haobo Ma"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec4761c8dcc68961566aad197a82aebe52d260a2ac06feea7f5e6fe2bcdf6859.json","authors":"Wenlin Zhang, Mazhar Ali, Haobo Ma","evidence":{"source_url":"https://arxiv.org/abs/2611.04568"},"id":"literature:arxiv:2611.04568","identifiers":{"arxiv":"2611.04568"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Time-invariant negativity and persistent bound entanglement in a cyclic two-qutrit family","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3A2611.04568"},{"aliases":["Atsushi Matsuo; Kiyokazu Nagatomo","hep-th/9704060"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1278d0df1c6c4ff5c759e85e57b4c36f11b4a1aadcfeddbfe82fdf54ffd4b8a9.json","authors":"Atsushi Matsuo; Kiyokazu Nagatomo","evidence":{"source_url":"https://arxiv.org/abs/hep-th/9704060"},"id":"literature:arxiv:hep-th/9704060","identifiers":{"arxiv":"hep-th/9704060"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Note on Free Bosonic Vertex Algebra and its Conformal Vectors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Ahep-th%2F9704060","year":1997},{"aliases":["Atsushi Matsuo; Kiyokazu Nagatomo","hep-th/9706118"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/681e48f65873d0e24e07bb9c1368d262a02f5638775da9c9866e3aee59a741d0.json","authors":"Atsushi Matsuo; Kiyokazu Nagatomo","evidence":{"source_url":"https://arxiv.org/abs/hep-th/9706118"},"id":"literature:arxiv:hep-th/9706118","identifiers":{"arxiv":"hep-th/9706118"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On axioms for a vertex algebra and the locality of quantum fields","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Ahep-th%2F9706118","year":1997},{"aliases":["Luis Báez-Duarte","math/0202141"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ceb835b8e18132052ed2c0df950d58bca6dd076a6c2f9d1677106af9313ad6e3.json","authors":"Luis Báez-Duarte","evidence":{"source_url":"https://arxiv.org/abs/math/0202141"},"id":"literature:arxiv:math/0202141","identifiers":{"arxiv":"math/0202141"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A strengthening of the Nyman-Beurling criterion for the Riemann Hypothesis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Amath%2F0202141","year":2002},{"aliases":["Jean-Louis Loday","math/0212126"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0aad2028a996ed8be1fcb8f83908836fb861cf588e5251a18c1ef1079d980cc7.json","authors":"Jean-Louis Loday","evidence":{"source_url":"https://arxiv.org/abs/math/0212126"},"id":"literature:arxiv:math/0212126","identifiers":{"arxiv":"math/0212126"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Realization of the Stasheff polytope","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Amath%2F0212126","year":2002},{"aliases":["Luis Báez-Duarte","math/0307215"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/34fd27c11ce9fde1e60caa4f6efefdc5b18e45a1ada592e2fb6b0a8d25e0ccf2.json","authors":"Luis Báez-Duarte","evidence":{"source_url":"https://arxiv.org/abs/math/0307215"},"id":"literature:arxiv:math/0307215","identifiers":{"arxiv":"math/0307215"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A new necessary and sufficient condition for the Riemann hypothesis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Amath%2F0307215","year":2003},{"aliases":["Jeffrey C. Lagarias","math/0404394"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c03af2a2a1c4662c34317cc2818a97d27d998cbf74ea5738d8acdf9c5344ffb9.json","authors":"Jeffrey C. Lagarias","evidence":{"source_url":"https://arxiv.org/abs/math/0404394"},"id":"literature:arxiv:math/0404394","identifiers":{"arxiv":"math/0404394"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Li Coefficients for Automorphic L-Functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Amath%2F0404394","year":2004},{"aliases":["Carl de Boor","math/0502036"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a912ecfc5aa383e923129a9d30a2afc121b84658775517739d85879ce96b6b3f.json","authors":"Carl de Boor","evidence":{"source_url":"https://arxiv.org/abs/math/0502036"},"id":"literature:arxiv:math/0502036","identifiers":{"arxiv":"math/0502036"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Factorization and Reflexivity on Fock spaces","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Amath%2F9404209","year":1994},{"aliases":["Masato Koashi and Nobuyuki Imoto","quant-ph/0101144"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7926edb6e790963b8158ddda0c1bcdab062ec16549c69ff0dc18388b340f1636.json","authors":"Masato Koashi and Nobuyuki Imoto","evidence":{"source_url":"https://arxiv.org/abs/quant-ph/0101144"},"id":"literature:arxiv:quant-ph/0101144","identifiers":{"arxiv":"quant-ph/0101144"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Operations that do not disturb partially known quantum states","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Aquant-ph%2F0101144","year":2002},{"aliases":["Denes Petz","quant-ph/0106125"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/80a25bc794afb987da4fbb60377e745816811d4168a42852930ad642333c8b0f.json","authors":"Denes Petz","evidence":{"source_url":"https://arxiv.org/abs/quant-ph/0106125"},"id":"literature:arxiv:quant-ph/0106125","identifiers":{"arxiv":"quant-ph/0106125"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Covariance and Fisher information in quantum mechanics","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Aquant-ph%2F0106125","year":2001},{"aliases":["Dorit Aharonov; Wim van Dam; Julia Kempe; Zeph Landau; Seth Lloyd; Oded Regev","quant-ph/0405098"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c88c84610062a8d5bca66b56ad5a65fa8f15d5218ac4e6d456f2a18835678cf.json","authors":"Dorit Aharonov; Wim van Dam; Julia Kempe; Zeph Landau; Seth Lloyd; Oded Regev","evidence":{"source_url":"https://arxiv.org/abs/quant-ph/0405098"},"id":"literature:arxiv:quant-ph/0405098","identifiers":{"arxiv":"quant-ph/0405098"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Aquant-ph%2F0405098","year":2004},{"aliases":["O. Krüger; R. F. Werner","quant-ph/0504166"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cec27fcd2c260a994c92b9a4a83a4d127f98ea05b5da722bd179c5bc9a72df87.json","authors":"O. Krüger; R. F. Werner","evidence":{"source_url":"https://arxiv.org/abs/quant-ph/0504166"},"id":"literature:arxiv:quant-ph/0504166","identifiers":{"arxiv":"quant-ph/0504166"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Distinguishability and Accessible Information in Quantum Theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aarxiv%3Aquant-ph%2F9601020","year":1996},{"aliases":["10.1002/9781118033067","Thomas Koshy"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d53bbcc0f24e7924e7bf0d28fad17e652a5f1313d493ebb4ed7591c31b51227a.json","authors":"Thomas Koshy","evidence":{"source_url":"https://doi.org/10.1002/9781118033067"},"id":"literature:doi:10.1002/9781118033067","identifiers":{"doi":"10.1002/9781118033067"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Fibonacci and Lucas Numbers with Applications","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1002%2F9781118033067","year":2001},{"aliases":["10.1002/ctpp.70090"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3dc11382e8db172fd3fd0828e758a6bfe183a1ecb0e290804029b518f5f4de5f.json","evidence":{"source_url":"https://doi.org/10.1002/ctpp.70090"},"id":"literature:doi:10.1002/ctpp.70090","identifiers":{"doi":"10.1002/ctpp.70090"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1002/ctpp.70090","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1002%2Fctpp.70090"},{"aliases":["10.1002/jgt.22724"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c110355c2188e9b55fd371cda3a2d34fed517a788171b5e65a586e669cefe63.json","evidence":{"source_url":"https://doi.org/10.1002/jgt.22724"},"id":"literature:doi:10.1002/jgt.22724","identifiers":{"doi":"10.1002/jgt.22724"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1002/prop.202000044","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1002%2Fprop.202000044"},{"aliases":["10.1006/eujc.1998.0211","Sylvain Gravier and Michel Mollard and Charles Payan"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be1df5cbb700359150bfdda02f49d4409b4c9d3dbb67facfdec35358a0a9fab1.json","authors":"Sylvain Gravier and Michel Mollard and Charles Payan","evidence":{"source_url":"https://doi.org/10.1006/eujc.1998.0211"},"id":"literature:doi:10.1006/eujc.1998.0211","identifiers":{"doi":"10.1006/eujc.1998.0211"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Non-existence of 3-Dimensional Tiling in the Lee Metric","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1006%2Feujc.1998.0211","year":1998},{"aliases":["10.1007/978-0-8176-8134-0","Steven G. Krantz and Harold R. Parks"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9191cb43a3fb69ee6088ca9c4db8c03b4cd4b25106f95150f79b44801c39bedc.json","authors":"Steven G. Krantz and Harold R. Parks","evidence":{"source_url":"https://doi.org/10.1007/978-0-8176-8134-0"},"id":"literature:doi:10.1007/978-0-8176-8134-0","identifiers":{"doi":"10.1007/978-0-8176-8134-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Primer of Real Analytic Functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-0-8176-8134-0","year":2002},{"aliases":["10.1007/978-1-4613-8566-0","Emil Grosswald"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f4e24203579a6587f24f2edc41aaf2262dc7fa91463767cf3a558522973170f.json","authors":"Emil Grosswald","evidence":{"source_url":"https://doi.org/10.1007/978-1-4613-8566-0"},"id":"literature:doi:10.1007/978-1-4613-8566-0","identifiers":{"doi":"10.1007/978-1-4613-8566-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Representations of Integers as Sums of Squares","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-1-4613-8566-0","year":1985},{"aliases":["10.1007/978-1-4757-5579-4","Tom M. Apostol"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8583874606ecffa3d06c0f864507323b7bdcd229e68593cdd618b2b3c6c8f233.json","authors":"Tom M. Apostol","evidence":{"source_url":"https://doi.org/10.1007/978-1-4757-5579-4"},"id":"literature:doi:10.1007/978-1-4757-5579-4","identifiers":{"doi":"10.1007/978-1-4757-5579-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Introduction to Analytic Number Theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-1-4757-5579-4","year":1976},{"aliases":["10.1007/978-1-4757-6568-7","J. H. Conway and N. J. A. Sloane"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/137143a4b7d1ace94c0bf46768af45104f2bec7b73dbe85574f817127ae2f75b.json","authors":"J. H. Conway and N. J. A. Sloane","evidence":{"source_url":"https://doi.org/10.1007/978-1-4757-6568-7"},"id":"literature:doi:10.1007/978-1-4757-6568-7","identifiers":{"doi":"10.1007/978-1-4757-6568-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Sphere Packings, Lattices and Groups","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-1-4757-6568-7","year":1999},{"aliases":["10.1007/978-3-031-33180-0_14"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8d21b902dece38cf76f9590b773d004e572c77cf0b054fc0abd21978f5d785d.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-031-33180-0_14"},"id":"literature:doi:10.1007/978-3-031-33180-0_14","identifiers":{"doi":"10.1007/978-3-031-33180-0_14"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-031-33180-0_14","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-031-33180-0_14"},{"aliases":["10.1007/978-3-031-55601-2_5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/403be0ec60235a86ba5edd9d78f2778a3ce638ce9a14809ee653ba1fe30b409b.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-031-55601-2_5"},"id":"literature:doi:10.1007/978-3-031-55601-2_5","identifiers":{"doi":"10.1007/978-3-031-55601-2_5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-031-55601-2_5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-031-55601-2_5"},{"aliases":["10.1007/978-3-031-66497-7_1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01bccbe37f1f92c5388192a524865faa31bdcb916fba7cd3a035feae727130db.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-031-66497-7_1"},"id":"literature:doi:10.1007/978-3-031-66497-7_1","identifiers":{"doi":"10.1007/978-3-031-66497-7_1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-031-66497-7_1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-031-66497-7_1"},{"aliases":["10.1007/978-3-032-09524-4_5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52c7dcdd907df40e996a6218a3f5214c642627a398a13a1388a3e548c900a9b1.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-032-09524-4_5"},"id":"literature:doi:10.1007/978-3-032-09524-4_5","identifiers":{"doi":"10.1007/978-3-032-09524-4_5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-032-09524-4_5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-032-09524-4_5"},{"aliases":["10.1007/978-3-032-17801-5_6"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4d89b61fe1eb7a6c9570857d803e730189f1f5e5aa38724b7f6486db973308d.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-032-17801-5_6"},"id":"literature:doi:10.1007/978-3-032-17801-5_6","identifiers":{"doi":"10.1007/978-3-032-17801-5_6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-032-17801-5_6","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-032-17801-5_6"},{"aliases":["10.1007/978-3-0348-0142-3_16"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9572dcee54a056b1531ccd342f1f6e365bd00c3289c8d1037c33d6c0d368bf0e.json","evidence":{"source_url":"https://doi.org/10.1007/978-3-0348-0142-3_16"},"id":"literature:doi:10.1007/978-3-0348-0142-3_16","identifiers":{"doi":"10.1007/978-3-0348-0142-3_16"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/978-3-0348-0142-3_16","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-0348-0142-3_16"},{"aliases":["10.1007/978-3-642-01642-4","Luis Ribes; Pavel Zalesskii"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83451d90db635baad3030e05dca4a20d69c1ee92747429ae0b9132d5f04658ee.json","authors":"Luis Ribes; Pavel Zalesskii","evidence":{"source_url":"https://doi.org/10.1007/978-3-642-01642-4"},"id":"literature:doi:10.1007/978-3-642-01642-4","identifiers":{"doi":"10.1007/978-3-642-01642-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Profinite Groups","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-642-01642-4","year":2010},{"aliases":["10.1007/978-3-642-39286-3_23","Gérald Tenenbaum"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1776f5e31310926e583dd75b4dc7bf3d6c411d60e8427d3f82fe2e541095249b.json","authors":"Gérald Tenenbaum","evidence":{"source_url":"https://doi.org/10.1007/978-3-642-39286-3_23"},"id":"literature:doi:10.1007/978-3-642-39286-3_23","identifiers":{"doi":"10.1007/978-3-642-39286-3_23"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some of Erdős' Unconventional Problems in Number Theory, Thirty-four Years Later","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2F978-3-642-39286-3_23","year":2013},{"aliases":["10.1007/bf00417500","B. S. Cirel'son"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b4ee2a1d214826c804841011e53f2fad45ca37d90b64d1179bb2b9eb98fc7bd.json","authors":"B. S. Cirel'son","evidence":{"source_url":"https://doi.org/10.1007/bf00417500"},"id":"literature:doi:10.1007/bf00417500","identifiers":{"doi":"10.1007/bf00417500"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum generalizations of Bell's inequality","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbf00417500","year":1980},{"aliases":["10.1007/bf01206656","Adolf Hurwitz"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e1351abd5b193e79d511da9b4467b96c2e9fe3230957f3db039bc91544b0ddb.json","authors":"Adolf Hurwitz","evidence":{"source_url":"https://doi.org/10.1007/bf01206656"},"id":"literature:doi:10.1007/bf01206656","identifiers":{"doi":"10.1007/bf01206656"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Ueber die angenaeherte Darstellung der Irrationalzahlen durch rationale Brueche","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbf01206656","year":1891},{"aliases":["10.1007/bf01447779","Leopold Fejer"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52fffa82e949c586a5919bb217629ed9a33932562398aa0da704a1d5d61b041e.json","authors":"Leopold Fejer","evidence":{"source_url":"https://doi.org/10.1007/bf01447779"},"id":"literature:doi:10.1007/bf01447779","identifiers":{"doi":"10.1007/bf01447779"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Untersuchungen uber Fouriersche Reihen","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbf01447779","year":1903},{"aliases":["10.1007/bf01588971","G. L. Nemhauser; L. A. Wolsey; M. L. Fisher"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0165d2ddc167ab2ec2c06aae49930a6710848d96d70bcb49ca4e2bb37fe2026f.json","authors":"G. L. Nemhauser; L. A. Wolsey; M. L. Fisher","evidence":{"source_url":"https://doi.org/10.1007/bf01588971"},"id":"literature:doi:10.1007/bf01588971","identifiers":{"doi":"10.1007/bf01588971"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"An Analysis of Approximations for Maximizing Submodular Set Functions—I","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbf01588971","year":1978},{"aliases":["10.1007/bf01877590","Ole J. Heilmann and Elliott H. Lieb"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/698a528c9d206923d729a3f99136c5bf2070b77f7416cef7204c168eebfac38c.json","authors":"Ole J. Heilmann and Elliott H. Lieb","evidence":{"source_url":"https://doi.org/10.1007/bf01877590"},"id":"literature:doi:10.1007/bf01877590","identifiers":{"doi":"10.1007/bf01877590"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Theory of monomer-dimer systems","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbf01877590","year":1972},{"aliases":["10.1007/bfb0080769","F. William Lawvere"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4eb5671ff25d0a0b8055f1bc81bdde271c429baa9ad1804c5dbae85975f1c690.json","authors":"F. William Lawvere","evidence":{"source_url":"https://doi.org/10.1007/bfb0080769"},"id":"literature:doi:10.1007/bfb0080769","identifiers":{"doi":"10.1007/bfb0080769"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Diagonal arguments and cartesian closed categories","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fbfb0080769","year":1969},{"aliases":["10.1007/jhep02(2024)025"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7beddd95719de0f8c1bb70e4f50ce84b66c5d027dd07d7bb51a5738e001c3443.json","evidence":{"source_url":"https://doi.org/10.1007/jhep02(2024)025"},"id":"literature:doi:10.1007/jhep02(2024)025","identifiers":{"doi":"10.1007/jhep02(2024)025"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep02(2024)025","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep02%282024%29025"},{"aliases":["10.1007/jhep03(2022)028"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73852cb75cdc9c6f04b61e780d0754705611e02c97fc6ba924dc721ab98b9435.json","evidence":{"source_url":"https://doi.org/10.1007/jhep03(2022)028"},"id":"literature:doi:10.1007/jhep03(2022)028","identifiers":{"doi":"10.1007/jhep03(2022)028"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep03(2022)028","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep03%282022%29028"},{"aliases":["10.1007/jhep03(2023)154"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18eeb0ad5af33e0e40f23604eb9d87f5e23501ffc04164a70d72300be3721eb5.json","evidence":{"source_url":"https://doi.org/10.1007/jhep03(2023)154"},"id":"literature:doi:10.1007/jhep03(2023)154","identifiers":{"doi":"10.1007/jhep03(2023)154"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep03(2023)154","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep03%282023%29154"},{"aliases":["10.1007/jhep03(2023)175"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81e41735eca8b9141e5b8c9db90d5986c5a94fafd4d22f78c160770d8c9ae1f3.json","evidence":{"source_url":"https://doi.org/10.1007/jhep03(2023)175"},"id":"literature:doi:10.1007/jhep03(2023)175","identifiers":{"doi":"10.1007/jhep03(2023)175"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep03(2023)175","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep03%282023%29175"},{"aliases":["10.1007/jhep05(2021)074"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0e6990685b168e53e49717524064e73ec0a20cdab25ab8a1e0d020f016693db.json","evidence":{"source_url":"https://doi.org/10.1007/jhep05(2021)074"},"id":"literature:doi:10.1007/jhep05(2021)074","identifiers":{"doi":"10.1007/jhep05(2021)074"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep05(2021)074","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep05%282021%29074"},{"aliases":["10.1007/jhep08(2020)161"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/674a8e418c4df1924c1bd8ca6dbf14ff38b0224f36896b61e32991ff73a6ad0d.json","evidence":{"source_url":"https://doi.org/10.1007/jhep08(2020)161"},"id":"literature:doi:10.1007/jhep08(2020)161","identifiers":{"doi":"10.1007/jhep08(2020)161"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep08(2020)161","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep08%282020%29161"},{"aliases":["10.1007/jhep11(2025)081"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/011ecd0b127e41b3ce3f92ec5b74b6a11bd1795cefbd426697470a2b59058c9e.json","evidence":{"source_url":"https://doi.org/10.1007/jhep11(2025)081"},"id":"literature:doi:10.1007/jhep11(2025)081","identifiers":{"doi":"10.1007/jhep11(2025)081"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep11(2025)081","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep11%282025%29081"},{"aliases":["10.1007/jhep12(2023)123"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5152cfbc2ca460a3a918d731672971d0659e39269564be0184488ee33667e255.json","evidence":{"source_url":"https://doi.org/10.1007/jhep12(2023)123"},"id":"literature:doi:10.1007/jhep12(2023)123","identifiers":{"doi":"10.1007/jhep12(2023)123"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/jhep12(2023)123","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fjhep12%282023%29123"},{"aliases":["10.1007/s00010-018-0545-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff947bbd5629b3666d76198bdd045a5f9d632eec1789770ce8b26b8140bfb3e7.json","evidence":{"source_url":"https://doi.org/10.1007/s00010-018-0545-8"},"id":"literature:doi:10.1007/s00010-018-0545-8","identifiers":{"doi":"10.1007/s00010-018-0545-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00010-018-0545-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00010-018-0545-8"},{"aliases":["10.1007/s00023-021-01071-y"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c5d1930879d4e7abbe0ab2455cac1548c987764d2ee54c792ae74d54f9866620.json","evidence":{"source_url":"https://doi.org/10.1007/s00023-021-01071-y"},"id":"literature:doi:10.1007/s00023-021-01071-y","identifiers":{"doi":"10.1007/s00023-021-01071-y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00023-021-01071-y","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00023-021-01071-y"},{"aliases":["10.1007/s00023-022-01187-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43f3db9c3fda9b1332f8f345ea1d5d5637ed58a66a973c01c0c98fe55ec0e2ed.json","evidence":{"source_url":"https://doi.org/10.1007/s00023-022-01187-9"},"id":"literature:doi:10.1007/s00023-022-01187-9","identifiers":{"doi":"10.1007/s00023-022-01187-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00023-022-01187-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00023-022-01187-9"},{"aliases":["10.1007/s00023-023-01325-x"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/755a9f875d19407124cadfcb58adc9df28e907ea69d8d95957398deb0fe55c04.json","evidence":{"source_url":"https://doi.org/10.1007/s00023-023-01325-x"},"id":"literature:doi:10.1007/s00023-023-01325-x","identifiers":{"doi":"10.1007/s00023-023-01325-x"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00023-023-01325-x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00023-023-01325-x"},{"aliases":["10.1007/s00023-023-01339-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0fadc22154b40d8c46fcb540db1157649496f8171083220315a20de3b01d36f3.json","evidence":{"source_url":"https://doi.org/10.1007/s00023-023-01339-5"},"id":"literature:doi:10.1007/s00023-023-01339-5","identifiers":{"doi":"10.1007/s00023-023-01339-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00023-023-01339-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00023-023-01339-5"},{"aliases":["10.1007/s00023-025-01587-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/648695c52cc9de77ac19b97b05f93c8e772029717cbeea072c5c76b441d8f175.json","evidence":{"source_url":"https://doi.org/10.1007/s00023-025-01587-7"},"id":"literature:doi:10.1007/s00023-025-01587-7","identifiers":{"doi":"10.1007/s00023-025-01587-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00023-025-01587-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00023-025-01587-7"},{"aliases":["10.1007/s00025-025-02421-6"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/551bcd9e8ed0efc81b7c614996f15e1e53e8f8ad106ced121ec1901a61421cc5.json","evidence":{"source_url":"https://doi.org/10.1007/s00025-025-02421-6"},"id":"literature:doi:10.1007/s00025-025-02421-6","identifiers":{"doi":"10.1007/s00025-025-02421-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00025-025-02421-6","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00025-025-02421-6"},{"aliases":["10.1007/s00025-025-02431-4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6efe9cec1688e78d39f115d676efd3d22e4998f5f4ae645bf1c7000e4511bfa0.json","evidence":{"source_url":"https://doi.org/10.1007/s00025-025-02431-4"},"id":"literature:doi:10.1007/s00025-025-02431-4","identifiers":{"doi":"10.1007/s00025-025-02431-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00025-025-02431-4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00025-025-02431-4"},{"aliases":["10.1007/s00025-026-02697-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fae3bfab23097650a030ac5efa5b756bb7e9ebba40f3830a2fca267ed9ffc015.json","evidence":{"source_url":"https://doi.org/10.1007/s00025-026-02697-2"},"id":"literature:doi:10.1007/s00025-026-02697-2","identifiers":{"doi":"10.1007/s00025-026-02697-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00025-026-02697-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00025-026-02697-2"},{"aliases":["10.1007/s00026-006-0289-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/743ca83b57d27d5523ffb61830f2370b58a0e08760c018c30a7c8a068d0914aa.json","evidence":{"source_url":"https://doi.org/10.1007/s00026-006-0289-3"},"id":"literature:doi:10.1007/s00026-006-0289-3","identifiers":{"doi":"10.1007/s00026-006-0289-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00026-006-0289-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00026-006-0289-3"},{"aliases":["10.1007/s00026-014-0242-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce4af35fbfe5f8f8c31f09e422c2eecae0decceb8e895af57197ecdca47e849a.json","evidence":{"source_url":"https://doi.org/10.1007/s00026-014-0242-9"},"id":"literature:doi:10.1007/s00026-014-0242-9","identifiers":{"doi":"10.1007/s00026-014-0242-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00026-014-0242-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00026-014-0242-9"},{"aliases":["10.1007/s00031-024-09863-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/111de409077ba712271a94b572410cecfb34d3b989e9fc6aac7f54907237a680.json","evidence":{"source_url":"https://doi.org/10.1007/s00031-024-09863-2"},"id":"literature:doi:10.1007/s00031-024-09863-2","identifiers":{"doi":"10.1007/s00031-024-09863-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00031-024-09863-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00031-024-09863-2"},{"aliases":["10.1007/s00182-022-00824-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bfe48d82c967f302dc72ba98d6e71f5601e7c8e57fdedce3327190c9446f3c7e.json","evidence":{"source_url":"https://doi.org/10.1007/s00182-022-00824-1"},"id":"literature:doi:10.1007/s00182-022-00824-1","identifiers":{"doi":"10.1007/s00182-022-00824-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00182-022-00824-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00182-022-00824-1"},{"aliases":["10.1007/s00182-024-00910-6"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9189b8591f45f082dd409a823f7cb8e26a8da30d25908ac79a6af62bc1201472.json","evidence":{"source_url":"https://doi.org/10.1007/s00182-024-00910-6"},"id":"literature:doi:10.1007/s00182-024-00910-6","identifiers":{"doi":"10.1007/s00182-024-00910-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00182-024-00910-6","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00182-024-00910-6"},{"aliases":["10.1007/s00209-022-03000-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c6a3d403765efce1ac2f24e8539f016ebf823fb6962b626f72e404587f4c2942.json","evidence":{"source_url":"https://doi.org/10.1007/s00209-022-03000-z"},"id":"literature:doi:10.1007/s00209-022-03000-z","identifiers":{"doi":"10.1007/s00209-022-03000-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00209-022-03000-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00209-022-03000-z"},{"aliases":["10.1007/s00220-024-04980-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44bd7bf8b6976f65dda558b69321607b069bec4a157171d2f97d8be5e84d0244.json","evidence":{"source_url":"https://doi.org/10.1007/s00220-024-04980-1"},"id":"literature:doi:10.1007/s00220-024-04980-1","identifiers":{"doi":"10.1007/s00220-024-04980-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00220-024-04980-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00220-024-04980-1"},{"aliases":["10.1007/s00220-025-05533-w"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24d48119652af0a0ae218fbf60c803e9fef865a2459f27ea4417bd7f474168e4.json","evidence":{"source_url":"https://doi.org/10.1007/s00220-025-05533-w"},"id":"literature:doi:10.1007/s00220-025-05533-w","identifiers":{"doi":"10.1007/s00220-025-05533-w"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00220-025-05533-w","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00220-025-05533-w"},{"aliases":["10.1007/s00220-026-05675-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32923bee11879dda6b3d421bc32659a32fa1e0c61d0316724b579ec44409d6d3.json","evidence":{"source_url":"https://doi.org/10.1007/s00220-026-05675-5"},"id":"literature:doi:10.1007/s00220-026-05675-5","identifiers":{"doi":"10.1007/s00220-026-05675-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00220-026-05675-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00220-026-05675-5"},{"aliases":["10.1007/s002220050232","Richard E. Borcherds"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf81f98ad2c4eef57c5970d8e0a81b76d3fc2d18ee770d8acc2b3034612d3800.json","authors":"Richard E. Borcherds","evidence":{"source_url":"https://doi.org/10.1007/s002220050232"},"id":"literature:doi:10.1007/s002220050232","identifiers":{"doi":"10.1007/s002220050232"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Automorphic forms with singularities on Grassmannians","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs002220050232","year":1998},{"aliases":["10.1007/s00233-026-10614-w"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/76b442b12e0495eed24f5dfc1e3caad2931c7662fe1634da362037f49bfc377c.json","evidence":{"source_url":"https://doi.org/10.1007/s00233-026-10614-w"},"id":"literature:doi:10.1007/s00233-026-10614-w","identifiers":{"doi":"10.1007/s00233-026-10614-w"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00233-026-10614-w","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00233-026-10614-w"},{"aliases":["10.1007/s00373-024-02798-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea22caa2b256c5dfadf1c5fb94bea99b463d00a9ee2f0f01a0dcfd5eaa581d11.json","evidence":{"source_url":"https://doi.org/10.1007/s00373-024-02798-1"},"id":"literature:doi:10.1007/s00373-024-02798-1","identifiers":{"doi":"10.1007/s00373-024-02798-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00373-024-02798-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00373-024-02798-1"},{"aliases":["10.1007/s00373-025-02936-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6a024ed09f8d0ef28dbd1596a029aeac148c8222fedbd945057150e9062e689.json","evidence":{"source_url":"https://doi.org/10.1007/s00373-025-02936-3"},"id":"literature:doi:10.1007/s00373-025-02936-3","identifiers":{"doi":"10.1007/s00373-025-02936-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s00373-025-02936-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs00373-025-02936-3"},{"aliases":["10.1007/s10114-026-4502-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fe3f522c60e1b2f5cd44c16ddf4569fc1ae9d7c25f25c66fb72f10427a1746c.json","evidence":{"source_url":"https://doi.org/10.1007/s10114-026-4502-5"},"id":"literature:doi:10.1007/s10114-026-4502-5","identifiers":{"doi":"10.1007/s10114-026-4502-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10114-026-4502-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10114-026-4502-5"},{"aliases":["10.1007/s10474-018-0848-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f62b3e3cf0f84668dfa90aa4a29f7a7bf0e8c7fa0093a630bdc06321e636657a.json","evidence":{"source_url":"https://doi.org/10.1007/s10474-018-0848-1"},"id":"literature:doi:10.1007/s10474-018-0848-1","identifiers":{"doi":"10.1007/s10474-018-0848-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10474-018-0848-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10474-018-0848-1"},{"aliases":["10.1007/s10476-017-0503-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a98e9e3770e8934c386e2bc5f338efeeb3ee0e28165112da1c4f8cb1849ad271.json","evidence":{"source_url":"https://doi.org/10.1007/s10476-017-0503-z"},"id":"literature:doi:10.1007/s10476-017-0503-z","identifiers":{"doi":"10.1007/s10476-017-0503-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10476-017-0503-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10476-017-0503-z"},{"aliases":["10.1007/s10659-024-10087-4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cdcdbf5f669a5c2177d6937686f20a4aa4659bc6989afd167f516aae8b21843c.json","evidence":{"source_url":"https://doi.org/10.1007/s10659-024-10087-4"},"id":"literature:doi:10.1007/s10659-024-10087-4","identifiers":{"doi":"10.1007/s10659-024-10087-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10659-024-10087-4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10659-024-10087-4"},{"aliases":["10.1007/s10701-015-9983-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/377f48df07695066f0169db89a3dfa21ffda1a18dedd61380c26e2ab60aba381.json","evidence":{"source_url":"https://doi.org/10.1007/s10701-015-9983-5"},"id":"literature:doi:10.1007/s10701-015-9983-5","identifiers":{"doi":"10.1007/s10701-015-9983-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10701-015-9983-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10701-015-9983-5"},{"aliases":["10.1007/s10701-017-0098-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55f0dfa817fff588398acf2637a9bf0cf02cff11300c9d1efbd8ca0071008608.json","evidence":{"source_url":"https://doi.org/10.1007/s10701-017-0098-z"},"id":"literature:doi:10.1007/s10701-017-0098-z","identifiers":{"doi":"10.1007/s10701-017-0098-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10701-017-0098-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10701-017-0098-z"},{"aliases":["10.1007/s10773-018-3890-4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/817e0934478890b6ab1ec031bb6b2f847fce83375a615c1ae1e74c27e40a1659.json","evidence":{"source_url":"https://doi.org/10.1007/s10773-018-3890-4"},"id":"literature:doi:10.1007/s10773-018-3890-4","identifiers":{"doi":"10.1007/s10773-018-3890-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10773-018-3890-4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10773-018-3890-4"},{"aliases":["10.1007/s10773-022-05219-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bbff09e36c369420e318ba783350df7687f7aa985e9fd069dae9dea1853a85ea.json","evidence":{"source_url":"https://doi.org/10.1007/s10773-022-05219-2"},"id":"literature:doi:10.1007/s10773-022-05219-2","identifiers":{"doi":"10.1007/s10773-022-05219-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10773-022-05219-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10773-022-05219-2"},{"aliases":["10.1007/s10801-008-0139-y"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f17534bb24c5ee587c4ac02d210689baff27510fc5b9304d4fd25eea3e8c3ccc.json","evidence":{"source_url":"https://doi.org/10.1007/s10801-008-0139-y"},"id":"literature:doi:10.1007/s10801-008-0139-y","identifiers":{"doi":"10.1007/s10801-008-0139-y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10801-008-0139-y","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10801-008-0139-y"},{"aliases":["10.1007/s10801-021-01028-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01432560df2e9b30c6d0d6f6ab544461bbd2f2ede1158d752955c043c64844db.json","evidence":{"source_url":"https://doi.org/10.1007/s10801-021-01028-8"},"id":"literature:doi:10.1007/s10801-021-01028-8","identifiers":{"doi":"10.1007/s10801-021-01028-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10801-021-01028-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10801-021-01028-8"},{"aliases":["10.1007/s10801-024-01308-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39c6b04def904f2dc47c1572491771d64f80025eefa43698c4eb505b519cf368.json","evidence":{"source_url":"https://doi.org/10.1007/s10801-024-01308-z"},"id":"literature:doi:10.1007/s10801-024-01308-z","identifiers":{"doi":"10.1007/s10801-024-01308-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10801-024-01308-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10801-024-01308-z"},{"aliases":["10.1007/s10878-024-01123-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f8f2d38287b8fe7d88498d88441ee4ec5b9435b2df3274cb4b890967ad235d92.json","evidence":{"source_url":"https://doi.org/10.1007/s10878-024-01123-z"},"id":"literature:doi:10.1007/s10878-024-01123-z","identifiers":{"doi":"10.1007/s10878-024-01123-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10878-024-01123-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10878-024-01123-z"},{"aliases":["10.1007/s10946-017-9629-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/839fd40e8c48ebd62e7cf6c70c2bc3cd2961a0c76bb8e66cd529aeba8cdb577c.json","evidence":{"source_url":"https://doi.org/10.1007/s10946-017-9629-5"},"id":"literature:doi:10.1007/s10946-017-9629-5","identifiers":{"doi":"10.1007/s10946-017-9629-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10946-017-9629-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10946-017-9629-5"},{"aliases":["10.1007/s10955-017-1876-x"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/54cd8d0db48e313a66a230dece0f1eba25122c283bc0e9b34c9b1d531d912082.json","evidence":{"source_url":"https://doi.org/10.1007/s10955-017-1876-x"},"id":"literature:doi:10.1007/s10955-017-1876-x","identifiers":{"doi":"10.1007/s10955-017-1876-x"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10955-017-1876-x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10955-017-1876-x"},{"aliases":["10.1007/s10955-023-03183-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b2f2e211c716d9c14e26b69b5bb6ca01312ebe9ef763a55544c5e7d3a0d28056.json","evidence":{"source_url":"https://doi.org/10.1007/s10955-023-03183-7"},"id":"literature:doi:10.1007/s10955-023-03183-7","identifiers":{"doi":"10.1007/s10955-023-03183-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10955-023-03183-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10955-023-03183-7"},{"aliases":["10.1007/s10955-025-03423-y"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0493f4209b6ca4a9a414514f4ede3d00a6a7b19b31df7c830314b8b65780a4ff.json","evidence":{"source_url":"https://doi.org/10.1007/s10955-025-03423-y"},"id":"literature:doi:10.1007/s10955-025-03423-y","identifiers":{"doi":"10.1007/s10955-025-03423-y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10955-025-03423-y","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10955-025-03423-y"},{"aliases":["10.1007/s10958-016-2958-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e0d0febead6ac10c758a7200640e515fbdf2531fe38976b2e99965495de543f.json","evidence":{"source_url":"https://doi.org/10.1007/s10958-016-2958-z"},"id":"literature:doi:10.1007/s10958-016-2958-z","identifiers":{"doi":"10.1007/s10958-016-2958-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10958-016-2958-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10958-016-2958-z"},{"aliases":["10.1007/s10958-017-3573-3","Vladimir Derkach and Ivan Kovalyov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ab2807526b8ceaa7f662a504942a5aecf7f4d78dd7d7723cd441fcd410cee87.json","authors":"Vladimir Derkach and Ivan Kovalyov","evidence":{"source_url":"https://doi.org/10.1007/s10958-017-3573-3"},"id":"literature:doi:10.1007/s10958-017-3573-3","identifiers":{"doi":"10.1007/s10958-017-3573-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"An operator approach to the indefinite Stieltjes moment problem","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10958-017-3573-3","year":2017},{"aliases":["10.1007/s10958-026-08499-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f31e8cb0371735987f9e31d7d5dff806ac06a94d4324867f6555b009b414af95.json","evidence":{"source_url":"https://doi.org/10.1007/s10958-026-08499-9"},"id":"literature:doi:10.1007/s10958-026-08499-9","identifiers":{"doi":"10.1007/s10958-026-08499-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s10958-026-08499-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs10958-026-08499-9"},{"aliases":["10.1007/s11005-025-02026-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/764f56d09cc20adb4186ac0ff7c51ca6c1626bffcda73cddcba21c7b800ffce8.json","evidence":{"source_url":"https://doi.org/10.1007/s11005-025-02026-8"},"id":"literature:doi:10.1007/s11005-025-02026-8","identifiers":{"doi":"10.1007/s11005-025-02026-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11005-025-02026-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11005-025-02026-8"},{"aliases":["10.1007/s11040-025-09513-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a1ec9cf854a9955cc0be4cab2a15d49331c5c4dcdbf6b4c9e9368c928014cdb.json","evidence":{"source_url":"https://doi.org/10.1007/s11040-025-09513-3"},"id":"literature:doi:10.1007/s11040-025-09513-3","identifiers":{"doi":"10.1007/s11040-025-09513-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11040-025-09513-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11040-025-09513-3"},{"aliases":["10.1007/s11040-026-09569-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6980e4d3db6dc281b49ee8bcb5a413c2ad3a78b310867dbbe116e4232be226c.json","evidence":{"source_url":"https://doi.org/10.1007/s11040-026-09569-9"},"id":"literature:doi:10.1007/s11040-026-09569-9","identifiers":{"doi":"10.1007/s11040-026-09569-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11040-026-09569-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11040-026-09569-9"},{"aliases":["10.1007/s11083-026-09735-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b5a4b7631dfa3f2aee7f59ca1fecc6d51dbeba4623a366cb8215df6277fdc2a.json","evidence":{"source_url":"https://doi.org/10.1007/s11083-026-09735-2"},"id":"literature:doi:10.1007/s11083-026-09735-2","identifiers":{"doi":"10.1007/s11083-026-09735-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11083-026-09735-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11083-026-09735-2"},{"aliases":["10.1007/s11128-013-0639-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e99ff72a539edd8dad66a3202968e7948a0160dce4cf4d857249989579b0c1aa.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-013-0639-0"},"id":"literature:doi:10.1007/s11128-013-0639-0","identifiers":{"doi":"10.1007/s11128-013-0639-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-013-0639-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-013-0639-0"},{"aliases":["10.1007/s11128-017-1701-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95e744e3494610e4e2004b825bcf5b09eb4d0e982d8596ebd285f358a939304e.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-017-1701-0"},"id":"literature:doi:10.1007/s11128-017-1701-0","identifiers":{"doi":"10.1007/s11128-017-1701-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-017-1701-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-017-1701-0"},{"aliases":["10.1007/s11128-020-02676-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f52fa9c7c5d79976a13fbc5da942a9778ca969d908e9c7098b8a7393b7d85d86.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-020-02676-8"},"id":"literature:doi:10.1007/s11128-020-02676-8","identifiers":{"doi":"10.1007/s11128-020-02676-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-020-02676-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-020-02676-8"},{"aliases":["10.1007/s11128-021-03394-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/107e4f9f1f50fcb18a44041d1be855725b69197dbaa797f47459c909983ade41.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-021-03394-5"},"id":"literature:doi:10.1007/s11128-021-03394-5","identifiers":{"doi":"10.1007/s11128-021-03394-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-021-03394-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-021-03394-5"},{"aliases":["10.1007/s11128-024-04309-w"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/08d8a1fc8dadcfd1cb4af1570e4c728c72bf2100ad0903888d0c147c15d79260.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-024-04309-w"},"id":"literature:doi:10.1007/s11128-024-04309-w","identifiers":{"doi":"10.1007/s11128-024-04309-w"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-024-04309-w","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-024-04309-w"},{"aliases":["10.1007/s11128-024-04467-x"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4010296302a552cca5c19c670ff42a5be178d15e63cbdee4fe6bec9166434566.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-024-04467-x"},"id":"literature:doi:10.1007/s11128-024-04467-x","identifiers":{"doi":"10.1007/s11128-024-04467-x"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-024-04467-x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-024-04467-x"},{"aliases":["10.1007/s11128-026-05258-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0712e3f81d29c6ef2f54406cff3938bb497a34750ddf2fb2d337c77486c1931a.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-026-05258-2"},"id":"literature:doi:10.1007/s11128-026-05258-2","identifiers":{"doi":"10.1007/s11128-026-05258-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-026-05258-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-026-05258-2"},{"aliases":["10.1007/s11128-026-05335-6"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4839e3481ab75c5e4b2f8e9e13e3b095942f36f1af580d6cb6d95a4314d6aad0.json","evidence":{"source_url":"https://doi.org/10.1007/s11128-026-05335-6"},"id":"literature:doi:10.1007/s11128-026-05335-6","identifiers":{"doi":"10.1007/s11128-026-05335-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11128-026-05335-6","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11128-026-05335-6"},{"aliases":["10.1007/s11139-022-00602-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f14700f7896d360cf6a2fd76c92ade79e57156587d6231abfdd6b4c915407550.json","evidence":{"source_url":"https://doi.org/10.1007/s11139-022-00602-3"},"id":"literature:doi:10.1007/s11139-022-00602-3","identifiers":{"doi":"10.1007/s11139-022-00602-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11139-022-00602-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11139-022-00602-3"},{"aliases":["10.1007/s11139-025-01305-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/006e89e70b30411408785260f71a90954eaf97dee1df0dc790f85ff0ecd368a2.json","evidence":{"source_url":"https://doi.org/10.1007/s11139-025-01305-1"},"id":"literature:doi:10.1007/s11139-025-01305-1","identifiers":{"doi":"10.1007/s11139-025-01305-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11139-025-01305-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11139-025-01305-1"},{"aliases":["10.1007/s11139-026-01345-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d2a94a551fff0507d31277c93188743ed24b79f3156753c1631d17e013a2b2e.json","evidence":{"source_url":"https://doi.org/10.1007/s11139-026-01345-1"},"id":"literature:doi:10.1007/s11139-026-01345-1","identifiers":{"doi":"10.1007/s11139-026-01345-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11139-026-01345-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11139-026-01345-1"},{"aliases":["10.1007/s11786-018-0384-y"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/346870d58dbec0d4f60fa8464f0a6e1a1a406b4a2bbd34ba9edcb1345d216964.json","evidence":{"source_url":"https://doi.org/10.1007/s11786-018-0384-y"},"id":"literature:doi:10.1007/s11786-018-0384-y","identifiers":{"doi":"10.1007/s11786-018-0384-y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s11786-018-0384-y","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs11786-018-0384-y"},{"aliases":["10.1007/s12095-021-00505-y","Pierrick Meaux"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb84fc4c8e905bd6decfd57ca351b9514eab9447cedaa53640981daf01a4b860.json","authors":"Pierrick Meaux","evidence":{"source_url":"https://doi.org/10.1007/s12095-021-00505-y"},"id":"literature:doi:10.1007/s12095-021-00505-y","identifiers":{"doi":"10.1007/s12095-021-00505-y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Fast Algebraic Immunity of Threshold Functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs12095-021-00505-y","year":2021},{"aliases":["10.1007/s13324-025-01136-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/10949eaefcd4e5200f0f40b2c62d6b5ed1223c489cac666de05edf9b8f35d894.json","evidence":{"source_url":"https://doi.org/10.1007/s13324-025-01136-9"},"id":"literature:doi:10.1007/s13324-025-01136-9","identifiers":{"doi":"10.1007/s13324-025-01136-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s13324-025-01136-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs13324-025-01136-9"},{"aliases":["10.1007/s13540-023-00238-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/378984b0aaff29270e9803a42bcd6ce3cb570fc5845eb7dff4404283cd892b11.json","evidence":{"source_url":"https://doi.org/10.1007/s13540-023-00238-0"},"id":"literature:doi:10.1007/s13540-023-00238-0","identifiers":{"doi":"10.1007/s13540-023-00238-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s13540-023-00238-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs13540-023-00238-0"},{"aliases":["10.1007/s40314-022-01957-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74ee7407558ed212a29d4e1d4d63298f906fec7d8411188bedc336cc93bce9ef.json","evidence":{"source_url":"https://doi.org/10.1007/s40314-022-01957-5"},"id":"literature:doi:10.1007/s40314-022-01957-5","identifiers":{"doi":"10.1007/s40314-022-01957-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40314-022-01957-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40314-022-01957-5"},{"aliases":["10.1007/s40314-025-03163-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6dc18711360c2caa8d465a2923b635e3b28df9e75c4af5545f9dbb66b3376d4.json","evidence":{"source_url":"https://doi.org/10.1007/s40314-025-03163-5"},"id":"literature:doi:10.1007/s40314-025-03163-5","identifiers":{"doi":"10.1007/s40314-025-03163-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40314-025-03163-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40314-025-03163-5"},{"aliases":["10.1007/s40315-015-0128-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38feabd7478d26d46eac84ac0d53ceebc53397304863966d4b21b3bfd440b9f1.json","evidence":{"source_url":"https://doi.org/10.1007/s40315-015-0128-9"},"id":"literature:doi:10.1007/s40315-015-0128-9","identifiers":{"doi":"10.1007/s40315-015-0128-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40315-015-0128-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40315-015-0128-9"},{"aliases":["10.1007/s40509-016-0093-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b36e7ebc9cdc913afa6bde41e431d39237f2b2c8106a839643872521a1784f7.json","evidence":{"source_url":"https://doi.org/10.1007/s40509-016-0093-7"},"id":"literature:doi:10.1007/s40509-016-0093-7","identifiers":{"doi":"10.1007/s40509-016-0093-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40509-016-0093-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40509-016-0093-7"},{"aliases":["10.1007/s40509-026-00402-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69042c601075b37225b3f83c77f8673244f64f980f786cbe727a6e8d0934b04f.json","evidence":{"source_url":"https://doi.org/10.1007/s40509-026-00402-0"},"id":"literature:doi:10.1007/s40509-026-00402-0","identifiers":{"doi":"10.1007/s40509-026-00402-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40509-026-00402-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40509-026-00402-0"},{"aliases":["10.1007/s40590-023-00583-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c6431801c06ed24d48e0ad7364dc2d1fb6df8f8cd56fa3d9b39808ada661891.json","evidence":{"source_url":"https://doi.org/10.1007/s40590-023-00583-3"},"id":"literature:doi:10.1007/s40590-023-00583-3","identifiers":{"doi":"10.1007/s40590-023-00583-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40590-023-00583-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40590-023-00583-3"},{"aliases":["10.1007/s40879-025-00822-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3403af723737086f6db36a66d7f307c342449b5497bb39a255d6085d1fb7533e.json","evidence":{"source_url":"https://doi.org/10.1007/s40879-025-00822-z"},"id":"literature:doi:10.1007/s40879-025-00822-z","identifiers":{"doi":"10.1007/s40879-025-00822-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s40879-025-00822-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs40879-025-00822-z"},{"aliases":["10.1007/s41478-026-01192-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d11655489e6614925c05801fccfc18ecefcf0e214fdc62789fbdf8d5312a2558.json","evidence":{"source_url":"https://doi.org/10.1007/s41478-026-01192-9"},"id":"literature:doi:10.1007/s41478-026-01192-9","identifiers":{"doi":"10.1007/s41478-026-01192-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s41478-026-01192-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs41478-026-01192-9"},{"aliases":["10.1007/s43036-025-00481-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb71461b58fa7c8ba195e77fc123becd65394e820df0f260f92d4a940df33049.json","evidence":{"source_url":"https://doi.org/10.1007/s43036-025-00481-7"},"id":"literature:doi:10.1007/s43036-025-00481-7","identifiers":{"doi":"10.1007/s43036-025-00481-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1007/s43036-025-00481-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1007%2Fs43036-025-00481-7"},{"aliases":["10.1016/0012-365x(94)00108-u","Paul Erdős and András Sárközy and Vera T. Sós"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2c6904b60fdc00de3a51933f091865449598a2e6eefcdd148fb7196eb20e63d0.json","authors":"Paul Erdős and András Sárközy and Vera T. Sós","evidence":{"source_url":"https://doi.org/10.1016/0012-365x(94)00108-u"},"id":"literature:doi:10.1016/0012-365x(94)00108-u","identifiers":{"doi":"10.1016/0012-365x(94)00108-u"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On additive properties of general sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2F0012-365x%2894%2900108-u","year":1994},{"aliases":["10.1016/0016-0032(76)90081-8","Jan C. Willems"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c76429f03850e9a305f60946a8ebda867d09b982e2baae605df371a7feab873.json","authors":"Jan C. Willems","evidence":{"source_url":"https://doi.org/10.1016/0016-0032(76)90081-8"},"id":"literature:doi:10.1016/0016-0032(76)90081-8","identifiers":{"doi":"10.1016/0016-0032(76)90081-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Realization of systems with internal passivity and symmetry constraints","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2F0016-0032%2876%2990081-8","year":1976},{"aliases":["10.1016/0095-8956(75)90041-6","Vaclav Chvatal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/67e233bc0c71e3bc6dad1e3ed77c84c967b2ae88a7cd9a2804d13e7cebb5bd58.json","authors":"Vaclav Chvatal","evidence":{"source_url":"https://doi.org/10.1016/0095-8956(75)90041-6"},"id":"literature:doi:10.1016/0095-8956(75)90041-6","identifiers":{"doi":"10.1016/0095-8956(75)90041-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On certain polytopes associated with graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2F0095-8956%2875%2990041-6","year":1975},{"aliases":["10.1016/0375-9601(87)90075-2","Lawrence J. Landau"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b2bf4262382d47c94d221c473db092c385bd1346fc31f23546a247aadb7e3244.json","authors":"Lawrence J. Landau","evidence":{"source_url":"https://doi.org/10.1016/0375-9601(87)90075-2"},"id":"literature:doi:10.1016/0375-9601(87)90075-2","identifiers":{"doi":"10.1016/0375-9601(87)90075-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the violation of Bell's inequality in quantum theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2F0375-9601%2887%2990075-2","year":1987},{"aliases":["10.1016/b978-0-12-092750-0.50008-9","Gerald Berman and K. D. Fryer"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a66d542f162f5efe788ad3d6de4ae20cf357ec79f9e19f33578945c32180179f.json","authors":"Gerald Berman and K. D. Fryer","evidence":{"source_url":"https://doi.org/10.1016/b978-0-12-092750-0.50008-9"},"id":"literature:doi:10.1016/b978-0-12-092750-0.50008-9","identifiers":{"doi":"10.1016/b978-0-12-092750-0.50008-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Inclusion-Exclusion Principle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fb978-0-12-092750-0.50008-9","year":1972},{"aliases":["10.1016/c2009-0-22289-6","Gerard J. Murphy"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8011bdf83abb0be39566ffb48b18f6c29053b7c79b85668301e9c4cf7af8a44f.json","authors":"Gerard J. Murphy","evidence":{"source_url":"https://doi.org/10.1016/c2009-0-22289-6"},"id":"literature:doi:10.1016/c2009-0-22289-6","identifiers":{"doi":"10.1016/c2009-0-22289-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"C*-Algebras and Operator Theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fc2009-0-22289-6","year":1990},{"aliases":["10.1016/j.aam.2022.102417"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2e9301d5f4db343c4433ccf0f724d717243ce26c92e2b1263d4aa59c4c8fdb9.json","evidence":{"source_url":"https://doi.org/10.1016/j.aam.2022.102417"},"id":"literature:doi:10.1016/j.aam.2022.102417","identifiers":{"doi":"10.1016/j.aam.2022.102417"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.aam.2022.102417","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.aam.2022.102417"},{"aliases":["10.1016/j.aam.2023.102621"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/979a0e23fdb141402b6737bbfb998174443bf207aedf75ab53e4bfa078c5c2e8.json","evidence":{"source_url":"https://doi.org/10.1016/j.aam.2023.102621"},"id":"literature:doi:10.1016/j.aam.2023.102621","identifiers":{"doi":"10.1016/j.aam.2023.102621"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.aam.2023.102621","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.aam.2023.102621"},{"aliases":["10.1016/j.aop.2016.12.024"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/041aa450b703d5c70dec0215886365b8f4d4d14bc4b6a69232beb3eb118001bf.json","evidence":{"source_url":"https://doi.org/10.1016/j.aop.2016.12.024"},"id":"literature:doi:10.1016/j.aop.2016.12.024","identifiers":{"doi":"10.1016/j.aop.2016.12.024"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.aop.2016.12.024","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.aop.2016.12.024"},{"aliases":["10.1016/j.aop.2023.169265"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5351c70e9e37e660e40fcf3f8649a8450b3c325d30c9c7d0e025808f3cf74d52.json","evidence":{"source_url":"https://doi.org/10.1016/j.aop.2023.169265"},"id":"literature:doi:10.1016/j.aop.2023.169265","identifiers":{"doi":"10.1016/j.aop.2023.169265"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.aop.2023.169265","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.aop.2023.169265"},{"aliases":["10.1016/j.chaos.2016.10.004"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0080e7d2c1b0cb2c6182d36217bcd82b63350d00cf0759d77a1ff4acc6a250c7.json","evidence":{"source_url":"https://doi.org/10.1016/j.chaos.2016.10.004"},"id":"literature:doi:10.1016/j.chaos.2016.10.004","identifiers":{"doi":"10.1016/j.chaos.2016.10.004"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.chaos.2016.10.004","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.chaos.2016.10.004"},{"aliases":["10.1016/j.chaos.2026.118260"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2297b5294509e8b220fa2839b0a1ab47032ab793caf93f450471a69bb985f74d.json","evidence":{"source_url":"https://doi.org/10.1016/j.chaos.2026.118260"},"id":"literature:doi:10.1016/j.chaos.2026.118260","identifiers":{"doi":"10.1016/j.chaos.2026.118260"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.chaos.2026.118260","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.chaos.2026.118260"},{"aliases":["10.1016/j.dam.2022.08.027"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/880de70b7093769ebb04c04819d6dc7f3ed009db77b43d5d0cde77fecbd5ff4b.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2022.08.027"},"id":"literature:doi:10.1016/j.dam.2022.08.027","identifiers":{"doi":"10.1016/j.dam.2022.08.027"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2022.08.027","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2022.08.027"},{"aliases":["10.1016/j.dam.2023.01.030"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e1249294a203e7ef23e8b2ee078647d917b6772dc536eed632da824a340aad93.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2023.01.030"},"id":"literature:doi:10.1016/j.dam.2023.01.030","identifiers":{"doi":"10.1016/j.dam.2023.01.030"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2023.01.030","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2023.01.030"},{"aliases":["10.1016/j.dam.2023.10.010"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c50853c16c61c300a41c2fe57b1c36b7cedea9545cbea4310a0d69bd0611184.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2023.10.010"},"id":"literature:doi:10.1016/j.dam.2023.10.010","identifiers":{"doi":"10.1016/j.dam.2023.10.010"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2023.10.010","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2023.10.010"},{"aliases":["10.1016/j.dam.2024.05.004"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/291e20f64068cd16d3bab34c61bfd754505e477c9658479219d16d1d34ae33c4.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2024.05.004"},"id":"literature:doi:10.1016/j.dam.2024.05.004","identifiers":{"doi":"10.1016/j.dam.2024.05.004"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2024.05.004","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2024.05.004"},{"aliases":["10.1016/j.dam.2025.07.017"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39f7e8d510516405d7f29e91fc8216e12f1c2dacbd91c50ce84b4b9c21c6a7ac.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2025.07.017"},"id":"literature:doi:10.1016/j.dam.2025.07.017","identifiers":{"doi":"10.1016/j.dam.2025.07.017"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2025.07.017","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2025.07.017"},{"aliases":["10.1016/j.dam.2025.10.041"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/425ca08c8bd2b44fb5722e9f1441b36bf5c6d9138728a452c08a50a4ffc72722.json","evidence":{"source_url":"https://doi.org/10.1016/j.dam.2025.10.041"},"id":"literature:doi:10.1016/j.dam.2025.10.041","identifiers":{"doi":"10.1016/j.dam.2025.10.041"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.dam.2025.10.041","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.dam.2025.10.041"},{"aliases":["10.1016/j.disc.2024.114323"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23193eb59e31df0c823fd389fd50f3f0c66a2ba603c1ba0fac4f8ed7230be465.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2024.114323"},"id":"literature:doi:10.1016/j.disc.2024.114323","identifiers":{"doi":"10.1016/j.disc.2024.114323"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2024.114323","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2024.114323"},{"aliases":["10.1016/j.disc.2024.114345"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f0238c23131bf16faffd4644d376a51d433c0b96e9451ce04630418b3bd04137.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2024.114345"},"id":"literature:doi:10.1016/j.disc.2024.114345","identifiers":{"doi":"10.1016/j.disc.2024.114345"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2024.114345","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2024.114345"},{"aliases":["10.1016/j.disc.2025.114399"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/72a88127dddc25e35b303236f300c99a8cf8953567c8e479b24b77ae0ed04460.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2025.114399"},"id":"literature:doi:10.1016/j.disc.2025.114399","identifiers":{"doi":"10.1016/j.disc.2025.114399"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2025.114399","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2025.114399"},{"aliases":["10.1016/j.disc.2025.114834"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9e1958a03fd63cbc62c83976958589359984910f7cd4ce4107f44760f9711e87.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2025.114834"},"id":"literature:doi:10.1016/j.disc.2025.114834","identifiers":{"doi":"10.1016/j.disc.2025.114834"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2025.114834","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2025.114834"},{"aliases":["10.1016/j.disc.2025.114959"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5cdf16e53b4f42e9df50588d389bd5dc484bd954c95b5a39751c4d41875e111a.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2025.114959"},"id":"literature:doi:10.1016/j.disc.2025.114959","identifiers":{"doi":"10.1016/j.disc.2025.114959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2025.114959","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2025.114959"},{"aliases":["10.1016/j.disc.2025.114964"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/073adb805309f2454a03907b0e243b7e79c34bce27133a9ef9493385e500f5c3.json","evidence":{"source_url":"https://doi.org/10.1016/j.disc.2025.114964"},"id":"literature:doi:10.1016/j.disc.2025.114964","identifiers":{"doi":"10.1016/j.disc.2025.114964"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.disc.2025.114964","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.disc.2025.114964"},{"aliases":["10.1016/j.ejc.2005.04.010","Richard Ehrenborg and Gábor Hetyei"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb65a654b8f363e19f8e352771ee86def9eed800baf6ba30a53243cbcd448fa3.json","authors":"Richard Ehrenborg and Gábor Hetyei","evidence":{"source_url":"https://doi.org/10.1016/j.ejc.2005.04.010"},"id":"literature:doi:10.1016/j.ejc.2005.04.010","identifiers":{"doi":"10.1016/j.ejc.2005.04.010"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The topology of the independence complex","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.ejc.2005.04.010","year":2006},{"aliases":["10.1016/j.ejc.2024.103942"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/21ef5bffc7d77ae5833664dd2d73028e0e5a3aab36a4281d572c014d2b8159db.json","evidence":{"source_url":"https://doi.org/10.1016/j.ejc.2024.103942"},"id":"literature:doi:10.1016/j.ejc.2024.103942","identifiers":{"doi":"10.1016/j.ejc.2024.103942"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.ejc.2024.103942","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.ejc.2024.103942"},{"aliases":["10.1016/j.exco.2023.100115"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/720b6218de3fd1d6e75449982461813d57567f246e40e13fad9cc8d0e9a82a39.json","evidence":{"source_url":"https://doi.org/10.1016/j.exco.2023.100115"},"id":"literature:doi:10.1016/j.exco.2023.100115","identifiers":{"doi":"10.1016/j.exco.2023.100115"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.exco.2023.100115","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.exco.2023.100115"},{"aliases":["10.1016/j.exmath.2011.07.001"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7dfdaedef6965520232a869180f0a2bceac7830828e58cfa0f21f2d0e46bd13.json","evidence":{"source_url":"https://doi.org/10.1016/j.exmath.2011.07.001"},"id":"literature:doi:10.1016/j.exmath.2011.07.001","identifiers":{"doi":"10.1016/j.exmath.2011.07.001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.exmath.2011.07.001","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.exmath.2011.07.001"},{"aliases":["10.1016/j.exmath.2025.125742"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3d82475b8577861e351b57177a4dc5cebb50c511e501c550bec8ebac9a2a0de.json","evidence":{"source_url":"https://doi.org/10.1016/j.exmath.2025.125742"},"id":"literature:doi:10.1016/j.exmath.2025.125742","identifiers":{"doi":"10.1016/j.exmath.2025.125742"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.exmath.2025.125742","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.exmath.2025.125742"},{"aliases":["10.1016/j.geomphys.2013.06.009"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68f89a3f4a006f092ade822ad47f12635ea9db8e3d1177b97240ba07bcf1f538.json","evidence":{"source_url":"https://doi.org/10.1016/j.geomphys.2013.06.009"},"id":"literature:doi:10.1016/j.geomphys.2013.06.009","identifiers":{"doi":"10.1016/j.geomphys.2013.06.009"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.geomphys.2013.06.009","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.geomphys.2013.06.009"},{"aliases":["10.1016/j.geomphys.2026.105993"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43f726130ab43301fca6c8a3c7d3cbf690a80f99782023d82c1097a7f7743a0a.json","evidence":{"source_url":"https://doi.org/10.1016/j.geomphys.2026.105993"},"id":"literature:doi:10.1016/j.geomphys.2026.105993","identifiers":{"doi":"10.1016/j.geomphys.2026.105993"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.geomphys.2026.105993","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.geomphys.2026.105993"},{"aliases":["10.1016/j.jalgebra.2019.12.029"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7cf6bcf81d95a97361ad988a3a67eb4f6e00fea67b6cafcd39b1703b54746a93.json","evidence":{"source_url":"https://doi.org/10.1016/j.jalgebra.2019.12.029"},"id":"literature:doi:10.1016/j.jalgebra.2019.12.029","identifiers":{"doi":"10.1016/j.jalgebra.2019.12.029"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jalgebra.2019.12.029","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jalgebra.2019.12.029"},{"aliases":["10.1016/j.jalgebra.2026.03.018"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52d0eb45d6dc98eaa072fdfa859f1c4af87a8269fefc563ee079e30f1ade2e44.json","evidence":{"source_url":"https://doi.org/10.1016/j.jalgebra.2026.03.018"},"id":"literature:doi:10.1016/j.jalgebra.2026.03.018","identifiers":{"doi":"10.1016/j.jalgebra.2026.03.018"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jalgebra.2026.03.018","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jalgebra.2026.03.018"},{"aliases":["10.1016/j.jalgebra.2026.07.040"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/848f887aada6646a48a5bbfce6585c6d0468d1dc67d709d1d68c19b90cdf70f9.json","evidence":{"source_url":"https://doi.org/10.1016/j.jalgebra.2026.07.040"},"id":"literature:doi:10.1016/j.jalgebra.2026.07.040","identifiers":{"doi":"10.1016/j.jalgebra.2026.07.040"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jalgebra.2026.07.040","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jalgebra.2026.07.040"},{"aliases":["10.1016/j.jat.2008.10.002"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8cda457f1ca1f033bda78a839fbb0586ba643134c0fd80d5babb2a3d936aa084.json","evidence":{"source_url":"https://doi.org/10.1016/j.jat.2008.10.002"},"id":"literature:doi:10.1016/j.jat.2008.10.002","identifiers":{"doi":"10.1016/j.jat.2008.10.002"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jat.2008.10.002","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jat.2008.10.002"},{"aliases":["10.1016/j.jcta.2013.05.006"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4226e6993422f9322d6d5cee2f9d0e03a4baa748ba7d296033298b7c64246d1.json","evidence":{"source_url":"https://doi.org/10.1016/j.jcta.2013.05.006"},"id":"literature:doi:10.1016/j.jcta.2013.05.006","identifiers":{"doi":"10.1016/j.jcta.2013.05.006"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jcta.2013.05.006","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jcta.2013.05.006"},{"aliases":["10.1016/j.jlamp.2026.101133"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8cc691ab0632d6cf612d1c8cba08eb96beee090d2d2e61da2f9301c4912e0233.json","evidence":{"source_url":"https://doi.org/10.1016/j.jlamp.2026.101133"},"id":"literature:doi:10.1016/j.jlamp.2026.101133","identifiers":{"doi":"10.1016/j.jlamp.2026.101133"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jlamp.2026.101133","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jlamp.2026.101133"},{"aliases":["10.1016/j.jmaa.2009.10.057"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07cc9c318e95c76d516d89080f6879138fb1b163b5bf5f39a5b1f66a7f3da2fe.json","evidence":{"source_url":"https://doi.org/10.1016/j.jmaa.2009.10.057"},"id":"literature:doi:10.1016/j.jmaa.2009.10.057","identifiers":{"doi":"10.1016/j.jmaa.2009.10.057"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jmaa.2009.10.057","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jmaa.2009.10.057"},{"aliases":["10.1016/j.jmaa.2013.01.015"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b6a564ebf45cd48421093921a2aa280b94dd8da297e40fa18187f24889131099.json","evidence":{"source_url":"https://doi.org/10.1016/j.jmaa.2013.01.015"},"id":"literature:doi:10.1016/j.jmaa.2013.01.015","identifiers":{"doi":"10.1016/j.jmaa.2013.01.015"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jmaa.2013.01.015","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jmaa.2013.01.015"},{"aliases":["10.1016/j.jmaa.2017.11.041"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/535ab6dff0bc3dc667ac2fa33fe9b73cb72418ccb990cd53eee3076c3c422490.json","evidence":{"source_url":"https://doi.org/10.1016/j.jmaa.2017.11.041"},"id":"literature:doi:10.1016/j.jmaa.2017.11.041","identifiers":{"doi":"10.1016/j.jmaa.2017.11.041"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jmaa.2017.11.041","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jmaa.2017.11.041"},{"aliases":["10.1016/j.jmaa.2025.129588"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ddc676600db30b860f24a1ff1b42b266117dd3912f33bbb7360a978236f07bc8.json","evidence":{"source_url":"https://doi.org/10.1016/j.jmaa.2025.129588"},"id":"literature:doi:10.1016/j.jmaa.2025.129588","identifiers":{"doi":"10.1016/j.jmaa.2025.129588"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jmaa.2025.129588","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jmaa.2025.129588"},{"aliases":["10.1016/j.jnt.2015.11.014","Piotr Miska"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32824224603ca391715d3a84a670b78535ad175899537477d96d397febbc9eea.json","authors":"Piotr Miska","evidence":{"source_url":"https://doi.org/10.1016/j.jnt.2015.11.014"},"id":"literature:doi:10.1016/j.jnt.2015.11.014","identifiers":{"doi":"10.1016/j.jnt.2015.11.014"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Arithmetic Properties of the Sequence of Derangements and its Generalizations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jnt.2015.11.014","year":2016},{"aliases":["10.1016/j.jnt.2020.04.007"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1f544969a94f19f792a7a622aa20dc5ec947f82d4eb36c8ea65640de1c5ea21.json","evidence":{"source_url":"https://doi.org/10.1016/j.jnt.2020.04.007"},"id":"literature:doi:10.1016/j.jnt.2020.04.007","identifiers":{"doi":"10.1016/j.jnt.2020.04.007"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jnt.2020.04.007","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jnt.2020.04.007"},{"aliases":["10.1016/j.jocs.2022.101853"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b8b89d6d585e6315c39b88cc127dd7c350d657c09948f885eafa6097a66a0d6f.json","evidence":{"source_url":"https://doi.org/10.1016/j.jocs.2022.101853"},"id":"literature:doi:10.1016/j.jocs.2022.101853","identifiers":{"doi":"10.1016/j.jocs.2022.101853"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jocs.2022.101853","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jocs.2022.101853"},{"aliases":["10.1016/j.jsc.2024.102352"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca5ad37d3887016f32afe1ff00b2155be0abfd525d3288211bf2e67622a4fee6.json","evidence":{"source_url":"https://doi.org/10.1016/j.jsc.2024.102352"},"id":"literature:doi:10.1016/j.jsc.2024.102352","identifiers":{"doi":"10.1016/j.jsc.2024.102352"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.jsc.2024.102352","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.jsc.2024.102352"},{"aliases":["10.1016/j.laa.2016.04.012"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e07cf782d9814b8e3394396c35fe4f16bafb62a85c3b32b2ff65bac93f6bf5b6.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2016.04.012"},"id":"literature:doi:10.1016/j.laa.2016.04.012","identifiers":{"doi":"10.1016/j.laa.2016.04.012"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2016.04.012","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2016.04.012"},{"aliases":["10.1016/j.laa.2016.06.030"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d51eb331a3faa3199130c607952575f75557b31bda92b23d4fdff91ec3019355.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2016.06.030"},"id":"literature:doi:10.1016/j.laa.2016.06.030","identifiers":{"doi":"10.1016/j.laa.2016.06.030"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2016.06.030","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2016.06.030"},{"aliases":["10.1016/j.laa.2017.06.015"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e83c044b4182fc7f17a6796c5fbab2671396e9bf7dc218bf916cfa159eecea2.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2017.06.015"},"id":"literature:doi:10.1016/j.laa.2017.06.015","identifiers":{"doi":"10.1016/j.laa.2017.06.015"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2017.06.015","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2017.06.015"},{"aliases":["10.1016/j.laa.2020.07.032","Minghua Lin and Gord Sinnamon"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ade15623453be4a185e15a95723b203eed8ecf39c08701f4b40abfc1cd95e566.json","authors":"Minghua Lin and Gord Sinnamon","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2020.07.032"},"id":"literature:doi:10.1016/j.laa.2020.07.032","identifiers":{"doi":"10.1016/j.laa.2020.07.032"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Revisiting a sharpened version of Hadamard's determinant inequality","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2020.07.032","year":2020},{"aliases":["10.1016/j.laa.2020.08.027"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1275ca25dd7e0e462332902c08dd2dd59ec684770177ffe0b52c6786a355d544.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2020.08.027"},"id":"literature:doi:10.1016/j.laa.2020.08.027","identifiers":{"doi":"10.1016/j.laa.2020.08.027"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2020.08.027","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2020.08.027"},{"aliases":["10.1016/j.laa.2022.05.004"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/296df2dbd7c6476d8b48ce289b1f82c69211b34986abb72fcc59ca3082d6c7a8.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2022.05.004"},"id":"literature:doi:10.1016/j.laa.2022.05.004","identifiers":{"doi":"10.1016/j.laa.2022.05.004"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2022.05.004","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2022.05.004"},{"aliases":["10.1016/j.laa.2023.10.014"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/024b81b3f7a9975e2058e0e4b3f87cd9f31f28ecec39ac0d935767e240e16683.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2023.10.014"},"id":"literature:doi:10.1016/j.laa.2023.10.014","identifiers":{"doi":"10.1016/j.laa.2023.10.014"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2023.10.014","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2023.10.014"},{"aliases":["10.1016/j.laa.2024.05.025"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7562b1896cb93c917ba81875c976545e37c62bc4c06a287e4b5580182446f0f0.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2024.05.025"},"id":"literature:doi:10.1016/j.laa.2024.05.025","identifiers":{"doi":"10.1016/j.laa.2024.05.025"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2024.05.025","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2024.05.025"},{"aliases":["10.1016/j.laa.2024.07.013"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9600baef56331a01f09eb4e62482116f3426bc96880ac5a849975fa9e5918c4a.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2024.07.013"},"id":"literature:doi:10.1016/j.laa.2024.07.013","identifiers":{"doi":"10.1016/j.laa.2024.07.013"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2024.07.013","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2024.07.013"},{"aliases":["10.1016/j.laa.2024.11.016"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d4bf50ffe34476c08e78954c1bf1ed9cc6175009f1d96c31892066c3a8a9dba.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2024.11.016"},"id":"literature:doi:10.1016/j.laa.2024.11.016","identifiers":{"doi":"10.1016/j.laa.2024.11.016"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2024.11.016","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2024.11.016"},{"aliases":["10.1016/j.laa.2025.10.001"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c769339ce3c86587129a6dd80a672c4c6879ca52be10d29a8e5660fe41ad2182.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2025.10.001"},"id":"literature:doi:10.1016/j.laa.2025.10.001","identifiers":{"doi":"10.1016/j.laa.2025.10.001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2025.10.001","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2025.10.001"},{"aliases":["10.1016/j.laa.2025.10.022"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aaabffabd47349792c289da98bef0dbb67c1985a22b512668ee44f1384d13ec5.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2025.10.022"},"id":"literature:doi:10.1016/j.laa.2025.10.022","identifiers":{"doi":"10.1016/j.laa.2025.10.022"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2025.10.022","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2025.10.022"},{"aliases":["10.1016/j.laa.2026.01.003"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/caf2928d33a3e8444daefcd81031c69a7fc270989694d3df3ba496a181dfee4a.json","evidence":{"source_url":"https://doi.org/10.1016/j.laa.2026.01.003"},"id":"literature:doi:10.1016/j.laa.2026.01.003","identifiers":{"doi":"10.1016/j.laa.2026.01.003"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.laa.2026.01.003","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.laa.2026.01.003"},{"aliases":["10.1016/j.nuclphysb.2007.07.001"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1667921d05997d4c7c2ef215168e6f1a6b34e7487b239e78b5a7674cd59610d3.json","evidence":{"source_url":"https://doi.org/10.1016/j.nuclphysb.2007.07.001"},"id":"literature:doi:10.1016/j.nuclphysb.2007.07.001","identifiers":{"doi":"10.1016/j.nuclphysb.2007.07.001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.nuclphysb.2007.07.001","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.nuclphysb.2007.07.001"},{"aliases":["10.1016/j.nuclphysb.2025.117268"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d170e7f927a5220ac1f6656652a001d9370eabed63ced82a229689516be75ab4.json","evidence":{"source_url":"https://doi.org/10.1016/j.nuclphysb.2025.117268"},"id":"literature:doi:10.1016/j.nuclphysb.2025.117268","identifiers":{"doi":"10.1016/j.nuclphysb.2025.117268"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.nuclphysb.2025.117268","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.nuclphysb.2025.117268"},{"aliases":["10.1016/j.physa.2015.02.061"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd609bbef7e4aa689af245e93e435a0ca3ae51e9d7429fec1d0d57c284f1c003.json","evidence":{"source_url":"https://doi.org/10.1016/j.physa.2015.02.061"},"id":"literature:doi:10.1016/j.physa.2015.02.061","identifiers":{"doi":"10.1016/j.physa.2015.02.061"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physa.2015.02.061","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physa.2015.02.061"},{"aliases":["10.1016/j.physa.2015.05.106"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fc5693935cc9ee20060b86ebadba71121ac98cb2e0e45f8e43f5a4240a7bedb7.json","evidence":{"source_url":"https://doi.org/10.1016/j.physa.2015.05.106"},"id":"literature:doi:10.1016/j.physa.2015.05.106","identifiers":{"doi":"10.1016/j.physa.2015.05.106"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physa.2015.05.106","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physa.2015.05.106"},{"aliases":["10.1016/j.physa.2016.06.131"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/438683ceb672918163ecec33999224c2fd186e4703795a1333dc12ba3b1d0422.json","evidence":{"source_url":"https://doi.org/10.1016/j.physa.2016.06.131"},"id":"literature:doi:10.1016/j.physa.2016.06.131","identifiers":{"doi":"10.1016/j.physa.2016.06.131"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physa.2016.06.131","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physa.2016.06.131"},{"aliases":["10.1016/j.physa.2023.129090"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5ed10d445b0dce9c4884413147966375251b214b5fbf9f799a48802268f2b9b.json","evidence":{"source_url":"https://doi.org/10.1016/j.physa.2023.129090"},"id":"literature:doi:10.1016/j.physa.2023.129090","identifiers":{"doi":"10.1016/j.physa.2023.129090"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physa.2023.129090","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physa.2023.129090"},{"aliases":["10.1016/j.physa.2023.129252"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23a7ee63ff86baf1fb6d4dc85f9970dc17a4660b2b4603609609617eabfa307b.json","evidence":{"source_url":"https://doi.org/10.1016/j.physa.2023.129252"},"id":"literature:doi:10.1016/j.physa.2023.129252","identifiers":{"doi":"10.1016/j.physa.2023.129252"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physa.2023.129252","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physa.2023.129252"},{"aliases":["10.1016/j.physd.2024.134469"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/378c2664a7133ec8385a4699a695bbdf74c4e6c74db56891676c2b30591b42a0.json","evidence":{"source_url":"https://doi.org/10.1016/j.physd.2024.134469"},"id":"literature:doi:10.1016/j.physd.2024.134469","identifiers":{"doi":"10.1016/j.physd.2024.134469"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physd.2024.134469","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physd.2024.134469"},{"aliases":["10.1016/j.physleta.2006.10.055","Mark W. Coffey"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6ce94c7234c4b366ba0982bed4d00cb03fe51b031f0ff11e5597028ecb720a9.json","authors":"Mark W. Coffey","evidence":{"source_url":"https://doi.org/10.1016/j.physleta.2006.10.055"},"id":"literature:doi:10.1016/j.physleta.2006.10.055","identifiers":{"doi":"10.1016/j.physleta.2006.10.055"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Theta and Riemann xi function representations from harmonic oscillator eigensolutions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physleta.2006.10.055","year":2007},{"aliases":["10.1016/j.physleta.2013.05.047"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/547934d64ef105568e88945ff824294280fc834a5ac6fec7a20ab5dd31823370.json","evidence":{"source_url":"https://doi.org/10.1016/j.physleta.2013.05.047"},"id":"literature:doi:10.1016/j.physleta.2013.05.047","identifiers":{"doi":"10.1016/j.physleta.2013.05.047"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physleta.2013.05.047","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physleta.2013.05.047"},{"aliases":["10.1016/j.physleta.2016.03.046"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6285996492556baa066008f83f3f19f9781db733beb5c1367d4dd731af751bd5.json","evidence":{"source_url":"https://doi.org/10.1016/j.physleta.2016.03.046"},"id":"literature:doi:10.1016/j.physleta.2016.03.046","identifiers":{"doi":"10.1016/j.physleta.2016.03.046"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.physleta.2016.03.046","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.physleta.2016.03.046"},{"aliases":["10.1016/j.rinp.2024.107430"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/47b30d55d7e896161a3423fe7df248d221ec8922761e06183456e908653e7ad0.json","evidence":{"source_url":"https://doi.org/10.1016/j.rinp.2024.107430"},"id":"literature:doi:10.1016/j.rinp.2024.107430","identifiers":{"doi":"10.1016/j.rinp.2024.107430"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.rinp.2024.107430","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.rinp.2024.107430"},{"aliases":["10.1016/j.tcs.2021.08.016"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/88d6be377357781eae9520447441f506260d890bbbd01715215989db03698aff.json","evidence":{"source_url":"https://doi.org/10.1016/j.tcs.2021.08.016"},"id":"literature:doi:10.1016/j.tcs.2021.08.016","identifiers":{"doi":"10.1016/j.tcs.2021.08.016"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.tcs.2021.08.016","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.tcs.2021.08.016"},{"aliases":["10.1016/j.tcs.2025.115391"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/27054ad0773ea149707ec11b460ae008ad294eb7e2dcdf9f2072dbff9c36d0b3.json","evidence":{"source_url":"https://doi.org/10.1016/j.tcs.2025.115391"},"id":"literature:doi:10.1016/j.tcs.2025.115391","identifiers":{"doi":"10.1016/j.tcs.2025.115391"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.tcs.2025.115391","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.tcs.2025.115391"},{"aliases":["10.1016/j.tcs.2026.115965"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85b8dbd9c3f7b68b902e37cae02e69a6e7624f4b7b621016a1916bfd7e8fe5c0.json","evidence":{"source_url":"https://doi.org/10.1016/j.tcs.2026.115965"},"id":"literature:doi:10.1016/j.tcs.2026.115965","identifiers":{"doi":"10.1016/j.tcs.2026.115965"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/j.tcs.2026.115965","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fj.tcs.2026.115965"},{"aliases":["10.1016/s0034-4877(24)00012-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d460a075301e9c03d7e9d0a6968bd27ccdb5ea4d054fa7e7c77b796bd094cfd5.json","evidence":{"source_url":"https://doi.org/10.1016/s0034-4877(24)00012-0"},"id":"literature:doi:10.1016/s0034-4877(24)00012-0","identifiers":{"doi":"10.1016/s0034-4877(24)00012-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/s0034-4877(24)00012-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fs0034-4877%2824%2900012-0"},{"aliases":["10.1016/s0079-8169(08)x6136-7","Igor B. Frenkel, James Lepowsky, and Arne Meurman"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fa16b5dd8803b154474caaff544fc6f2ebc30f989af7781a165c4b467414d7a.json","authors":"Igor B. Frenkel, James Lepowsky, and Arne Meurman","evidence":{"source_url":"https://doi.org/10.1016/s0079-8169(08)x6136-7"},"id":"literature:doi:10.1016/s0079-8169(08)x6136-7","identifiers":{"doi":"10.1016/s0079-8169(08)x6136-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Vertex Operator Algebras and the Monster","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fs0079-8169%2808%29x6136-7","year":1988},{"aliases":["10.1016/s0375-9601(00)00480-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e8df8a9cebe56608b29815cf6643c6f0dad2fdc244b80050a647493fb41630ba.json","evidence":{"source_url":"https://doi.org/10.1016/s0375-9601(00)00480-1"},"id":"literature:doi:10.1016/s0375-9601(00)00480-1","identifiers":{"doi":"10.1016/s0375-9601(00)00480-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1016/s0375-9601(00)00480-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1016%2Fs0375-9601%2800%2900480-1"},{"aliases":["10.1017/9781316809976","Mark M. Wilde"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8635f6a9433ce6dd5209f44b28ea4dcbb0d41f5171b5855c49184f7a4ddf50e2.json","authors":"Mark M. Wilde","evidence":{"source_url":"https://doi.org/10.1017/9781316809976"},"id":"literature:doi:10.1017/9781316809976","identifiers":{"doi":"10.1017/9781316809976"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Information Theory","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2F9781316809976","year":2017},{"aliases":["10.1017/9781316848142","John Watrous"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/329cb77365ff77ce42503de065b7be6593a374271a29b91c51bd660ad8f457c7.json","authors":"John Watrous","evidence":{"source_url":"https://doi.org/10.1017/9781316848142"},"id":"literature:doi:10.1017/9781316848142","identifiers":{"doi":"10.1017/9781316848142"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Theory of Quantum Information","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2F9781316848142","year":2018},{"aliases":["10.1017/cbo9780511525790","Rudolf Lidl; Harald Niederreiter"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/feec36d2e090df516da9fa3a5263a9b15091adba5839aefb4ed79701b531a635.json","authors":"Rudolf Lidl; Harald Niederreiter","evidence":{"source_url":"https://doi.org/10.1017/cbo9780511525790"},"id":"literature:doi:10.1017/cbo9780511525790","identifiers":{"doi":"10.1017/cbo9780511525790"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Finite Fields","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fcbo9780511525790","year":1997},{"aliases":["10.1017/cbo9780511987045","J. H. van Lint and R. M. Wilson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce1c3f38fd5a087e6f8723990a97eb2a497363f6c8a93626c9988eb374cec3c7.json","authors":"J. H. van Lint and R. M. Wilson","evidence":{"source_url":"https://doi.org/10.1017/cbo9780511987045"},"id":"literature:doi:10.1017/cbo9780511987045","identifiers":{"doi":"10.1017/cbo9780511987045"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Course in Combinatorics","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fcbo9780511987045","year":2001},{"aliases":["10.1017/cbo9781139020411","Roger A. Horn and Charles R. Johnson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5b6d0d17cc96351a52a19ffd95521290f0f6aa14659fc072c7c857df96bf449.json","authors":"Roger A. Horn and Charles R. Johnson","evidence":{"source_url":"https://doi.org/10.1017/cbo9781139020411"},"id":"literature:doi:10.1017/cbo9781139020411","identifiers":{"doi":"10.1017/cbo9781139020411"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Matrix Analysis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fcbo9781139020411","year":2012},{"aliases":["10.1017/cbo9781139644136","Charles A. Weibel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a7382e55b77de5ea5884bcb4f221d2140a96bfc0bb1f7d065ad80bf356b6d90.json","authors":"Charles A. Weibel","evidence":{"source_url":"https://doi.org/10.1017/cbo9781139644136"},"id":"literature:doi:10.1017/cbo9781139644136","identifiers":{"doi":"10.1017/cbo9781139644136"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"An Introduction to Homological Algebra","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fcbo9781139644136","year":1994},{"aliases":["10.1017/cbo9781139644136.004","Charles A. Weibel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16f264226b2f2be9e6c455573656a8e40d4f05163dceae2826244dee42f3ce10.json","authors":"Charles A. Weibel","evidence":{"source_url":"https://doi.org/10.1017/cbo9781139644136.004"},"id":"literature:doi:10.1017/cbo9781139644136.004","identifiers":{"doi":"10.1017/cbo9781139644136.004"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Tor and Ext","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fcbo9781139644136.004","year":1994},{"aliases":["10.1017/fms.2025.10165"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b127d268c772132c58177930b1288ac31d7c2a27e6f8573610e92a8def761058.json","evidence":{"source_url":"https://doi.org/10.1017/fms.2025.10165"},"id":"literature:doi:10.1017/fms.2025.10165","identifiers":{"doi":"10.1017/fms.2025.10165"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1017/fms.2025.10165","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Ffms.2025.10165"},{"aliases":["10.1017/s0017089500009757","J. H. E. Cohn"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ed8157b36e5f7613599a031897a1019eb63f2cdb67d86c02cba2d6e1558656f.json","authors":"J. H. E. Cohn","evidence":{"source_url":"https://doi.org/10.1017/s0017089500009757"},"id":"literature:doi:10.1017/s0017089500009757","identifiers":{"doi":"10.1017/s0017089500009757"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Diophantine equation x^2+3=y^n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fs0017089500009757","year":1993},{"aliases":["10.1017/s0305004100010112","R. E. A. C. Paley and A. Zygmund"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e535c1620d76de1d6c65c9d1370a09658d2d4d86d7fd93495c993c23baaf84eb.json","authors":"R. E. A. C. Paley and A. Zygmund","evidence":{"source_url":"https://doi.org/10.1017/s0305004100010112"},"id":"literature:doi:10.1017/s0305004100010112","identifiers":{"doi":"10.1017/s0305004100010112"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A note on analytic functions in the unit circle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fs0305004100010112","year":1932},{"aliases":["10.1017/s0305004100030401","R. Penrose"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa73238758c03f4b5d857546811545c8ecc3628e9383dbb9e7e05ba46053345e.json","authors":"R. Penrose","evidence":{"source_url":"https://doi.org/10.1017/s0305004100030401"},"id":"literature:doi:10.1017/s0305004100030401","identifiers":{"doi":"10.1017/s0305004100030401"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A generalized inverse for matrices","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1017%2Fs0305004100030401","year":1955},{"aliases":["10.1038/ncomms8689"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b18991e7fa0fbb1882f7d8817a9a8b5e05a0db1bd12c821edd9dd8827b642c54.json","evidence":{"source_url":"https://doi.org/10.1038/ncomms8689"},"id":"literature:doi:10.1038/ncomms8689","identifiers":{"doi":"10.1038/ncomms8689"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/ncomms8689","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fncomms8689"},{"aliases":["10.1038/s41467-019-10333-7","Kaonan Micadei, John P. S. Peterson, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, Gabriel T. Landi, Tiago B. Batalhão, Roberto M. Serra, Eric Lutz"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d115b6d488662b9919bc0302fa747c9d06328a36ac19af51c57e3aa1fbdb217.json","authors":"Kaonan Micadei, John P. S. Peterson, Alexandre M. Souza, Roberto S. Sarthour, Ivan S. Oliveira, Gabriel T. Landi, Tiago B. Batalhão, Roberto M. Serra, Eric Lutz","evidence":{"source_url":"https://doi.org/10.1038/s41467-019-10333-7"},"id":"literature:doi:10.1038/s41467-019-10333-7","identifiers":{"doi":"10.1038/s41467-019-10333-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Reversing the direction of heat flow using quantum correlations","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41467-019-10333-7","year":2019},{"aliases":["10.1038/s41467-021-22275-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3464af129a62b49948cf6e18e30ffb50e580dd27f09432d9677f644c76759d3c.json","evidence":{"source_url":"https://doi.org/10.1038/s41467-021-22275-0"},"id":"literature:doi:10.1038/s41467-021-22275-0","identifiers":{"doi":"10.1038/s41467-021-22275-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41467-021-22275-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41467-021-22275-0"},{"aliases":["10.1038/s41467-022-33563-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1556d0a451100cc2dd0fca43b43645c3bf4ee73da3e60115d20547269c0f65f.json","evidence":{"source_url":"https://doi.org/10.1038/s41467-022-33563-8"},"id":"literature:doi:10.1038/s41467-022-33563-8","identifiers":{"doi":"10.1038/s41467-022-33563-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41467-022-33563-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41467-022-33563-8"},{"aliases":["10.1038/s41534-024-00956-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f6c28fe5a199f71cf6838e91b7515850d61b180df4c5bb2b62348450b66312b.json","evidence":{"source_url":"https://doi.org/10.1038/s41534-024-00956-0"},"id":"literature:doi:10.1038/s41534-024-00956-0","identifiers":{"doi":"10.1038/s41534-024-00956-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41534-024-00956-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41534-024-00956-0"},{"aliases":["10.1038/s41534-025-01161-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b12c4f0268b3ad079407eeed7f7626dd73566e6c026ce5aa333f52e2e13fba9a.json","evidence":{"source_url":"https://doi.org/10.1038/s41534-025-01161-3"},"id":"literature:doi:10.1038/s41534-025-01161-3","identifiers":{"doi":"10.1038/s41534-025-01161-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41534-025-01161-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41534-025-01161-3"},{"aliases":["10.1038/s41598-021-82303-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d8557bf73e1f48b4ab97e3f68f0479b085aabc7ec2f2cbb74f4bf7a5e4e2ffe.json","evidence":{"source_url":"https://doi.org/10.1038/s41598-021-82303-3"},"id":"literature:doi:10.1038/s41598-021-82303-3","identifiers":{"doi":"10.1038/s41598-021-82303-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41598-021-82303-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41598-021-82303-3"},{"aliases":["10.1038/s41598-021-85273-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/da63b8a72dbbd8599aacc9724515c16ad1da8ad1c757773f160b2f639580929c.json","evidence":{"source_url":"https://doi.org/10.1038/s41598-021-85273-8"},"id":"literature:doi:10.1038/s41598-021-85273-8","identifiers":{"doi":"10.1038/s41598-021-85273-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41598-021-85273-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41598-021-85273-8"},{"aliases":["10.1038/s41598-021-93161-4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d03939fad88dae0eac987fbd20613d7b33e6385270da162ebe453063aa4b718b.json","evidence":{"source_url":"https://doi.org/10.1038/s41598-021-93161-4"},"id":"literature:doi:10.1038/s41598-021-93161-4","identifiers":{"doi":"10.1038/s41598-021-93161-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41598-021-93161-4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41598-021-93161-4"},{"aliases":["10.1038/s41598-023-29438-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/753bb4fe0daf7988f5afd7c127483c86fc893664b99d01ea0e404c9c359dbf9e.json","evidence":{"source_url":"https://doi.org/10.1038/s41598-023-29438-7"},"id":"literature:doi:10.1038/s41598-023-29438-7","identifiers":{"doi":"10.1038/s41598-023-29438-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41598-023-29438-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41598-023-29438-7"},{"aliases":["10.1038/s41598-023-39529-0"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4abf229792fcd6a1a73729c26314c30e76b730231a723a91a2a84834f0fd6e41.json","evidence":{"source_url":"https://doi.org/10.1038/s41598-023-39529-0"},"id":"literature:doi:10.1038/s41598-023-39529-0","identifiers":{"doi":"10.1038/s41598-023-39529-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1038/s41598-023-39529-0","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1038%2Fs41598-023-39529-0"},{"aliases":["10.1051/ita/2011121"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b641cedb038cf2c6c56939541dd2687be8a1ad395c1f982442b3558a7eee2bd5.json","evidence":{"source_url":"https://doi.org/10.1051/ita/2011121"},"id":"literature:doi:10.1051/ita/2011121","identifiers":{"doi":"10.1051/ita/2011121"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1051/ita/2011121","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1051%2Fita%2F2011121"},{"aliases":["10.1051/ita/2026032","Haobo Ma, Wenlin Zhang"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db299b79c5994bbc14f03b4597600a5272f4728b4daa1f90cc447aa6cfd3bfef.json","authors":"Haobo Ma, Wenlin Zhang","evidence":{"source_url":"https://doi.org/10.1051/ita/2026032"},"id":"literature:doi:10.1051/ita/2026032","identifiers":{"doi":"10.1051/ita/2026032"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Canonical Zeckendorf Normalization and sharp iteration depth of the Berstel Adder","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1051%2Fita%2F2026032"},{"aliases":["10.1063/1.1459754","H. Barnum and E. Knill"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d173996e0e8d2355471c9a2105b8c7b05b1e96106fe339c02740a111cc3c209.json","authors":"H. Barnum and E. Knill","evidence":{"source_url":"https://doi.org/10.1063/1.1459754"},"id":"literature:doi:10.1063/1.1459754","identifiers":{"doi":"10.1063/1.1459754"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Reversing quantum dynamics with near-optimal quantum and classical fidelity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.1459754","year":2002},{"aliases":["10.1063/1.1834693"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a5d43bc6e41462b8104a37aff3856fc8ed485408f4006ed718cc22f1d30b7ca.json","evidence":{"source_url":"https://doi.org/10.1063/1.1834693"},"id":"literature:doi:10.1063/1.1834693","identifiers":{"doi":"10.1063/1.1834693"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/1.1834693","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.1834693"},{"aliases":["10.1063/1.3234186"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/afb817d2206ee3c02df59251f7b9327ff5c5aed9bb9591a6190e816be78b893b.json","evidence":{"source_url":"https://doi.org/10.1063/1.3234186"},"id":"literature:doi:10.1063/1.3234186","identifiers":{"doi":"10.1063/1.3234186"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/1.3234186","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.3234186"},{"aliases":["10.1063/1.4882935"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0c3be9e3107c91671145eae63d51ecd8e4277c2bf0b666b4401938b56c83292.json","evidence":{"source_url":"https://doi.org/10.1063/1.4882935"},"id":"literature:doi:10.1063/1.4882935","identifiers":{"doi":"10.1063/1.4882935"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/1.4882935","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.4882935"},{"aliases":["10.1063/1.4936215"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4ffae08cd14db5d0d1a2ad447107054f4f10abd44146231b1e009b7af3ee027.json","evidence":{"source_url":"https://doi.org/10.1063/1.4936215"},"id":"literature:doi:10.1063/1.4936215","identifiers":{"doi":"10.1063/1.4936215"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/1.4936215","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.4936215"},{"aliases":["10.1063/1.5122836"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9e8eae834ff60176ec78b94c70c40030303b7adc9dbf244d362a65f2447a6ab.json","evidence":{"source_url":"https://doi.org/10.1063/1.5122836"},"id":"literature:doi:10.1063/1.5122836","identifiers":{"doi":"10.1063/1.5122836"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/1.5122836","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F1.5122836"},{"aliases":["10.1063/5.0031760"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa2680486831b19f40397b197d2916975a687d731c4520eb3bcea44cf4cfb976.json","evidence":{"source_url":"https://doi.org/10.1063/5.0031760"},"id":"literature:doi:10.1063/5.0031760","identifiers":{"doi":"10.1063/5.0031760"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0031760","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0031760"},{"aliases":["10.1063/5.0046581"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/88ef8f1fb79b569d412ff4a09802f543cd6e7f11fd01a2d13aba5a8e2a4ddd04.json","evidence":{"source_url":"https://doi.org/10.1063/5.0046581"},"id":"literature:doi:10.1063/5.0046581","identifiers":{"doi":"10.1063/5.0046581"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0046581","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0046581"},{"aliases":["10.1063/5.0054438"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69952ce51d9080b663e66e2624a8bb41d846fb1b1fee8e78f766183973c4a7ff.json","evidence":{"source_url":"https://doi.org/10.1063/5.0054438"},"id":"literature:doi:10.1063/5.0054438","identifiers":{"doi":"10.1063/5.0054438"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0054438","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0054438"},{"aliases":["10.1063/5.0075919"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6dbc17f4b32de2c063213b4123bf5215e436a74f72549e9638ec06dc215bb1ce.json","evidence":{"source_url":"https://doi.org/10.1063/5.0075919"},"id":"literature:doi:10.1063/5.0075919","identifiers":{"doi":"10.1063/5.0075919"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0075919","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0075919"},{"aliases":["10.1063/5.0085774"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8504dfaeb14602e0e9466565d011af297c0ddb5a081e9b15c3dc7c194920f3a8.json","evidence":{"source_url":"https://doi.org/10.1063/5.0085774"},"id":"literature:doi:10.1063/5.0085774","identifiers":{"doi":"10.1063/5.0085774"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0085774","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0085774"},{"aliases":["10.1063/5.0110267"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6b8105aaf250077a9659778e639034d99bb05beb9da680d9c9afe6ccd536bbd8.json","evidence":{"source_url":"https://doi.org/10.1063/5.0110267"},"id":"literature:doi:10.1063/5.0110267","identifiers":{"doi":"10.1063/5.0110267"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0110267","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0110267"},{"aliases":["10.1063/5.0231812"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8a0d2757f2706f709780e4a8d18d80c5bdd726d52ded4d0c30196679ccaa5806.json","evidence":{"source_url":"https://doi.org/10.1063/5.0231812"},"id":"literature:doi:10.1063/5.0231812","identifiers":{"doi":"10.1063/5.0231812"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0231812","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0231812"},{"aliases":["10.1063/5.0274921"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28208e433c13818b52d7acf629e612594c7fb6a77727d7d18054df472478c1a4.json","evidence":{"source_url":"https://doi.org/10.1063/5.0274921"},"id":"literature:doi:10.1063/5.0274921","identifiers":{"doi":"10.1063/5.0274921"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1063/5.0274921","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1063%2F5.0274921"},{"aliases":["10.1073/pnas.46.4.570","Julian Schwinger"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7de16ccaa1459d74e22547f2c461515e6b0b219473b78f2de0f4a7a973f78e73.json","authors":"Julian Schwinger","evidence":{"source_url":"https://doi.org/10.1073/pnas.46.4.570"},"id":"literature:doi:10.1073/pnas.46.4.570","identifiers":{"doi":"10.1073/pnas.46.4.570"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Unitary Operator Bases","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1073%2Fpnas.46.4.570","year":1960},{"aliases":["10.1073/pnas.78.2.689","Robert L. Griess Jr."],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d683e7eb6abb0fce0d61edbad28733f75b792d1f7159c2dc09361f6c8c142ca.json","authors":"Robert L. Griess Jr.","evidence":{"source_url":"https://doi.org/10.1073/pnas.78.2.689"},"id":"literature:doi:10.1073/pnas.78.2.689","identifiers":{"doi":"10.1073/pnas.78.2.689"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A construction of F1 as automorphisms of a 196,883-dimensional algebra","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1073%2Fpnas.78.2.689","year":1981},{"aliases":["10.1080/00029890.1970.11992654","Solomon W. Golomb"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b5d9317283298f16f7ea76b5aac521b599594a750a7a0d7c7708359e36ff1d2.json","authors":"Solomon W. Golomb","evidence":{"source_url":"https://doi.org/10.1080/00029890.1970.11992654"},"id":"literature:doi:10.1080/00029890.1970.11992654","identifiers":{"doi":"10.1080/00029890.1970.11992654"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Powerful numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F00029890.1970.11992654","year":1970},{"aliases":["10.1080/00150517.2017.12427731"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ced457d67652afbb001ae5e8a516c20aff3d22115dc369cf5f877fd1ce5da791.json","evidence":{"source_url":"https://doi.org/10.1080/00150517.2017.12427731"},"id":"literature:doi:10.1080/00150517.2017.12427731","identifiers":{"doi":"10.1080/00150517.2017.12427731"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/00150517.2017.12427731","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F00150517.2017.12427731"},{"aliases":["10.1080/00150517.2022.12427444"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb4a2c5beb0a4891110438b54bce16e9875c1d7e97ca6592eb4603d0139848c7.json","evidence":{"source_url":"https://doi.org/10.1080/00150517.2022.12427444"},"id":"literature:doi:10.1080/00150517.2022.12427444","identifiers":{"doi":"10.1080/00150517.2022.12427444"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/00150517.2022.12427444","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F00150517.2022.12427444"},{"aliases":["10.1080/00150517.2025.2556152"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1f906f12f7bd22e8f70601e50e1d3b8ddc589c8d0664a18b58bdc287301184b2.json","evidence":{"source_url":"https://doi.org/10.1080/00150517.2025.2556152"},"id":"literature:doi:10.1080/00150517.2025.2556152","identifiers":{"doi":"10.1080/00150517.2025.2556152"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/00150517.2025.2556152","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F00150517.2025.2556152"},{"aliases":["10.1080/00927872.2025.2498055"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9264ceb0b30a6eec1e1812f3283b28af315604fdbf28629a568c61cc66b0cdff.json","evidence":{"source_url":"https://doi.org/10.1080/00927872.2025.2498055"},"id":"literature:doi:10.1080/00927872.2025.2498055","identifiers":{"doi":"10.1080/00927872.2025.2498055"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/00927872.2025.2498055","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F00927872.2025.2498055"},{"aliases":["10.1080/01630563.2022.2128818"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07312935fd311b92c741d11f9ce21c724816c9c64a2d03bfa2dd11989c1cb9f8.json","evidence":{"source_url":"https://doi.org/10.1080/01630563.2022.2128818"},"id":"literature:doi:10.1080/01630563.2022.2128818","identifiers":{"doi":"10.1080/01630563.2022.2128818"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/01630563.2022.2128818","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F01630563.2022.2128818"},{"aliases":["10.1080/03081080701535898"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c86dc83048ad166feafe423a19c3ad5b8e1913c26e42b631ba186d07f9afd3f.json","evidence":{"source_url":"https://doi.org/10.1080/03081080701535898"},"id":"literature:doi:10.1080/03081080701535898","identifiers":{"doi":"10.1080/03081080701535898"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/03081080701535898","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F03081080701535898"},{"aliases":["10.1080/03081087.2018.1442810"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aeaca1e1903097c06f9cd8cfc2f495ab11722269663f34e74611754225483fe0.json","evidence":{"source_url":"https://doi.org/10.1080/03081087.2018.1442810"},"id":"literature:doi:10.1080/03081087.2018.1442810","identifiers":{"doi":"10.1080/03081087.2018.1442810"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/03081087.2018.1442810","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F03081087.2018.1442810"},{"aliases":["10.1080/03081087.2020.1860887"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/30394f74048192b67e5a28627741a28bff3c7aa20ab76c750c9f255f445eed35.json","evidence":{"source_url":"https://doi.org/10.1080/03081087.2020.1860887"},"id":"literature:doi:10.1080/03081087.2020.1860887","identifiers":{"doi":"10.1080/03081087.2020.1860887"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/03081087.2020.1860887","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F03081087.2020.1860887"},{"aliases":["10.1080/03081087.2023.2236282"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40a80ae8e9410ceba920c78eea2685d318f135480e20cb13c446c6beaf93f45e.json","evidence":{"source_url":"https://doi.org/10.1080/03081087.2023.2236282"},"id":"literature:doi:10.1080/03081087.2023.2236282","identifiers":{"doi":"10.1080/03081087.2023.2236282"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/03081087.2023.2236282","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F03081087.2023.2236282"},{"aliases":["10.1080/09728600.2024.2389070"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66bddb0f12f67fa478d42024c6d8e5bc482e5b4d2c2502fab0b24fd8d394f448.json","evidence":{"source_url":"https://doi.org/10.1080/09728600.2024.2389070"},"id":"literature:doi:10.1080/09728600.2024.2389070","identifiers":{"doi":"10.1080/09728600.2024.2389070"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/09728600.2024.2389070","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F09728600.2024.2389070"},{"aliases":["10.1080/09728600.2025.2479147"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c9cf71352a48b523cbbdfddc4486faefba0a32538a0f3981215f246b6ce4a5f1.json","evidence":{"source_url":"https://doi.org/10.1080/09728600.2025.2479147"},"id":"literature:doi:10.1080/09728600.2025.2479147","identifiers":{"doi":"10.1080/09728600.2025.2479147"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/09728600.2025.2479147","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F09728600.2025.2479147"},{"aliases":["10.1080/10236198.2014.917635"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/561cb341831f5c85c6c975125088b4fb9c6cffe26eafa5a93e9e51830b6ca50e.json","evidence":{"source_url":"https://doi.org/10.1080/10236198.2014.917635"},"id":"literature:doi:10.1080/10236198.2014.917635","identifiers":{"doi":"10.1080/10236198.2014.917635"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/10236198.2014.917635","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F10236198.2014.917635"},{"aliases":["10.1080/10586458.2021.1982427"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d965e05a779f5ba76b02653eb900c7fe44a1b3211b8df28836d3a08e1bf89518.json","evidence":{"source_url":"https://doi.org/10.1080/10586458.2021.1982427"},"id":"literature:doi:10.1080/10586458.2021.1982427","identifiers":{"doi":"10.1080/10586458.2021.1982427"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/10586458.2021.1982427","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F10586458.2021.1982427"},{"aliases":["10.1080/10652469.2013.764874"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9812b555b8037bf3827362e02c016b66f4e555fd728a6702ddf3d35851df78a4.json","evidence":{"source_url":"https://doi.org/10.1080/10652469.2013.764874"},"id":"literature:doi:10.1080/10652469.2013.764874","identifiers":{"doi":"10.1080/10652469.2013.764874"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1080/10652469.2013.764874","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1080%2F10652469.2013.764874"},{"aliases":["10.1088/1361-6544/ae64a2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef2bd08e48d82de2a50919f88e3a567307c9e05f1f2d147bbb67a0e19ea8af6f.json","evidence":{"source_url":"https://doi.org/10.1088/1361-6544/ae64a2"},"id":"literature:doi:10.1088/1361-6544/ae64a2","identifiers":{"doi":"10.1088/1361-6544/ae64a2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1361-6544/ae64a2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1361-6544%2Fae64a2"},{"aliases":["10.1088/1367-2630/18/4/043018"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0302de41b05ca2873193cc097d9c705847526b206381a04caca480d7039e15b.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/18/4/043018"},"id":"literature:doi:10.1088/1367-2630/18/4/043018","identifiers":{"doi":"10.1088/1367-2630/18/4/043018"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/18/4/043018","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2F18%2F4%2F043018"},{"aliases":["10.1088/1367-2630/aa5015"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/360f01c64f88b00135e8de77ebc401e1dcfab47343f98b73f7409384f34c5d44.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/aa5015"},"id":"literature:doi:10.1088/1367-2630/aa5015","identifiers":{"doi":"10.1088/1367-2630/aa5015"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/aa5015","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Faa5015"},{"aliases":["10.1088/1367-2630/aac057"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2ffaf10bd9ed4d41b8ffe5369f445a1a1fef626601d4261c601493b258e9820a.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/aac057"},"id":"literature:doi:10.1088/1367-2630/aac057","identifiers":{"doi":"10.1088/1367-2630/aac057"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/aac057","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Faac057"},{"aliases":["10.1088/1367-2630/abcc9b"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ff2b7decf6d8217990c638e57a4452898df5849d1939ff444e43f78ef8a43bd.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/abcc9b"},"id":"literature:doi:10.1088/1367-2630/abcc9b","identifiers":{"doi":"10.1088/1367-2630/abcc9b"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/abcc9b","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fabcc9b"},{"aliases":["10.1088/1367-2630/ac10a5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4daab373e2b461725f6ee969a1385d42674159d3bebffa272a99da3ced437e96.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/ac10a5"},"id":"literature:doi:10.1088/1367-2630/ac10a5","identifiers":{"doi":"10.1088/1367-2630/ac10a5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/ac10a5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fac10a5"},{"aliases":["10.1088/1367-2630/acdf78"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d8acd669f8082dfbd198d5618fb318dd8f143ce20f6ac470700eedc0fa6ea9c.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/acdf78"},"id":"literature:doi:10.1088/1367-2630/acdf78","identifiers":{"doi":"10.1088/1367-2630/acdf78"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/acdf78","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Facdf78"},{"aliases":["10.1088/1367-2630/acf152"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95e08d75f70e8f3c4e114478f5d4f601af57739396eeba6b4c66aca66d56aeb6.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/acf152"},"id":"literature:doi:10.1088/1367-2630/acf152","identifiers":{"doi":"10.1088/1367-2630/acf152"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/acf152","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Facf152"},{"aliases":["10.1088/1367-2630/ad7167"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f5d78cd41050ddba76fc76ef8de3aae36536bb2aa1487ec700835701de2b3eb.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/ad7167"},"id":"literature:doi:10.1088/1367-2630/ad7167","identifiers":{"doi":"10.1088/1367-2630/ad7167"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/ad7167","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fad7167"},{"aliases":["10.1088/1367-2630/ad9bdf"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/359ff73fe1869db6484dcc8ad9f2b141d44a47533137d1079079ab1acd9052e2.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/ad9bdf"},"id":"literature:doi:10.1088/1367-2630/ad9bdf","identifiers":{"doi":"10.1088/1367-2630/ad9bdf"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/ad9bdf","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fad9bdf"},{"aliases":["10.1088/1367-2630/adda73"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4500a41450b73fbc82496029af9141c0f839dae0debb7fd72862df51e48fc3ea.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/adda73"},"id":"literature:doi:10.1088/1367-2630/adda73","identifiers":{"doi":"10.1088/1367-2630/adda73"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/adda73","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fadda73"},{"aliases":["10.1088/1367-2630/adf87c"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/347dcba9fa04dbce5381ef5c1c156b73b7da38fc113a3586cfc4009851cb8808.json","evidence":{"source_url":"https://doi.org/10.1088/1367-2630/adf87c"},"id":"literature:doi:10.1088/1367-2630/adf87c","identifiers":{"doi":"10.1088/1367-2630/adf87c"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1367-2630/adf87c","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1367-2630%2Fadf87c"},{"aliases":["10.1088/1402-4896/abafcb"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aed81d3a578a8efb35fe7bd4a157c6dab40a4fff8c76f587ec6b2b3df8843dce.json","evidence":{"source_url":"https://doi.org/10.1088/1402-4896/abafcb"},"id":"literature:doi:10.1088/1402-4896/abafcb","identifiers":{"doi":"10.1088/1402-4896/abafcb"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1402-4896/abafcb","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1402-4896%2Fabafcb"},{"aliases":["10.1088/1402-4896/acbcf7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0a73059fa15938ec79351e2240ea1ce4d6f940aa3b77d78cf547bc87b8e8a07.json","evidence":{"source_url":"https://doi.org/10.1088/1402-4896/acbcf7"},"id":"literature:doi:10.1088/1402-4896/acbcf7","identifiers":{"doi":"10.1088/1402-4896/acbcf7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1402-4896/acbcf7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1402-4896%2Facbcf7"},{"aliases":["10.1088/1402-4896/adcc66"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad78d8f32b275dc91644dc3c5d992bdada28095cf6799bbafe9236a712d59917.json","evidence":{"source_url":"https://doi.org/10.1088/1402-4896/adcc66"},"id":"literature:doi:10.1088/1402-4896/adcc66","identifiers":{"doi":"10.1088/1402-4896/adcc66"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1402-4896/adcc66","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1402-4896%2Fadcc66"},{"aliases":["10.1088/1402-4896/adda94"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55171870eb2cf47311cf4306f581df8bca2a2a01605ae5e01df3834283cf5457.json","evidence":{"source_url":"https://doi.org/10.1088/1402-4896/adda94"},"id":"literature:doi:10.1088/1402-4896/adda94","identifiers":{"doi":"10.1088/1402-4896/adda94"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1402-4896/adda94","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1402-4896%2Fadda94"},{"aliases":["10.1088/1402-4896/ae9148"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/188cc0ab22cd550e26485de68f619dab15a6c351ba9c982e5112a80c725c369d.json","evidence":{"source_url":"https://doi.org/10.1088/1402-4896/ae9148"},"id":"literature:doi:10.1088/1402-4896/ae9148","identifiers":{"doi":"10.1088/1402-4896/ae9148"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1402-4896/ae9148","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1402-4896%2Fae9148"},{"aliases":["10.1088/1572-9494/acc92f"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6adf34f50b8ae9b3b1988dfedd33a780808d6eea5187e04309ba3896f62a0d4b.json","evidence":{"source_url":"https://doi.org/10.1088/1572-9494/acc92f"},"id":"literature:doi:10.1088/1572-9494/acc92f","identifiers":{"doi":"10.1088/1572-9494/acc92f"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1572-9494/acc92f","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1572-9494%2Facc92f"},{"aliases":["10.1088/1572-9494/adbdbd"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85cc7572336ace2318bdf23dcd45dd21cd4143cef528bc2afcac6216d84f2ade.json","evidence":{"source_url":"https://doi.org/10.1088/1572-9494/adbdbd"},"id":"literature:doi:10.1088/1572-9494/adbdbd","identifiers":{"doi":"10.1088/1572-9494/adbdbd"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1572-9494/adbdbd","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1572-9494%2Fadbdbd"},{"aliases":["10.1088/1742-5468/ab3721"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52f0495c133b80f76a58cef8a3940d5bc2e344c954b8478114083e8c9dd6464e.json","evidence":{"source_url":"https://doi.org/10.1088/1742-5468/ab3721"},"id":"literature:doi:10.1088/1742-5468/ab3721","identifiers":{"doi":"10.1088/1742-5468/ab3721"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-5468/ab3721","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-5468%2Fab3721"},{"aliases":["10.1088/1742-5468/ab5d08"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b011676c7a0eb5ff8be79a5c27a634930cb3107ad7d22aec53e807314cd9af3.json","evidence":{"source_url":"https://doi.org/10.1088/1742-5468/ab5d08"},"id":"literature:doi:10.1088/1742-5468/ab5d08","identifiers":{"doi":"10.1088/1742-5468/ab5d08"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-5468/ab5d08","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-5468%2Fab5d08"},{"aliases":["10.1088/1742-5468/acdce7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2dc715005dc9d606e61b3bd173f955f67f84f6bb04eebf9cd8a0beff874ebda1.json","evidence":{"source_url":"https://doi.org/10.1088/1742-5468/acdce7"},"id":"literature:doi:10.1088/1742-5468/acdce7","identifiers":{"doi":"10.1088/1742-5468/acdce7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-5468/acdce7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-5468%2Facdce7"},{"aliases":["10.1088/1742-5468/ae2092"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b99805385e66e6c4e1211e702d1c7baa2109bf648228b89681d059650d6baa7.json","evidence":{"source_url":"https://doi.org/10.1088/1742-5468/ae2092"},"id":"literature:doi:10.1088/1742-5468/ae2092","identifiers":{"doi":"10.1088/1742-5468/ae2092"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-5468/ae2092","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-5468%2Fae2092"},{"aliases":["10.1088/1742-5468/ae80b5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b5d643ad85846d83d082447a352fa04eb7800c164f11ba3a6ee20efd4fd1770f.json","evidence":{"source_url":"https://doi.org/10.1088/1742-5468/ae80b5"},"id":"literature:doi:10.1088/1742-5468/ae80b5","identifiers":{"doi":"10.1088/1742-5468/ae80b5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-5468/ae80b5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-5468%2Fae80b5"},{"aliases":["10.1088/1742-6596/2912/1/012043"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e30658955649e7463b5363570a299d148fe8e57e7271c2cffcb6ce51e431f9f6.json","evidence":{"source_url":"https://doi.org/10.1088/1742-6596/2912/1/012043"},"id":"literature:doi:10.1088/1742-6596/2912/1/012043","identifiers":{"doi":"10.1088/1742-6596/2912/1/012043"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1742-6596/2912/1/012043","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1742-6596%2F2912%2F1%2F012043"},{"aliases":["10.1088/1751-8113/42/24/245303","Man-Duen Choi and Nathaniel Johnston and David W. Kribs"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef4522e1bf7d6adadb49a7583b418fafbef3ed64ac8a63d81bbd1cfde019e37a.json","authors":"Man-Duen Choi and Nathaniel Johnston and David W. Kribs","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/42/24/245303"},"id":"literature:doi:10.1088/1751-8113/42/24/245303","identifiers":{"doi":"10.1088/1751-8113/42/24/245303"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The multiplicative domain in quantum error correction","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F42%2F24%2F245303","year":2009},{"aliases":["10.1088/1751-8113/43/4/045003"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb5d359d82d3416070ecef245aa03c3c3413a54fe9d5511446e973b725a58f5f.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/43/4/045003"},"id":"literature:doi:10.1088/1751-8113/43/4/045003","identifiers":{"doi":"10.1088/1751-8113/43/4/045003"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/43/4/045003","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F43%2F4%2F045003"},{"aliases":["10.1088/1751-8113/45/36/365201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce05a51894e8a1ae823908729275b463672047969748746f3d8ef63944320c8f.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/45/36/365201"},"id":"literature:doi:10.1088/1751-8113/45/36/365201","identifiers":{"doi":"10.1088/1751-8113/45/36/365201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/45/36/365201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F45%2F36%2F365201"},{"aliases":["10.1088/1751-8113/46/44/445001"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/126b46fb9fd01f2904a9ea0c98fb3f03e345623366f7161df30fef9068dc7d9a.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/46/44/445001"},"id":"literature:doi:10.1088/1751-8113/46/44/445001","identifiers":{"doi":"10.1088/1751-8113/46/44/445001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/46/44/445001","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F46%2F44%2F445001"},{"aliases":["10.1088/1751-8113/47/49/495302"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8971a57b9eaa48e39fa06e0622d82b50389834c2ee0c66bf78075aa9ebc69146.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/47/49/495302"},"id":"literature:doi:10.1088/1751-8113/47/49/495302","identifiers":{"doi":"10.1088/1751-8113/47/49/495302"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/47/49/495302","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F47%2F49%2F495302"},{"aliases":["10.1088/1751-8113/48/39/395303"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1fb652eca75e14ed23f3d12c295c3e710a9df74b0092bf888457b20164ec7e4.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/48/39/395303"},"id":"literature:doi:10.1088/1751-8113/48/39/395303","identifiers":{"doi":"10.1088/1751-8113/48/39/395303"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/48/39/395303","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F48%2F39%2F395303"},{"aliases":["10.1088/1751-8113/49/14/143001","John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, Paul Skrzypczyk"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef485727ef06c93ca1f08627896d9927b12a94ff29ad0d81d9af854b4e64ff36.json","authors":"John Goold, Marcus Huber, Arnau Riera, Lídia del Rio, Paul Skrzypczyk","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/49/14/143001"},"id":"literature:doi:10.1088/1751-8113/49/14/143001","identifiers":{"doi":"10.1088/1751-8113/49/14/143001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The role of quantum information in thermodynamics — a topical review","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F49%2F14%2F143001","year":2016},{"aliases":["10.1088/1751-8113/49/9/095201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/21d247f7894a72917a257ad7b5b203478c1a1af7e3e37fd99255ae464cff5d25.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8113/49/9/095201"},"id":"literature:doi:10.1088/1751-8113/49/9/095201","identifiers":{"doi":"10.1088/1751-8113/49/9/095201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8113/49/9/095201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8113%2F49%2F9%2F095201"},{"aliases":["10.1088/1751-8121/aa5840"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5341860330b1944e43e0d5fe3194937b4a6d313de55f5020c8d9c8b86667f828.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/aa5840"},"id":"literature:doi:10.1088/1751-8121/aa5840","identifiers":{"doi":"10.1088/1751-8121/aa5840"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/aa5840","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Faa5840"},{"aliases":["10.1088/1751-8121/aac8e8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b79e7f21d7ab9cfd418af8bd3160e8ba91b0c3e2c2ae6d3bb3be5a25c803eeab.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/aac8e8"},"id":"literature:doi:10.1088/1751-8121/aac8e8","identifiers":{"doi":"10.1088/1751-8121/aac8e8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/aac8e8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Faac8e8"},{"aliases":["10.1088/1751-8121/aad3dc"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/431c8f1a6fce07f5079159676e3ff88906d352a94ed0d112644d095c7279d3c8.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/aad3dc"},"id":"literature:doi:10.1088/1751-8121/aad3dc","identifiers":{"doi":"10.1088/1751-8121/aad3dc"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/aad3dc","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Faad3dc"},{"aliases":["10.1088/1751-8121/ab0774"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85e32f8a2217ec01d0fd0351edc2820fb25175e7e41ab1263e51cfecddb98a4b.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ab0774"},"id":"literature:doi:10.1088/1751-8121/ab0774","identifiers":{"doi":"10.1088/1751-8121/ab0774"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ab0774","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fab0774"},{"aliases":["10.1088/1751-8121/ab30f7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3e1bdfaccdca511df4602f5096da0532f28454779306707645f89617f2ca05b2.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ab30f7"},"id":"literature:doi:10.1088/1751-8121/ab30f7","identifiers":{"doi":"10.1088/1751-8121/ab30f7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ab30f7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fab30f7"},{"aliases":["10.1088/1751-8121/ab8d52"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a8e766c12b95675773fb9acd5f90bfbd21416cba84677ea24cc1468e1a09b4e1.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ab8d52"},"id":"literature:doi:10.1088/1751-8121/ab8d52","identifiers":{"doi":"10.1088/1751-8121/ab8d52"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ab8d52","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fab8d52"},{"aliases":["10.1088/1751-8121/ac37e5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/393ec04f9f55970ad8aebcc8c9ee23f2167f69ee97ad215399fb0971f78c600f.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ac37e5"},"id":"literature:doi:10.1088/1751-8121/ac37e5","identifiers":{"doi":"10.1088/1751-8121/ac37e5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ac37e5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fac37e5"},{"aliases":["10.1088/1751-8121/ac6d2c"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff91914c1d26147002e3dfd4832b1db9a091d98a37018f94e71ace7ae7837a4a.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ac6d2c"},"id":"literature:doi:10.1088/1751-8121/ac6d2c","identifiers":{"doi":"10.1088/1751-8121/ac6d2c"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ac6d2c","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fac6d2c"},{"aliases":["10.1088/1751-8121/ac7017"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9776c803f03320cf77a0efe43a107a9e87c9ae0ee556ae49c0199d9ec887e659.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ac7017"},"id":"literature:doi:10.1088/1751-8121/ac7017","identifiers":{"doi":"10.1088/1751-8121/ac7017"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ac7017","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fac7017"},{"aliases":["10.1088/1751-8121/acaa16"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c43afefc16a6d431a00463bbf27b5cee698314b15e4c00738054bd742e22c9b.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/acaa16"},"id":"literature:doi:10.1088/1751-8121/acaa16","identifiers":{"doi":"10.1088/1751-8121/acaa16"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/acaa16","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Facaa16"},{"aliases":["10.1088/1751-8121/ace56d"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2373e881d11691c5a6e35ab18821682527048527c6db4776a31b8395c7c4d4e9.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ace56d"},"id":"literature:doi:10.1088/1751-8121/ace56d","identifiers":{"doi":"10.1088/1751-8121/ace56d"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ace56d","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Face56d"},{"aliases":["10.1088/1751-8121/ad5edd"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c6935203610c859c983576cf88d319a7f466085e84125d636ee4f2e8dd66117b.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ad5edd"},"id":"literature:doi:10.1088/1751-8121/ad5edd","identifiers":{"doi":"10.1088/1751-8121/ad5edd"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ad5edd","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fad5edd"},{"aliases":["10.1088/1751-8121/ad6653"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e9a4481751d066e644338d7551e01e747524449f89971eef6f741ea552398dbb.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ad6653"},"id":"literature:doi:10.1088/1751-8121/ad6653","identifiers":{"doi":"10.1088/1751-8121/ad6653"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ad6653","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fad6653"},{"aliases":["10.1088/1751-8121/ad7dc2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad6fbe219ba5b2d4b07fc668a2d757c4ce5a228b3c324c8fe8ba225a2dfb31fb.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ad7dc2"},"id":"literature:doi:10.1088/1751-8121/ad7dc2","identifiers":{"doi":"10.1088/1751-8121/ad7dc2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ad7dc2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fad7dc2"},{"aliases":["10.1088/1751-8121/ad8607"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/866ecf9f29faaa9bf3dcab7c34f4ac772708e24e47708bd2b11001f0efcb11f5.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ad8607"},"id":"literature:doi:10.1088/1751-8121/ad8607","identifiers":{"doi":"10.1088/1751-8121/ad8607"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ad8607","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fad8607"},{"aliases":["10.1088/1751-8121/adc645"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2b8e9efd365dee4bee54f988d9e33175556157672c889e1ec018699c8a0781e.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/adc645"},"id":"literature:doi:10.1088/1751-8121/adc645","identifiers":{"doi":"10.1088/1751-8121/adc645"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/adc645","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fadc645"},{"aliases":["10.1088/1751-8121/ae05da"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/75afe9c6a9b3ac591505549f52fd4510bd7771bae658421d4d0ac01a00748d09.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ae05da"},"id":"literature:doi:10.1088/1751-8121/ae05da","identifiers":{"doi":"10.1088/1751-8121/ae05da"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ae05da","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fae05da"},{"aliases":["10.1088/1751-8121/ae1644"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5239c573bbe52dad7910494578f3da2d040716cd7f25af4e172e516814ba7364.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ae1644"},"id":"literature:doi:10.1088/1751-8121/ae1644","identifiers":{"doi":"10.1088/1751-8121/ae1644"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ae1644","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fae1644"},{"aliases":["10.1088/1751-8121/ae58d2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ddf188c67babbcde4f6fd47ca568d507344c48aa9fe87e2e45569942d235658e.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ae58d2"},"id":"literature:doi:10.1088/1751-8121/ae58d2","identifiers":{"doi":"10.1088/1751-8121/ae58d2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ae58d2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fae58d2"},{"aliases":["10.1088/1751-8121/ae7683"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a3a49aac15606684fb17c72c1f7f5a58c311db1e54d15f8ba8efa405a31ff17.json","evidence":{"source_url":"https://doi.org/10.1088/1751-8121/ae7683"},"id":"literature:doi:10.1088/1751-8121/ae7683","identifiers":{"doi":"10.1088/1751-8121/ae7683"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/1751-8121/ae7683","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F1751-8121%2Fae7683"},{"aliases":["10.1088/2058-9565/adcbce"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e2c31261f6862d151c20c4cc370ee2f9450fd74a9fe06b54ad2e0b20a9690660.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/adcbce"},"id":"literature:doi:10.1088/2058-9565/adcbce","identifiers":{"doi":"10.1088/2058-9565/adcbce"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/adcbce","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fadcbce"},{"aliases":["10.1088/2058-9565/adfd0d"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44f170361f6e7eac11278b9b8f9f284c191d9867e7caee3721b428b8d5f2fe80.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/adfd0d"},"id":"literature:doi:10.1088/2058-9565/adfd0d","identifiers":{"doi":"10.1088/2058-9565/adfd0d"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/adfd0d","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fadfd0d"},{"aliases":["10.1088/2058-9565/ae2200"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b7e6b6d353103a888150e40d3784f69358be0ce587ce796c448abdfc5159e7ed.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/ae2200"},"id":"literature:doi:10.1088/2058-9565/ae2200","identifiers":{"doi":"10.1088/2058-9565/ae2200"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/ae2200","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fae2200"},{"aliases":["10.1088/2058-9565/ae30a3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/03df5882b340b0cffde5e98767051874446d5e2c0d86dbc4f750457a935827b1.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/ae30a3"},"id":"literature:doi:10.1088/2058-9565/ae30a3","identifiers":{"doi":"10.1088/2058-9565/ae30a3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/ae30a3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fae30a3"},{"aliases":["10.1088/2058-9565/ae4d80"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/765d08058751922390fa50185c93a28da15bd01833eddfa4a58cda576aabd932.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/ae4d80"},"id":"literature:doi:10.1088/2058-9565/ae4d80","identifiers":{"doi":"10.1088/2058-9565/ae4d80"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/ae4d80","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fae4d80"},{"aliases":["10.1088/2058-9565/ae6bb1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/840a44d70ab44c78996ef4cb86e9877b6b68d314de2a0e271a19109ce2f6a456.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/ae6bb1"},"id":"literature:doi:10.1088/2058-9565/ae6bb1","identifiers":{"doi":"10.1088/2058-9565/ae6bb1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/ae6bb1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Fae6bb1"},{"aliases":["10.1088/2058-9565/aea92d"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ad1dd81c2a010f111d4ee4dfa200e984edb69014de632efc7e9a0b4b6af9c971.json","evidence":{"source_url":"https://doi.org/10.1088/2058-9565/aea92d"},"id":"literature:doi:10.1088/2058-9565/aea92d","identifiers":{"doi":"10.1088/2058-9565/aea92d"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2058-9565/aea92d","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2058-9565%2Faea92d"},{"aliases":["10.1088/2632-072x/aeb36a"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4df25ac1853b46e76fd1a8361b75595628a11d9b26d7070bd63eecf1ab42f95.json","evidence":{"source_url":"https://doi.org/10.1088/2632-072x/aeb36a"},"id":"literature:doi:10.1088/2632-072x/aeb36a","identifiers":{"doi":"10.1088/2632-072x/aeb36a"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1088/2632-072x/aeb36a","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1088%2F2632-072x%2Faeb36a"},{"aliases":["10.1090/conm/609/12101"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d7b34333130bc287836325c4de692183b669f060cd6887c15668ef694494b68.json","evidence":{"source_url":"https://doi.org/10.1090/conm/609/12101"},"id":"literature:doi:10.1090/conm/609/12101","identifiers":{"doi":"10.1090/conm/609/12101"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/conm/609/12101","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fconm%2F609%2F12101"},{"aliases":["10.1090/proc/14433"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce6eff7602ced3834f3303cf9f392f67f078f9f2acac463f8153375f13855179.json","evidence":{"source_url":"https://doi.org/10.1090/proc/14433"},"id":"literature:doi:10.1090/proc/14433","identifiers":{"doi":"10.1090/proc/14433"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/proc/14433","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fproc%2F14433"},{"aliases":["10.1090/proc/17480"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/754fe6629eebc03adf88616242dd3a817f7f2c28f6b08c0e20eeb32ba6ceadd0.json","evidence":{"source_url":"https://doi.org/10.1090/proc/17480"},"id":"literature:doi:10.1090/proc/17480","identifiers":{"doi":"10.1090/proc/17480"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/proc/17480","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fproc%2F17480"},{"aliases":["10.1090/proc/17861"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ae85615e7498cff9f3200153a392bddb6212668befe28fa95867ccfa97df9ad9.json","evidence":{"source_url":"https://doi.org/10.1090/proc/17861"},"id":"literature:doi:10.1090/proc/17861","identifiers":{"doi":"10.1090/proc/17861"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/proc/17861","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fproc%2F17861"},{"aliases":["10.1090/proc/17897"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a672e35b2fba169c25b2dadf0ee4814af47f054adeb70e1b68097f37d13ac152.json","evidence":{"source_url":"https://doi.org/10.1090/proc/17897"},"id":"literature:doi:10.1090/proc/17897","identifiers":{"doi":"10.1090/proc/17897"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/proc/17897","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fproc%2F17897"},{"aliases":["10.1090/s0002-9904-1949-09315-1","H. Chatland"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dea95d817757ac37446ae6ac279380ff690dead2a3f0d5dcb91c374699a4ff92.json","authors":"H. Chatland","evidence":{"source_url":"https://doi.org/10.1090/s0002-9904-1949-09315-1"},"id":"literature:doi:10.1090/s0002-9904-1949-09315-1","identifiers":{"doi":"10.1090/s0002-9904-1949-09315-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Euclidean algorithm in quadratic number fields","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fs0002-9904-1949-09315-1","year":1949},{"aliases":["10.1090/s0002-9947-1913-1500940-6","T. H. Gronwall"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16a688b5289a1361904a0359af364f5bb2794a9c911e23883fd979e9a10a45ad.json","authors":"T. H. Gronwall","evidence":{"source_url":"https://doi.org/10.1090/s0002-9947-1913-1500940-6"},"id":"literature:doi:10.1090/s0002-9947-1913-1500940-6","identifiers":{"doi":"10.1090/s0002-9947-1913-1500940-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some asymptotic expressions in the theory of numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fs0002-9947-1913-1500940-6","year":1913},{"aliases":["10.1090/s0002-9947-1936-1501858-0","Alonzo Church and J. B. Rosser"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/672d021ba9bdf3c971606525e09a8c522a0bfd4704413a6ab797a20c5ef19e2c.json","authors":"Alonzo Church and J. B. Rosser","evidence":{"source_url":"https://doi.org/10.1090/s0002-9947-1936-1501858-0"},"id":"literature:doi:10.1090/s0002-9947-1936-1501858-0","identifiers":{"doi":"10.1090/s0002-9947-1936-1501858-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Some Properties of Conversion","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fs0002-9947-1936-1501858-0","year":1936},{"aliases":["10.1090/s0002-9947-1953-0053041-6","Henry Gordon Rice"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b58104952ef8e2eb79c697e16fd5068e264056aced6630664aa80cc35ac29ca1.json","authors":"Henry Gordon Rice","evidence":{"source_url":"https://doi.org/10.1090/s0002-9947-1953-0053041-6"},"id":"literature:doi:10.1090/s0002-9947-1953-0053041-6","identifiers":{"doi":"10.1090/s0002-9947-1953-0053041-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Classes of recursively enumerable sets and their decision problems","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fs0002-9947-1953-0053041-6","year":1953},{"aliases":["10.1090/s1061-0022-2011-01148-3"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f8cff6a9c5579a524c1b07882b28ef9927157f588469264e287c4c7ac555838.json","evidence":{"source_url":"https://doi.org/10.1090/s1061-0022-2011-01148-3"},"id":"literature:doi:10.1090/s1061-0022-2011-01148-3","identifiers":{"doi":"10.1090/s1061-0022-2011-01148-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1090/s1061-0022-2011-01148-3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1090%2Fs1061-0022-2011-01148-3"},{"aliases":["10.1103/1qyd-59z7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/517f42099ac4121dca424e5ca3ea3edea8f1aaec3166c3cbd2a67086f577c20e.json","evidence":{"source_url":"https://doi.org/10.1103/1qyd-59z7"},"id":"literature:doi:10.1103/1qyd-59z7","identifiers":{"doi":"10.1103/1qyd-59z7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/1qyd-59z7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F1qyd-59z7"},{"aliases":["10.1103/1sd2-rxmy"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/531a35dd529cee92d760898a2aa9bfef4a03938fdd92771fac30f2f09341a67f.json","evidence":{"source_url":"https://doi.org/10.1103/1sd2-rxmy"},"id":"literature:doi:10.1103/1sd2-rxmy","identifiers":{"doi":"10.1103/1sd2-rxmy"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/1sd2-rxmy","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F1sd2-rxmy"},{"aliases":["10.1103/21vm-n8gp"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc808b923e0c19402a46a452eb6346057376e02e110a8a5a6d8e3c4e15cc664a.json","evidence":{"source_url":"https://doi.org/10.1103/21vm-n8gp"},"id":"literature:doi:10.1103/21vm-n8gp","identifiers":{"doi":"10.1103/21vm-n8gp"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/21vm-n8gp","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F21vm-n8gp"},{"aliases":["10.1103/6kv3-hr92"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2d5012556dff313c94e8a4299a8aef25fedd5f62acb87de5b3a74361a2a20b96.json","evidence":{"source_url":"https://doi.org/10.1103/6kv3-hr92"},"id":"literature:doi:10.1103/6kv3-hr92","identifiers":{"doi":"10.1103/6kv3-hr92"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/6kv3-hr92","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F6kv3-hr92"},{"aliases":["10.1103/82gn-78lv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/394071555bc4810a0779897dec2e6b44b1eefea8fbbce9e4c3eae09d69ab4755.json","evidence":{"source_url":"https://doi.org/10.1103/82gn-78lv"},"id":"literature:doi:10.1103/82gn-78lv","identifiers":{"doi":"10.1103/82gn-78lv"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/82gn-78lv","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F82gn-78lv"},{"aliases":["10.1103/9fkf-hm8l"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4166845714dbae1e7476d636655702de16de82643f65573c2fe6396d5c2446a8.json","evidence":{"source_url":"https://doi.org/10.1103/9fkf-hm8l"},"id":"literature:doi:10.1103/9fkf-hm8l","identifiers":{"doi":"10.1103/9fkf-hm8l"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/9fkf-hm8l","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2F9fkf-hm8l"},{"aliases":["10.1103/bz3j-9njg"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46f46313e13ead298b43f8daa32fe205135c687d0036e2e328c206b527d807ef.json","evidence":{"source_url":"https://doi.org/10.1103/bz3j-9njg"},"id":"literature:doi:10.1103/bz3j-9njg","identifiers":{"doi":"10.1103/bz3j-9njg"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/bz3j-9njg","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fbz3j-9njg"},{"aliases":["10.1103/cw25-pqx1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b2d1af54377c9ce9ea9d67607bdbac404ed67955c8e6876d21cb6a6af151f4e.json","evidence":{"source_url":"https://doi.org/10.1103/cw25-pqx1"},"id":"literature:doi:10.1103/cw25-pqx1","identifiers":{"doi":"10.1103/cw25-pqx1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/cw25-pqx1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fcw25-pqx1"},{"aliases":["10.1103/fj8w-v1bb"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1c9d6c6ea76b00c91ced5e432f4e8558f18393d60aa629e4ca7c5c73d882583.json","evidence":{"source_url":"https://doi.org/10.1103/fj8w-v1bb"},"id":"literature:doi:10.1103/fj8w-v1bb","identifiers":{"doi":"10.1103/fj8w-v1bb"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/fj8w-v1bb","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Ffj8w-v1bb"},{"aliases":["10.1103/gmjw-rb13"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4e54683550628c2c8a6630c3137b19c665c12d267702b270192bfab6b126d2a.json","evidence":{"source_url":"https://doi.org/10.1103/gmjw-rb13"},"id":"literature:doi:10.1103/gmjw-rb13","identifiers":{"doi":"10.1103/gmjw-rb13"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/gmjw-rb13","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fgmjw-rb13"},{"aliases":["10.1103/grdp-jph9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4883cb1cf3514f381bcca34dabf6d0b583269b4a33e1ea61e031b38b19f2165.json","evidence":{"source_url":"https://doi.org/10.1103/grdp-jph9"},"id":"literature:doi:10.1103/grdp-jph9","identifiers":{"doi":"10.1103/grdp-jph9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/grdp-jph9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fgrdp-jph9"},{"aliases":["10.1103/jtlj-qs3y"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/320896d2aa8bbea7d68b4a418547dfff8c83a418e4f245464ec49d45968d06c5.json","evidence":{"source_url":"https://doi.org/10.1103/jtlj-qs3y"},"id":"literature:doi:10.1103/jtlj-qs3y","identifiers":{"doi":"10.1103/jtlj-qs3y"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/jtlj-qs3y","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fjtlj-qs3y"},{"aliases":["10.1103/physrev.34.163","H. P. Robertson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b7348bce1c6b607083f2c86b02598208c41e070e340ff42726e0aa082ab11156.json","authors":"H. P. Robertson","evidence":{"source_url":"https://doi.org/10.1103/physrev.34.163"},"id":"literature:doi:10.1103/physrev.34.163","identifiers":{"doi":"10.1103/physrev.34.163"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Uncertainty Principle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrev.34.163","year":1929},{"aliases":["10.1103/physrev.75.486","Freeman J. Dyson"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81eb2c39c46e23c1d7579b6d18e99371051e8ff5605fef0988c0af4298e5960b.json","authors":"Freeman J. Dyson","evidence":{"source_url":"https://doi.org/10.1103/physrev.75.486"},"id":"literature:doi:10.1103/physrev.75.486","identifiers":{"doi":"10.1103/physrev.75.486"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Radiation Theories of Tomonaga, Schwinger, and Feynman","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrev.75.486","year":1949},{"aliases":["10.1103/physreva.100.032324"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c1e5ac6db44bdc84b25c19b1b3ff19f06c1c8b6ff465230df38e98ecbc27e83.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.100.032324"},"id":"literature:doi:10.1103/physreva.100.032324","identifiers":{"doi":"10.1103/physreva.100.032324"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.100.032324","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.100.032324"},{"aliases":["10.1103/physreva.100.042103"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d4d7b94769e7bae2fa8c427f70d6e2e7c4ec62b17ce2cc1e113ab0aff428df6.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.100.042103"},"id":"literature:doi:10.1103/physreva.100.042103","identifiers":{"doi":"10.1103/physreva.100.042103"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.100.042103","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.100.042103"},{"aliases":["10.1103/physreva.100.062327"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/202d714fd27fae1849af019f5f9c7867ee85ae68489b8e500c158ab2364ab929.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.100.062327"},"id":"literature:doi:10.1103/physreva.100.062327","identifiers":{"doi":"10.1103/physreva.100.062327"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.100.062327","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.100.062327"},{"aliases":["10.1103/physreva.100.062329"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/41499644a646e1c47b29d99fbb56e1bed3025d2d6918e19a891367baa3e245a8.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.100.062329"},"id":"literature:doi:10.1103/physreva.100.062329","identifiers":{"doi":"10.1103/physreva.100.062329"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.100.062329","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.100.062329"},{"aliases":["10.1103/physreva.101.062110"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/282ae1b9eadeee950225666620d288ad8973be4fd334dcf9088fd0c0d3aad8f5.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.101.062110"},"id":"literature:doi:10.1103/physreva.101.062110","identifiers":{"doi":"10.1103/physreva.101.062110"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.101.062110","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.101.062110"},{"aliases":["10.1103/physreva.101.062323"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cd2463447548df626e8c76e9276632a96ed4763db79e9341b343016a2b2d199.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.101.062323"},"id":"literature:doi:10.1103/physreva.101.062323","identifiers":{"doi":"10.1103/physreva.101.062323"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.101.062323","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.101.062323"},{"aliases":["10.1103/physreva.102.042404"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dfbc637efcfb1da1f18a1583e33401560fb48198cb4c271e2708d2daee672e42.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.102.042404"},"id":"literature:doi:10.1103/physreva.102.042404","identifiers":{"doi":"10.1103/physreva.102.042404"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.102.042404","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.102.042404"},{"aliases":["10.1103/physreva.102.042416"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24fdfd4c959cac9f9320f89cfabc82822811edcc545791219970db0c648eb29b.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.102.042416"},"id":"literature:doi:10.1103/physreva.102.042416","identifiers":{"doi":"10.1103/physreva.102.042416"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.102.042416","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.102.042416"},{"aliases":["10.1103/physreva.103.022203"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0ad9523dbdef6c38ab8645fba01e9c8e1037f3b2d4e7afbf8ea69ebffa1b2721.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.103.022203"},"id":"literature:doi:10.1103/physreva.103.022203","identifiers":{"doi":"10.1103/physreva.103.022203"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.103.022203","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.103.022203"},{"aliases":["10.1103/physreva.103.042426"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/271ba6439c9e22d519855027c8aadc7c2c1e3c845e400b7cef8a61ef2bcad60f.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.103.042426"},"id":"literature:doi:10.1103/physreva.103.042426","identifiers":{"doi":"10.1103/physreva.103.042426"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.103.042426","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.103.042426"},{"aliases":["10.1103/physreva.103.052422"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/100098ab97a62ff5ec59511cfabc77d5b8901df0d61b33b57a8433b16fd8c69a.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.103.052422"},"id":"literature:doi:10.1103/physreva.103.052422","identifiers":{"doi":"10.1103/physreva.103.052422"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.103.052422","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.103.052422"},{"aliases":["10.1103/physreva.104.022206"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d257a0951dab43c570b9653a759fa44bbe35bfdb1d407a9176aef752d4a9dab3.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.104.022206"},"id":"literature:doi:10.1103/physreva.104.022206","identifiers":{"doi":"10.1103/physreva.104.022206"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.104.022206","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.104.022206"},{"aliases":["10.1103/physreva.104.032414"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b21f88f941380f010e0f154acc0cd27d332e0ce81ed25fe3a24b218158baf716.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.104.032414"},"id":"literature:doi:10.1103/physreva.104.032414","identifiers":{"doi":"10.1103/physreva.104.032414"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.104.032414","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.104.032414"},{"aliases":["10.1103/physreva.104.042420"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e7f0f760de2cf0607023e61d0b6b12215e212356068319084c0e273d83a4c2bd.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.104.042420"},"id":"literature:doi:10.1103/physreva.104.042420","identifiers":{"doi":"10.1103/physreva.104.042420"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.104.042420","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.104.042420"},{"aliases":["10.1103/physreva.104.042423"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6d2ef587823e126f1e79ae4722aee7fb8b9a46daa0f3822e8ea3af09523c5ec0.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.104.042423"},"id":"literature:doi:10.1103/physreva.104.042423","identifiers":{"doi":"10.1103/physreva.104.042423"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.104.042423","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.104.042423"},{"aliases":["10.1103/physreva.104.052216"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af5c354769183fcade2d07ca477208aef0b4894cb2a89ec1c8e6f45d701d9e9d.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.104.052216"},"id":"literature:doi:10.1103/physreva.104.052216","identifiers":{"doi":"10.1103/physreva.104.052216"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.104.052216","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.104.052216"},{"aliases":["10.1103/physreva.105.022419"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a65bf96b1d0c2daeedb0726c6caf0b0e9f8efd08f8ae9a05cb04f9c8c616a27.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.105.022419"},"id":"literature:doi:10.1103/physreva.105.022419","identifiers":{"doi":"10.1103/physreva.105.022419"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.105.022419","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.105.022419"},{"aliases":["10.1103/physreva.106.012606"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/948d0a78df79c88a1adb1f18d5d78673ff2a93bfe4af38ba9f81ad7c1fc2fdad.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.106.012606"},"id":"literature:doi:10.1103/physreva.106.012606","identifiers":{"doi":"10.1103/physreva.106.012606"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.106.012606","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.106.012606"},{"aliases":["10.1103/physreva.106.022218"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0612d0d9cf9c42f10a9e486ea8c6df737e44534b955a363116b1648e4a930b70.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.106.022218"},"id":"literature:doi:10.1103/physreva.106.022218","identifiers":{"doi":"10.1103/physreva.106.022218"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.106.022218","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.106.022218"},{"aliases":["10.1103/physreva.106.052417"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/54f85d8c88233e21f75999eca54731ba12272f727db63ffbf1b925aa7bfff639.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.106.052417"},"id":"literature:doi:10.1103/physreva.106.052417","identifiers":{"doi":"10.1103/physreva.106.052417"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.106.052417","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.106.052417"},{"aliases":["10.1103/physreva.106.062413"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc715cfbf081335cc63914a14a79dfe00824e115b093410ecb9ef40d710a84c9.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.106.062413"},"id":"literature:doi:10.1103/physreva.106.062413","identifiers":{"doi":"10.1103/physreva.106.062413"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.106.062413","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.106.062413"},{"aliases":["10.1103/physreva.107.022412"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9202d90307e8223d4897372b6f78043be2d51a78fcf1ef0d30b046a84bc7eb3f.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.107.022412"},"id":"literature:doi:10.1103/physreva.107.022412","identifiers":{"doi":"10.1103/physreva.107.022412"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.107.022412","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.107.022412"},{"aliases":["10.1103/physreva.107.052405"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5257733ce701376f82f525695cd2a83fd94c1156cb582fdb6c95aaaad8c19996.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.107.052405"},"id":"literature:doi:10.1103/physreva.107.052405","identifiers":{"doi":"10.1103/physreva.107.052405"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.107.052405","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.107.052405"},{"aliases":["10.1103/physreva.107.l040401"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d658f27328d2994c913b75f9cd89cbbcfe86b0cb5847bf6a8e54c555b3a277d1.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.107.l040401"},"id":"literature:doi:10.1103/physreva.107.l040401","identifiers":{"doi":"10.1103/physreva.107.l040401"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.107.l040401","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.107.l040401"},{"aliases":["10.1103/physreva.109.022205"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b5ae89f582cb2529a1f34a190b479d9fdfbec9c4cea9f83edc45c763e427557.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.109.022205"},"id":"literature:doi:10.1103/physreva.109.022205","identifiers":{"doi":"10.1103/physreva.109.022205"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.109.022205","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.109.022205"},{"aliases":["10.1103/physreva.109.042402"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6356e218a65036e52ba725548edc1b13c7f8c827b548d11826e4577b35243b6e.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.109.042402"},"id":"literature:doi:10.1103/physreva.109.042402","identifiers":{"doi":"10.1103/physreva.109.042402"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.109.042402","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.109.042402"},{"aliases":["10.1103/physreva.109.052424"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bcbaffa0ce5418df304adb451fd473179914dd9b504731456f544923092a8b17.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.109.052424"},"id":"literature:doi:10.1103/physreva.109.052424","identifiers":{"doi":"10.1103/physreva.109.052424"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.109.052424","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.109.052424"},{"aliases":["10.1103/physreva.109.052620"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a5fad35ae668a7cb2f7c847dec3db50d6650b7c7ae2cb060fdbf727ce972d83e.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.109.052620"},"id":"literature:doi:10.1103/physreva.109.052620","identifiers":{"doi":"10.1103/physreva.109.052620"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.109.052620","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.109.052620"},{"aliases":["10.1103/physreva.110.022211"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e2580fb9949f629be9cea095beb286918e01c5eef1622ee028b9fe6ff149c036.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.022211"},"id":"literature:doi:10.1103/physreva.110.022211","identifiers":{"doi":"10.1103/physreva.110.022211"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.110.022211","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.022211"},{"aliases":["10.1103/physreva.110.022435"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5db8325a8e502896fed46f11face8fdbecfa10c0a9eab2a05ca29389d093212.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.022435"},"id":"literature:doi:10.1103/physreva.110.022435","identifiers":{"doi":"10.1103/physreva.110.022435"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.110.022435","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.022435"},{"aliases":["10.1103/physreva.110.042430"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/489f8681576035a06cef51cc303a8f694e8b85189c2843aa81db696b736bc6e4.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.042430"},"id":"literature:doi:10.1103/physreva.110.042430","identifiers":{"doi":"10.1103/physreva.110.042430"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.110.042430","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.042430"},{"aliases":["10.1103/physreva.110.052438","David Raveh and Rafael I. Nepomechie"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa4c412995e7a42d6135f18db637a9bc1e4e68a68819698a32cd299663c6973b.json","authors":"David Raveh and Rafael I. Nepomechie","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.052438"},"id":"literature:doi:10.1103/physreva.110.052438","identifiers":{"doi":"10.1103/physreva.110.052438"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Dicke states as matrix product states","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.052438","year":2024},{"aliases":["10.1103/physreva.110.062417"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d57b5dc38cbd1f8500a388a2cc2bf88d4a639484e6db0ea289ff0ba7e1dd8390.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.062417"},"id":"literature:doi:10.1103/physreva.110.062417","identifiers":{"doi":"10.1103/physreva.110.062417"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.110.062417","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.062417"},{"aliases":["10.1103/physreva.110.l060201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7f5e0d63290171103464a907289c2ca41e4899736b70788764ba2fed1bcb334.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.110.l060201"},"id":"literature:doi:10.1103/physreva.110.l060201","identifiers":{"doi":"10.1103/physreva.110.l060201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.110.l060201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.110.l060201"},{"aliases":["10.1103/physreva.111.022407"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9946f184dd9f8034798254d9775b508e849e67182e399f680d527f6c03c6f71d.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.111.022407"},"id":"literature:doi:10.1103/physreva.111.022407","identifiers":{"doi":"10.1103/physreva.111.022407"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.111.022407","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.111.022407"},{"aliases":["10.1103/physreva.111.042418"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d87c7d375183e96c0a8857110bafc0e46c478ab8ac9bb7dcec8a6165fc0ba8f0.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.111.042418"},"id":"literature:doi:10.1103/physreva.111.042418","identifiers":{"doi":"10.1103/physreva.111.042418"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.111.042418","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.111.042418"},{"aliases":["10.1103/physreva.111.052443"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af8f7ec2a857ee7f1564d771cd3990921d9c98221a61f8255c3db84fe3e77a3b.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.111.052443"},"id":"literature:doi:10.1103/physreva.111.052443","identifiers":{"doi":"10.1103/physreva.111.052443"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.111.052443","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.111.052443"},{"aliases":["10.1103/physreva.111.062401"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/484bcf857104624c5d219d29730295eef5ff57b3fb1e4d9fa41b73c66af47d5b.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.111.062401"},"id":"literature:doi:10.1103/physreva.111.062401","identifiers":{"doi":"10.1103/physreva.111.062401"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.111.062401","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.111.062401"},{"aliases":["10.1103/physreva.55.900","E. Knill and R. Laflamme"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db8e817977b8a4267d3972276cc7eddd1d9f3210fd998b9c0b75212465cb96da.json","authors":"E. Knill and R. Laflamme","evidence":{"source_url":"https://doi.org/10.1103/physreva.55.900"},"id":"literature:doi:10.1103/physreva.55.900","identifiers":{"doi":"10.1103/physreva.55.900"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Theory of quantum error-correcting codes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.55.900","year":1997},{"aliases":["10.1103/physreva.65.052302"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3fffd0d03b144a59533460b6c172d71f199c2ef29a4fb18e046cd9d2012f976.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.65.052302"},"id":"literature:doi:10.1103/physreva.65.052302","identifiers":{"doi":"10.1103/physreva.65.052302"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.65.052302","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.65.052302"},{"aliases":["10.1103/physreva.74.022316"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9107487a4685780895c12438cb2ce2246bbb15d7c26154183d49c3305da3b20b.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.74.022316"},"id":"literature:doi:10.1103/physreva.74.022316","identifiers":{"doi":"10.1103/physreva.74.022316"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.74.022316","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.74.022316"},{"aliases":["10.1103/physreva.76.022336"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce51d5a7a61fd3b3998c4569ff8d4d8642bb5f4ba14694d25a9e6618f74be969.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.76.022336"},"id":"literature:doi:10.1103/physreva.76.022336","identifiers":{"doi":"10.1103/physreva.76.022336"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.76.022336","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.76.022336"},{"aliases":["10.1103/physreva.76.042303","Cédric Bény and Achim Kempf and David W. Kribs"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bd85840c679fc5f7c3c63c8b701d640c3f1be784e677d1a5639e678bd4d55cd.json","authors":"Cédric Bény and Achim Kempf and David W. Kribs","evidence":{"source_url":"https://doi.org/10.1103/physreva.76.042303"},"id":"literature:doi:10.1103/physreva.76.042303","identifiers":{"doi":"10.1103/physreva.76.042303"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Error Correction of Observables","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.76.042303","year":2007},{"aliases":["10.1103/physreva.79.042308"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a01433474018704b5ff11252fef9df40f187b58f6c0ced662e24adef848a649f.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.79.042308"},"id":"literature:doi:10.1103/physreva.79.042308","identifiers":{"doi":"10.1103/physreva.79.042308"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.79.042308","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.79.042308"},{"aliases":["10.1103/physreva.79.052311"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2e8dd625588b9b9d987e3a1dd1280515e8f31173a782095eb1b21f4dc9374845.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.79.052311"},"id":"literature:doi:10.1103/physreva.79.052311","identifiers":{"doi":"10.1103/physreva.79.052311"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.79.052311","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.79.052311"},{"aliases":["10.1103/physreva.82.042314"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/039dd3548649bafe219833504857c2ca9d9b7d96f733e66fddd7c107b23b6dcf.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.82.042314"},"id":"literature:doi:10.1103/physreva.82.042314","identifiers":{"doi":"10.1103/physreva.82.042314"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.82.042314","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.82.042314"},{"aliases":["10.1103/physreva.85.052113"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f05139107405988de5259fadecc89721df45c04113b37da4ba296016b1145b6.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.85.052113"},"id":"literature:doi:10.1103/physreva.85.052113","identifiers":{"doi":"10.1103/physreva.85.052113"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.85.052113","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.85.052113"},{"aliases":["10.1103/physreva.89.052335"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6fad4da65e4bbf13e8a385ea706d2db6c810dd55bd80898aff99bac3c630a87f.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.89.052335"},"id":"literature:doi:10.1103/physreva.89.052335","identifiers":{"doi":"10.1103/physreva.89.052335"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.89.052335","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.89.052335"},{"aliases":["10.1103/physreva.90.024302"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1fc546c91deee0d0155f0f48158e3b3a401e589f1e2cbf5fdf5662a1bfba48c8.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.90.024302"},"id":"literature:doi:10.1103/physreva.90.024302","identifiers":{"doi":"10.1103/physreva.90.024302"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.90.024302","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.90.024302"},{"aliases":["10.1103/physreva.93.012110"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c91e9c13e66ad6053f176149e395827e1f2e22b27e9acd5fd9fce65c7f077df7.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.93.012110"},"id":"literature:doi:10.1103/physreva.93.012110","identifiers":{"doi":"10.1103/physreva.93.012110"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.93.012110","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.93.012110"},{"aliases":["10.1103/physreva.93.032114"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85e25b43b5d7f1c6c6abd441697a0f0bf5c0d03a006f0e6c2607636f6142642d.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.93.032114"},"id":"literature:doi:10.1103/physreva.93.032114","identifiers":{"doi":"10.1103/physreva.93.032114"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.93.032114","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.93.032114"},{"aliases":["10.1103/physreva.93.052323"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2aa3d963591669d716acf91036c787a81f9138297b1962dba4b6d9302f872d64.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.93.052323"},"id":"literature:doi:10.1103/physreva.93.052323","identifiers":{"doi":"10.1103/physreva.93.052323"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.93.052323","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.93.052323"},{"aliases":["10.1103/physreva.94.022341"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32b3e73c16876c4017c0696752202df318ac52cb003599c1f2deee69e2bbda6d.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.94.022341"},"id":"literature:doi:10.1103/physreva.94.022341","identifiers":{"doi":"10.1103/physreva.94.022341"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.94.022341","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.94.022341"},{"aliases":["10.1103/physreva.94.042105"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be1ecba77f8efefbc0947cb7f9736b7feba6495f3cb7feabbf1a45e115c81e60.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.94.042105"},"id":"literature:doi:10.1103/physreva.94.042105","identifiers":{"doi":"10.1103/physreva.94.042105"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.94.042105","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.94.042105"},{"aliases":["10.1103/physreva.95.012133"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4554c8c35830c6b6e115ad46cad466f11aae97ce91cdd04a4d969ed99f9bdaa.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.95.012133"},"id":"literature:doi:10.1103/physreva.95.012133","identifiers":{"doi":"10.1103/physreva.95.012133"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.95.012133","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.95.012133"},{"aliases":["10.1103/physreva.95.042337"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6fe2933ac81cbe7f5770993f82c81d9d3356c2f4a52d70220bc3fdac465045b1.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.95.042337"},"id":"literature:doi:10.1103/physreva.95.042337","identifiers":{"doi":"10.1103/physreva.95.042337"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.95.042337","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.95.042337"},{"aliases":["10.1103/physreva.95.052315"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8b38706f0b8a05c15f8fe1bc861cf42f03f3f796760dd83cafc56febf1bf5a5e.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.95.052315"},"id":"literature:doi:10.1103/physreva.95.052315","identifiers":{"doi":"10.1103/physreva.95.052315"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.95.052315","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.95.052315"},{"aliases":["10.1103/physreva.96.012128"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c7a145caba897fc235b602560c5ba755c8117b192e436c4b43b31f6f80c00962.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.96.012128"},"id":"literature:doi:10.1103/physreva.96.012128","identifiers":{"doi":"10.1103/physreva.96.012128"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.96.012128","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.96.012128"},{"aliases":["10.1103/physreva.98.052333"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b9d41039aaa2d1da3e4d3d4d7cd505350bc37f510d0fe1da33ba9298c72fe43.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.98.052333"},"id":"literature:doi:10.1103/physreva.98.052333","identifiers":{"doi":"10.1103/physreva.98.052333"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.98.052333","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.98.052333"},{"aliases":["10.1103/physreva.98.062107"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/454faa2cb3bb073307d81fd4dba2e4686aa37beb565a6292f1d5d52b2937c394.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.98.062107"},"id":"literature:doi:10.1103/physreva.98.062107","identifiers":{"doi":"10.1103/physreva.98.062107"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.98.062107","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.98.062107"},{"aliases":["10.1103/physreva.99.032312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/333de53d8259252bbabc53f33f687f76117bcd43d5074af3f0df5ce12eadbf57.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.99.032312"},"id":"literature:doi:10.1103/physreva.99.032312","identifiers":{"doi":"10.1103/physreva.99.032312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.99.032312","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.99.032312"},{"aliases":["10.1103/physreva.99.042319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7376da5fabd6071619b1fc797de758d6d9a5733e655db099acc32ad63ae74ab8.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.99.042319"},"id":"literature:doi:10.1103/physreva.99.042319","identifiers":{"doi":"10.1103/physreva.99.042319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.99.042319","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.99.042319"},{"aliases":["10.1103/physreva.99.062312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e0304a3a0dbb18a5414f090d72bfbd935bff9bc55a82faf05da1cf2638d407c.json","evidence":{"source_url":"https://doi.org/10.1103/physreva.99.062312"},"id":"literature:doi:10.1103/physreva.99.062312","identifiers":{"doi":"10.1103/physreva.99.062312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreva.99.062312","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreva.99.062312"},{"aliases":["10.1103/physrevapplied.23.034079"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9168558a7b0e1f813d81f8b23a3a5e0b3a8977b0b6ed6dbf3f54989ebd6a231e.json","evidence":{"source_url":"https://doi.org/10.1103/physrevapplied.23.034079"},"id":"literature:doi:10.1103/physrevapplied.23.034079","identifiers":{"doi":"10.1103/physrevapplied.23.034079"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevapplied.23.034079","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevapplied.23.034079"},{"aliases":["10.1103/physrevb.104.155117"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c834a8c69662523df10a9af41e1e8b2858aa6c5b3cf0731bf54e82cec215853b.json","evidence":{"source_url":"https://doi.org/10.1103/physrevb.104.155117"},"id":"literature:doi:10.1103/physrevb.104.155117","identifiers":{"doi":"10.1103/physrevb.104.155117"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevb.104.155117","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevb.104.155117"},{"aliases":["10.1103/physrevb.107.045412"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51c80e816648684275b4899583b23a1de99df4f35c8dcdf8d2535615c2a1bb2d.json","evidence":{"source_url":"https://doi.org/10.1103/physrevb.107.045412"},"id":"literature:doi:10.1103/physrevb.107.045412","identifiers":{"doi":"10.1103/physrevb.107.045412"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevb.107.045412","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevb.107.045412"},{"aliases":["10.1103/physrevb.99.045155"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9b1e2fb290128cfb6e0356c9493407cdcf25e9def5839c3b0bbc4f3ec617869.json","evidence":{"source_url":"https://doi.org/10.1103/physrevb.99.045155"},"id":"literature:doi:10.1103/physrevb.99.045155","identifiers":{"doi":"10.1103/physrevb.99.045155"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevb.99.045155","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevb.99.045155"},{"aliases":["10.1103/physrevd.103.026020"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b94436f369eabc021379b4420f48a8f8275a893bf7c5213c831375bc8e6f5ad8.json","evidence":{"source_url":"https://doi.org/10.1103/physrevd.103.026020"},"id":"literature:doi:10.1103/physrevd.103.026020","identifiers":{"doi":"10.1103/physrevd.103.026020"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevd.103.026020","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevd.103.026020"},{"aliases":["10.1103/physreve.100.062139"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/33284994dd1928374a4a8c343a7f52d6923f6cf65f294ce615ed990a6ff787a5.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.100.062139"},"id":"literature:doi:10.1103/physreve.100.062139","identifiers":{"doi":"10.1103/physreve.100.062139"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.100.062139","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.100.062139"},{"aliases":["10.1103/physreve.106.024119"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/067d8b17a4b34f638a5ca16bca9622c2cdab09b0441064ce64c697ef3fecf066.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.106.024119"},"id":"literature:doi:10.1103/physreve.106.024119","identifiers":{"doi":"10.1103/physreve.106.024119"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.106.024119","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.106.024119"},{"aliases":["10.1103/physreve.106.l022101"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01c886f9a6ad51d4d3f2bbf6a8b8b5f0425b7aecbdfce7204e7ffef03135a0d7.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.106.l022101"},"id":"literature:doi:10.1103/physreve.106.l022101","identifiers":{"doi":"10.1103/physreve.106.l022101"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.106.l022101","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.106.l022101"},{"aliases":["10.1103/physreve.108.054119"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/631869aedd41ce2bd1c22f16ad1f16be5f828f58bc6b9faeb51d3e0ff146d539.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.108.054119"},"id":"literature:doi:10.1103/physreve.108.054119","identifiers":{"doi":"10.1103/physreve.108.054119"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.108.054119","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.108.054119"},{"aliases":["10.1103/physreve.108.l052103"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd7e6987fbcf0f58281a7e686a6638e754fe081d3d07442e2208a70062a4faa0.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.108.l052103"},"id":"literature:doi:10.1103/physreve.108.l052103","identifiers":{"doi":"10.1103/physreve.108.l052103"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.108.l052103","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.108.l052103"},{"aliases":["10.1103/physreve.109.044138"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f2fdcf54e1c02a317b1d77b8c9fae520f1aea458781079576628ae61f60679e.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.109.044138"},"id":"literature:doi:10.1103/physreve.109.044138","identifiers":{"doi":"10.1103/physreve.109.044138"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.109.044138","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.109.044138"},{"aliases":["10.1103/physreve.109.l042101"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f0204e298cc7a72bfcd3f42be45fc0df311e0a54d92495a0f9cab7bca16b360b.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.109.l042101"},"id":"literature:doi:10.1103/physreve.109.l042101","identifiers":{"doi":"10.1103/physreve.109.l042101"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.109.l042101","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.109.l042101"},{"aliases":["10.1103/physreve.93.062310"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d4a0e69d6e5cd25c301d1d60b43cb7e5e06d5a106d641b886a2d3c708c2c58a.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.93.062310"},"id":"literature:doi:10.1103/physreve.93.062310","identifiers":{"doi":"10.1103/physreve.93.062310"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.93.062310","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.93.062310"},{"aliases":["10.1103/physreve.97.032136"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/712f1e6d8f4de6d6424748ec7f9f0dd23691733ed29ddc29908b6989b4672666.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.97.032136"},"id":"literature:doi:10.1103/physreve.97.032136","identifiers":{"doi":"10.1103/physreve.97.032136"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.97.032136","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.97.032136"},{"aliases":["10.1103/physreve.99.022201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d59fc339e9331521f87f74c018db266a0dc0e3ab67f83c535b270d3dfbe805ae.json","evidence":{"source_url":"https://doi.org/10.1103/physreve.99.022201"},"id":"literature:doi:10.1103/physreve.99.022201","identifiers":{"doi":"10.1103/physreve.99.022201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physreve.99.022201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysreve.99.022201"},{"aliases":["10.1103/physrevlett.115.188103"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/298a0c38d4db17b2007ecb01dcc3cd88a6305f178b1f73717e1c257cfeb3b0df.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.115.188103"},"id":"literature:doi:10.1103/physrevlett.115.188103","identifiers":{"doi":"10.1103/physrevlett.115.188103"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.115.188103","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.115.188103"},{"aliases":["10.1103/physrevlett.118.080402"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/042cf5c727167a95392b955746923b48b7c5bbc35e596aa36a834bc06e274633.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.118.080402"},"id":"literature:doi:10.1103/physrevlett.118.080402","identifiers":{"doi":"10.1103/physrevlett.118.080402"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.118.080402","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.118.080402"},{"aliases":["10.1103/physrevlett.128.250501"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b40dce0523c2c0e359127e1ae7e1871193052307e264011d6c92d92478dbc9ef.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.128.250501"},"id":"literature:doi:10.1103/physrevlett.128.250501","identifiers":{"doi":"10.1103/physrevlett.128.250501"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.128.250501","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.128.250501"},{"aliases":["10.1103/physrevlett.133.110202"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/98ae599c65024955a11ec6bf8e90e6d00ce50055d173ae949babe8f40f4c54a3.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.133.110202"},"id":"literature:doi:10.1103/physrevlett.133.110202","identifiers":{"doi":"10.1103/physrevlett.133.110202"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.133.110202","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.133.110202"},{"aliases":["10.1103/physrevlett.133.230201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/487934bd8e50d2eed0471976598272cd3854a807d2f64e4c790404b1845d6dbc.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.133.230201"},"id":"literature:doi:10.1103/physrevlett.133.230201","identifiers":{"doi":"10.1103/physrevlett.133.230201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.133.230201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.133.230201"},{"aliases":["10.1103/physrevlett.133.240201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04716e9ade9af5dbc4239310cee091bb3615684ad2fda3c5cc9e4bb56f664da5.json","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.133.240201"},"id":"literature:doi:10.1103/physrevlett.133.240201","identifiers":{"doi":"10.1103/physrevlett.133.240201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevlett.133.240201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.133.240201"},{"aliases":["10.1103/physrevlett.83.648","Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/031bf092ae75cfbb3fdaaaf5b2e4903b03110797c25225c9f79091a0fdff41f4.json","authors":"Richard Cleve, Daniel Gottesman, and Hoi-Kwong Lo","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.83.648"},"id":"literature:doi:10.1103/physrevlett.83.648","identifiers":{"doi":"10.1103/physrevlett.83.648"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"How to share a quantum secret","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.83.648","year":1999},{"aliases":["10.1103/physrevlett.98.080502","Samuel L. Braunstein and Arun K. Pati"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dab3861021644ce292833b13d2dad168b366dd38221e3ac96e344dc4aab370f3.json","authors":"Samuel L. Braunstein and Arun K. Pati","evidence":{"source_url":"https://doi.org/10.1103/physrevlett.98.080502"},"id":"literature:doi:10.1103/physrevlett.98.080502","identifiers":{"doi":"10.1103/physrevlett.98.080502"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum Information Cannot Be Completely Hidden in Correlations: Implications for the Black-Hole Information Paradox","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevlett.98.080502","year":2007},{"aliases":["10.1103/physrevresearch.2.013220"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97c91bb29f52eff05bd8fc0ea5d48390bf9dc3f0eb8887ac529ead6a0e485f15.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.2.013220"},"id":"literature:doi:10.1103/physrevresearch.2.013220","identifiers":{"doi":"10.1103/physrevresearch.2.013220"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.2.013220","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.2.013220"},{"aliases":["10.1103/physrevresearch.2.043122"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b9e7b26b2d525ca4f70b0d37ea3c347466ff45d3fcc916ed03ce2abf6966bb58.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.2.043122"},"id":"literature:doi:10.1103/physrevresearch.2.043122","identifiers":{"doi":"10.1103/physrevresearch.2.043122"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.2.043122","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.2.043122"},{"aliases":["10.1103/physrevresearch.3.023101"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/656827ace83187f9104e7172ed73ab129c11b01357c26fa291bbf70941e91341.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.3.023101"},"id":"literature:doi:10.1103/physrevresearch.3.023101","identifiers":{"doi":"10.1103/physrevresearch.3.023101"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.3.023101","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.3.023101"},{"aliases":["10.1103/physrevresearch.3.033028"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ef1b569e61d4df5fb1cfc424e40c3ec0b17daad073dc7d5d1dd7ea35a547a00a.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.3.033028"},"id":"literature:doi:10.1103/physrevresearch.3.033028","identifiers":{"doi":"10.1103/physrevresearch.3.033028"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.3.033028","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.3.033028"},{"aliases":["10.1103/physrevresearch.3.043073"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f4ee850b1e74ad61249eb1faebb87c7c55e4bccaa4eed01cd98964770bb29c6.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.3.043073"},"id":"literature:doi:10.1103/physrevresearch.3.043073","identifiers":{"doi":"10.1103/physrevresearch.3.043073"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.3.043073","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.3.043073"},{"aliases":["10.1103/physrevresearch.4.023041"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66acc047b35b64396df2410881183b3cfbf123200aab335efc2000ae98b567e7.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.4.023041"},"id":"literature:doi:10.1103/physrevresearch.4.023041","identifiers":{"doi":"10.1103/physrevresearch.4.023041"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.4.023041","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.4.023041"},{"aliases":["10.1103/physrevresearch.4.043070"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c41acaed44f6402c6d64e0985686fddf198543c11c41922ccea18277615a886.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.4.043070"},"id":"literature:doi:10.1103/physrevresearch.4.043070","identifiers":{"doi":"10.1103/physrevresearch.4.043070"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.4.043070","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.4.043070"},{"aliases":["10.1103/physrevresearch.5.023086"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66b149b808b5825d23c2b1f2891e3c0f758c376cbe1e8f4685079a9d6b4958f2.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.5.023086"},"id":"literature:doi:10.1103/physrevresearch.5.023086","identifiers":{"doi":"10.1103/physrevresearch.5.023086"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.5.023086","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.5.023086"},{"aliases":["10.1103/physrevresearch.5.043033"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/560cdf95cc402b38a4876fca8b8f3dc865aed10554a163e9b7b35946024ce70e.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.5.043033"},"id":"literature:doi:10.1103/physrevresearch.5.043033","identifiers":{"doi":"10.1103/physrevresearch.5.043033"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.5.043033","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.5.043033"},{"aliases":["10.1103/physrevresearch.6.033043"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f19414b206681d7c1bf810276d551c1e62498d401409bdf25238563cc598c475.json","evidence":{"source_url":"https://doi.org/10.1103/physrevresearch.6.033043"},"id":"literature:doi:10.1103/physrevresearch.6.033043","identifiers":{"doi":"10.1103/physrevresearch.6.033043"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/physrevresearch.6.033043","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fphysrevresearch.6.033043"},{"aliases":["10.1103/prxquantum.2.020334"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bb51fa7c693937bd1161ddf8822543da5c7c0bfea28c2ac48af0f3e95328855.json","evidence":{"source_url":"https://doi.org/10.1103/prxquantum.2.020334"},"id":"literature:doi:10.1103/prxquantum.2.020334","identifiers":{"doi":"10.1103/prxquantum.2.020334"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/prxquantum.2.020334","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fprxquantum.2.020334"},{"aliases":["10.1103/prxquantum.3.020307"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/64ffb8990e514bb8f590b6345a43f2c433830f11c4e487fc418c5617cf0a870e.json","evidence":{"source_url":"https://doi.org/10.1103/prxquantum.3.020307"},"id":"literature:doi:10.1103/prxquantum.3.020307","identifiers":{"doi":"10.1103/prxquantum.3.020307"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/prxquantum.3.020307","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fprxquantum.3.020307"},{"aliases":["10.1103/prxquantum.4.040304"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/27eeb2d15193ee66dc366016629873e208aaaaa5929d100fcca7aa77a78717d0.json","evidence":{"source_url":"https://doi.org/10.1103/prxquantum.4.040304"},"id":"literature:doi:10.1103/prxquantum.4.040304","identifiers":{"doi":"10.1103/prxquantum.4.040304"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/prxquantum.4.040304","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fprxquantum.4.040304"},{"aliases":["10.1103/prxquantum.5.020319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a871dba0fa2e1c6ed234255684342f7a5843aae60cb13469169db33e72e114a.json","evidence":{"source_url":"https://doi.org/10.1103/prxquantum.5.020319"},"id":"literature:doi:10.1103/prxquantum.5.020319","identifiers":{"doi":"10.1103/prxquantum.5.020319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/prxquantum.5.020319","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fprxquantum.5.020319"},{"aliases":["10.1103/qgcx-wxpd"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1dd9d4171edb278aafe4ae20e012bf16f42ff4a6903cc0416aade45134180f6a.json","evidence":{"source_url":"https://doi.org/10.1103/qgcx-wxpd"},"id":"literature:doi:10.1103/qgcx-wxpd","identifiers":{"doi":"10.1103/qgcx-wxpd"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/qgcx-wxpd","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fqgcx-wxpd"},{"aliases":["10.1103/revmodphys.75.715","Wojciech H. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Decoherence, einselection, and the quantum origins of the classical","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Frevmodphys.75.715","year":2003},{"aliases":["10.1103/rrmd-fqx6"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f0654064fee294ee2ea88c1f6643e8a794051238a32abbba7aa5d33653ecc17.json","evidence":{"source_url":"https://doi.org/10.1103/rrmd-fqx6"},"id":"literature:doi:10.1103/rrmd-fqx6","identifiers":{"doi":"10.1103/rrmd-fqx6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/rrmd-fqx6","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Frrmd-fqx6"},{"aliases":["10.1103/s6mc-lnvl"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aab706c9ada3dbf7d5fa4958c99279962813f9a13a4504e27adcbd2914272361.json","evidence":{"source_url":"https://doi.org/10.1103/s6mc-lnvl"},"id":"literature:doi:10.1103/s6mc-lnvl","identifiers":{"doi":"10.1103/s6mc-lnvl"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/s6mc-lnvl","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fs6mc-lnvl"},{"aliases":["10.1103/s9sp-2fd7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9daf5f4874caf124834715b63295a49c3c817e5c2a910a11e16d99fce294f89f.json","evidence":{"source_url":"https://doi.org/10.1103/s9sp-2fd7"},"id":"literature:doi:10.1103/s9sp-2fd7","identifiers":{"doi":"10.1103/s9sp-2fd7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/s9sp-2fd7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fs9sp-2fd7"},{"aliases":["10.1103/w561-h3z5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c18927b767790a5caef1ae2b5ec92c9a7f406090642d833e5eee65c492d3dcf7.json","evidence":{"source_url":"https://doi.org/10.1103/w561-h3z5"},"id":"literature:doi:10.1103/w561-h3z5","identifiers":{"doi":"10.1103/w561-h3z5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/w561-h3z5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fw561-h3z5"},{"aliases":["10.1103/z7jn-4try"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4783a7eb691b75ab58b27a0bb4172d3e5d9f01b1f1c9176f000d12d3789896ad.json","evidence":{"source_url":"https://doi.org/10.1103/z7jn-4try"},"id":"literature:doi:10.1103/z7jn-4try","identifiers":{"doi":"10.1103/z7jn-4try"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1103/z7jn-4try","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1103%2Fz7jn-4try"},{"aliases":["10.1109/isit44484.2020.9174425"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c65eb1fb193a613c88bc4540de325aac59e96306c720da069756b9df8164cff1.json","evidence":{"source_url":"https://doi.org/10.1109/isit44484.2020.9174425"},"id":"literature:doi:10.1109/isit44484.2020.9174425","identifiers":{"doi":"10.1109/isit44484.2020.9174425"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/isit44484.2020.9174425","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Fisit44484.2020.9174425"},{"aliases":["10.1109/isit54713.2023.10206941"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06a71ad26cd317f6b8447465963e79196806854a9013c822343d486b30648d61.json","evidence":{"source_url":"https://doi.org/10.1109/isit54713.2023.10206941"},"id":"literature:doi:10.1109/isit54713.2023.10206941","identifiers":{"doi":"10.1109/isit54713.2023.10206941"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/isit54713.2023.10206941","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Fisit54713.2023.10206941"},{"aliases":["10.1109/jsait.2025.3582156"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a5b75cd13162af2e0fd465812b8bb1bc99b16eb70e060f2e479224ffb0999fdc.json","evidence":{"source_url":"https://doi.org/10.1109/jsait.2025.3582156"},"id":"literature:doi:10.1109/jsait.2025.3582156","identifiers":{"doi":"10.1109/jsait.2025.3582156"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/jsait.2025.3582156","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Fjsait.2025.3582156"},{"aliases":["10.1109/lcsys.2023.3292049"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29931a64f1979960bac05060fe33bed9c4864566ee3d95ad81de801a72143ab5.json","evidence":{"source_url":"https://doi.org/10.1109/lcsys.2023.3292049"},"id":"literature:doi:10.1109/lcsys.2023.3292049","identifiers":{"doi":"10.1109/lcsys.2023.3292049"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/lcsys.2023.3292049","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Flcsys.2023.3292049"},{"aliases":["10.1109/tit.2005.851738"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f3ece1d6a523417a76e25f9e7ff8c1cdfed5b1507175faa74a0c1994d557c0c7.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2005.851738"},"id":"literature:doi:10.1109/tit.2005.851738","identifiers":{"doi":"10.1109/tit.2005.851738"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2005.851738","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2005.851738"},{"aliases":["10.1109/tit.2016.2641447"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4244e9d9036a00fc6bb59ad284067b755275414bce4c47d52fe0c9ef33c34e5.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2016.2641447"},"id":"literature:doi:10.1109/tit.2016.2641447","identifiers":{"doi":"10.1109/tit.2016.2641447"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2016.2641447","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2016.2641447"},{"aliases":["10.1109/tit.2022.3218540"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46e63ee8a94ff19c71b12aa5ffe5ced996acc80e74c5ada5706c63f0cbdf9532.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2022.3218540"},"id":"literature:doi:10.1109/tit.2022.3218540","identifiers":{"doi":"10.1109/tit.2022.3218540"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2022.3218540","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2022.3218540"},{"aliases":["10.1109/tit.2023.3274434"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca11e740a714d57e18096c63c89a3b26afddff2f8b6ef7d59a76707e88bf9f17.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2023.3274434"},"id":"literature:doi:10.1109/tit.2023.3274434","identifiers":{"doi":"10.1109/tit.2023.3274434"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2023.3274434","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2023.3274434"},{"aliases":["10.1109/tit.2024.3435412"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/efd5e7b47966ce20b7203d47b93240343d7cad815fd669d39e0d6f89a46e72f2.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2024.3435412"},"id":"literature:doi:10.1109/tit.2024.3435412","identifiers":{"doi":"10.1109/tit.2024.3435412"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2024.3435412","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2024.3435412"},{"aliases":["10.1109/tit.2025.3543276"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b285b93a1d7dae6dec626e8b7e04fb5f0acd530c6d45f7a0ef0d3fcc2cc53e46.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2025.3543276"},"id":"literature:doi:10.1109/tit.2025.3543276","identifiers":{"doi":"10.1109/tit.2025.3543276"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2025.3543276","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2025.3543276"},{"aliases":["10.1109/tit.2025.3581849"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c451dac8f8b902e48c3ec88e3f958d882fd2b9fe28cb4a99a4a83a303fd3c47.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2025.3581849"},"id":"literature:doi:10.1109/tit.2025.3581849","identifiers":{"doi":"10.1109/tit.2025.3581849"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2025.3581849","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2025.3581849"},{"aliases":["10.1109/tit.2026.3700031"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb06da880c004d3fbf9e7ef847e92e1b66d86231b38a62cdfc6567210c75c212.json","evidence":{"source_url":"https://doi.org/10.1109/tit.2026.3700031"},"id":"literature:doi:10.1109/tit.2026.3700031","identifiers":{"doi":"10.1109/tit.2026.3700031"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1109/tit.2026.3700031","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1109%2Ftit.2026.3700031"},{"aliases":["10.1112/s0010437x05001739","Yann Bugeaud, Maurice Mignotte, Samir Siksek"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52a8fa851d1ef9ec5decd2337ae027a25149ebf1b248efd83e9cceb48e1803e7.json","authors":"Yann Bugeaud, Maurice Mignotte, Samir Siksek","evidence":{"source_url":"https://doi.org/10.1112/s0010437x05001739"},"id":"literature:doi:10.1112/s0010437x05001739","identifiers":{"doi":"10.1112/s0010437x05001739"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Classical and modular approaches to exponential Diophantine equations II. The Lebesgue-Nagell equation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1112%2Fs0010437x05001739","year":2006},{"aliases":["10.1116/5.0137579"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6aedaa1590dc3fc22a5fa2757ef8e65a4c14cde279f202a7018f920589ac303f.json","evidence":{"source_url":"https://doi.org/10.1116/5.0137579"},"id":"literature:doi:10.1116/5.0137579","identifiers":{"doi":"10.1116/5.0137579"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1116/5.0137579","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1116%2F5.0137579"},{"aliases":["10.1126/sciadv.adq4467"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3a812e4875129fbe1552495ddc3fa63a514842e682dc25dc5e8994dac977056c.json","evidence":{"source_url":"https://doi.org/10.1126/sciadv.adq4467"},"id":"literature:doi:10.1126/sciadv.adq4467","identifiers":{"doi":"10.1126/sciadv.adq4467"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1126/sciadv.adq4467","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1126%2Fsciadv.adq4467"},{"aliases":["10.1134/s0081543811080098","Michael Baake; Ulrike Schlaegel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9e1fc8e90227de6f8a83e1c6858d40f6fd3a912d753032d977c457d9be576188.json","authors":"Michael Baake; Ulrike Schlaegel","evidence":{"source_url":"https://doi.org/10.1134/s0081543811080098"},"id":"literature:doi:10.1134/s0081543811080098","identifiers":{"doi":"10.1134/s0081543811080098"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Peano-Baker series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1134%2Fs0081543811080098","year":2011},{"aliases":["10.1134/s1055134424010012"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b066012bad33ff13606f25c06d6186a995e8c5adba5eb9b7cbeb49bed8c562b9.json","evidence":{"source_url":"https://doi.org/10.1134/s1055134424010012"},"id":"literature:doi:10.1134/s1055134424010012","identifiers":{"doi":"10.1134/s1055134424010012"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1134/s1055134424010012","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1134%2Fs1055134424010012"},{"aliases":["10.1137/19m1296525"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e0eabb68a7f45784b502424d0dc328472a6e1b6fd1ccf59eb385ddea4e12b8e.json","evidence":{"source_url":"https://doi.org/10.1137/19m1296525"},"id":"literature:doi:10.1137/19m1296525","identifiers":{"doi":"10.1137/19m1296525"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1137/19m1296525","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1137%2F19m1296525"},{"aliases":["10.1137/25m1760209"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70eb37198864edd5a27f7e3e963174a35d09d0a8d823e8ff9eddf7e050df8202.json","evidence":{"source_url":"https://doi.org/10.1137/25m1760209"},"id":"literature:doi:10.1137/25m1760209","identifiers":{"doi":"10.1137/25m1760209"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1137/25m1760209","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1137%2F25m1760209"},{"aliases":["10.1140/epjs/s11734-026-02447-z"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/19b0541005ff7d19e4ec3b789aa227f92170d323105f4173c3fac70d74ed091e.json","evidence":{"source_url":"https://doi.org/10.1140/epjs/s11734-026-02447-z"},"id":"literature:doi:10.1140/epjs/s11734-026-02447-z","identifiers":{"doi":"10.1140/epjs/s11734-026-02447-z"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1140/epjs/s11734-026-02447-z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1140%2Fepjs%2Fs11734-026-02447-z"},{"aliases":["10.1140/epjst/e2018-800068-8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4066c6dc3088be1114946018a4b778b1381bf0308e046afbeaa5b852f1ada216.json","evidence":{"source_url":"https://doi.org/10.1140/epjst/e2018-800068-8"},"id":"literature:doi:10.1140/epjst/e2018-800068-8","identifiers":{"doi":"10.1140/epjst/e2018-800068-8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1140/epjst/e2018-800068-8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1140%2Fepjst%2Fe2018-800068-8"},{"aliases":["10.1142/s0129167x09005509"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b2731e39b9db8c48895cbd8229e129d6e482ce0ba706dc0c6bf3c2070da16bd.json","evidence":{"source_url":"https://doi.org/10.1142/s0129167x09005509"},"id":"literature:doi:10.1142/s0129167x09005509","identifiers":{"doi":"10.1142/s0129167x09005509"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0129167x09005509","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0129167x09005509"},{"aliases":["10.1142/s0129167x22500094"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be533e7bf9c704375cf2c6cdf8d2117da5493ecaef39bec93cd8f7988ab7b47d.json","evidence":{"source_url":"https://doi.org/10.1142/s0129167x22500094"},"id":"literature:doi:10.1142/s0129167x22500094","identifiers":{"doi":"10.1142/s0129167x22500094"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0129167x22500094","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0129167x22500094"},{"aliases":["10.1142/s0217732319501025"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f4d72c5fa6892a5f2e2e16c93d3cc6e8ae9165cd93b6e17226c07d1eab7568d.json","evidence":{"source_url":"https://doi.org/10.1142/s0217732319501025"},"id":"literature:doi:10.1142/s0217732319501025","identifiers":{"doi":"10.1142/s0217732319501025"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0217732319501025","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0217732319501025"},{"aliases":["10.1142/s0217751x22500932"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69561cf690298880af8d5c97375894ad604402ac38993f7d865f6753509d6bc9.json","evidence":{"source_url":"https://doi.org/10.1142/s0217751x22500932"},"id":"literature:doi:10.1142/s0217751x22500932","identifiers":{"doi":"10.1142/s0217751x22500932"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0217751x22500932","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0217751x22500932"},{"aliases":["10.1142/s0217979216502568"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/487ebe0ae6fce5c55696c1250c101330577a3d687c1c82a141ef8f1b1f5ff78b.json","evidence":{"source_url":"https://doi.org/10.1142/s0217979216502568"},"id":"literature:doi:10.1142/s0217979216502568","identifiers":{"doi":"10.1142/s0217979216502568"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0217979216502568","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0217979216502568"},{"aliases":["10.1142/s0219199716500449"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66f02ef427f946b1a7a090cbc1a769e6ac11c231ba39672fcb0c18b453f9f5a7.json","evidence":{"source_url":"https://doi.org/10.1142/s0219199716500449"},"id":"literature:doi:10.1142/s0219199716500449","identifiers":{"doi":"10.1142/s0219199716500449"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219199716500449","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219199716500449"},{"aliases":["10.1142/s0219493721400013","Michael Baake, Natalie Priebe Frank, and Uwe Grimm"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb2dd76c844e11620ff071a0aabb2a8a9338880db71f7c90ee7e238a2cb8fddf.json","authors":"Michael Baake, Natalie Priebe Frank, and Uwe Grimm","evidence":{"source_url":"https://doi.org/10.1142/s0219493721400013"},"id":"literature:doi:10.1142/s0219493721400013","identifiers":{"doi":"10.1142/s0219493721400013"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219749919500461","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219749919500461"},{"aliases":["10.1142/s0219749920400055"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c147b9aa756ea147cc0071b51c7be142eb36d94758047a8139da348649eb50a.json","evidence":{"source_url":"https://doi.org/10.1142/s0219749920400055"},"id":"literature:doi:10.1142/s0219749920400055","identifiers":{"doi":"10.1142/s0219749920400055"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219749920400055","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219749920400055"},{"aliases":["10.1142/s0219749923500399"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95e3d8d9ca697d63774a8826bcbb57e098ebdf092213952a9fd69ae3519be2d6.json","evidence":{"source_url":"https://doi.org/10.1142/s0219749923500399"},"id":"literature:doi:10.1142/s0219749923500399","identifiers":{"doi":"10.1142/s0219749923500399"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219749923500399","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219749923500399"},{"aliases":["10.1142/s0219749924400033"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec001f94bcb1bc2e8a36298ea5be496cc348d002076b34f1089bb1b4ee630526.json","evidence":{"source_url":"https://doi.org/10.1142/s0219749924400033"},"id":"literature:doi:10.1142/s0219749924400033","identifiers":{"doi":"10.1142/s0219749924400033"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219749924400033","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219749924400033"},{"aliases":["10.1142/s0219749924500357"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3ad2641bfef1a912a75b93917ed29bc6d9aed21c492415fafa267c445d30568.json","evidence":{"source_url":"https://doi.org/10.1142/s0219749924500357"},"id":"literature:doi:10.1142/s0219749924500357","identifiers":{"doi":"10.1142/s0219749924500357"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s0219749924500357","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs0219749924500357"},{"aliases":["10.1142/s1230161214500139"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c27001960bf82d6c07d46c301c5bce4e7ea397d7e8c447c7583c2ec1deea1e64.json","evidence":{"source_url":"https://doi.org/10.1142/s1230161214500139"},"id":"literature:doi:10.1142/s1230161214500139","identifiers":{"doi":"10.1142/s1230161214500139"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s1230161214500139","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs1230161214500139"},{"aliases":["10.1142/s1230161224500124"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/291594dcdb2ee6496fcb0540b899858338dbcf3dc6cf9eb4eb182fd5755341c2.json","evidence":{"source_url":"https://doi.org/10.1142/s1230161224500124"},"id":"literature:doi:10.1142/s1230161224500124","identifiers":{"doi":"10.1142/s1230161224500124"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s1230161224500124","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs1230161224500124"},{"aliases":["10.1142/s1793042124500635"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/094b375ef0b55ed5e14233b31e71e8b84128bb19fd7610336ae6276d4338d4da.json","evidence":{"source_url":"https://doi.org/10.1142/s1793042124500635"},"id":"literature:doi:10.1142/s1793042124500635","identifiers":{"doi":"10.1142/s1793042124500635"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1142/s1793042124500635","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1142%2Fs1793042124500635"},{"aliases":["10.1143/jpsj.5.435","Tosio Kato"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/987a227fd511c92d35e9c4c563ca37b9350155ff5b44ae8e6aba8cb4b93ccd08.json","authors":"Tosio Kato","evidence":{"source_url":"https://doi.org/10.1143/jpsj.5.435"},"id":"literature:doi:10.1143/jpsj.5.435","identifiers":{"doi":"10.1143/jpsj.5.435"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the Adiabatic Theorem of Quantum Mechanics","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1143%2Fjpsj.5.435","year":1950},{"aliases":["10.1145/3655028"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be1f1c650124009f89f0e3651327cd28458c15f5e42450fe2b1d30ad4e19be11.json","evidence":{"source_url":"https://doi.org/10.1145/3655028"},"id":"literature:doi:10.1145/3655028","identifiers":{"doi":"10.1145/3655028"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1145/3655028","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1145%2F3655028"},{"aliases":["10.1155/2014/851213"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7bbc74ecc709024c050e80390385b0163d0c903e499007baa1aab5094ea5d3f0.json","evidence":{"source_url":"https://doi.org/10.1155/2014/851213"},"id":"literature:doi:10.1155/2014/851213","identifiers":{"doi":"10.1155/2014/851213"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1155/2014/851213","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1155%2F2014%2F851213"},{"aliases":["10.1155/2018/2507265"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ccb2e936b4317f4672ef744bf5de64802d6de6415f5d99a841627823a0cbacf0.json","evidence":{"source_url":"https://doi.org/10.1155/2018/2507265"},"id":"literature:doi:10.1155/2018/2507265","identifiers":{"doi":"10.1155/2018/2507265"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1155/2018/2507265","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1155%2F2018%2F2507265"},{"aliases":["10.1186/s13660-026-03505-9"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6348a42d1a442c8ad62a84884b58aa5e367982cf7123a37d8c2e4678199af18.json","evidence":{"source_url":"https://doi.org/10.1186/s13660-026-03505-9"},"id":"literature:doi:10.1186/s13660-026-03505-9","identifiers":{"doi":"10.1186/s13660-026-03505-9"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1186/s13660-026-03505-9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1186%2Fs13660-026-03505-9"},{"aliases":["10.1201/9780429262913-ch12","Peter Winkler"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b16b03d00da360be9ccc3fac6780214a9da7a11d75dba20e9d8434996b08a412.json","authors":"Peter Winkler","evidence":{"source_url":"https://doi.org/10.1201/9780429262913-ch12"},"id":"literature:doi:10.1201/9780429262913-ch12","identifiers":{"doi":"10.1201/9780429262913-ch12"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Pigeonhole Principle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1201%2F9780429262913-ch12","year":2020},{"aliases":["10.1201/9781003462002","Ian Stewart and David Tall"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff74bf1663b8793dbd887c034c5a00644633609c38ba1c06c1ee43016ca214b2.json","authors":"Ian Stewart and David Tall","evidence":{"source_url":"https://doi.org/10.1201/9781003462002"},"id":"literature:doi:10.1201/9781003462002","identifiers":{"doi":"10.1201/9781003462002"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Algebraic Number Theory and Fermat's Last Theorem","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1201%2F9781003462002","year":2025},{"aliases":["10.1214/17-ecp82"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c844dc3c098d38fa9e0eb440e949949c81850b75fed1fe4d66e1894b8fc4cb69.json","evidence":{"source_url":"https://doi.org/10.1214/17-ecp82"},"id":"literature:doi:10.1214/17-ecp82","identifiers":{"doi":"10.1214/17-ecp82"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1214/17-ecp82","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1214%2F17-ecp82"},{"aliases":["10.1214/23-aop1637"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7aec1628c793a53d0d5cb75d491adcf00c8c527c61e149a740c05ac2144d025c.json","evidence":{"source_url":"https://doi.org/10.1214/23-aop1637"},"id":"literature:doi:10.1214/23-aop1637","identifiers":{"doi":"10.1214/23-aop1637"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1214/23-aop1637","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1214%2F23-aop1637"},{"aliases":["10.1214/aop/1176995765","Janos Galambos"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1f2d1dfeb1a2a6a114b36ef25f3b2cc2fc190154304fd1e36a3add6cb2748531.json","authors":"Janos Galambos","evidence":{"source_url":"https://doi.org/10.1214/aop/1176995765"},"id":"literature:doi:10.1214/aop/1176995765","identifiers":{"doi":"10.1214/aop/1176995765"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Bonferroni Inequalities","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1214%2Faop%2F1176995765","year":1977},{"aliases":["10.1215/s0012-7094-97-08601-4","Hakan Hedenmalm, Peter Lindqvist, and Kristian Seip"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a5e72f2fb8c2d25928a4cf851df9cf346dbf60317ebe15bb63b7ccc480ef1fb2.json","authors":"Hakan Hedenmalm, Peter Lindqvist, and Kristian Seip","evidence":{"source_url":"https://doi.org/10.1215/s0012-7094-97-08601-4"},"id":"literature:doi:10.1215/s0012-7094-97-08601-4","identifiers":{"doi":"10.1215/s0012-7094-97-08601-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A Hilbert space of Dirichlet series and systems of dilated functions in L2(0,1)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1215%2Fs0012-7094-97-08601-4","year":1997},{"aliases":["10.1216/rmj-1985-15-2-353","Paul Erdős"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/134ccc354638d346ab81152f2577279dde5b130cec26c0a5733ae554970856c1.json","authors":"Paul Erdős","evidence":{"source_url":"https://doi.org/10.1216/rmj-1985-15-2-353"},"id":"literature:doi:10.1216/rmj-1985-15-2-353","identifiers":{"doi":"10.1216/rmj-1985-15-2-353"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Problems and results on consecutive integers and prime factors of binomial coefficients","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1216%2Frmj-1985-15-2-353","year":1985},{"aliases":["10.1216/rmj-2013-43-5-1707","Pieter Moree"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7ff0ae0bb39f065d683d2891f23116dec12520fa1c52e44a6ca344ff6a6ab87.json","authors":"Pieter Moree","evidence":{"source_url":"https://doi.org/10.1216/rmj-2013-43-5-1707"},"id":"literature:doi:10.1216/rmj-2013-43-5-1707","identifiers":{"doi":"10.1216/rmj-2013-43-5-1707"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Moser's mathemagical work on the equation 1^k + 2^k + ... + (m−1)^k = m^k","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1216%2Frmj-2013-43-5-1707","year":2013},{"aliases":["10.1287/moor.2019.0990"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28b41be8a8d61d9979e91552d3ee33723106ef0dade0052a85019f581cc6abe9.json","evidence":{"source_url":"https://doi.org/10.1287/moor.2019.0990"},"id":"literature:doi:10.1287/moor.2019.0990","identifiers":{"doi":"10.1287/moor.2019.0990"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1287/moor.2019.0990","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1287%2Fmoor.2019.0990"},{"aliases":["10.13001/1081-3810.3082"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff15d4600d1f7ff80c9efcf58ac0c958c096b90f8f65f08a15c7b0631f667033.json","evidence":{"source_url":"https://doi.org/10.13001/1081-3810.3082"},"id":"literature:doi:10.13001/1081-3810.3082","identifiers":{"doi":"10.13001/1081-3810.3082"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/1081-3810.3082","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2F1081-3810.3082"},{"aliases":["10.13001/1081-3810.3108"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06c2bd2a13320558251dd7c194cb76b13fab2be5d3a6b26bb195da902e85cd0e.json","evidence":{"source_url":"https://doi.org/10.13001/1081-3810.3108"},"id":"literature:doi:10.13001/1081-3810.3108","identifiers":{"doi":"10.13001/1081-3810.3108"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/1081-3810.3108","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2F1081-3810.3108"},{"aliases":["10.13001/ela.2022.5943"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f92f2ace219386c8f57a9e9c4e7f497a7f0da68c1d25dede42a2dce0720eef0f.json","evidence":{"source_url":"https://doi.org/10.13001/ela.2022.5943"},"id":"literature:doi:10.13001/ela.2022.5943","identifiers":{"doi":"10.13001/ela.2022.5943"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/ela.2022.5943","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2Fela.2022.5943"},{"aliases":["10.13001/ela.2022.6721"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a783bf4a9106a0b4e7475119cefdc3d086a2e9111ab92028c8aeecd540a35cf.json","evidence":{"source_url":"https://doi.org/10.13001/ela.2022.6721"},"id":"literature:doi:10.13001/ela.2022.6721","identifiers":{"doi":"10.13001/ela.2022.6721"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/ela.2022.6721","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2Fela.2022.6721"},{"aliases":["10.13001/ela.2024.8283"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5b4b0043c2bb0a164b0bd5ad857b4429b3839407458b29aa67754397c3ef541.json","evidence":{"source_url":"https://doi.org/10.13001/ela.2024.8283"},"id":"literature:doi:10.13001/ela.2024.8283","identifiers":{"doi":"10.13001/ela.2024.8283"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/ela.2024.8283","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2Fela.2024.8283"},{"aliases":["10.13001/ela.2025.9201"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6b9df0687157518d6b0d1bb434f1f649aefac0b9ed86cd5372d5845c3444bfe.json","evidence":{"source_url":"https://doi.org/10.13001/ela.2025.9201"},"id":"literature:doi:10.13001/ela.2025.9201","identifiers":{"doi":"10.13001/ela.2025.9201"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/ela.2025.9201","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2Fela.2025.9201"},{"aliases":["10.13001/ela.2026.10523"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5fc898189717e9b4ae0e5d81adf43a03419df397aa87385f32404a0dca8b7c54.json","evidence":{"source_url":"https://doi.org/10.13001/ela.2026.10523"},"id":"literature:doi:10.13001/ela.2026.10523","identifiers":{"doi":"10.13001/ela.2026.10523"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.13001/ela.2026.10523","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.13001%2Fela.2026.10523"},{"aliases":["10.1307/mmj/1028989917","Frank Harary"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6ac87049c4d122b2e0724399ae4b35cdb5254a63a4e7ca47232d7fac255f331e.json","authors":"Frank Harary","evidence":{"source_url":"https://doi.org/10.1307/mmj/1028989917"},"id":"literature:doi:10.1307/mmj/1028989917","identifiers":{"doi":"10.1307/mmj/1028989917"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the notion of balance of a signed graph","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1307%2Fmmj%2F1028989917","year":1953},{"aliases":["10.1364/aop.1.000238","Stephen M. Barnett and Sarah Croke"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9834817c58466601aab2a8648bd0a6a54d1e547f3b5ee2d3681f81d478c6b80.json","authors":"Stephen M. Barnett and Sarah Croke","evidence":{"source_url":"https://doi.org/10.1364/aop.1.000238"},"id":"literature:doi:10.1364/aop.1.000238","identifiers":{"doi":"10.1364/aop.1.000238"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Quantum state discrimination","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1364%2Faop.1.000238","year":2009},{"aliases":["10.1371/journal.pone.0297862"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/44822cca7d361370db6ebbc5a44579b8ca7bc9b8a7b17b467e40f056975074a3.json","evidence":{"source_url":"https://doi.org/10.1371/journal.pone.0297862"},"id":"literature:doi:10.1371/journal.pone.0297862","identifiers":{"doi":"10.1371/journal.pone.0297862"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.1371/journal.pone.0297862","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1371%2Fjournal.pone.0297862"},{"aliases":["10.1512/iumj.1957.6.56050","Andrew M. Gleason"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ee96c107aac8023857156e2080db0e22550552ff1da8bc9c5ec950802cad3af.json","authors":"Andrew M. Gleason","evidence":{"source_url":"https://doi.org/10.1512/iumj.1957.6.56050"},"id":"literature:doi:10.1512/iumj.1957.6.56050","identifiers":{"doi":"10.1512/iumj.1957.6.56050"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Measures on the Closed Subspaces of a Hilbert Space","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1512%2Fiumj.1957.6.56050","year":1957},{"aliases":["10.1515/9781400876457","David Vernon Widder"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2870b0841357ad3ea1ace47c4fec35cac8364a8a7d91115b28605ea31479bd95.json","authors":"David Vernon Widder","evidence":{"source_url":"https://doi.org/10.1515/9781400876457"},"id":"literature:doi:10.1515/9781400876457","identifiers":{"doi":"10.1515/9781400876457"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Laplace Transform","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1515%2F9781400876457","year":1941},{"aliases":["10.15381/pesquimat.v26i2.24218"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40da996977ec4a24fc86cee050cfc64262ce9a964447f469af23d93e7eb61ff7.json","evidence":{"source_url":"https://doi.org/10.15381/pesquimat.v26i2.24218"},"id":"literature:doi:10.15381/pesquimat.v26i2.24218","identifiers":{"doi":"10.15381/pesquimat.v26i2.24218"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.15381/pesquimat.v26i2.24218","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.15381%2Fpesquimat.v26i2.24218"},{"aliases":["10.1561/2400000003","Neal Parikh; Stephen Boyd"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bfee9ece5b102af048e121a99daae2f462e4d13aea78e105f2da25d5d05ccb66.json","authors":"Neal Parikh; Stephen Boyd","evidence":{"source_url":"https://doi.org/10.1561/2400000003"},"id":"literature:doi:10.1561/2400000003","identifiers":{"doi":"10.1561/2400000003"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Proximal Algorithms","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.1561%2F2400000003","year":2014},{"aliases":["10.15672/hujms.1564485","Radica Bojičić; M. D. Petković; Paul Barry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e72bbf6ab975718f25faa3ceec5f69b7ce3a39a0252b180b309a98123aab8be.json","authors":"Radica Bojičić; M. D. Petković; Paul Barry","evidence":{"source_url":"https://doi.org/10.15672/hujms.1564485"},"id":"literature:doi:10.15672/hujms.1564485","identifiers":{"doi":"10.15672/hujms.1564485"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Hankel transform of linear combination of three consecutive Catalan numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.15672%2Fhujms.1564485","year":2025},{"aliases":["10.15807/jorsj.67.84"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29bc358d1841b8f75516d0a57bef22f0c7a202463673d3edffd1b5f97e0874cc.json","evidence":{"source_url":"https://doi.org/10.15807/jorsj.67.84"},"id":"literature:doi:10.15807/jorsj.67.84","identifiers":{"doi":"10.15807/jorsj.67.84"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.15807/jorsj.67.84","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.15807%2Fjorsj.67.84"},{"aliases":["10.18452/33769"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7774cdb12f83b786cb798699e54eda43ad0b1faea4fce1bccf36349684f329c.json","evidence":{"source_url":"https://doi.org/10.18452/33769"},"id":"literature:doi:10.18452/33769","identifiers":{"doi":"10.18452/33769"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.18452/33769","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.18452%2F33769"},{"aliases":["10.2140/ent.2026.5.243"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71ab96d890af6622e8f14f70020aea0255afdba6f7c6d113441750aa631406f5.json","evidence":{"source_url":"https://doi.org/10.2140/ent.2026.5.243"},"id":"literature:doi:10.2140/ent.2026.5.243","identifiers":{"doi":"10.2140/ent.2026.5.243"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2140/ent.2026.5.243","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2140%2Fent.2026.5.243"},{"aliases":["10.2140/involve.2022.15.525"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e7711dda7210998eb5b1b2e54efbc9e4f80d80594df198c5c332ba0cb725a6d.json","evidence":{"source_url":"https://doi.org/10.2140/involve.2022.15.525"},"id":"literature:doi:10.2140/involve.2022.15.525","identifiers":{"doi":"10.2140/involve.2022.15.525"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2140/involve.2022.15.525","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2140%2Finvolve.2022.15.525"},{"aliases":["10.2140/involve.2026.19.443"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3685ab12672c8b85ed541340f4e0e05e07a9fa122abb2ca960cc8b5512b8cb25.json","evidence":{"source_url":"https://doi.org/10.2140/involve.2026.19.443"},"id":"literature:doi:10.2140/involve.2026.19.443","identifiers":{"doi":"10.2140/involve.2026.19.443"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2140/involve.2026.19.443","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2140%2Finvolve.2026.19.443"},{"aliases":["10.2140/moscow.2023.12.181"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/34058e6fb6ed254e376875118d74ef8908b34c10a49c87b1536148a82871ac4a.json","evidence":{"source_url":"https://doi.org/10.2140/moscow.2023.12.181"},"id":"literature:doi:10.2140/moscow.2023.12.181","identifiers":{"doi":"10.2140/moscow.2023.12.181"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2140/moscow.2023.12.181","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2140%2Fmoscow.2023.12.181"},{"aliases":["10.21468/scipostphys.16.5.129"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/005384b24355da1aacc20154b3b09f6e1f5f30a51fe06ff94c87ded433b2c57c.json","evidence":{"source_url":"https://doi.org/10.21468/scipostphys.16.5.129"},"id":"literature:doi:10.21468/scipostphys.16.5.129","identifiers":{"doi":"10.21468/scipostphys.16.5.129"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.21468/scipostphys.16.5.129","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.21468%2Fscipostphys.16.5.129"},{"aliases":["10.21468/scipostphys.17.5.132"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8af82cde70b5d13c76b525de7c819cdc7ad0dc1f617e749bd18ba77b509cf97f.json","evidence":{"source_url":"https://doi.org/10.21468/scipostphys.17.5.132"},"id":"literature:doi:10.21468/scipostphys.17.5.132","identifiers":{"doi":"10.21468/scipostphys.17.5.132"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.21468/scipostphys.17.5.132","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.21468%2Fscipostphys.17.5.132"},{"aliases":["10.21954/ou.ro.00106631"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1519364450b87d2247cf6c8fc7bca7ef35e4c82095beafb7f6ce29c3f1e001bd.json","evidence":{"source_url":"https://doi.org/10.21954/ou.ro.00106631"},"id":"literature:doi:10.21954/ou.ro.00106631","identifiers":{"doi":"10.21954/ou.ro.00106631"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.21954/ou.ro.00106631","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.21954%2Fou.ro.00106631"},{"aliases":["10.22199/issn.0717-6279-5801"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a33d7eb6e878edbe74f2cc912eb2ce3b9ca3b9a6c9e63c2459df71f7b3a7adfb.json","evidence":{"source_url":"https://doi.org/10.22199/issn.0717-6279-5801"},"id":"literature:doi:10.22199/issn.0717-6279-5801","identifiers":{"doi":"10.22199/issn.0717-6279-5801"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22199/issn.0717-6279-5801","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22199%2Fissn.0717-6279-5801"},{"aliases":["10.22331/q-2017-12-14-39"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce20ec1ed0162e3cfaba574cb6f42d62e7560b39d06de7d1f9b372e3f868113d.json","evidence":{"source_url":"https://doi.org/10.22331/q-2017-12-14-39"},"id":"literature:doi:10.22331/q-2017-12-14-39","identifiers":{"doi":"10.22331/q-2017-12-14-39"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2017-12-14-39","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2017-12-14-39"},{"aliases":["10.22331/q-2019-05-20-144"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b4af6e61c3a0d41930d32da949dde6bdeb9ab6b2d7231df06498d32ae44a628a.json","evidence":{"source_url":"https://doi.org/10.22331/q-2019-05-20-144"},"id":"literature:doi:10.22331/q-2019-05-20-144","identifiers":{"doi":"10.22331/q-2019-05-20-144"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2019-05-20-144","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2019-05-20-144"},{"aliases":["10.22331/q-2019-10-14-194"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40ad957eb14feb1a53a50b35264d674fe6fea44aeb6f25529b2df02d649c3723.json","evidence":{"source_url":"https://doi.org/10.22331/q-2019-10-14-194"},"id":"literature:doi:10.22331/q-2019-10-14-194","identifiers":{"doi":"10.22331/q-2019-10-14-194"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2019-10-14-194","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2019-10-14-194"},{"aliases":["10.22331/q-2020-10-21-346"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74d52e346cf9534a0b0582adca69f63f446125561534f930682403a563b89105.json","evidence":{"source_url":"https://doi.org/10.22331/q-2020-10-21-346"},"id":"literature:doi:10.22331/q-2020-10-21-346","identifiers":{"doi":"10.22331/q-2020-10-21-346"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2020-10-21-346","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2020-10-21-346"},{"aliases":["10.22331/q-2021-06-29-488"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/397ac74ce2bbfe87890e502062fa23a134b920fcb9bd241fc5c3b8be92fb77de.json","evidence":{"source_url":"https://doi.org/10.22331/q-2021-06-29-488"},"id":"literature:doi:10.22331/q-2021-06-29-488","identifiers":{"doi":"10.22331/q-2021-06-29-488"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2021-06-29-488","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2021-06-29-488"},{"aliases":["10.22331/q-2022-02-16-654"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0d240bd28e9e8610e67ef408f0407dd45b62ff79747de3034674106d9f604b0.json","evidence":{"source_url":"https://doi.org/10.22331/q-2022-02-16-654"},"id":"literature:doi:10.22331/q-2022-02-16-654","identifiers":{"doi":"10.22331/q-2022-02-16-654"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2022-02-16-654","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2022-02-16-654"},{"aliases":["10.22331/q-2023-02-09-915"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cebe810b29d79addab7565da7f040cc679b17b04b358c9d0c8dc480f9d93039d.json","evidence":{"source_url":"https://doi.org/10.22331/q-2023-02-09-915"},"id":"literature:doi:10.22331/q-2023-02-09-915","identifiers":{"doi":"10.22331/q-2023-02-09-915"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2024-10-04-1492","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2024-10-04-1492"},{"aliases":["10.22331/q-2025-01-20-1601"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e7975518f52bb60dd88b4ee16f14c457ebe1c5cc85d0a568499b48e5086f345.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-01-20-1601"},"id":"literature:doi:10.22331/q-2025-01-20-1601","identifiers":{"doi":"10.22331/q-2025-01-20-1601"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-01-20-1601","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-01-20-1601"},{"aliases":["10.22331/q-2025-02-04-1622"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d024f0b276a79d2cbaadb9f3ab66eb845f259c85052f1fda978f49c938c3e7bd.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-02-04-1622"},"id":"literature:doi:10.22331/q-2025-02-04-1622","identifiers":{"doi":"10.22331/q-2025-02-04-1622"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-02-04-1622","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-02-04-1622"},{"aliases":["10.22331/q-2025-02-20-1641"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/511ff92b900f1945eac115b51546a56f43a155e787d7f095262ac198e640b3ce.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-02-20-1641"},"id":"literature:doi:10.22331/q-2025-02-20-1641","identifiers":{"doi":"10.22331/q-2025-02-20-1641"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-02-20-1641","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-02-20-1641"},{"aliases":["10.22331/q-2025-04-18-1718"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b12f6b3aacd5b3919a20ebd0af09305c5b83c020e82b2b57c388e6068e78ffd2.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-04-18-1718"},"id":"literature:doi:10.22331/q-2025-04-18-1718","identifiers":{"doi":"10.22331/q-2025-04-18-1718"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-04-18-1718","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-04-18-1718"},{"aliases":["10.22331/q-2025-05-15-1745"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce0a967b513a254fba7fba866d21fd13da61302ea4bd043349002dd2584aeb1d.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-05-15-1745"},"id":"literature:doi:10.22331/q-2025-05-15-1745","identifiers":{"doi":"10.22331/q-2025-05-15-1745"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-05-15-1745","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-05-15-1745"},{"aliases":["10.22331/q-2025-08-27-1837"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4cd75d33e0d684dc32c4daf62c67dfd75e9cadb35bd1d46741c78b30cda0e9f.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-08-27-1837"},"id":"literature:doi:10.22331/q-2025-08-27-1837","identifiers":{"doi":"10.22331/q-2025-08-27-1837"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-08-27-1837","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-08-27-1837"},{"aliases":["10.22331/q-2025-09-15-1858"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cd554427a76e95328f0604631846364544fade4826eb317b30d8d3529f4d724.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-09-15-1858"},"id":"literature:doi:10.22331/q-2025-09-15-1858","identifiers":{"doi":"10.22331/q-2025-09-15-1858"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-09-15-1858","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-09-15-1858"},{"aliases":["10.22331/q-2025-09-30-1867"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b2330d677e4c52af71859eaa4258c8a287a0ac95010e2fb7d8e7df6d19a83e43.json","evidence":{"source_url":"https://doi.org/10.22331/q-2025-09-30-1867"},"id":"literature:doi:10.22331/q-2025-09-30-1867","identifiers":{"doi":"10.22331/q-2025-09-30-1867"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2025-09-30-1867","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2025-09-30-1867"},{"aliases":["10.22331/q-2026-05-21-2111"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1eed964deadfbcec175365eec10e0dcfb621ef000d0d4adcea79510e7de911b3.json","evidence":{"source_url":"https://doi.org/10.22331/q-2026-05-21-2111"},"id":"literature:doi:10.22331/q-2026-05-21-2111","identifiers":{"doi":"10.22331/q-2026-05-21-2111"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2026-05-21-2111","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2026-05-21-2111"},{"aliases":["10.22331/q-2026-07-23-2174"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84b84c561e31d2d2ef99aa9cc8329b07a23ad0403f9dd70ca6e087932b748037.json","evidence":{"source_url":"https://doi.org/10.22331/q-2026-07-23-2174"},"id":"literature:doi:10.22331/q-2026-07-23-2174","identifiers":{"doi":"10.22331/q-2026-07-23-2174"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2026-07-23-2174","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2026-07-23-2174"},{"aliases":["10.22331/q-2026-10-06-2228"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2246044c7a51c3878ac81f7ddf7c905cc4e80813ee4192060f47dc91f48b22d.json","evidence":{"source_url":"https://doi.org/10.22331/q-2026-10-06-2228"},"id":"literature:doi:10.22331/q-2026-10-06-2228","identifiers":{"doi":"10.22331/q-2026-10-06-2228"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.22331/q-2026-10-06-2228","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.22331%2Fq-2026-10-06-2228"},{"aliases":["10.2298/fil2513551l"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/acb4df01b1154530c76381c0da6b4c944e71c727656b14ef30983abbdccce9c2.json","evidence":{"source_url":"https://doi.org/10.2298/fil2513551l"},"id":"literature:doi:10.2298/fil2513551l","identifiers":{"doi":"10.2298/fil2513551l"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2298/fil2513551l","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2298%2Ffil2513551l"},{"aliases":["10.2307/1968867","M. H. A. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On Theories with a Combinatorial Definition of \"Equivalence\"","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2307%2F1968867","year":1942},{"aliases":["10.2307/1990319","Leonidas Alaoglu and Paul Erdos"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a2485ec2043c30949e31094cc65e72f8fe4e7e9cee556a9efd49fb16b52a202.json","authors":"Leonidas Alaoglu and Paul Erdos","evidence":{"source_url":"https://doi.org/10.2307/1990319"},"id":"literature:doi:10.2307/1990319","identifiers":{"doi":"10.2307/1990319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On highly composite and similar numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2307%2F1990319","year":1944},{"aliases":["10.2307/2267778","Stephen Cole Kleene"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/adf47d765d66dc9b9b30832943c6d5ac4e8c4c99cb422ebd0f9cf9a957b186e2.json","authors":"Stephen Cole Kleene","evidence":{"source_url":"https://doi.org/10.2307/2267778"},"id":"literature:doi:10.2307/2267778","identifiers":{"doi":"10.2307/2267778"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On notation for ordinal numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2307%2F2267778","year":1938},{"aliases":["10.2307/2371431","Marston Morse and Gustav A. Hedlund"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5d77bc0b88161435206200b5b29b3bcc27ce70a5bdd017d89f35cc81ca1fcb93.json","authors":"Marston Morse and Gustav A. Hedlund","evidence":{"source_url":"https://doi.org/10.2307/2371431"},"id":"literature:doi:10.2307/2371431","identifiers":{"doi":"10.2307/2371431"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Symbolic Dynamics II. Sturmian Trajectories","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2307%2F2371431","year":1940},{"aliases":["10.24330/ieja.1060709"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/adacc69e7fa1289efc8e701549e40fa630b61d0dbec1aeef9ca716b470de3203.json","evidence":{"source_url":"https://doi.org/10.24330/ieja.1060709"},"id":"literature:doi:10.24330/ieja.1060709","identifiers":{"doi":"10.24330/ieja.1060709"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.24330/ieja.1060709","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.24330%2Fieja.1060709"},{"aliases":["10.2478/amsil-2025-0014"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a2026daf52e24f79454c163a4a756f1fbc770614c864e7bd0cd3a7a03d38bb81.json","evidence":{"source_url":"https://doi.org/10.2478/amsil-2025-0014"},"id":"literature:doi:10.2478/amsil-2025-0014","identifiers":{"doi":"10.2478/amsil-2025-0014"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2478/amsil-2025-0014","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2478%2Famsil-2025-0014"},{"aliases":["10.26421/qic17.13-14-7"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9e9b3903bee01cb1050ae77b2ff141ac54ca5302bbc27dc87cef97405db09e71.json","evidence":{"source_url":"https://doi.org/10.26421/qic17.13-14-7"},"id":"literature:doi:10.26421/qic17.13-14-7","identifiers":{"doi":"10.26421/qic17.13-14-7"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.26421/qic17.13-14-7","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.26421%2Fqic17.13-14-7"},{"aliases":["10.26421/qic21.5-6-5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4982d033cb42d3bf9ac986ed77eb64bf1edd3b605ed7d2ffc2d87b4489c5c76.json","evidence":{"source_url":"https://doi.org/10.26421/qic21.5-6-5"},"id":"literature:doi:10.26421/qic21.5-6-5","identifiers":{"doi":"10.26421/qic21.5-6-5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.26421/qic21.5-6-5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.26421%2Fqic21.5-6-5"},{"aliases":["10.26421/qic24.1-2-2"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ff679ea1ce523d407a24234d1443a6568aff747d79e289986b35f0a5315c88c.json","evidence":{"source_url":"https://doi.org/10.26421/qic24.1-2-2"},"id":"literature:doi:10.26421/qic24.1-2-2","identifiers":{"doi":"10.26421/qic24.1-2-2"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.26421/qic24.1-2-2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.26421%2Fqic24.1-2-2"},{"aliases":["10.26421/qic7.1-2-6","Ashwin Nayak and Pranab Sen"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c746fff8e08e6881c73c5ab555b1b3343898e04733b8757ce0ffe070dd201906.json","authors":"Ashwin Nayak and Pranab Sen","evidence":{"source_url":"https://doi.org/10.26421/qic7.1-2-6"},"id":"literature:doi:10.26421/qic7.1-2-6","identifiers":{"doi":"10.26421/qic7.1-2-6"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Invertible Quantum Operations and Perfect Encryption of Quantum States","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.26421%2Fqic7.1-2-6","year":2007},{"aliases":["10.26493/1855-3974.2894.b07"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/697da317db91078a8a0e72dc0b8ce657c6be84446b7083e00d96cae396a376fd.json","evidence":{"source_url":"https://doi.org/10.26493/1855-3974.2894.b07"},"id":"literature:doi:10.26493/1855-3974.2894.b07","identifiers":{"doi":"10.26493/1855-3974.2894.b07"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.26493/1855-3974.2894.b07","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.26493%2F1855-3974.2894.b07"},{"aliases":["10.2989/16073606.2025.2574046"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0e9bbb872c3ebd78c78388257c2315223349f50b6a0cdc881754de4ab562557.json","evidence":{"source_url":"https://doi.org/10.2989/16073606.2025.2574046"},"id":"literature:doi:10.2989/16073606.2025.2574046","identifiers":{"doi":"10.2989/16073606.2025.2574046"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.2989/16073606.2025.2574046","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.2989%2F16073606.2025.2574046"},{"aliases":["10.30757/alea.v19-64"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c4bf6b132bf68471df6ec2aa0f2f1ab06a2de750423daf349d8871cdb772369a.json","evidence":{"source_url":"https://doi.org/10.30757/alea.v19-64"},"id":"literature:doi:10.30757/alea.v19-64","identifiers":{"doi":"10.30757/alea.v19-64"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.30757/alea.v19-64","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.30757%2Falea.v19-64"},{"aliases":["10.3389/frqst.2022.1090022"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8590eb0f896812c262bb26aa25f090e13c272569c6dc6a28c24bde28c0eeb79b.json","evidence":{"source_url":"https://doi.org/10.3389/frqst.2022.1090022"},"id":"literature:doi:10.3389/frqst.2022.1090022","identifiers":{"doi":"10.3389/frqst.2022.1090022"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3389/frqst.2022.1090022","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3389%2Ffrqst.2022.1090022"},{"aliases":["10.3390/e15062340"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c30b8dba6a946dcda8557ef82b703564b01cea1908a81f5aae2f8b0098ace4ba.json","evidence":{"source_url":"https://doi.org/10.3390/e15062340"},"id":"literature:doi:10.3390/e15062340","identifiers":{"doi":"10.3390/e15062340"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e15062340","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe15062340"},{"aliases":["10.3390/e23121594"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61b6a125526bf11bfb5b52082fadde4b524cb5552a73c054351b98e2a9c8a28a.json","evidence":{"source_url":"https://doi.org/10.3390/e23121594"},"id":"literature:doi:10.3390/e23121594","identifiers":{"doi":"10.3390/e23121594"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e23121594","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe23121594"},{"aliases":["10.3390/e24060805"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a49b0100897aaacbd8ea9303a49eb0c162c034b9c0d3647c76c6cb98fb2cbab.json","evidence":{"source_url":"https://doi.org/10.3390/e24060805"},"id":"literature:doi:10.3390/e24060805","identifiers":{"doi":"10.3390/e24060805"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e24060805","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe24060805"},{"aliases":["10.3390/e24081146"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1573aabe5d7ddeab701f4cd16dbcb257a6f84d4f25db006f1d0ceb35d06beb7.json","evidence":{"source_url":"https://doi.org/10.3390/e24081146"},"id":"literature:doi:10.3390/e24081146","identifiers":{"doi":"10.3390/e24081146"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e24081146","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe24081146"},{"aliases":["10.3390/e24091292"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/51aba0332a17f1f5f1b5250a14d1b9121998dc3a9e63d577db8323c2beabebd5.json","evidence":{"source_url":"https://doi.org/10.3390/e24091292"},"id":"literature:doi:10.3390/e24091292","identifiers":{"doi":"10.3390/e24091292"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e24091292","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe24091292"},{"aliases":["10.3390/e26030217"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c03dd57b3788e33331c0f84e3c242e45f86647ba74ebb81e0aad0fae970e2df7.json","evidence":{"source_url":"https://doi.org/10.3390/e26030217"},"id":"literature:doi:10.3390/e26030217","identifiers":{"doi":"10.3390/e26030217"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e26030217","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe26030217"},{"aliases":["10.3390/e26100816"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43402e97b4921503adfca8e3fd07632c80546e5e8c683b2299bc951aad1913bc.json","evidence":{"source_url":"https://doi.org/10.3390/e26100816"},"id":"literature:doi:10.3390/e26100816","identifiers":{"doi":"10.3390/e26100816"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e26100816","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe26100816"},{"aliases":["10.3390/e27040331"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c1ad489de7d2e2805248acf10ed09ac6df04f03248a7ff5ebb6a978dedc50fe8.json","evidence":{"source_url":"https://doi.org/10.3390/e27040331"},"id":"literature:doi:10.3390/e27040331","identifiers":{"doi":"10.3390/e27040331"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/e27040331","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fe27040331"},{"aliases":["10.3390/foundations5020019"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b9d643e723d5ce6a4cd22d8e237d9328c9a2b757ea9e34f1e320004d5ae1ab4.json","evidence":{"source_url":"https://doi.org/10.3390/foundations5020019"},"id":"literature:doi:10.3390/foundations5020019","identifiers":{"doi":"10.3390/foundations5020019"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/foundations5020019","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Ffoundations5020019"},{"aliases":["10.3390/math13132080"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/090e3cd2ad2a89a47029bd0c77f70a26b341ce9df32c165ebce270843fcb072e.json","evidence":{"source_url":"https://doi.org/10.3390/math13132080"},"id":"literature:doi:10.3390/math13132080","identifiers":{"doi":"10.3390/math13132080"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/math13132080","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fmath13132080"},{"aliases":["10.3390/math14111983"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/253c9e9f13d68eac09860064b8f1aaa09d8df0ce2c73d613b5caa7e5851348bc.json","evidence":{"source_url":"https://doi.org/10.3390/math14111983"},"id":"literature:doi:10.3390/math14111983","identifiers":{"doi":"10.3390/math14111983"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/math14111983","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fmath14111983"},{"aliases":["10.3390/math9030217"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3a50b64653f5d9b89eded32a39eee662df8752162c8419a3342f51e079e77c9.json","evidence":{"source_url":"https://doi.org/10.3390/math9030217"},"id":"literature:doi:10.3390/math9030217","identifiers":{"doi":"10.3390/math9030217"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/math9030217","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fmath9030217"},{"aliases":["10.3390/sym18010186"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebf38796a8ea2b2ea7b48cc247804a5dce61ecb2353abc1b18891d18efd24eba.json","evidence":{"source_url":"https://doi.org/10.3390/sym18010186"},"id":"literature:doi:10.3390/sym18010186","identifiers":{"doi":"10.3390/sym18010186"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3390/sym18010186","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3390%2Fsym18010186"},{"aliases":["10.37236/12040"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ae1317ed612651f7d76688c988ff6b446fd739d7290885584cd00b13e2f2f470.json","evidence":{"source_url":"https://doi.org/10.37236/12040"},"id":"literature:doi:10.37236/12040","identifiers":{"doi":"10.37236/12040"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.37236/12040","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37236%2F12040"},{"aliases":["10.37236/14401"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7ca8955b10168e74db1dcf860b43f4547965dda82b41cc4964f8aaf84132849.json","evidence":{"source_url":"https://doi.org/10.37236/14401"},"id":"literature:doi:10.37236/14401","identifiers":{"doi":"10.37236/14401"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.37236/14401","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37236%2F14401"},{"aliases":["10.37236/15173"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90714336c55c9540f81b0652ccc5df4ab481bd779a68cc51a722c77a89ad6a98.json","evidence":{"source_url":"https://doi.org/10.37236/15173"},"id":"literature:doi:10.37236/15173","identifiers":{"doi":"10.37236/15173"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.37236/15173","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37236%2F15173"},{"aliases":["10.37236/4771"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3e9fed116abee2474c025247ba6b1c307b31f3c8ae3da929116afd2d79d6360.json","evidence":{"source_url":"https://doi.org/10.37236/4771"},"id":"literature:doi:10.37236/4771","identifiers":{"doi":"10.37236/4771"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.37236/4771","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37236%2F4771"},{"aliases":["10.37236/6915"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/225d143cfa8e39f3df02c0e5f52c4693a19337da3b41f210b4ce6ce8c3c13ed7.json","evidence":{"source_url":"https://doi.org/10.37236/6915"},"id":"literature:doi:10.37236/6915","identifiers":{"doi":"10.37236/6915"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.37236/6915","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37236%2F6915"},{"aliases":["10.37398/jsr.2021.650210","Pintoo R. Jaiswar"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/555e0dc21dbe3b998b5fab99c956e14c5aa3f280dd6f8e7e9e170e4bec3092c9.json","authors":"Pintoo R. Jaiswar","evidence":{"source_url":"https://doi.org/10.37398/jsr.2021.650210"},"id":"literature:doi:10.37398/jsr.2021.650210","identifiers":{"doi":"10.37398/jsr.2021.650210"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Identity Theorem in Complex Analysis","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.37398%2Fjsr.2021.650210","year":2021},{"aliases":["10.3842/sigma.2014.095"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43808fad4703e963f80e1b3f1b672e3f908cc3f68695e0c391234092b113b309.json","evidence":{"source_url":"https://doi.org/10.3842/sigma.2014.095"},"id":"literature:doi:10.3842/sigma.2014.095","identifiers":{"doi":"10.3842/sigma.2014.095"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3842/sigma.2014.095","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3842%2Fsigma.2014.095"},{"aliases":["10.3842/sigma.2016.053"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cb594e6920e4241f7488cefb84d941cb372054497d74a3f984199d1a260d3a89.json","evidence":{"source_url":"https://doi.org/10.3842/sigma.2016.053"},"id":"literature:doi:10.3842/sigma.2016.053","identifiers":{"doi":"10.3842/sigma.2016.053"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3842/sigma.2016.053","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3842%2Fsigma.2016.053"},{"aliases":["10.3842/sigma.2022.099"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5230ab7c9167022eb4a86b77610ab9fe96e0a08f977deda5175ac846c584fd4a.json","evidence":{"source_url":"https://doi.org/10.3842/sigma.2022.099"},"id":"literature:doi:10.3842/sigma.2022.099","identifiers":{"doi":"10.3842/sigma.2022.099"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3842/sigma.2022.099","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3842%2Fsigma.2022.099"},{"aliases":["10.3934/math.2020224"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/643b523aaf4ecd141b9e6f87e147f58dca5f9ca43abca2833db988607e4f6cc6.json","evidence":{"source_url":"https://doi.org/10.3934/math.2020224"},"id":"literature:doi:10.3934/math.2020224","identifiers":{"doi":"10.3934/math.2020224"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.3934/math.2020224","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3934%2Fmath.2020224"},{"aliases":["10.3934/nhm.2026030","Waseem Ahmad Khan and Francesco Aldo Costabile and Khidir Shaib Mohamed and Ugur Duran and Abdulghani Muhyi and Azhar Iqbal and Wei Sin Koh"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bc5e6d54bca50aef3a2a350968c3875f790477fa76af4a3d245dcbe55db28608.json","authors":"Waseem Ahmad Khan and Francesco Aldo Costabile and Khidir Shaib Mohamed and Ugur Duran and Abdulghani Muhyi and Azhar Iqbal and Wei Sin Koh","evidence":{"source_url":"https://doi.org/10.3934/nhm.2026030"},"id":"literature:doi:10.3934/nhm.2026030","identifiers":{"doi":"10.3934/nhm.2026030"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Algebraic properties of central Bell-based type 2 Bernoulli and Euler polynomials of complex variable","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.3934%2Fnhm.2026030","year":2026},{"aliases":["10.4064/aa118-3-4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a8669e0a3f1e7ec5c52c80fcf4a0352e941b7fe698e977b9a289fa1914f1b4c.json","evidence":{"source_url":"https://doi.org/10.4064/aa118-3-4"},"id":"literature:doi:10.4064/aa118-3-4","identifiers":{"doi":"10.4064/aa118-3-4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4064/aa118-3-4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4064%2Faa118-3-4"},{"aliases":["10.4064/aa200515-10-3","Dan Romik"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7319d88721ca6c911b6e2a530fa516bec1b472ac945e3a72d4bd28633ee04620.json","authors":"Dan Romik","evidence":{"source_url":"https://doi.org/10.4064/aa200515-10-3"},"id":"literature:doi:10.4064/aa200515-10-3","identifiers":{"doi":"10.4064/aa200515-10-3"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Orthogonal polynomial expansions for the Riemann xi function in the Hermite, Meixner–Pollaczek, and continuous Hahn bases","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4064%2Faa200515-10-3","year":2021},{"aliases":["10.4064/aa220613-19-1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23a96a0a09fd43e08357853a0ebe3635c16af00809c3444d61f09f7180e57b8c.json","evidence":{"source_url":"https://doi.org/10.4064/aa220613-19-1"},"id":"literature:doi:10.4064/aa220613-19-1","identifiers":{"doi":"10.4064/aa220613-19-1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4064/aa220613-19-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4064%2Faa220613-19-1"},{"aliases":["10.4064/bc112-0-16"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6eab1b5a0b974a048edb44f1b723a7be0dcd92d1708323a35d5ec3675f1cb52.json","evidence":{"source_url":"https://doi.org/10.4064/bc112-0-16"},"id":"literature:doi:10.4064/bc112-0-16","identifiers":{"doi":"10.4064/bc112-0-16"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4064/bc112-0-16","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4064%2Fbc112-0-16"},{"aliases":["10.4153/cmb-1986-050-0","Marc A. Berger; Alexander Felzenbaum; Aviezri Fraenkel"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e055e1b5f0eb7ba14e8e4419fbfbe55cbcebd8737e37b884f2d17ee62b695646.json","authors":"Marc A. Berger; Alexander Felzenbaum; Aviezri Fraenkel","evidence":{"source_url":"https://doi.org/10.4153/cmb-1986-050-0"},"id":"literature:doi:10.4153/cmb-1986-050-0","identifiers":{"doi":"10.4153/cmb-1986-050-0"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Herzog-Schonheim Conjecture for Finite Nilpotent Groups","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4153%2Fcmb-1986-050-0","year":1986},{"aliases":["10.4171/aihpd/129"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9dc5a388a0fd8e070ca0e72756326465fd1cb90a596557bb06c7b269dc110e1.json","evidence":{"source_url":"https://doi.org/10.4171/aihpd/129"},"id":"literature:doi:10.4171/aihpd/129","identifiers":{"doi":"10.4171/aihpd/129"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4171/aihpd/129","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4171%2Faihpd%2F129"},{"aliases":["10.4171/mems/23"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca37e5e7f6b940d6cf7bf6df596d843e5eee6926f90551c669d02d2f3e0c1516.json","evidence":{"source_url":"https://doi.org/10.4171/mems/23"},"id":"literature:doi:10.4171/mems/23","identifiers":{"doi":"10.4171/mems/23"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4171/mems/23","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4171%2Fmems%2F23"},{"aliases":["10.4204/eptcs.343.1"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4476960df395791b62a68392acb8518e69e132df84cb82cc9d81c93ef165f2d.json","evidence":{"source_url":"https://doi.org/10.4204/eptcs.343.1"},"id":"literature:doi:10.4204/eptcs.343.1","identifiers":{"doi":"10.4204/eptcs.343.1"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4204/eptcs.343.1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4204%2Feptcs.343.1"},{"aliases":["10.4204/eptcs.445.17"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/adbac77339e7111132e0a635ddd8c19001857fa438487b9d0f4ad75c6f186a02.json","evidence":{"source_url":"https://doi.org/10.4204/eptcs.445.17"},"id":"literature:doi:10.4204/eptcs.445.17","identifiers":{"doi":"10.4204/eptcs.445.17"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4204/eptcs.445.17","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4204%2Feptcs.445.17"},{"aliases":["10.4230/lipics.fsttcs.2024.32"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01e68372034c709ace7698d7701d59f6393b60eca00c6ebeefe0dab3ece6e70c.json","evidence":{"source_url":"https://doi.org/10.4230/lipics.fsttcs.2024.32"},"id":"literature:doi:10.4230/lipics.fsttcs.2024.32","identifiers":{"doi":"10.4230/lipics.fsttcs.2024.32"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4230/lipics.fsttcs.2024.32","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4230%2Flipics.fsttcs.2024.32"},{"aliases":["10.4230/lipics.stacs.2026.66"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c0680ddf412117ca75eb035be896bab795c52b5f92862a1371184d662cbe705d.json","evidence":{"source_url":"https://doi.org/10.4230/lipics.stacs.2026.66"},"id":"literature:doi:10.4230/lipics.stacs.2026.66","identifiers":{"doi":"10.4230/lipics.stacs.2026.66"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4230/lipics.stacs.2026.66","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4230%2Flipics.stacs.2026.66"},{"aliases":["10.4230/lipics.tqc.2017.5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5ba1dbb07190e2e0af3f9aa3facbb61f522ac020e6caf036affaa80a60a2685.json","evidence":{"source_url":"https://doi.org/10.4230/lipics.tqc.2017.5"},"id":"literature:doi:10.4230/lipics.tqc.2017.5","identifiers":{"doi":"10.4230/lipics.tqc.2017.5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4230/lipics.tqc.2017.5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4230%2Flipics.tqc.2017.5"},{"aliases":["10.4310/joc.2023.v14.n4.a8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d254b8770eb6c430fca98d36b96119516cabcccca8bc86fefcd939a71061f60b.json","evidence":{"source_url":"https://doi.org/10.4310/joc.2023.v14.n4.a8"},"id":"literature:doi:10.4310/joc.2023.v14.n4.a8","identifiers":{"doi":"10.4310/joc.2023.v14.n4.a8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4310/joc.2023.v14.n4.a8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4310%2Fjoc.2023.v14.n4.a8"},{"aliases":["10.4418/2019.74.1.8"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6a96555c0df4e35372ddcf553b49f19ea5c34a45acff03cf9e5b99b3a17878ce.json","evidence":{"source_url":"https://doi.org/10.4418/2019.74.1.8"},"id":"literature:doi:10.4418/2019.74.1.8","identifiers":{"doi":"10.4418/2019.74.1.8"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.4418/2019.74.1.8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.4418%2F2019.74.1.8"},{"aliases":["10.46298/afm.14253","Martin Dvorak and Vladimir Kolmogorov"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18ba9597e2d87b5eb4f7e023077d94c59495eead908c23e4b3cd61b80fe81ef3.json","authors":"Martin Dvorak and Vladimir Kolmogorov","evidence":{"source_url":"https://doi.org/10.46298/afm.14253"},"id":"literature:doi:10.46298/afm.14253","identifiers":{"doi":"10.46298/afm.14253"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Duality theory in linear optimization and its extensions -- formally verified","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46298%2Fafm.14253","year":2026},{"aliases":["10.46298/dmtcs.12273"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5361f410c585d02ef78eaf522b7d3ca5c23e30f710df8b69382d90e7ce951f88.json","evidence":{"source_url":"https://doi.org/10.46298/dmtcs.12273"},"id":"literature:doi:10.46298/dmtcs.12273","identifiers":{"doi":"10.46298/dmtcs.12273"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.46298/dmtcs.12273","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46298%2Fdmtcs.12273"},{"aliases":["10.46298/dmtcs.17199"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6ba9aba4cd8eb0031efbbe4d9da6b90a92ac021f1fb8b2a70171e2784d74319.json","evidence":{"source_url":"https://doi.org/10.46298/dmtcs.17199"},"id":"literature:doi:10.46298/dmtcs.17199","identifiers":{"doi":"10.46298/dmtcs.17199"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.46298/dmtcs.17199","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46298%2Fdmtcs.17199"},{"aliases":["10.46298/dmtcs.7134"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a537c7c799a1dd4b60501d1f4120353e8878b60103670c944805dcb623e0d21.json","evidence":{"source_url":"https://doi.org/10.46298/dmtcs.7134"},"id":"literature:doi:10.46298/dmtcs.7134","identifiers":{"doi":"10.46298/dmtcs.7134"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.46298/dmtcs.7134","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46298%2Fdmtcs.7134"},{"aliases":["10.46571/jci.2023.2.4"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/426e41272675fc9e7ce3d543ea5407feb733a76a50409b51749d4e22d4672324.json","evidence":{"source_url":"https://doi.org/10.46571/jci.2023.2.4"},"id":"literature:doi:10.46571/jci.2023.2.4","identifiers":{"doi":"10.46571/jci.2023.2.4"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.46571/jci.2023.2.4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46571%2Fjci.2023.2.4"},{"aliases":["10.46787/pump.v9i.6304"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65245ae1a527000528075670ec276789647f3275ef23bb191537ddcaa27df6ff.json","evidence":{"source_url":"https://doi.org/10.46787/pump.v9i.6304"},"id":"literature:doi:10.46787/pump.v9i.6304","identifiers":{"doi":"10.46787/pump.v9i.6304"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.46787/pump.v9i.6304","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.46787%2Fpump.v9i.6304"},{"aliases":["10.47443/dml.2025.176"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/200558745a869a68f5a6775e9ef76a58df9e1478ab0a28fb5f5f7259957a10fa.json","evidence":{"source_url":"https://doi.org/10.47443/dml.2025.176"},"id":"literature:doi:10.47443/dml.2025.176","identifiers":{"doi":"10.47443/dml.2025.176"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.47443/dml.2025.176","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.47443%2Fdml.2025.176"},{"aliases":["10.48550/arxiv.math/0312418","N. J. A. Sloane; James A. Sellers"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/415031ca95c3a5a2891cb91f9c410d649e38ac469b6da2f5263acbf06764a863.json","authors":"N. J. A. Sloane; James A. Sellers","evidence":{"source_url":"https://doi.org/10.48550/arxiv.math/0312418"},"id":"literature:doi:10.48550/arxiv.math/0312418","identifiers":{"doi":"10.48550/arxiv.math/0312418"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On Non-Squashing Partitions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.48550%2Farxiv.math%2F0312418","year":2003},{"aliases":["10.48550/arxiv.math/0408263","Herbert S. Wilf"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b249b9eb5d41da2c4c350823e4ea3e1fc43b9efca0024358e4a30df4f5cb0446.json","authors":"Herbert S. Wilf","evidence":{"source_url":"https://doi.org/10.48550/arxiv.math/0408263"},"id":"literature:doi:10.48550/arxiv.math/0408263","identifiers":{"doi":"10.48550/arxiv.math/0408263"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"The Redheffer matrix of a partially ordered set","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.48550%2Farxiv.math%2F0408263","year":2004},{"aliases":["10.5281/zenodo.10160456"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5fd74e607e396a1f96444bdc91b3a2f8519c85d55fedf018c27a22a7adaa28d6.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.10160456"},"id":"literature:doi:10.5281/zenodo.10160456","identifiers":{"doi":"10.5281/zenodo.10160456"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.10160456","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.10160456"},{"aliases":["10.5281/zenodo.13331499"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3292f7a45715e06020bcda6e3849162271b7a8bb6f1ab060eb355b5d8cac7cb4.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.13331499"},"id":"literature:doi:10.5281/zenodo.13331499","identifiers":{"doi":"10.5281/zenodo.13331499"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.13331499","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.13331499"},{"aliases":["10.5281/zenodo.14679256"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/29a2b317869a01e34b4d21a54e875b179905c64823bd3573428f89817f1c8190.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.14679256"},"id":"literature:doi:10.5281/zenodo.14679256","identifiers":{"doi":"10.5281/zenodo.14679256"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.14679256","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.14679256"},{"aliases":["10.5281/zenodo.15283699"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2e2489d21f144cbb94758f1c3d1a3d624398faaa5252d09cb4817249d9863b21.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.15283699"},"id":"literature:doi:10.5281/zenodo.15283699","identifiers":{"doi":"10.5281/zenodo.15283699"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.15283699","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.15283699"},{"aliases":["10.5281/zenodo.17144266"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/af663a8755a376b4546637a2cb67975931caf8007e2be5875e0d05f25efacfb4.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.17144266"},"id":"literature:doi:10.5281/zenodo.17144266","identifiers":{"doi":"10.5281/zenodo.17144266"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.17144266","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.17144266"},{"aliases":["10.5281/zenodo.17535229"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60b5d56b93467ec2331ad35024bcf0e5666f14c875d69415dc062732e061da37.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.17535229"},"id":"literature:doi:10.5281/zenodo.17535229","identifiers":{"doi":"10.5281/zenodo.17535229"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.17535229","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.17535229"},{"aliases":["10.5281/zenodo.18154061"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f265081dd36d2e149531185f97f2dc2cca7f574286d60fdb70194b4341bed2f3.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.18154061"},"id":"literature:doi:10.5281/zenodo.18154061","identifiers":{"doi":"10.5281/zenodo.18154061"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.18154061","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.18154061"},{"aliases":["10.5281/zenodo.18154119"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1ded91d3daf01fb5d5fd39a56d6a788cf1059cfcddb5649c9fa6c23c262c82a.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.18154119"},"id":"literature:doi:10.5281/zenodo.18154119","identifiers":{"doi":"10.5281/zenodo.18154119"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.18154119","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.18154119"},{"aliases":["10.5281/zenodo.18154216"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43001d5a65c813b60aa626731ca063238ed1e401b1133f041f905a5e8e0b12d1.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.18154216"},"id":"literature:doi:10.5281/zenodo.18154216","identifiers":{"doi":"10.5281/zenodo.18154216"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.18154216","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.18154216"},{"aliases":["10.5281/zenodo.18305140"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/806224ca49dbca5f27111ead3f247f5022ee77026c4eed3345890f7b4c4e49a4.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.18305140"},"id":"literature:doi:10.5281/zenodo.18305140","identifiers":{"doi":"10.5281/zenodo.18305140"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.18305140","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.18305140"},{"aliases":["10.5281/zenodo.19949535"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f5b1bcf8f36dcd0b393c83aac199193dfa8b17e7ea46a71e85059629288508bb.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.19949535"},"id":"literature:doi:10.5281/zenodo.19949535","identifiers":{"doi":"10.5281/zenodo.19949535"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.19949535","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.19949535"},{"aliases":["10.5281/zenodo.8399694"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4248f5d16ff1826471327f575ff1064ceb7b2c5b80d270c4d023c6dce557480.json","evidence":{"source_url":"https://doi.org/10.5281/zenodo.8399694"},"id":"literature:doi:10.5281/zenodo.8399694","identifiers":{"doi":"10.5281/zenodo.8399694"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5281/zenodo.8399694","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5281%2Fzenodo.8399694"},{"aliases":["10.53656/math2024-2-1-pro"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/247225f05d986d47e23fa93a1d8f3b222efe7522070b728c196f344f0da63afc.json","evidence":{"source_url":"https://doi.org/10.53656/math2024-2-1-pro"},"id":"literature:doi:10.53656/math2024-2-1-pro","identifiers":{"doi":"10.53656/math2024-2-1-pro"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.53656/math2024-2-1-pro","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.53656%2Fmath2024-2-1-pro"},{"aliases":["10.54550/eca2025v5s3r22"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7877793e08c8cfd7518278204e374a4369c1bd401bd79591a8865abed2e4250b.json","evidence":{"source_url":"https://doi.org/10.54550/eca2025v5s3r22"},"id":"literature:doi:10.54550/eca2025v5s3r22","identifiers":{"doi":"10.54550/eca2025v5s3r22"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.54550/eca2025v5s3r22","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.54550%2Feca2025v5s3r22"},{"aliases":["10.54550/eca2026v6s4r33"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2aa183934c5ced17543dba278110ddbd9cade0d38a814cd8c361fd2a4e8169ae.json","evidence":{"source_url":"https://doi.org/10.54550/eca2026v6s4r33"},"id":"literature:doi:10.54550/eca2026v6s4r33","identifiers":{"doi":"10.54550/eca2026v6s4r33"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.54550/eca2026v6s4r33","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.54550%2Feca2026v6s4r33"},{"aliases":["10.5506/aphyspolbsupp.12.75"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e84a4bd2bbc9bdcf83955206bd70fdf5bb99fbbd04e2853bbfb81ea87397f2a.json","evidence":{"source_url":"https://doi.org/10.5506/aphyspolbsupp.12.75"},"id":"literature:doi:10.5506/aphyspolbsupp.12.75","identifiers":{"doi":"10.5506/aphyspolbsupp.12.75"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5506/aphyspolbsupp.12.75","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5506%2Faphyspolbsupp.12.75"},{"aliases":["10.5614/ejgta.2025.13.1.5"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de7a883c59a7b9a2a5c89f090e859fe77f276231e4dc3b557f6ab5e844e6de82.json","evidence":{"source_url":"https://doi.org/10.5614/ejgta.2025.13.1.5"},"id":"literature:doi:10.5614/ejgta.2025.13.1.5","identifiers":{"doi":"10.5614/ejgta.2025.13.1.5"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.5614/ejgta.2025.13.1.5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5614%2Fejgta.2025.13.1.5"},{"aliases":["10.5802/aif.2233","Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c401ee53cd04388c0b0d985bb03735fadfa5fd802f952894b695f413bfd62511.json","authors":"Guy Barat, Valérie Berthé, Pierre Liardet, Jörg Thuswaldner","evidence":{"source_url":"https://doi.org/10.5802/aif.2233"},"id":"literature:doi:10.5802/aif.2233","identifiers":{"doi":"10.5802/aif.2233"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Dynamical directions in numeration","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5802%2Faif.2233","year":2006},{"aliases":["10.5802/aif.2311","Jeffrey C. Lagarias"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2444e04874157be74e4fadacf995f61c5fddbf7072e289b2a85d77c1bc293970.json","authors":"Jeffrey C. Lagarias","evidence":{"source_url":"https://doi.org/10.5802/aif.2311"},"id":"literature:doi:10.5802/aif.2311","identifiers":{"doi":"10.5802/aif.2311"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Li Coefficients for Automorphic L-Functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.5802%2Faif.2311","year":2007},{"aliases":["10.61091/um124-06"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f7e910147186a42d74a846dcba2eccca1db90465bf9a6f23df0bc1d4f55ad8f.json","evidence":{"source_url":"https://doi.org/10.61091/um124-06"},"id":"literature:doi:10.61091/um124-06","identifiers":{"doi":"10.61091/um124-06"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.61091/um124-06","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.61091%2Fum124-06"},{"aliases":["10.61091/um128-09"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6396a3c2a2a9b4ccfd8b4ef303afc31a76330aa593c06a63a5d1e852f6e37564.json","evidence":{"source_url":"https://doi.org/10.61091/um128-09"},"id":"literature:doi:10.61091/um128-09","identifiers":{"doi":"10.61091/um128-09"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.61091/um128-09","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.61091%2Fum128-09"},{"aliases":["10.61091/um128-18"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02dd05a13a679b7faaba07dc5142baa9330a112d8c48e539e8215e0e474480f8.json","evidence":{"source_url":"https://doi.org/10.61091/um128-18"},"id":"literature:doi:10.61091/um128-18","identifiers":{"doi":"10.61091/um128-18"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.61091/um128-18","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.61091%2Fum128-18"},{"aliases":["10.62072/acm.2026.09011"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f315f8d50ca4c9b68bca7a2f2d776305747e50c31529a822a899a6a303ecdeb.json","evidence":{"source_url":"https://doi.org/10.62072/acm.2026.09011"},"id":"literature:doi:10.62072/acm.2026.09011","identifiers":{"doi":"10.62072/acm.2026.09011"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.62072/acm.2026.09011","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.62072%2Facm.2026.09011"},{"aliases":["10.7151/dmgaa.1461"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3aab24629c2819f8c739a959d0db3ea3036c2fd66d2819d9bb9530557e1e553d.json","evidence":{"source_url":"https://doi.org/10.7151/dmgaa.1461"},"id":"literature:doi:10.7151/dmgaa.1461","identifiers":{"doi":"10.7151/dmgaa.1461"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.7151/dmgaa.1461","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7151%2Fdmgaa.1461"},{"aliases":["10.7151/dmgt.2491"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/69ae12aba7544df8dcde259761ea837a6d377f8b6cf98629976747267303bb51.json","evidence":{"source_url":"https://doi.org/10.7151/dmgt.2491"},"id":"literature:doi:10.7151/dmgt.2491","identifiers":{"doi":"10.7151/dmgt.2491"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.7151/dmgt.2491","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7151%2Fdmgt.2491"},{"aliases":["10.7151/dmgt.2580"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6397772824042b9fa23ae60efe2377c1d315dd9e6b393c68963cb36d1d08a754.json","evidence":{"source_url":"https://doi.org/10.7151/dmgt.2580"},"id":"literature:doi:10.7151/dmgt.2580","identifiers":{"doi":"10.7151/dmgt.2580"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.7151/dmgt.2580","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7151%2Fdmgt.2580"},{"aliases":["10.7151/dmgt.2603"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a616e3cb1026e96242df0e167287c10e796c125c738cbf34c3292919e24d5e9f.json","evidence":{"source_url":"https://doi.org/10.7151/dmgt.2603"},"id":"literature:doi:10.7151/dmgt.2603","identifiers":{"doi":"10.7151/dmgt.2603"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.7151/dmgt.2603","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7151%2Fdmgt.2603"},{"aliases":["10.7151/dmgt.2619","Alexander Bastien and Omid Khormali"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95a083eb6d3c8ec11fb599590c151df9cdf92868eca6425631c5e55fcd1ccdf9.json","authors":"Alexander Bastien and Omid Khormali","evidence":{"source_url":"https://doi.org/10.7151/dmgt.2619"},"id":"literature:doi:10.7151/dmgt.2619","identifiers":{"doi":"10.7151/dmgt.2619"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"On the regularity, planarity and edge bounds of link-irregular graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7151%2Fdmgt.2619","year":2026},{"aliases":["10.7153/mia-2018-21-81"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ef7cadf03e3be3b5ac025a148d3fbdf03107c688b359152649bbf06fe41c203.json","evidence":{"source_url":"https://doi.org/10.7153/mia-2018-21-81"},"id":"literature:doi:10.7153/mia-2018-21-81","identifiers":{"doi":"10.7153/mia-2018-21-81"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"10.7153/mia-2018-21-81","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Adoi%3A10.7153%2Fmia-2018-21-81"},{"aliases":["10.7153/mia-2022-25-64"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0089fefc98129415ed6710ba12e2691ee7f41a142cd661e483321727a57e7817.json","evidence":{"source_url":"https://doi.org/10.7153/mia-2022-25-64"},"id":"literature:doi:10.7153/mia-2022-25-64","identifiers":{"doi":"10.7153/mia-2022-25-64"},"kind":"literature","status":"external-literature","summary":"External research source. 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Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A001222, the Ω/ω binomial convolution conjecture (x+y)^Ω(n) = Σ_{d|n} x^Ω(d) (x+y)^{Ω(n/d)−ω(n/d)} y^{ω(n/d)}","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA001222","year":2018},{"aliases":["N. J. A. Sloane; Peter Bala","https://oeis.org/A002000"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d66a99748dcb4d26cb3422be9f801ab3b2d411c317fa7576bdbe88e7afb5bba.json","authors":"N. J. A. Sloane; Peter Bala","evidence":{"source_url":"https://oeis.org/A002000"},"id":"literature:url:https://oeis.org/A002000","identifiers":{"url":"https://oeis.org/A002000"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A002000, the cubic recurrence a(n+1) = a(n)(a(n)^2 - 3) from 7, with Bala's 3-adic congruence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA002000","year":2022},{"aliases":["N. J. A. Sloane; Gary W. Adamson","https://oeis.org/A002487"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d38baf78c84a86fbf066996e921c46fde2cea5341cf258da9b537a1682680a6c.json","authors":"N. J. A. Sloane; Gary W. Adamson","evidence":{"source_url":"https://oeis.org/A002487"},"id":"literature:url:https://oeis.org/A002487","identifiers":{"url":"https://oeis.org/A002487"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A002487, Stern's diatomic series, with the Adamson Farey-tree inverse conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA002487","year":2023},{"aliases":["Richard R. Forberg; OEIS Foundation Inc.","https://oeis.org/A002620/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16282543c072eaf77202ba7260b9dad87d0ed3fe35808213f30a281fa32aa645.json","authors":"Richard R. Forberg; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A002620/internal"},"id":"literature:url:https://oeis.org/A002620/internal","identifiers":{"url":"https://oeis.org/A002620/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Forberg's empirical exponential formula for OEIS A002620","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA002620%2Finternal","year":2013},{"aliases":["Mikhail Kurkov","https://oeis.org/A003319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/42cbfad93b3ef10fdc2d9c29f11545a35b6f9a5fba12be68cafdeade6268f35c.json","authors":"Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A003319"},"id":"literature:url:https://oeis.org/A003319","identifiers":{"url":"https://oeis.org/A003319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A003319 and Kurkov's A370380/A370381 double-array conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA003319","year":2024},{"aliases":["N. J. A. Sloane; Michel Marcus","https://oeis.org/A003558"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7089772e1caf6232b4394cab1b1af2594124382a3bc0a94135c81f59607ebbbc.json","authors":"N. J. A. Sloane; Michel Marcus","evidence":{"source_url":"https://oeis.org/A003558"},"id":"literature:url:https://oeis.org/A003558","identifiers":{"url":"https://oeis.org/A003558"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A003558, least m with 2^m = ±1 mod 2n+1, with Marcus's tent-map cycle-length conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA003558","year":2025},{"aliases":["Peter Bala","https://oeis.org/A004123"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b8813f6742499dfa3a2acd502d664996c58027c96757008ef6bf0d0be1a3e14f.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A004123"},"id":"literature:url:https://oeis.org/A004123","identifiers":{"url":"https://oeis.org/A004123"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A004123, number of generalized weak orders on n points","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA004123","year":2022},{"aliases":["N. J. A. Sloane; Ralf Stephan","https://oeis.org/A005590"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70328b6a9ac253b37e2b597d0f94187cad82e7ade123c75f3d7f67cac4d344ef.json","authors":"N. J. A. Sloane; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A005590"},"id":"literature:url:https://oeis.org/A005590","identifiers":{"url":"https://oeis.org/A005590"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A005590, a(0)=0, a(1)=1, a(2n)=a(n), a(2n+1)=a(n+1)-a(n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA005590","year":2003},{"aliases":["N. J. A. Sloane; Paul D. Hanna; Mikhail Kurkov","https://oeis.org/A006068"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9ec52ddda8d5e06e86f8eceb65e5ce5ffb857b22ca4bd5ab74b66d2329372ae2.json","authors":"N. J. A. Sloane; Paul D. Hanna; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A006068"},"id":"literature:url:https://oeis.org/A006068","identifiers":{"url":"https://oeis.org/A006068"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A006068, inverse Gray code, with Kurkov's recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA006068","year":2023},{"aliases":["Werner Schulte","https://oeis.org/A006472"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0ff17a2c86392bf61dc55c975a57a37d03678aec13c02a4d9bd2d87a81cee1d0.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A006472"},"id":"literature:url:https://oeis.org/A006472","identifiers":{"url":"https://oeis.org/A006472"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A006472, n!(n-1)!/2^(n-1), with the primality criterion n | 2 a(n-1) + 4 iff n is prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA006472","year":2020},{"aliases":["Davide Rotondo","https://oeis.org/A006972"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e10c27d73d6ed5ba22366e8ec9c7471ecc5205e703d68028b9c5ceddb559c14c.json","authors":"Davide Rotondo","evidence":{"source_url":"https://oeis.org/A006972"},"id":"literature:url:https://oeis.org/A006972","identifiers":{"url":"https://oeis.org/A006972"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A006972, Lucas-Carmichael numbers, with Rotondo's sufficient condition for three distinct odd prime factors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA006972","year":2020},{"aliases":["Werner Schulte","https://oeis.org/A008287"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e583dd9e2c9bf35decd3ccd4873be463e9b695fd58c02d25d562c8f6487138f9.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A008287"},"id":"literature:url:https://oeis.org/A008287","identifiers":{"url":"https://oeis.org/A008287"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A008287, quadrinomial coefficients, with the conjectured expansion T(n,k) = Sum_{j=0..k} (-2)^j binomial(n,j) binomial(3n-2j,k-j)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA008287","year":2015},{"aliases":["N. J. A. Sloane; Gary Detlefs","https://oeis.org/A008365"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe1ca7ce3e9925f462f84df9130872d807cba7c7347540eef56bbc42ab7fae4f.json","authors":"N. J. A. Sloane; Gary Detlefs","evidence":{"source_url":"https://oeis.org/A008365"},"id":"literature:url:https://oeis.org/A008365","identifiers":{"url":"https://oeis.org/A008365"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A008365, 13-rough numbers: positive integers that have no prime factors less than 13","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA008365","year":2011},{"aliases":["Ivan N. Ianakiev; Olivier Gérard","https://oeis.org/A008474"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/951d9fa02e112eb6439a484a51ee7969c9dc635077f3789f0db4676271d10c52.json","authors":"Ivan N. Ianakiev; Olivier Gérard","evidence":{"source_url":"https://oeis.org/A008474"},"id":"literature:url:https://oeis.org/A008474","identifiers":{"url":"https://oeis.org/A008474"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A008474, If n = Product (p_j^k_j) then a(n) = Sum (p_j + k_j)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA008474","year":2014},{"aliases":["N. J. A. Sloane; Lechoslaw Ratajczak","https://oeis.org/A008590"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3829c2f19075f18bd06f9b092eab5f8d83ececa20ec771cd259107d4592a1512.json","authors":"N. J. A. Sloane; Lechoslaw Ratajczak","evidence":{"source_url":"https://oeis.org/A008590"},"id":"literature:url:https://oeis.org/A008590","identifiers":{"url":"https://oeis.org/A008590"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A008590, Multiples of 8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA008590","year":2017},{"aliases":["Amarnath Murthy; David W. Wilson","https://oeis.org/A018800"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/385f9196e4fead20d2f99c6afc554457ae1f92fa046fb68dca6836356979f22d.json","authors":"Amarnath Murthy; David W. Wilson","evidence":{"source_url":"https://oeis.org/A018800"},"id":"literature:url:https://oeis.org/A018800","identifiers":{"url":"https://oeis.org/A018800"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A018800, Smallest prime that begins with n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA018800","year":2002},{"aliases":["OEIS Foundation Inc.; Vaclav Kotesovec","https://oeis.org/A022629"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fc80626b15f512122ae94b3043b61983804f5ca68a3bbb4f062fffb8f8256e44.json","authors":"OEIS Foundation Inc.; Vaclav Kotesovec","evidence":{"source_url":"https://oeis.org/A022629"},"id":"literature:url:https://oeis.org/A022629","identifiers":{"url":"https://oeis.org/A022629"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A022629 and the weighted distinct-partition logarithmic conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA022629","year":2018},{"aliases":["Olivier Gérard; Velin Yanev","https://oeis.org/A023887"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bc9a45fade2d5210c3e7acc1cb7b9422294a7ecc174a8bb5f394ecd8cd554f8.json","authors":"Olivier Gérard; Velin Yanev","evidence":{"source_url":"https://oeis.org/A023887"},"id":"literature:url:https://oeis.org/A023887","identifiers":{"url":"https://oeis.org/A023887"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A023887, sigma_n(n), with Yanev's sigma_m identity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA023887","year":2017},{"aliases":["Amiram Eldar; Thomas Ordowski; OEIS Foundation Inc.","https://oeis.org/A025529"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81ac36f9f2e24955d9a7af0cf9871dd061be8e67c58b2c423868e36eaab75d25.json","authors":"Amiram Eldar; Thomas Ordowski; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A025529"},"id":"literature:url:https://oeis.org/A025529","identifiers":{"url":"https://oeis.org/A025529"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A025529: the LCM-weighted harmonic sum and composite-solution conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA025529","year":2019},{"aliases":["Gus Wiseman","https://oeis.org/A026010"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/75d29d026ed2b865d7fa04cfe5971cba8ed582906b6a215b85a723e36b157998.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A026010"},"id":"literature:url:https://oeis.org/A026010","identifiers":{"url":"https://oeis.org/A026010"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A026010, a(n) = number of (s(0), s(1), ..., s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,...,n and s(0) = 2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA026010","year":2018},{"aliases":["R. J. Mathar","https://oeis.org/A026023"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77441d1dbdb1edbb3df312e5641d28db1b4703f7648c10c63ff3debf0d091aa0.json","authors":"R. J. Mathar","evidence":{"source_url":"https://oeis.org/A026023"},"id":"literature:url:https://oeis.org/A026023","identifiers":{"url":"https://oeis.org/A026023"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A026023, walks on the nonnegative integers from 3: recurrence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA026023","year":2012},{"aliases":["R. J. Mathar and Clark Kimberling","https://oeis.org/A027182"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8b74a80e156b2f2965c13d5d46579350a206e1fdfc9a6f94912e68def3502a9.json","authors":"R. J. Mathar and Clark Kimberling","evidence":{"source_url":"https://oeis.org/A027182"},"id":"literature:url:https://oeis.org/A027182","identifiers":{"url":"https://oeis.org/A027182"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A027182: self-convolution of the four-diamondized Pascal row","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA027182","year":2026},{"aliases":["David W. Wilson; Ralf Stephan; Velin Yanev; Alois P. Heinz; Henry Bottomley","https://oeis.org/A030101"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7566604696e5c909cdc03eb946d23a8f3c8bc46b608528a1d3e7b4bfa092a5be.json","authors":"David W. Wilson; Ralf Stephan; Velin Yanev; Alois P. Heinz; Henry Bottomley","evidence":{"source_url":"https://oeis.org/A030101"},"id":"literature:url:https://oeis.org/A030101","identifiers":{"url":"https://oeis.org/A030101"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A030101, binary digit reversal, with Yanev's position identity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA030101","year":2017},{"aliases":["Clark Kimberling","https://oeis.org/A035513"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/952f68abd118a19eba58fa510e3c9f761f84b047345fa0c364198d8c957728e8.json","authors":"Clark Kimberling","evidence":{"source_url":"https://oeis.org/A035513"},"id":"literature:url:https://oeis.org/A035513","identifiers":{"url":"https://oeis.org/A035513"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A035513: Wythoff array read by falling antidiagonals","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA035513","year":2025},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A036236"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9fdb370f924d86daf6b0fd7a795ba52e59085590646b337ab492b1595c844c41.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A036236"},"id":"literature:url:https://oeis.org/A036236","identifiers":{"url":"https://oeis.org/A036236"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A036236","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA036236","year":2026},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A038867"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a188792b40b3d4981b4db904806bc0a92e973a10a5e1b2ad64ca035dd3ae75f4.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A038867"},"id":"literature:url:https://oeis.org/A038867","identifiers":{"url":"https://oeis.org/A038867"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A038867","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA038867","year":2026},{"aliases":["Labos Elemer; Jaroslav Krizek","https://oeis.org/A046528"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9f0613f08324d2f89cd1cc4d978c1a74afa2bbfd2253e64b9b0c9af6fa0f13e.json","authors":"Labos Elemer; Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A046528"},"id":"literature:url:https://oeis.org/A046528","identifiers":{"url":"https://oeis.org/A046528"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A046528, Numbers that are a product of distinct Mersenne primes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA046528","year":2013},{"aliases":["N. J. A. Sloane; Clark Kimberling","https://oeis.org/A049473"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43f0b6a2d9ba529a25752be12a9f7e18677337c1c59a863815c55f40c6ca461d.json","authors":"N. J. A. Sloane; Clark Kimberling","evidence":{"source_url":"https://oeis.org/A049473"},"id":"literature:url:https://oeis.org/A049473","identifiers":{"url":"https://oeis.org/A049473"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A049473, Nearest integer to n/sqrt(2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA049473","year":2014},{"aliases":["Labos Elemer; Benoit Cloitre","https://oeis.org/A049591"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24259afae3baa1fcec7685e9f92a678e9f413d6ec05009f261078168ab146a01.json","authors":"Labos Elemer; Benoit Cloitre","evidence":{"source_url":"https://oeis.org/A049591"},"id":"literature:url:https://oeis.org/A049591","identifiers":{"url":"https://oeis.org/A049591"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A049591, Odd primes p such that p+2 is composite","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA049591","year":2002},{"aliases":["Labos Elemer; Velin Yanev","https://oeis.org/A051903"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/378805d1a2f7576f917dd7f28e011467da0865a88b689729c9ffcf468c31a064.json","authors":"Labos Elemer; Velin Yanev","evidence":{"source_url":"https://oeis.org/A051903"},"id":"literature:url:https://oeis.org/A051903","identifiers":{"url":"https://oeis.org/A051903"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A051903, maximum exponent in the prime factorization of n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA051903","year":2017},{"aliases":["Harry Richman","https://oeis.org/A053871"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6c609dbc51e3cd7445727ecbd8a61ab9c0ff9e6f2597eec9f06ff65ae73c151e.json","authors":"Harry Richman","evidence":{"source_url":"https://oeis.org/A053871"},"id":"literature:url:https://oeis.org/A053871","identifiers":{"url":"https://oeis.org/A053871"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A053871, a(n) = 2*(n-1)*(a(n-1) + a(n-2)), starting a(0) = 1; a(1) = 0; congruence conjecture of Harry Richman","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA053871","year":2023},{"aliases":["Clark Kimberling; Ralf Stephan","https://oeis.org/A054096"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7072aa3e831c6d1ae1ad0cacecb64f59a7d0d45a5f68c4678b5c803f39112e7a.json","authors":"Clark Kimberling; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A054096"},"id":"literature:url:https://oeis.org/A054096","identifiers":{"url":"https://oeis.org/A054096"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A054096, T(n,2), array T as in A054090","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA054096","year":2004},{"aliases":["Marc LeBrun; Ralf Stephan","https://oeis.org/A059894"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/127e15bc9c08bee0a7312c7b89fc1d61ebbf2fd7c0acba62e97d424ee7e792f2.json","authors":"Marc LeBrun; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A059894"},"id":"literature:url:https://oeis.org/A059894","identifiers":{"url":"https://oeis.org/A059894"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A059894, complement and reverse all but the most significant bit","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA059894","year":2003},{"aliases":["Werner Schulte","https://oeis.org/A060693"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71753bd9babf853cb651623848767f234f8405dbe3b9f16c6aa6517166182c42.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A060693"},"id":"literature:url:https://oeis.org/A060693","identifiers":{"url":"https://oeis.org/A060693"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A060693, Triangle read by rows: T(n, k) is the number of Schröder paths from (0,0) to (2n,0) having k peaks","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA060693","year":2017},{"aliases":["Gus Wiseman","https://oeis.org/A062319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8aa468d0dfcefca93a1aad28c27822700a4bbf340e19ff9dfec7a0417cffefeb.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A062319"},"id":"literature:url:https://oeis.org/A062319","identifiers":{"url":"https://oeis.org/A062319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A062319, Number of divisors of n^n, or of A000312(n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA062319","year":2021},{"aliases":["Peter Bala","https://oeis.org/A064618"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a5b0a08172d01968d1d4d1a163b20353d55cfcd9bf7f84f1d379512913ad18cf.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A064618"},"id":"literature:url:https://oeis.org/A064618","identifiers":{"url":"https://oeis.org/A064618"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A064618, Stirling transform of (n!)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA064618","year":2018},{"aliases":["OEIS Foundation Inc.; Antti Karttunen; David A. Corneth","https://oeis.org/A068012"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1788ad9fef527001caf980bc6f9b1f1971fb39e5572bf95f2dbcac5ebb7dba05.json","authors":"OEIS Foundation Inc.; Antti Karttunen; David A. Corneth","evidence":{"source_url":"https://oeis.org/A068012"},"id":"literature:url:https://oeis.org/A068012","identifiers":{"url":"https://oeis.org/A068012"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A068012 subset sum doubling conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA068012","year":2025},{"aliases":["R. J. Mathar","https://oeis.org/A068551"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f240d31d9f2e8f601ff0d2444ef35d3d1ad371a5c8181728e4baaeddf21e32e.json","authors":"R. J. Mathar","evidence":{"source_url":"https://oeis.org/A068551"},"id":"literature:url:https://oeis.org/A068551","identifiers":{"url":"https://oeis.org/A068551"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A068551, a(n) = 4^n - binomial(2n, n): recurrence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA068551","year":2012},{"aliases":["Benoit Cloitre","https://oeis.org/A072872"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52476653ead33038f72b7465e09006b71bb156672c2f9ef2a9ab6669ef8191c0.json","authors":"Benoit Cloitre","evidence":{"source_url":"https://oeis.org/A072872"},"id":"literature:url:https://oeis.org/A072872","identifiers":{"url":"https://oeis.org/A072872"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A072872, a(n) is the smallest positive number k such that n divides 2^k - k","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA072872","year":2002},{"aliases":["R. J. Mathar","https://oeis.org/A073759"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c5ed0cf65673747c725eb4c649583cde9fb7e26a2126b2c790d3ba258b97b82.json","authors":"R. J. Mathar","evidence":{"source_url":"https://oeis.org/A073759"},"id":"literature:url:https://oeis.org/A073759","identifiers":{"url":"https://oeis.org/A073759"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A073759: largest number neither a divisor of nor relatively prime to n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA073759","year":2026},{"aliases":["Benoit Cloitre; Michel Marcus; Peter Bala","https://oeis.org/A074724"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5493cc9535a8e328c4accc9ad20a1ee7d10b90d84a562ccb5ed590052b15512a.json","authors":"Benoit Cloitre; Michel Marcus; Peter Bala","evidence":{"source_url":"https://oeis.org/A074724"},"id":"literature:url:https://oeis.org/A074724","identifiers":{"url":"https://oeis.org/A074724"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A074724, highest power of 3 dividing F(4n), with the Marcus and Bala conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA074724","year":2022},{"aliases":["Benoit Cloitre","https://oeis.org/A076502"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3ce7721a2adfaf10e6abcad2a51560de0a388065051a54df732fe8615d09622.json","authors":"Benoit Cloitre","evidence":{"source_url":"https://oeis.org/A076502"},"id":"literature:url:https://oeis.org/A076502","identifiers":{"url":"https://oeis.org/A076502"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A076502, a(1)=1, a(n)=n-a(n-a(n-a(n-1)))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA076502","year":2002},{"aliases":["Werner Schulte","https://oeis.org/A077864"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8eea1db2a870d24b6ab241a9baeeddd5429e2331a0a1f8f94f1ba6cf66985ace.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A077864"},"id":"literature:url:https://oeis.org/A077864","identifiers":{"url":"https://oeis.org/A077864"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A077864, Expansion of (1-x)^(-1)/(1-x-2*x^2-x^3)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA077864","year":2015},{"aliases":["Clark Kimberling","https://oeis.org/A077866"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/076a46e46b1f2c03ed5f61428ad2f0ce1fdf7dc0510acf081f5f3f9ef1cdc276.json","authors":"Clark Kimberling","evidence":{"source_url":"https://oeis.org/A077866"},"id":"literature:url:https://oeis.org/A077866","identifiers":{"url":"https://oeis.org/A077866"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A077866: Expansion of (1-x)^(-1)/(1-x-2*x^2+2*x^3)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA077866","year":2022},{"aliases":["Benoit Cloitre","https://oeis.org/A079063"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/973b6c788bbac8c2af624dc242ce97e6eadc4cd8a96db05fa652ce773673a40a.json","authors":"Benoit Cloitre","evidence":{"source_url":"https://oeis.org/A079063"},"id":"literature:url:https://oeis.org/A079063","identifiers":{"url":"https://oeis.org/A079063"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A079063, Least k such that sqrt(prime(n+k))-sqrt(prime(n))>1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA079063","year":2003},{"aliases":["Leroy Quet; N. J. A. Sloane","https://oeis.org/A079278"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe40286957a16f4abcf8113a2b5e43eb3a599b4e32a619ce1597a622f5d2f9a1.json","authors":"Leroy Quet; N. J. A. Sloane","evidence":{"source_url":"https://oeis.org/A079278"},"id":"literature:url:https://oeis.org/A079278","identifiers":{"url":"https://oeis.org/A079278"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A079278, denominators of the rational iteration b(n) = b(n-1) + 1/(1 + 1/b(n-1))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA079278","year":2003},{"aliases":["OEIS Foundation Inc.; Joseph A. Stocke","https://oeis.org/A084068"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a66e3ed17450016e0df2339271bcfc5b24384831d81d1dbed79a88c27b83cfcd.json","authors":"OEIS Foundation Inc.; Joseph A. Stocke","evidence":{"source_url":"https://oeis.org/A084068"},"id":"literature:url:https://oeis.org/A084068","identifiers":{"url":"https://oeis.org/A084068"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A084068 gcd conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA084068","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A088713"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/512f04a959b67f5569ce44d4721c0abcd2eed02826db977560f8ce37d2749d27.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A088713"},"id":"literature:url:https://oeis.org/A088713","identifiers":{"url":"https://oeis.org/A088713"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A088713, A(x/A(x)) = 1/(1-x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA088713","year":2003},{"aliases":["Cino Hilliard","https://oeis.org/A089610"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59239256096442674a5c4aa0bf33d8031c65bed4820d7d2884ae0d299939fea7.json","authors":"Cino Hilliard","evidence":{"source_url":"https://oeis.org/A089610"},"id":"literature:url:https://oeis.org/A089610","identifiers":{"url":"https://oeis.org/A089610"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A089610, Number of primes between n^2 and (n+1/2)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA089610","year":2003},{"aliases":["Reinhard Zumkeller; Velin Yanev","https://oeis.org/A090079"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82a64c53bcc514db991a242983f91bd7cc21a04d16a6fb6b07f40e95f944ba97.json","authors":"Reinhard Zumkeller; Velin Yanev","evidence":{"source_url":"https://oeis.org/A090079"},"id":"literature:url:https://oeis.org/A090079","identifiers":{"url":"https://oeis.org/A090079"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A090079, binary run compression","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA090079","year":2016},{"aliases":["Benoit Cloitre","https://oeis.org/A090825"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/79f8bab2a6b7cedc9d4e1590ce0828648c94f0b49a4943e68d0413dd1a318729.json","authors":"Benoit Cloitre","evidence":{"source_url":"https://oeis.org/A090825"},"id":"literature:url:https://oeis.org/A090825","identifiers":{"url":"https://oeis.org/A090825"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A090825, nonprimes n such that (3/2)(1/n)(2n+1)(3^n+1)B(2n) is an integer","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA090825","year":2004},{"aliases":["Paul D. Hanna","https://oeis.org/A091190"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4e3b9185152784b6de1ec3ec670d3fe9dbfa8425a159bcf27736d9a4e9a0c751.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A091190"},"id":"literature:url:https://oeis.org/A091190","identifiers":{"url":"https://oeis.org/A091190"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A091190 and A091200: the Somos prime integrality family","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA091190","year":2004},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A091259"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f22c811d176fe370b9b0b88f85191ad6772fea6417b9c9ffa61e17ece5eca9b.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A091259"},"id":"literature:url:https://oeis.org/A091259","identifiers":{"url":"https://oeis.org/A091259"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A091259","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA091259","year":2024},{"aliases":["Jon Perry; Ralf Stephan","https://oeis.org/A091915"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b81726e3250e693db3a29b4c8a67e6254bd4bd6d53ccc38fd7397319c7cade5.json","authors":"Jon Perry; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A091915"},"id":"literature:url:https://oeis.org/A091915","identifiers":{"url":"https://oeis.org/A091915"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A091915, Maximum of even products of partitions of n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA091915","year":2004},{"aliases":["Farideh Firoozbakht","https://oeis.org/A092028"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8a0f57631a3b159af458640d9fdddaf8df7d0a810224bc2c7e021bbbe9700d3b.json","authors":"Farideh Firoozbakht","evidence":{"source_url":"https://oeis.org/A092028"},"id":"literature:url:https://oeis.org/A092028","identifiers":{"url":"https://oeis.org/A092028"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A092028, a(n) is the smallest m > 1 such that m divides n^m-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA092028","year":2004},{"aliases":["Jon Perry","https://oeis.org/A094802"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6faa9b08f3e3e1cd0997fe76a1c92a1aaccb76c6fb3bbe13897263d4c5f4702e.json","authors":"Jon Perry","evidence":{"source_url":"https://oeis.org/A094802"},"id":"literature:url:https://oeis.org/A094802","identifiers":{"url":"https://oeis.org/A094802"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A094802, a(n) = smallest k such that all of 1 through n divides k!","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA094802","year":2004},{"aliases":["Amarnath Murthy","https://oeis.org/A096127"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85e2acf52af0fc1b7619f339a6bca6233f9e11460b1d26d505164f09fe9b696b.json","authors":"Amarnath Murthy","evidence":{"source_url":"https://oeis.org/A096127"},"id":"literature:url:https://oeis.org/A096127","identifiers":{"url":"https://oeis.org/A096127"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A096127, a(n) is the largest k such that (n^2)!/(n!)^k is an integer","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA096127","year":2004},{"aliases":["Reinhard Zumkeller; Vladeta Jovovic","https://oeis.org/A097602"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f1817b0d3efe2b9475076342e72411002787449a758346c0e3e57261d368b63.json","authors":"Reinhard Zumkeller; Vladeta Jovovic","evidence":{"source_url":"https://oeis.org/A097602"},"id":"literature:url:https://oeis.org/A097602","identifiers":{"url":"https://oeis.org/A097602"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A097602, a(n+1) = a(n) + number of squares so far; a(1) = 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA097602","year":2004},{"aliases":["Wouter Meeussen","https://oeis.org/A098052"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a672e1d4414544a8ecc7621758140b20ddd790de2499cbefca9bb8eab7c696c0.json","authors":"Wouter Meeussen","evidence":{"source_url":"https://oeis.org/A098052"},"id":"literature:url:https://oeis.org/A098052","identifiers":{"url":"https://oeis.org/A098052"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A098052, T(n,k) counts the solid partitions of n that can be extended to a solid partition of n+1 in exactly (k+3) ways; with its twin A098530","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA098052","year":2004},{"aliases":["Jonathan Vos Post; Reinhard Zumkeller","https://oeis.org/A100952"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/543b4073897e1cc28a26a0d7a99bd255cfddf18efaae0deb2c0e994db733fbd0.json","authors":"Jonathan Vos Post; Reinhard Zumkeller","evidence":{"source_url":"https://oeis.org/A100952"},"id":"literature:url:https://oeis.org/A100952","identifiers":{"url":"https://oeis.org/A100952"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A100952, Numbers that cannot be written as p*q+r with three distinct primes p, q and r","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA100952","year":2004},{"aliases":["Benoit Cloitre; Philippe Deléham; Ralf Stephan","https://oeis.org/A103585"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8c322aa9ae3ff0b5803501c39cc168307476a2f5de30bbea11089eacc39df590.json","authors":"Benoit Cloitre; Philippe Deléham; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A103585"},"id":"literature:url:https://oeis.org/A103585","identifiers":{"url":"https://oeis.org/A103585"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A103585, numbers k with (A102370(k)-k)/2 = 1, read mod 4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA103585","year":2005},{"aliases":["Zak Seidov; Ralf Stephan","https://oeis.org/A104863"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4eb5d658ebb5d521ce47cae2a6ec44ee9a6547c3c51e225a226d8c76639d30e0.json","authors":"Zak Seidov; Ralf Stephan","evidence":{"source_url":"https://oeis.org/A104863"},"id":"literature:url:https://oeis.org/A104863","identifiers":{"url":"https://oeis.org/A104863"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A104863, a(n) = floor(sqrt(a(n-1)^2 + a(n-2)^2)), a(1)=10, a(2)=30","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA104863","year":2005},{"aliases":["Paul Barry","https://oeis.org/A105595"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c91c945df1aae016c90c18131bbfbe15eb32cb4bca9ca2ad1b6daa8a6928fbd4.json","authors":"Paul Barry","evidence":{"source_url":"https://oeis.org/A105595"},"id":"literature:url:https://oeis.org/A105595","identifiers":{"url":"https://oeis.org/A105595"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A105595, Row sums of number triangle A105594","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA105595","year":2005},{"aliases":["Paul D. Hanna","https://oeis.org/A107099"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/26f7bf7fdf396e2b545c3c23149846a9ea9c50298d976d6db382451657910de6.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A107099"},"id":"literature:url:https://oeis.org/A107099","identifiers":{"url":"https://oeis.org/A107099"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A107099, compositional half-iterate of x + 4*x^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA107099","year":2005},{"aliases":["Ralf Stephan","https://oeis.org/A107928"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d2f6a5d5788417d3b252ca7454a8f587cebb45b21d4f96801a7c8565addabfe.json","authors":"Ralf Stephan","evidence":{"source_url":"https://oeis.org/A107928"},"id":"literature:url:https://oeis.org/A107928","identifiers":{"url":"https://oeis.org/A107928"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A107928, a(n) is the numerator of harmonic mean of a(n-1) and a(n-2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA107928","year":2010},{"aliases":["R. J. Mathar","https://oeis.org/A108958"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf2c7e186bb343159150ec38a52cd4d44d34c4a5d9d35abddc4b2b205f84adaa.json","authors":"R. J. Mathar","evidence":{"source_url":"https://oeis.org/A108958"},"id":"literature:url:https://oeis.org/A108958","identifiers":{"url":"https://oeis.org/A108958"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A108958, number of unordered pairs of distinct length-n binary words having the same number of 1's: recurrence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA108958","year":2012},{"aliases":["OEIS Foundation Inc.; Alan Michael Gómez Calderón","https://oeis.org/A110037"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa78702e7ea3401cce08cce5a601890307be9698ffe6f8a5997b8c4af0fc4646.json","authors":"OEIS Foundation Inc.; Alan Michael Gómez Calderón","evidence":{"source_url":"https://oeis.org/A110037"},"id":"literature:url:https://oeis.org/A110037","identifiers":{"url":"https://oeis.org/A110037"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A110037 signed non-squashing partition parity and its paperfold difference conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA110037","year":2025},{"aliases":["Amarnath Murthy","https://oeis.org/A110454"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fd0ad5d3302d348a3983419184da01c7fb70bd47afc15ef6ef32538cdc91923b.json","authors":"Amarnath Murthy","evidence":{"source_url":"https://oeis.org/A110454"},"id":"literature:url:https://oeis.org/A110454","identifiers":{"url":"https://oeis.org/A110454"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A110454, Largest composite number obtained by concatenation of parts of a distinct partition of n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA110454","year":2005},{"aliases":["Lechoslaw Ratajczak","https://oeis.org/A110491"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c93fb38255a437ab15d73ad044cfa7a7eeff651f064c19f1c8c186ee8b2cb908.json","authors":"Lechoslaw Ratajczak","evidence":{"source_url":"https://oeis.org/A110491"},"id":"literature:url:https://oeis.org/A110491","identifiers":{"url":"https://oeis.org/A110491"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A110491, T-transform determinant conjecture for A093178","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA110491","year":2021},{"aliases":["Pierre Cami; Harvey P. Dale","https://oeis.org/A112041"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/364f1baa8cd3044066cb80de84e106def461a841276dc68f98ebadcfa96f4fea.json","authors":"Pierre Cami; Harvey P. Dale","evidence":{"source_url":"https://oeis.org/A112041"},"id":"literature:url:https://oeis.org/A112041","identifiers":{"url":"https://oeis.org/A112041"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A112041, Numbers k such that 1*k + 1, 3*k + 1, 9*k + 1, 27*k + 1 are all primes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA112041","year":2015},{"aliases":["Johannes W. Meijer","https://oeis.org/A112302"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83e5f84c2af55b128e087979ba2b7a85649c8b2475ac68b3fcf09c9a2652f226.json","authors":"Johannes W. Meijer","evidence":{"source_url":"https://oeis.org/A112302"},"id":"literature:url:https://oeis.org/A112302","identifiers":{"url":"https://oeis.org/A112302"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A112302, Lerch recurrence limit conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA112302","year":2016},{"aliases":["Jamie Morken","https://oeis.org/A112399/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3fa161bbc9757063a224b69af72dd6075997a3b6a7bf6b7afc3c13e36e002659.json","authors":"Jamie Morken","evidence":{"source_url":"https://oeis.org/A112399/internal"},"id":"literature:url:https://oeis.org/A112399/internal","identifiers":{"url":"https://oeis.org/A112399/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"First divisor-sum conjecture in OEIS A112399","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA112399%2Finternal","year":2026},{"aliases":["Creighton Dement","https://oeis.org/A113249"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81a26141c4fb50b16a813bfe71e024fe869bfcf5474771cf199331ad6a93ee3b.json","authors":"Creighton Dement","evidence":{"source_url":"https://oeis.org/A113249"},"id":"literature:url:https://oeis.org/A113249","identifiers":{"url":"https://oeis.org/A113249"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A113249: odd-index squares in a fourth-order recurrence family","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA113249","year":2005},{"aliases":["R. J. Mathar; Paul Barry; G. C. Greubel","https://oeis.org/A113409"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f3c168d7554e2cf1e69ce17556573d2f96eb485842187eb03246de9bc64ee8d1.json","authors":"R. J. Mathar; Paul Barry; G. C. Greubel","evidence":{"source_url":"https://oeis.org/A113409"},"id":"literature:url:https://oeis.org/A113409","identifiers":{"url":"https://oeis.org/A113409"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A113409, A transform of the central binomial coefficients A001405","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA113409","year":2012},{"aliases":["Cino Hilliard; Robert Israel","https://oeis.org/A115039"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52dd0e3f699dc631958828da3329dec78d42e528dcd3b09c7605b1e5f217aab8.json","authors":"Cino Hilliard; Robert Israel","evidence":{"source_url":"https://oeis.org/A115039"},"id":"literature:url:https://oeis.org/A115039","identifiers":{"url":"https://oeis.org/A115039"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A115039 and A115038, prime square perfect-power representation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA115039","year":2006},{"aliases":["Antti Karttunen","https://oeis.org/A115770"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55886ec2faab0f83f08a29ea572d3dc15b5fa3c2b630fce6a30386e3d993e5d1.json","authors":"Antti Karttunen","evidence":{"source_url":"https://oeis.org/A115770"},"id":"literature:url:https://oeis.org/A115770","identifiers":{"url":"https://oeis.org/A115770"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A115770: seven and five versus binary carryless multiplication","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA115770","year":2025},{"aliases":["OEIS Foundation","https://oeis.org/A115801"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/14abb778d6a8cf79fe13bfeeb4584a9a7b7ee964761e0117d5fd97058d197f2e.json","authors":"OEIS Foundation","evidence":{"source_url":"https://oeis.org/A115801"},"id":"literature:url:https://oeis.org/A115801","identifiers":{"url":"https://oeis.org/A115801"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A115801: ordinary and binary carryless multiplier conditions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA115801","year":2026},{"aliases":["OEIS Foundation","https://oeis.org/A115803"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7fb623569de2ad04f8f464a2f1e873ef0e7851f040ac6cf2c648a14963f6d8ba.json","authors":"OEIS Foundation","evidence":{"source_url":"https://oeis.org/A115803"},"id":"literature:url:https://oeis.org/A115803","identifiers":{"url":"https://oeis.org/A115803"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A115803 and A115805: carryless multiplier containment","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA115803","year":2026},{"aliases":["Antti Karttunen","https://oeis.org/A115809"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/73dabef82e20e2d654fea1efe09efc4d40ddfa4f31139f31cf0cf0d2e82ed1bb.json","authors":"Antti Karttunen","evidence":{"source_url":"https://oeis.org/A115809"},"id":"literature:url:https://oeis.org/A115809","identifiers":{"url":"https://oeis.org/A115809"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A115809: seventeen-times binary carryless multiplier condition","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA115809","year":2006},{"aliases":["Alexander Adamchuk","https://oeis.org/A116184"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8c2869285a5ab2509ada21f54a15796d081de6af4c1028729b30af89c9f519e.json","authors":"Alexander Adamchuk","evidence":{"source_url":"https://oeis.org/A116184"},"id":"literature:url:https://oeis.org/A116184","identifiers":{"url":"https://oeis.org/A116184"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A116184, Numbers n such that 37^3 divides the numerator of generalized harmonic number H(36,n) = Sum[ 1/k^n, {k,1,36} ]","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA116184","year":2007},{"aliases":["Zak Seidov","https://oeis.org/A119623"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cf197a6dcef0ac50ecf34947f67eeef4874bed7a651f2b45a13072603eeb4d2.json","authors":"Zak Seidov","evidence":{"source_url":"https://oeis.org/A119623"},"id":"literature:url:https://oeis.org/A119623","identifiers":{"url":"https://oeis.org/A119623"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A119623, Composite numbers for which the second elementary symmetric function of divisors is prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA119623","year":2006},{"aliases":["Paolo P. Lava; Giorgio Balzarotti; John W. Layman","https://oeis.org/A119690"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d6f1b2b16324212662b9ebdd8fafd12ebc7b94fa95e41655f984074aa987ff2c.json","authors":"Paolo P. Lava; Giorgio Balzarotti; John W. Layman","evidence":{"source_url":"https://oeis.org/A119690"},"id":"literature:url:https://oeis.org/A119690","identifiers":{"url":"https://oeis.org/A119690"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A119690, n! mod n*(n+1)/2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA119690","year":2010},{"aliases":["Peter Bala","https://oeis.org/A122399"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8fa5cd9c85efc0e5675f5d9d9006741f50cca2e2a9088eb528f557822a5e5e30.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A122399"},"id":"literature:url:https://oeis.org/A122399","identifiers":{"url":"https://oeis.org/A122399"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A122399 (Jovovic 2006), conjecture: a(n) mod p is purely periodic with period p - 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA122399","year":2022},{"aliases":["Paul D. Hanna","https://oeis.org/A124418"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/84653e70e36407c60ea1cb468b867cf8543d7b0121b63df0b60aaaac022705bb.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A124418"},"id":"literature:url:https://oeis.org/A124418","identifiers":{"url":"https://oeis.org/A124418"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A124418, Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} having exactly k blocks that contain both odd and even entries","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA124418","year":2006},{"aliases":["Peter Bala","https://oeis.org/A126156"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/603c761750e4dc1c0bf5db5ad4c4152d2a0cc2d8fe915d30d526353ee5696e6e.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A126156"},"id":"literature:url:https://oeis.org/A126156","identifiers":{"url":"https://oeis.org/A126156"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A126156: eventual periods dividing the Euler totient","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA126156","year":2026},{"aliases":["Paul D. Hanna; Mikhail Kurkov","https://oeis.org/A126347"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0ea1837c0ed62784faa6f2ee8984fefdd9ccad1418643b123e536368f2e6b4b.json","authors":"Paul D. Hanna; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A126347"},"id":"literature:url:https://oeis.org/A126347","identifiers":{"url":"https://oeis.org/A126347"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A126347 q-Bell coefficient triangle and Kurkov's reversed-row conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA126347","year":2025},{"aliases":["Thomas Ordowski","https://oeis.org/A126762"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32291062ee25ad738facb94b9dcfb0c5136da1f1f0a17d9638599470c9b619fa.json","authors":"Thomas Ordowski","evidence":{"source_url":"https://oeis.org/A126762"},"id":"literature:url:https://oeis.org/A126762","identifiers":{"url":"https://oeis.org/A126762"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A126762, a(n) is the least k > n such that the remainder when n^k is divided by k is n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA126762","year":2018},{"aliases":["Antti Karttunen","https://oeis.org/A129598"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db3640c3e0242aa00c7ab1d00ab0dd73dbf3dc7e4d473b3600321b8470acf7f8.json","authors":"Antti Karttunen","evidence":{"source_url":"https://oeis.org/A129598"},"id":"literature:url:https://oeis.org/A129598","identifiers":{"url":"https://oeis.org/A129598"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A129598, a(n) = n * A111089(n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA129598","year":2007},{"aliases":["Reinhard Zumkeller","https://oeis.org/A131853"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/24410c89ae1524a883b04cd4793c3b8934260f55214d7277c8fc44769d53a0a7.json","authors":"Reinhard Zumkeller","evidence":{"source_url":"https://oeis.org/A131853"},"id":"literature:url:https://oeis.org/A131853","identifiers":{"url":"https://oeis.org/A131853"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A131853, Numbers m such that z(m)=(0,0) with z as defined in A131851","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA131853","year":2007},{"aliases":["Mikhail Kurkov","https://oeis.org/A132393"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bf712932798ed50a2ce72d0a92da90797f6693f2854f864589785ccb752661ea.json","authors":"Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A132393"},"id":"literature:url:https://oeis.org/A132393","identifiers":{"url":"https://oeis.org/A132393"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A132393: Kurkov's nested sum for unsigned Stirling numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA132393","year":2026},{"aliases":["Klaus Brockhaus; Zak Seidov","https://oeis.org/A133901"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc6fae22228bb9d232488188a836b16abd4cb51bd16467e5d22279eeeec9e2e2.json","authors":"Klaus Brockhaus; Zak Seidov","evidence":{"source_url":"https://oeis.org/A133901"},"id":"literature:url:https://oeis.org/A133901","identifiers":{"url":"https://oeis.org/A133901"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A133901, Numbers in A128921 whose square is not a palindrome","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA133901","year":2007},{"aliases":["Thomas Ordowski","https://oeis.org/A133907"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97b66489548532ba6270f04055ca680f9de65d628adf13662fcac7300855de7d.json","authors":"Thomas Ordowski","evidence":{"source_url":"https://oeis.org/A133907"},"id":"literature:url:https://oeis.org/A133907","identifiers":{"url":"https://oeis.org/A133907"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A133907, least prime p with binomial(n+p,p) ≡ 1 (mod p), with the power-sum least-prime conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA133907","year":2018},{"aliases":["Felix Huber; Artur Jasinski","https://oeis.org/A134492"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55754d53ff96901a1692cfc9b365fcfb965297b498ff19f223357f2b015a770a.json","authors":"Felix Huber; Artur Jasinski","evidence":{"source_url":"https://oeis.org/A134492"},"id":"literature:url:https://oeis.org/A134492","identifiers":{"url":"https://oeis.org/A134492"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A134492, a(n) = Fibonacci(6*n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA134492","year":2023},{"aliases":["Harry J. Smith; Jaroslav Krizek","https://oeis.org/A136162"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f70883e453854f76b57dee5d6fcea96d7831e686a5736964d7940d02c1be7bf.json","authors":"Harry J. Smith; Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A136162"},"id":"literature:url:https://oeis.org/A136162","identifiers":{"url":"https://oeis.org/A136162"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A136162, List of prime quadruplets {p, p+2, p+6, p+8}","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA136162","year":2017},{"aliases":["Paul Curtz and Benedict W. J. Irwin","https://oeis.org/A138099"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bb5dd35da8f687fa65bd2ebe2fc3855feee12f22d0795ba18685cf9d3339d768.json","authors":"Paul Curtz and Benedict W. J. Irwin","evidence":{"source_url":"https://oeis.org/A138099"},"id":"literature:url:https://oeis.org/A138099","identifiers":{"url":"https://oeis.org/A138099"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A138099: irregular triangle and Irwin's hook-sum conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA138099","year":2016},{"aliases":["Vincenzo Librandi","https://oeis.org/A140869"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0664a004a83c2c81c1376f0cb5fb5f3a3bea4664eb24b97899e903947ec7db06.json","authors":"Vincenzo Librandi","evidence":{"source_url":"https://oeis.org/A140869"},"id":"literature:url:https://oeis.org/A140869","identifiers":{"url":"https://oeis.org/A140869"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A140869, Triangle read by rows where T(m,n) = floor((2mn+m+n-2)/2), m >= n >= 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA140869","year":2012},{"aliases":["Peter Bala","https://oeis.org/A144837"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97a4e84aff40803684bcc181a943df5c6cb9dc2691eb9ce8e75cb3312cdc387e.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A144837"},"id":"literature:url:https://oeis.org/A144837","identifiers":{"url":"https://oeis.org/A144837"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A144837, Lucas(5^n), with the 5-adic congruence conjecture a(n+1) ≡ a(n) (mod 5^(n+r+1))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA144837","year":2022},{"aliases":["Mats Granvik","https://oeis.org/A145110"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d3ec6a0d170823a08f288bab942411906db251a8274a6489614249d87ed3982.json","authors":"Mats Granvik","evidence":{"source_url":"https://oeis.org/A145110"},"id":"literature:url:https://oeis.org/A145110","identifiers":{"url":"https://oeis.org/A145110"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A145110, Number of elements in the Redheffer matrix that contribute to the Moebius function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA145110","year":2008},{"aliases":["OEIS Foundation Inc.; Peter Bala; Max Alekseyev","https://oeis.org/A146557"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66694fdd8ace853fba92d2630eabdd21a586c60447af83ff2df4987618a9db11.json","authors":"OEIS Foundation Inc.; Peter Bala; Max Alekseyev","evidence":{"source_url":"https://oeis.org/A146557"},"id":"literature:url:https://oeis.org/A146557","identifiers":{"url":"https://oeis.org/A146557"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A146557 collinear triple divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA146557","year":2025},{"aliases":["Andres Cicuttin","https://oeis.org/A152020"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43f2b57b878644f1463e1bb60553f19464b606fca11a2a57dbc78e915c56fdaf.json","authors":"Andres Cicuttin","evidence":{"source_url":"https://oeis.org/A152020"},"id":"literature:url:https://oeis.org/A152020","identifiers":{"url":"https://oeis.org/A152020"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A152020, Denominator of 8/(9*n^2) divided by 9","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA152020","year":2017},{"aliases":["Emeric Deutsch; R. J. Mathar","https://oeis.org/A152661"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/99252b79cb54b227a2bd4137c26c947641e44709b716decc2ba2256fbb897f5c.json","authors":"Emeric Deutsch; R. J. Mathar","evidence":{"source_url":"https://oeis.org/A152661"},"id":"literature:url:https://oeis.org/A152661","identifiers":{"url":"https://oeis.org/A152661"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A152661: permutations whose first two elements have the same parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA152661","year":2008},{"aliases":["Paul D. Hanna","https://oeis.org/A156216/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9b32d80c1119f42f7c9e14fbf574e9867ef450952b1f1e1aec6cc542aa416f02.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A156216/internal"},"id":"literature:url:https://oeis.org/A156216/internal","identifiers":{"url":"https://oeis.org/A156216/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A156216: diagonal ghost-power integrality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA156216%2Finternal","year":2009},{"aliases":["N. J. A. Sloane; Robert Israel","https://oeis.org/A160627"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c44293fe6b2ff36b98a954eeff883003eaa2c2907784f5e6831c9c9f20494320.json","authors":"N. J. A. Sloane; Robert Israel","evidence":{"source_url":"https://oeis.org/A160627"},"id":"literature:url:https://oeis.org/A160627","identifiers":{"url":"https://oeis.org/A160627"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A160627, Numerator of Laguerre(n, 4)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA160627","year":2018},{"aliases":["Velin Yanev","https://oeis.org/A163617"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3289cf255f2ab69e9eb62e9634841d366f1c2b6f215b1e5f20b35afa843b90bb.json","authors":"Velin Yanev","evidence":{"source_url":"https://oeis.org/A163617"},"id":"literature:url:https://oeis.org/A163617","identifiers":{"url":"https://oeis.org/A163617"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A163617, a(2*n) = 2*a(n), a(2*n + 1) = 2*a(n) + 2 + (-1)^n, for all n in Z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA163617","year":2016},{"aliases":["Vladimir Joseph Stephan Orlovsky; Bill McEachen","https://oeis.org/A165719"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6eacfb9c2583e10abab4ce5a16ab711ae95f6f9645f0a6eed1c75be6b8833763.json","authors":"Vladimir Joseph Stephan Orlovsky; Bill McEachen","evidence":{"source_url":"https://oeis.org/A165719"},"id":"literature:url:https://oeis.org/A165719","identifiers":{"url":"https://oeis.org/A165719"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A165719, Integers of the form k*(k+9)/8","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA165719","year":2009},{"aliases":["Paul D. Hanna","https://oeis.org/A167138"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/565d518d3faac1a36d96e1ab399876fb76e400f3cb5b5497b9b67b244dac56b8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A167138"},"id":"literature:url:https://oeis.org/A167138","identifiers":{"url":"https://oeis.org/A167138"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Hanna's logarithmic integer-series power closure conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA167138","year":2009},{"aliases":["Paul D. Hanna","https://oeis.org/A167540"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/26fe4d1633b38279e3411e90c319afa96ff3021e851949edfe6aa7257386e495.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A167540"},"id":"literature:url:https://oeis.org/A167540","identifiers":{"url":"https://oeis.org/A167540"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Hanna's logarithmic integer-series arithmetic-subsequence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA167540","year":2009},{"aliases":["T. D. Noe","https://oeis.org/A167857"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d91462e8878875180e0ccffd51b629f79b7718f20691aeb65f32d8e79bfc0f5f.json","authors":"T. D. Noe","evidence":{"source_url":"https://oeis.org/A167857"},"id":"literature:url:https://oeis.org/A167857","identifiers":{"url":"https://oeis.org/A167857"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A167857: Numbers whose divisors are represented by an integer polynomial","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA167857","year":2009},{"aliases":["R. J. Mathar; OEIS Foundation Inc.","https://oeis.org/A174193"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8899518d6911046dc41652607d81426920ff2bab60bce598e4d2909ca15986e9.json","authors":"R. J. Mathar; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A174193"},"id":"literature:url:https://oeis.org/A174193","identifiers":{"url":"https://oeis.org/A174193"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A174193 — an order-four recurrence for an algebraic generating series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA174193","year":2016},{"aliases":["Paul D. Hanna","https://oeis.org/A174470"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/313449c906b1fca717181327be40d8f7ddb0fb6ea5f1d0833497ed2d73434030.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A174470"},"id":"literature:url:https://oeis.org/A174470","identifiers":{"url":"https://oeis.org/A174470"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A174470: Thue/Roth weighted exponential integrality","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA174470","year":2010},{"aliases":["Charles R Greathouse IV; Zak Seidov","https://oeis.org/A175406"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1925d8db731562ab64aafcde5e2b313094248f7c5bc48be91f9ae8809bf1835f.json","authors":"Charles R Greathouse IV; Zak Seidov","evidence":{"source_url":"https://oeis.org/A175406"},"id":"literature:url:https://oeis.org/A175406","identifiers":{"url":"https://oeis.org/A175406"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A175406, The greatest integer k such that (1+1/n)^k <= 2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA175406","year":2012},{"aliases":["Juri-Stepan Gerasimov","https://oeis.org/A177680"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ee5d82f0a96276ce46ee96dfc5e7bf6f1c18659c57358bf7fe3d6432185ef712.json","authors":"Juri-Stepan Gerasimov","evidence":{"source_url":"https://oeis.org/A177680"},"id":"literature:url:https://oeis.org/A177680","identifiers":{"url":"https://oeis.org/A177680"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A177680, Numbers n such that 6n and 12n are both the average of twin prime pairs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA177680","year":2010},{"aliases":["Richard J. Mathar","https://oeis.org/A178294"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f82689197fb04562e3f31eb5147758eff8656715e7ce15d81d31e74f5ca9ecd.json","authors":"Richard J. Mathar","evidence":{"source_url":"https://oeis.org/A178294"},"id":"literature:url:https://oeis.org/A178294","identifiers":{"url":"https://oeis.org/A178294"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A178294, Number of collinear point triples in a 4 X 4 X 4 X... n-dimensional cubic grid","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA178294","year":2010},{"aliases":["Colin Barker","https://oeis.org/A181278"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ca4a95cfcfca1dab162a5c98d082b04c37ae417269b472d14b9669ef63292a52.json","authors":"Colin Barker","evidence":{"source_url":"https://oeis.org/A181278"},"id":"literature:url:https://oeis.org/A181278","identifiers":{"url":"https://oeis.org/A181278"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A181278, binary matrices with ordered rows and decreasing mod-2 Gram rows","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA181278","year":2018},{"aliases":["Paul Curtz","https://oeis.org/A189733"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/510489f72c8f70a52236c16ab8674a1a2e36f09ff90d9184ceb76bd17800e235.json","authors":"Paul Curtz","evidence":{"source_url":"https://oeis.org/A189733"},"id":"literature:url:https://oeis.org/A189733","identifiers":{"url":"https://oeis.org/A189733"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A189733, denominators of the first row of a rational difference array","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA189733","year":2011},{"aliases":["Clark Kimberling","https://oeis.org/A192023"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb69dcd258a5caa8a45c7b8266e7bc6af0de643ed29834d280ed9ad1a0f69f2f.json","authors":"Clark Kimberling","evidence":{"source_url":"https://oeis.org/A192023"},"id":"literature:url:https://oeis.org/A192023","identifiers":{"url":"https://oeis.org/A192023"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A192023, The Wiener index of the comb-shaped graph |_|_|...|_| with 2n (n>=1) nodes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA192023","year":2012},{"aliases":["Mats Granvik; Olivier Gérard; Michael Somos","https://oeis.org/A192763"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f768e8ee5211b6ebf79d71b36673ccbcadff98834e2bd0ed9e19617f2265d6a3.json","authors":"Mats Granvik; Olivier Gérard; Michael Somos","evidence":{"source_url":"https://oeis.org/A192763"},"id":"literature:url:https://oeis.org/A192763","identifiers":{"url":"https://oeis.org/A192763"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A192763 and the Granvik Mertens array","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA192763","year":2011},{"aliases":["John W. Layman","https://oeis.org/A193020"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ec1fb7b15f03ae27c7aa354ce517c52d937d88141134f7f97265bfbc3bf77e91.json","authors":"John W. Layman","evidence":{"source_url":"https://oeis.org/A193020"},"id":"literature:url:https://oeis.org/A193020","identifiers":{"url":"https://oeis.org/A193020"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A193020, distinct binary self-shuffles","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA193020","year":2011},{"aliases":["https://oeis.org/A193236/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66d56b51b4389ff9c1fe0979330d6184c6bff37dafefd6ed9ffa7d5c8d01a9af.json","evidence":{"source_url":"https://oeis.org/A193236/internal"},"id":"literature:url:https://oeis.org/A193236/internal","identifiers":{"url":"https://oeis.org/A193236/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A193236/internal","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA193236%2Finternal"},{"aliases":["Peter Bala; Paul D. Hanna","https://oeis.org/A193237/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/76d61675543ccced06854eeda06ff8f4e10fad8c796b5f93ca0e40e45f94122c.json","authors":"Peter Bala; Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A193237/internal"},"id":"literature:url:https://oeis.org/A193237/internal","identifiers":{"url":"https://oeis.org/A193237/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A193237 and A193236, congruences for triangular reciprocal cube roots","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA193237%2Finternal","year":2021},{"aliases":["Michel Lagneau","https://oeis.org/A193351"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ce2103d04208fdd7acc2aea7f2ef33b54daf9c42c38bb837fa578e3c2ffc1c84.json","authors":"Michel Lagneau","evidence":{"source_url":"https://oeis.org/A193351"},"id":"literature:url:https://oeis.org/A193351","identifiers":{"url":"https://oeis.org/A193351"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A193351, Numbers k such that A071324(k) is prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA193351","year":2012},{"aliases":["John W. Layman","https://oeis.org/A196226"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e6c087c8d64d60cefb7ccc485be2027403dde1fe6c76628831260bd5d19e4bfe.json","authors":"John W. Layman","evidence":{"source_url":"https://oeis.org/A196226"},"id":"literature:url:https://oeis.org/A196226","identifiers":{"url":"https://oeis.org/A196226"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A196226, divisor-sum residue and divisor-power congruence conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA196226","year":2011},{"aliases":["Paul D. Hanna","https://oeis.org/A204061"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e919e289d3b3c1bdfbb4b1fdc16ac9fa48944a405556f1f42016f2edeb838e5.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A204061"},"id":"literature:url:https://oeis.org/A204061","identifiers":{"url":"https://oeis.org/A204061"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A204061, G.f.: exp( Sum_{n>=1} A001333(n)^2 * x^n/n )","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA204061","year":2012},{"aliases":["Werner Schulte","https://oeis.org/A208342"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a27e6cda78a965087ad801adab890e822cfa57957b3a461477d82670ebacb6d.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A208342"},"id":"literature:url:https://oeis.org/A208342","identifiers":{"url":"https://oeis.org/A208342"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A208342, Triangle of coefficients of polynomials u(n,x) jointly generated with A208343","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA208342","year":2017},{"aliases":["Knud Werner","https://oeis.org/A208509"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eb91df1e7a17dce3acfd097589be8aa2c7cd6990c4b1059343d3327a5d4ce70b.json","authors":"Knud Werner","evidence":{"source_url":"https://oeis.org/A208509"},"id":"literature:url:https://oeis.org/A208509","identifiers":{"url":"https://oeis.org/A208509"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Conjectured coefficient formula in OEIS A208509","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA208509","year":2022},{"aliases":["R. H. Hardin; Colin Barker","https://oeis.org/A209723/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e50a19ecdcf87a5e4176f31bcfd92381cbfe193035ed0c5cb9ed469f4012a77a.json","authors":"R. H. Hardin; Colin Barker","evidence":{"source_url":"https://oeis.org/A209723/internal"},"id":"literature:url:https://oeis.org/A209723/internal","identifiers":{"url":"https://oeis.org/A209723/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A209723: five-column arrays with distinct clockwise edge differences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA209723%2Finternal","year":2012},{"aliases":["Paul D. Hanna","https://oeis.org/A219431"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0b2061a9da9ccc17285bbdc0abac18f837c6b225fd5aecc916d6549639a1cab1.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A219431"},"id":"literature:url:https://oeis.org/A219431","identifiers":{"url":"https://oeis.org/A219431"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A219431: logarithmic derivative of square-indexed overpartitions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA219431","year":2012},{"aliases":["Peter Bala","https://oeis.org/A220181"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b208b8b068b662c4358e27bb1bc68755cff30a8f5efe1a7f173a945dfa239657.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A220181"},"id":"literature:url:https://oeis.org/A220181","identifiers":{"url":"https://oeis.org/A220181"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A220181, e.g.f. Sum_{n>=0} (1 - exp(-n*x))^n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA220181","year":2022},{"aliases":["Chai Wah Wu; Alex Ratushnyak","https://oeis.org/A221018/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7e6a6ca9e63906ffec6eec25eb4d2f9a86cac3d7e00e3ff7ff6b6e9d99e242c8.json","authors":"Chai Wah Wu; Alex Ratushnyak","evidence":{"source_url":"https://oeis.org/A221018/internal"},"id":"literature:url:https://oeis.org/A221018/internal","identifiers":{"url":"https://oeis.org/A221018/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A221018, first FORMULA conjecture on adjacent-index Fibonacci divisibility","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA221018%2Finternal","year":2026},{"aliases":["Peter Bala","https://oeis.org/A221077"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e40557b506ae93a3ba0d81a7c25ff45b2702df1f1d03eabc104ddfcc46bde97f.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A221077"},"id":"literature:url:https://oeis.org/A221077","identifiers":{"url":"https://oeis.org/A221077"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A221077, e.g.f. Sum_{n>=0} tanh(n*x)^n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA221077","year":2022},{"aliases":["R. H. Hardin; Colin Barker","https://oeis.org/A222001"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c379fdee6665fa5f23defad62c9abeb427abf5a494d3c52d947d036f0bd9e724.json","authors":"R. H. Hardin; Colin Barker","evidence":{"source_url":"https://oeis.org/A222001"},"id":"literature:url:https://oeis.org/A222001","identifiers":{"url":"https://oeis.org/A222001"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A222001, Number of n X 3 arrays with permutation rows and compatible descent and lexicographic orders","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA222001","year":2013},{"aliases":["Paul D. Hanna","https://oeis.org/A222013"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/110ca3075f797716b300e12cd3526e48bc74e2a84117cde1fc9a8fed52402487.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A222013"},"id":"literature:url:https://oeis.org/A222013","identifiers":{"url":"https://oeis.org/A222013"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A222013, g.f. A(x) = Sum n! x^n A(x)^(n(n+1)/2) / Product (1 + k x A(x)^k), parity conjecture (Dec 2024)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA222013","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A222014"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b646ebb5c3991b6931669edb0efbf5aeed6257084eb66d036eb9a2f91cb1feb8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A222014"},"id":"literature:url:https://oeis.org/A222014","identifiers":{"url":"https://oeis.org/A222014"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A222014, factorial square-exponent product-sum generating function and parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA222014","year":2024},{"aliases":["T. D. Noe","https://oeis.org/A222759"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52395c590a57c44bdce1d28de51f7a59cdde156bb4c53cac14790cd2cbd4a4b8.json","authors":"T. D. Noe","evidence":{"source_url":"https://oeis.org/A222759"},"id":"literature:url:https://oeis.org/A222759","identifiers":{"url":"https://oeis.org/A222759"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A222759: first growth assertion for literal binomial remainders","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA222759","year":2013},{"aliases":["R. H. Hardin; Gus Wiseman","https://oeis.org/A222955"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6754c2a0504edac6eb13105f5f4a3c20ec821f61d13c03066fd98d3a2c6a6ad4.json","authors":"R. H. Hardin; Gus Wiseman","evidence":{"source_url":"https://oeis.org/A222955"},"id":"literature:url:https://oeis.org/A222955","identifiers":{"url":"https://oeis.org/A222955"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A222955, Number of n X 1 0..1 arrays with every row and column least squares fitting to a zero slope straight line","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA222955","year":2023},{"aliases":["Peter Bala","https://oeis.org/A224899"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2aa32dc13f65e6b0a4a8835dc722f18d70b25c72726c7c2bbc0d43afc8401c3f.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A224899"},"id":"literature:url:https://oeis.org/A224899","identifiers":{"url":"https://oeis.org/A224899"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A224899, e.g.f. Sum_{n>=0} sinh(n*x)^n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA224899","year":2022},{"aliases":["J. Lowell","https://oeis.org/A225004"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1a32327087f0188df016cae3bf4313180720f99fb6051387a4d7b17fd6d6e7de.json","authors":"J. Lowell","evidence":{"source_url":"https://oeis.org/A225004"},"id":"literature:url:https://oeis.org/A225004","identifiers":{"url":"https://oeis.org/A225004"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A225004, a(n) is the largest multiple of n with fewer than twice as many divisors as n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA225004","year":2013},{"aliases":["Paul D. Hanna","https://oeis.org/A226839"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b7a09b52f100c51fe74e0ac50a044939782877d1d60f98dbfc63482c848e9db5.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A226839"},"id":"literature:url:https://oeis.org/A226839","identifiers":{"url":"https://oeis.org/A226839"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A226839: triangular-exponent exponential generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA226839","year":2013},{"aliases":["Colin Barker; Alonso del Arte","https://oeis.org/A226857"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0fd20b2da8c16fd5d8d675b0776e470ca599d8e284ce85c74e1f0e5dd93537ab.json","authors":"Colin Barker; Alonso del Arte","evidence":{"source_url":"https://oeis.org/A226857"},"id":"literature:url:https://oeis.org/A226857","identifiers":{"url":"https://oeis.org/A226857"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A226857, Numbers that are both the sum of two Fibonacci numbers and the product of two Fibonacci numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA226857","year":2014},{"aliases":["Paul D. Hanna","https://oeis.org/A227051"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82e291aaedafa222dce3c9a7c9e2d269c388d2a86ef4887381e3fc0ebcd91c9b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A227051"},"id":"literature:url:https://oeis.org/A227051","identifiers":{"url":"https://oeis.org/A227051"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A227051: mixed trigonometric and hyperbolic reciprocal","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA227051","year":2013},{"aliases":["Robert Price","https://oeis.org/A228558"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/12d687974310b9245d38a9c61a1c0667f1bda9678c95d6e36cd33fdd985f04ac.json","authors":"Robert Price","evidence":{"source_url":"https://oeis.org/A228558"},"id":"literature:url:https://oeis.org/A228558","identifiers":{"url":"https://oeis.org/A228558"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A228558, Numbers k such that (17^k + 4^k)/21 is prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA228558","year":2013},{"aliases":["Zhi-Wei Sun","https://oeis.org/A228884/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3011226459dd130327c332de7f218066229e86fc5d860f9532552fd7a36640d.json","authors":"Zhi-Wei Sun","evidence":{"source_url":"https://oeis.org/A228884/internal"},"id":"literature:url:https://oeis.org/A228884/internal","identifiers":{"url":"https://oeis.org/A228884/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Determinant of the n X n matrix with (i,j)-entry equal to the greatest common divisor of i-j and n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA228884%2Finternal","year":2013},{"aliases":["Ralf Stephan","https://oeis.org/A229140"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8654f4c65f79abdea0282a031aa4f50a2fade947050033877a31669392f8487a.json","authors":"Ralf Stephan","evidence":{"source_url":"https://oeis.org/A229140"},"id":"literature:url:https://oeis.org/A229140","identifiers":{"url":"https://oeis.org/A229140"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A229140, Smallest k such that k^2 + l^2 = n-th number expressible as sum of two squares","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA229140","year":2013},{"aliases":["https://oeis.org/A231329"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cb5787d85183a611aa5b31d6903ae319960da4dfa58f4dab5c32c3112c27d3f.json","evidence":{"source_url":"https://oeis.org/A231329"},"id":"literature:url:https://oeis.org/A231329","identifiers":{"url":"https://oeis.org/A231329"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A231329","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA231329"},{"aliases":["Jaroslav Krizek","https://oeis.org/A231548"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9123e6ee526be9dc95d370b06cd440e6410bd38547fff8f9d95df39303f171d.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A231548"},"id":"literature:url:https://oeis.org/A231548","identifiers":{"url":"https://oeis.org/A231548"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A231548, Numbers n such that 2*n - 1 < sigma(n) - sigma(n-2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA231548","year":2013},{"aliases":["https://oeis.org/A232686"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff1bbea7b24c81559d3a7b980759628a743c179d6102c52bd82d48fd25f1b099.json","evidence":{"source_url":"https://oeis.org/A232686"},"id":"literature:url:https://oeis.org/A232686","identifiers":{"url":"https://oeis.org/A232686"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A232686","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA232686"},{"aliases":["Paul D. Hanna","https://oeis.org/A232694"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ced3db77a9e05650a6357fed3f375df49b937c572d555750119acf16beea64f3.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A232694"},"id":"literature:url:https://oeis.org/A232694","identifiers":{"url":"https://oeis.org/A232694"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Derivative-substitution exponential generating functions A232694, A232695, A232696 and A232686","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA232694","year":2013},{"aliases":["https://oeis.org/A232695"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e613bd9bf0a3aabfb50bb2f88fb9d0b6b24e3711d22abd60e763756008f9d7f4.json","evidence":{"source_url":"https://oeis.org/A232695"},"id":"literature:url:https://oeis.org/A232695","identifiers":{"url":"https://oeis.org/A232695"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A232695","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA232695"},{"aliases":["https://oeis.org/A232696"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09fea90aa1856320e0c8ddda2213137177f9c3a2f9c0692361632765ec0e57a2.json","evidence":{"source_url":"https://oeis.org/A232696"},"id":"literature:url:https://oeis.org/A232696","identifiers":{"url":"https://oeis.org/A232696"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A232696","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA232696"},{"aliases":["Wolfdieter Lang","https://oeis.org/A238474"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4540a58b21cf66e6e5dc833144268f32b4cdd19971d22d8fd12b6f20d4fd6838.json","authors":"Wolfdieter Lang","evidence":{"source_url":"https://oeis.org/A238474"},"id":"literature:url:https://oeis.org/A238474","identifiers":{"url":"https://oeis.org/A238474"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A238474: a signed factorial quotient","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA238474","year":2014},{"aliases":["Robert Ferreol; Thomas Ordowski","https://oeis.org/A239293"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2fa5ed68b6575fd727a9792ac73fe65e885c282f16871465825c4bd2d8e37140.json","authors":"Robert Ferreol; Thomas Ordowski","evidence":{"source_url":"https://oeis.org/A239293"},"id":"literature:url:https://oeis.org/A239293","identifiers":{"url":"https://oeis.org/A239293"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A239293, Smallest composite c > n such that n^c == n (mod c)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA239293","year":2018},{"aliases":["N. J. A. Sloane; Mikhail Kurkov","https://oeis.org/A239907"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8b95cf52719b1ff977bca6921cda7a30323ca63d9ff4834cb3a2b64608edad20.json","authors":"N. J. A. Sloane; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A239907"},"id":"literature:url:https://oeis.org/A239907","identifiers":{"url":"https://oeis.org/A239907"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A239907, alternating count of runs of ones, with Kurkov's A329320 identity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA239907","year":2021},{"aliases":["Jaroslav Krizek","https://oeis.org/A245786"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2706536219d10c0b03a85d3d9bd3c5576c067bfd1d0daf75f93f9dd487f1f0b5.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A245786"},"id":"literature:url:https://oeis.org/A245786","identifiers":{"url":"https://oeis.org/A245786"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A245786, Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA245786","year":2014},{"aliases":["Paul D. Hanna","https://oeis.org/A246056"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1ef386e6850dafad3935880440edc5d64a6af83c07c816212214f5e2ce736e0.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A246056"},"id":"literature:url:https://oeis.org/A246056","identifiers":{"url":"https://oeis.org/A246056"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A246056 and A246423, two binomial-square generating functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA246056","year":2014},{"aliases":["https://oeis.org/A246423"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/834e0c495b92e21f8fc3d9a4e409b8bf9c83b1cc4f5a24cdad7249bee4b96874.json","evidence":{"source_url":"https://oeis.org/A246423"},"id":"literature:url:https://oeis.org/A246423","identifiers":{"url":"https://oeis.org/A246423"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A246423","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA246423"},{"aliases":["Chai Wah Wu","https://oeis.org/A249621"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e988ed4bb1f3a138ccfc33d7c1cc68de53383f8ac0656c7d42ca05e26c038933.json","authors":"Chai Wah Wu","evidence":{"source_url":"https://oeis.org/A249621"},"id":"literature:url:https://oeis.org/A249621","identifiers":{"url":"https://oeis.org/A249621"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A249621, Squares that remain squares when prepended with 1 and appended with 25 in base 10","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA249621","year":2014},{"aliases":["Jaroslav Krizek","https://oeis.org/A249759"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f4cb1cbebae44c16718e57d10cff1d9d1816db881c1b6e8b5c7f1fa03be4f483.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A249759"},"id":"literature:url:https://oeis.org/A249759","identifiers":{"url":"https://oeis.org/A249759"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A249759, primes p for which sigma(p-1) is prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA249759","year":2014},{"aliases":["Felix Fröhlich; Chai Wah Wu","https://oeis.org/A254074"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e960b211bf9d1d59153e15351389f260240c5ad1e97b4f976ede8600dcfbe0d4.json","authors":"Felix Fröhlich; Chai Wah Wu","evidence":{"source_url":"https://oeis.org/A254074"},"id":"literature:url:https://oeis.org/A254074","identifiers":{"url":"https://oeis.org/A254074"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A254074, Numbers k such that the decimal expansions of both k and k^2 have 5 as the digit with the smallest value and 9 as the digit with the largest value","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA254074","year":2017},{"aliases":["Juri-Stepan Gerasimov","https://oeis.org/A254748"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82022573b511a5839d66848c425c94d7f22ecf56e1fdbb04178856fa6ce2c595.json","authors":"Juri-Stepan Gerasimov","evidence":{"source_url":"https://oeis.org/A254748"},"id":"literature:url:https://oeis.org/A254748","identifiers":{"url":"https://oeis.org/A254748"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A254748, Numbers without superdivisors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA254748","year":2015},{"aliases":["Gus Wiseman","https://oeis.org/A258409"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d71af187f887975f3d7165692710314104eed528bbc66020f2e1033a3cac921b.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A258409"},"id":"literature:url:https://oeis.org/A258409","identifiers":{"url":"https://oeis.org/A258409"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A258409, Greatest common divisor of all (d-1)'s, where the d's are the positive divisors of n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA258409","year":2019},{"aliases":["Paul D. Hanna; Peter Bala","https://oeis.org/A258899"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3394aa531f11ff162b77313aeea713dc81cb73c8d8040a985db300ab0bb466d6.json","authors":"Paul D. Hanna; Peter Bala","evidence":{"source_url":"https://oeis.org/A258899"},"id":"literature:url:https://oeis.org/A258899","identifiers":{"url":"https://oeis.org/A258899"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A258899 and A258903: divisor-factorial sequences and totient-period conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA258899","year":2025},{"aliases":["Robert G. Wilson v; Juri-Stepan Gerasimov","https://oeis.org/A260310"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9751eb35d257bf729e24bba075f894f2c847279fb35563b9131a6504db80f92b.json","authors":"Robert G. Wilson v; Juri-Stepan Gerasimov","evidence":{"source_url":"https://oeis.org/A260310"},"id":"literature:url:https://oeis.org/A260310","identifiers":{"url":"https://oeis.org/A260310"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A260310, Pairs with balanced sums of prime divisors (A008472) and inverse prime divisors (A069359), ordered by larger members","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA260310","year":2015},{"aliases":["M. F. Hasler; Jaroslav Krizek","https://oeis.org/A260407"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f446bae138e0c09102abbd4454d2d57ef1b92b9bf843fc57fe3c09f6cd8401d2.json","authors":"M. F. Hasler; Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A260407"},"id":"literature:url:https://oeis.org/A260407","identifiers":{"url":"https://oeis.org/A260407"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A260407, Numbers n such that (n-1)^2+1 divides 2^(n-1)-1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA260407","year":2016},{"aliases":["Robert Price","https://oeis.org/A266285"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a779545de359d1656b3a495c6533d18dec9161a146b19b609579f697d8d614a5.json","authors":"Robert Price","evidence":{"source_url":"https://oeis.org/A266285"},"id":"literature:url:https://oeis.org/A266285","identifiers":{"url":"https://oeis.org/A266285"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A266285, ON cells of the Rule 13 elementary cellular automaton: closed-form, recurrence, generating-function and parity conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA266285","year":2015},{"aliases":["Paul D. Hanna","https://oeis.org/A266489"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a2c7a93bc5a4d5b0049428555a4d1f9af6b4c329a4e99a70e4fbe1a441bfebfe.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A266489"},"id":"literature:url:https://oeis.org/A266489","identifiers":{"url":"https://oeis.org/A266489"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A266489, g.f. A(x) satisfies: [x^n] A(x/A(x)^n) = 0 for n>1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA266489","year":2016},{"aliases":["Robert Price","https://oeis.org/A266614"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f94999ecb2ebdce5c487a5b6faa6a479b888d759e4bfef358335a765f6da142.json","authors":"Robert Price","evidence":{"source_url":"https://oeis.org/A266614"},"id":"literature:url:https://oeis.org/A266614","identifiers":{"url":"https://oeis.org/A266614"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A266614, ON cells of the Rule 41 elementary cellular automaton: recurrence and generating-function conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA266614","year":2016},{"aliases":["Robert Price","https://oeis.org/A267681"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/101288f81c7c15537c89b3fe484ee2db64cb4d3f69b46e6033ae53fe80fd7c76.json","authors":"Robert Price","evidence":{"source_url":"https://oeis.org/A267681"},"id":"literature:url:https://oeis.org/A267681","identifiers":{"url":"https://oeis.org/A267681"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A267681 and A267680, rows of the Rule 201 elementary cellular automaton: recurrence, generating-function and closed-form conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA267681","year":2016},{"aliases":["R. J. Mathar","https://oeis.org/A270236"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f49b3a29690be7adb96f3ae887fa42e9ba9a4f5406aa42887b81260be941db8.json","authors":"R. J. Mathar","evidence":{"source_url":"https://oeis.org/A270236"},"id":"literature:url:https://oeis.org/A270236","identifiers":{"url":"https://oeis.org/A270236"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A270236, Triangle T(n,p): occurrences of p in the restricted growth functions of length n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA270236","year":2016},{"aliases":["Paul D. Hanna","https://oeis.org/A271931"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1d3a0269f2d7ee0ac7037969269040ad66c481724ad04305cbd330fc500f14f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A271931"},"id":"literature:url:https://oeis.org/A271931","identifiers":{"url":"https://oeis.org/A271931"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A271931, A271932 and A271933: prime substitution integrality","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA271931","year":2016},{"aliases":["Paul D. Hanna","https://oeis.org/A271960"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7dd26623457b0af3b7f0358216d735c36dd31c3a0f55d36922988de8c172e661.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A271960"},"id":"literature:url:https://oeis.org/A271960","identifiers":{"url":"https://oeis.org/A271960"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A271960: a squared compositional generating function and dyadic vanishing","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA271960","year":2016},{"aliases":["Andres Cicuttin","https://oeis.org/A272170"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6981526f1d08858df1b4c4c461d3e3070a69dfb0a7dd64b56870480831e29ced.json","authors":"Andres Cicuttin","evidence":{"source_url":"https://oeis.org/A272170"},"id":"literature:url:https://oeis.org/A272170","identifiers":{"url":"https://oeis.org/A272170"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A272170, Second most significant bit of Fibonacci numbers > 1 written in base 2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA272170","year":2016},{"aliases":["Robert Israel; Ridouane Oudra","https://oeis.org/A274010"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c2e127053d57fd07c9f89cb84e5f26a46ea455f05e1cde08ecbbad46a2a9bcf8.json","authors":"Robert Israel; Ridouane Oudra","evidence":{"source_url":"https://oeis.org/A274010"},"id":"literature:url:https://oeis.org/A274010","identifiers":{"url":"https://oeis.org/A274010"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A274010: the Boris Stechkin function and an adjacent divisor-count formula","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA274010","year":2020},{"aliases":["Paul D. Hanna","https://oeis.org/A274479"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c11813005607966be191ab9b2b612e965858d429b898372da7e9339f40fe96d2.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A274479"},"id":"literature:url:https://oeis.org/A274479","identifiers":{"url":"https://oeis.org/A274479"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A274479, a generating-function coefficient conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA274479","year":2016},{"aliases":["Paul D. Hanna","https://oeis.org/A277403"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8257c0fa07721b149fd48836c21bae7eacee097f808b7fe963cf8700dfd2165c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A277403"},"id":"literature:url:https://oeis.org/A277403","identifiers":{"url":"https://oeis.org/A277403"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A277403 — E.g.f. satisfies: A(x - Integral A(x) dx) = x + Integral A(x) dx.","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA277403","year":2016},{"aliases":["Vaclav Kotesovec","https://oeis.org/A277829"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/86a05b04843af44665ca8681faff48bd158db26719c28029ef9ebfdf9ee19eff.json","authors":"Vaclav Kotesovec","evidence":{"source_url":"https://oeis.org/A277829"},"id":"literature:url:https://oeis.org/A277829","identifiers":{"url":"https://oeis.org/A277829"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A277829 and A278770: squared-Catalan Hankel determinants","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA277829","year":2016},{"aliases":["Jaroslav Krizek","https://oeis.org/A280258"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0080feae6b931c402c818e47377570e762226628b21184aded8800690f85ffad.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A280258"},"id":"literature:url:https://oeis.org/A280258","identifiers":{"url":"https://oeis.org/A280258"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A280258, sum of phitorials over divisors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA280258","year":2017},{"aliases":["Antti Karttunen; Mikhail Kurkov","Mikhail Kurkov; Antti Karttunen; OEIS contributors","https://oeis.org/A284005"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4c9c8af61868be78df08da592f6be41cbf48abbf089e2600b7cd24eaace7afc4.json","authors":"Antti Karttunen; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A284005"},"id":"literature:url:https://oeis.org/A284005","identifiers":{"url":"https://oeis.org/A284005"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A284005, product of (1 + binary weight) over the binary shifts of n, with Kurkov's binomial conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA284005","year":2023},{"aliases":["Hector J. Partridge","https://oeis.org/A289832"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/09b8a263c694e91dcc3b30e07257e434c2c3ca72ae64b9541a945a273b8d924b.json","authors":"Hector J. Partridge","evidence":{"source_url":"https://oeis.org/A289832"},"id":"literature:url:https://oeis.org/A289832","identifiers":{"url":"https://oeis.org/A289832"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A289832, rectangles in rectangular lattice grids","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA289832","year":2017},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A290322"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a66454a9af068035ab9696879d527d76df20c4cae4174556b4dff4957c185b16.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A290322"},"id":"literature:url:https://oeis.org/A290322","identifiers":{"url":"https://oeis.org/A290322"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A290322","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA290322","year":2024},{"aliases":["Mats Granvik","https://oeis.org/A292273"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/92baddbf699d07d1b9edb21e09d21e84eaae15865f92402190d6be3f2e13de80.json","authors":"Mats Granvik","evidence":{"source_url":"https://oeis.org/A292273"},"id":"literature:url:https://oeis.org/A292273","identifiers":{"url":"https://oeis.org/A292273"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A292273, real-part formula for the even-supported signed Mobius function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA292273","year":2024},{"aliases":["https://oeis.org/A292394"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f43e8656802d40cf740bc542d3278eb296c07197476cdbf418a77a0c90d10bc0.json","evidence":{"source_url":"https://oeis.org/A292394"},"id":"literature:url:https://oeis.org/A292394","identifiers":{"url":"https://oeis.org/A292394"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A292394","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA292394"},{"aliases":["Joerg Arndt","https://oeis.org/A293452"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b51887b172cc821b94c377cc5ac40bf7bf164d44eee89465b51cb332317a7e46.json","authors":"Joerg Arndt","evidence":{"source_url":"https://oeis.org/A293452"},"id":"literature:url:https://oeis.org/A293452","identifiers":{"url":"https://oeis.org/A293452"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A293452, Triangle T(n,k) read by rows: T(n,k) is the number of iterations to reach a final state for an n X k lattice of sandpiles on a torus according to rules specified in A249872","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA293452","year":2017},{"aliases":["Peter Bala","https://oeis.org/A295462"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7ef275d319d7370b3f6204299add3b26bba552bdca6ee90b877f9987bb7a2349.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A295462"},"id":"literature:url:https://oeis.org/A295462","identifiers":{"url":"https://oeis.org/A295462"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A295462: binary-spacing parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA295462","year":2026},{"aliases":["Werner Schulte","https://oeis.org/A299406"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/79030dd1730ad9e38068fbf298b5fe0e7e2647d8dd624ba99dc2953d9fd225df.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A299406"},"id":"literature:url:https://oeis.org/A299406","identifiers":{"url":"https://oeis.org/A299406"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A299406, coefficients of zeta(s) zeta(6s) / (zeta(2s) zeta(3s)), with the Liouville–A210826 product conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA299406","year":2018},{"aliases":["Jaroslav Krizek","https://oeis.org/A300657"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71379a9e4763b7d91ea609524fb369389c94f91161b72b5bea0732b865577b4f.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A300657"},"id":"literature:url:https://oeis.org/A300657","identifiers":{"url":"https://oeis.org/A300657"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A300657, sum of divisor-sum residues","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA300657","year":2018},{"aliases":["https://oeis.org/A300732"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/31f9248a872aeb4072d8283de783cb3d4c9eade47b377bd86ed711a1b075550e.json","evidence":{"source_url":"https://oeis.org/A300732"},"id":"literature:url:https://oeis.org/A300732","identifiers":{"url":"https://oeis.org/A300732"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A300732","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA300732"},{"aliases":["https://oeis.org/A300733"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ff6640764ac8117fd47c26e0813446fec3db0d744530aac65eac8b1914b6aed1.json","evidence":{"source_url":"https://oeis.org/A300733"},"id":"literature:url:https://oeis.org/A300733","identifiers":{"url":"https://oeis.org/A300733"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A300733","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA300733"},{"aliases":["https://oeis.org/A300734"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/60b72b656650f498f50fc09e6e354bb299392164857e94ce1e7262a3ae9cdc65.json","evidence":{"source_url":"https://oeis.org/A300734"},"id":"literature:url:https://oeis.org/A300734","identifiers":{"url":"https://oeis.org/A300734"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A300734","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA300734"},{"aliases":["Gregory Gerard Wojnar","https://oeis.org/A301420"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4ced29acf9da0e4c46eb72831f243f6df78dfd4ba3dbd11ae56219597df148e4.json","authors":"Gregory Gerard Wojnar","evidence":{"source_url":"https://oeis.org/A301420"},"id":"literature:url:https://oeis.org/A301420","identifiers":{"url":"https://oeis.org/A301420"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A301420: positive coefficients of five-data generalized Chebyshev polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA301420","year":2018},{"aliases":["Peter Kagey; Colin Barker","https://oeis.org/A301976"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a567a0d2f45aa2d6bcbea1136bcb09fd6d4bfe3003c43ef643a970727237f665.json","authors":"Peter Kagey; Colin Barker","evidence":{"source_url":"https://oeis.org/A301976"},"id":"literature:url:https://oeis.org/A301976","identifiers":{"url":"https://oeis.org/A301976"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A301976, Number of no-leaf subgraphs of the 3 X n grid","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA301976","year":2018},{"aliases":["Jaroslav Krizek","https://oeis.org/A302975"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50fdf4744b6cd5117f53f83ee35fb3a29d065b85289162ca5100a66e4151398b.json","authors":"Jaroslav Krizek","evidence":{"source_url":"https://oeis.org/A302975"},"id":"literature:url:https://oeis.org/A302975","identifiers":{"url":"https://oeis.org/A302975"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A302975, a(n) = denominator of tau(n)^n / n^tau(n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA302975","year":2018},{"aliases":["Peter Bala","https://oeis.org/A305550"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/877aa68d954d1716bf3f555138e47a35b9b1103c8bd5d45eba0ad0580b9adc27.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A305550"},"id":"literature:url:https://oeis.org/A305550","identifiers":{"url":"https://oeis.org/A305550"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A305550, Expansion of e.g.f. Product_{k>=1} (1 + (exp(x) - 1)^k)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA305550","year":2022},{"aliases":["Thomas Ordowski","https://oeis.org/A306270"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4fe6bb2f762551d537070d35023a6a132a685a0a9a1bca29f6c9ba8c3e88d6b2.json","authors":"Thomas Ordowski","evidence":{"source_url":"https://oeis.org/A306270"},"id":"literature:url:https://oeis.org/A306270","identifiers":{"url":"https://oeis.org/A306270"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A306270, composite k with b^(k(k-1)) = 1 mod k^2 for all b coprime to k, with the conjecture that its semiprimes > 4 are the p(p^2-p+1) of A190275","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA306270","year":2020},{"aliases":["Pedro Caceres","https://oeis.org/A308090"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a4d104bd5cd1d3cf22ccaeaea22604a02f41e432f90ad63be5b9350355c56c20.json","authors":"Pedro Caceres","evidence":{"source_url":"https://oeis.org/A308090"},"id":"literature:url:https://oeis.org/A308090","identifiers":{"url":"https://oeis.org/A308090"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A308090, gcd(2^n + n!, 3^n + n!, n+1), with the conjectured primality criterion gcd = n+1 implies n+1 prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA308090","year":2019},{"aliases":["R. J. Mathar; OEIS Foundation Inc.","https://oeis.org/A308815"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d7bdace1654451b7eae7d88f8b5baf4af9007e904c8eeec32af80452f59a38bb.json","authors":"R. J. Mathar; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A308815"},"id":"literature:url:https://oeis.org/A308815","identifiers":{"url":"https://oeis.org/A308815"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A308815 — D-finite recurrence for alternating Catalan numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA308815","year":2020},{"aliases":["N. J. A. Sloane","https://oeis.org/A309221"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f68b1cef01247a8cbd930be666024c4488d12fe0803d2fccfcdd9bcdd5c630d2.json","authors":"N. J. A. Sloane","evidence":{"source_url":"https://oeis.org/A309221"},"id":"literature:url:https://oeis.org/A309221","identifiers":{"url":"https://oeis.org/A309221"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A309221, Expected number of distinct squares visited by a knight's random walk on an infinite chessboard after n steps, normalized to give an integer","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA309221","year":2019},{"aliases":["Wolfdieter Lang","https://oeis.org/A319197"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a6c044f56e4944e6368f46e10d2c9e8cc006b23480d1124f64f4bc0a20ee144.json","authors":"Wolfdieter Lang","evidence":{"source_url":"https://oeis.org/A319197"},"id":"literature:url:https://oeis.org/A319197","identifiers":{"url":"https://oeis.org/A319197"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A319197, factors of Fibonacci(2^(n-2)*3*m), with the dyadic divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA319197","year":2018},{"aliases":["Ivan N. Ianakiev","https://oeis.org/A319927"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e08d4fbab1d00f9814a07f471587cb8936aa03001252c1374ad47fa2b92cd8e4.json","authors":"Ivan N. Ianakiev","evidence":{"source_url":"https://oeis.org/A319927"},"id":"literature:url:https://oeis.org/A319927","identifiers":{"url":"https://oeis.org/A319927"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A319927, Numbers k such that the sum of the squares of the odd non-unitary divisors of k divides the sum of the squares of the non-unitary divisors of k","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA319927","year":2018},{"aliases":["Stefano Spezia; OEIS Foundation Inc.","https://oeis.org/A320043/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d76873800bf525125d0bf2b25f6b81ac6fec5d28d1df584ab610d8a39215fc7.json","authors":"Stefano Spezia; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A320043/internal"},"id":"literature:url:https://oeis.org/A320043/internal","identifiers":{"url":"https://oeis.org/A320043/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A320043: Row sums of the triangle A322550","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA320043%2Finternal","year":2018},{"aliases":["Werner Schulte","https://oeis.org/A322327"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/030973e3954cf4067fa7f0808d4f0af860a6377c50d201a94a33a40a4cfdad0a.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A322327"},"id":"literature:url:https://oeis.org/A322327","identifiers":{"url":"https://oeis.org/A322327"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A322327, A005361(n)·A034444(n), with the conjecture that A005361(n)·k^ω(n) is multiplicative with prime-power values k·e","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA322327","year":2018},{"aliases":["Stefano Spezia","https://oeis.org/A322844"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/78c34a8a416bf8d2a26255ef830591e57dc1aaf140e97c22b0cb9b2873c5c3f5.json","authors":"Stefano Spezia","evidence":{"source_url":"https://oeis.org/A322844"},"id":"literature:url:https://oeis.org/A322844","identifiers":{"url":"https://oeis.org/A322844"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A322844 and the serpentine matrix A317614","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA322844","year":2018},{"aliases":["Paul D. Hanna","https://oeis.org/A323689"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6f54b076fadfb32ec5a2ba06f2c649e03bc9c9564a0ecb2c6389a8e005fcbf1f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A323689"},"id":"literature:url:https://oeis.org/A323689","identifiers":{"url":"https://oeis.org/A323689"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A323689: Gaussian generating function and square parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA323689","year":2019},{"aliases":["Paul D. Hanna","https://oeis.org/A324300"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/45936eef7bde73b8f826a46c6699cf4ec2d36ef95d76a241a485b6bbfc71bb5f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A324300"},"id":"literature:url:https://oeis.org/A324300","identifiers":{"url":"https://oeis.org/A324300"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A324300: Gaussian generating function and square parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA324300","year":2019},{"aliases":["Colin Barker","https://oeis.org/A326244"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c7e689ae84549fae38457ffa01a014f072c40494dfd23b6d7ecff7b7bd2d26e5.json","authors":"Colin Barker","evidence":{"source_url":"https://oeis.org/A326244"},"id":"literature:url:https://oeis.org/A326244","identifiers":{"url":"https://oeis.org/A326244"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A326244, Number of labeled n-vertex simple graphs without crossing or nesting edges","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA326244","year":2019},{"aliases":["Colin Barker","https://oeis.org/A327692"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/710964fc402b3c4f4915f54d6d507eacfd9e8919b1885804f563ef8baaf6d123.json","authors":"Colin Barker","evidence":{"source_url":"https://oeis.org/A327692"},"id":"literature:url:https://oeis.org/A327692","identifiers":{"url":"https://oeis.org/A327692"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A327692, Number of length-n phone numbers that can be dialed by a chess knight on a 0-9 keypad that starts on any number and takes n-1 steps","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA327692","year":2019},{"aliases":["Gus Wiseman","https://oeis.org/A328959"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/47a2da0084b1feff3e9d0837325e1183338cfaf5c0acf73596fce575f5a554d1.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A328959"},"id":"literature:url:https://oeis.org/A328959","identifiers":{"url":"https://oeis.org/A328959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A328959, a(n) = sigma_0(n) - 2 - (omega(n) - 1) * nu(n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA328959","year":2019},{"aliases":["Peter Kagey","https://oeis.org/A329278"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d858b09a359b4e17081de300603a3ab8e1f4165fdffff1d932a8303e2c8a38f7.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A329278"},"id":"literature:url:https://oeis.org/A329278","identifiers":{"url":"https://oeis.org/A329278"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A329278, Irregular table read by rows. The n-th row is the permutation of {0, 1, 2, ..., 2^n-1} given by T(n,k) = k(k+1)/2 (mod 2^n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA329278","year":2019},{"aliases":["OEIS Foundation Inc.; Mikhail Kurkov","https://oeis.org/A329369"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cf6e034263e3ae935a763838944b7d798d9bc37457793911f9581d1470ae1a27.json","authors":"OEIS Foundation Inc.; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A329369"},"id":"literature:url:https://oeis.org/A329369","identifiers":{"url":"https://oeis.org/A329369"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A329369 sixth conjectured sum over the A373183 rows","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA329369","year":2024},{"aliases":["Peter Kagey","https://oeis.org/A329479/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50e12227c55a303ebd1f45500983da54d27b05360165f77590882c785a665a71.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A329479/internal"},"id":"literature:url:https://oeis.org/A329479/internal","identifiers":{"url":"https://oeis.org/A329479/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A329479: Binary null polynomials modulo three","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA329479%2Finternal","year":2019},{"aliases":["Stefano Spezia","https://oeis.org/A330700"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e84e0a3a4fb19dca6a58156ff0b190f1012d014f67969c7e076a23aaf317792a.json","authors":"Stefano Spezia","evidence":{"source_url":"https://oeis.org/A330700"},"id":"literature:url:https://oeis.org/A330700","identifiers":{"url":"https://oeis.org/A330700"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A330700, second exterior-trace conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA330700","year":2019},{"aliases":["Paul Barry","https://oeis.org/A331474"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cb7e4c33791b7436669b358bd5b79afc9480d7f20bb98640ef9cbe7a755e19b.json","authors":"Paul Barry","evidence":{"source_url":"https://oeis.org/A331474"},"id":"literature:url:https://oeis.org/A331474","identifiers":{"url":"https://oeis.org/A331474"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A331474, Hankel transform of A331473","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA331474","year":2020},{"aliases":["Peter Kagey","https://oeis.org/A331859"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c8dc4c24e7f364f7ddbf8015e749f2d132f20c24f6c7adc11a87c3c5530f8f60.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A331859"},"id":"literature:url:https://oeis.org/A331859","identifiers":{"url":"https://oeis.org/A331859"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A331859, The total number of elastic collisions between a block of mass n, a block of mass 1, and a wall","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA331859","year":2020},{"aliases":["Peter Kagey","https://oeis.org/A334595"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/88ffc857725ab8d58e2c1db7424c0092f797f8e5c71f8187810b79a256221da7.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A334595"},"id":"literature:url:https://oeis.org/A334595","identifiers":{"url":"https://oeis.org/A334595"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A334595: the right edge of a binary XOR triangle","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA334595","year":2020},{"aliases":["Altug Alkan","https://oeis.org/A335901"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ed61525163cd31d213994e7f41b3be4aa16b1b95a644298b1123e155aa2b4ec.json","authors":"Altug Alkan","evidence":{"source_url":"https://oeis.org/A335901"},"id":"literature:url:https://oeis.org/A335901","identifiers":{"url":"https://oeis.org/A335901"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A335901, a(n) = 2*a(floor((n-1)/a(n-1))) with a(1) = 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA335901","year":2020},{"aliases":["Altug Alkan","https://oeis.org/A335925"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/15f68e93802a4076ac43654dc6d6cf0a6260f233f964fa3bfeb3d8309066e60e.json","authors":"Altug Alkan","evidence":{"source_url":"https://oeis.org/A335925"},"id":"literature:url:https://oeis.org/A335925","identifiers":{"url":"https://oeis.org/A335925"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A335925, a(n) = a(floor((n-1)/a(n-1))) + 1 with a(1) = 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA335925","year":2020},{"aliases":["Ivan N. Ianakiev; Bernard Schott","https://oeis.org/A336687"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/27f79edd83953638c94fe200183abe08227bd6108a1136b72dec8c3b939da24f.json","authors":"Ivan N. Ianakiev; Bernard Schott","evidence":{"source_url":"https://oeis.org/A336687"},"id":"literature:url:https://oeis.org/A336687","identifiers":{"url":"https://oeis.org/A336687"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A336687, divisor compositions and a three-solution equality conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA336687","year":2020},{"aliases":["Peter Bala","https://oeis.org/A338040"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5baded413973ed6c950da02e6c669ba07dd9a11734ab0cab896792f467ca0711.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A338040"},"id":"literature:url:https://oeis.org/A338040","identifiers":{"url":"https://oeis.org/A338040"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A338040 (Kotesovec 2020), conjecture: a(n) mod p purely periodic with period p − 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA338040","year":2022},{"aliases":["Vaclav Kotesovec; Mikhail Kurkov","https://oeis.org/A338193"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0663b40b6fb193d07878049dd814e960dac1128c1079ea80edfd57813269f557.json","authors":"Vaclav Kotesovec; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A338193"},"id":"literature:url:https://oeis.org/A338193","identifiers":{"url":"https://oeis.org/A338193"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A338193, an integral-defined exponential generating function and Kurkov's recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA338193","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A338632"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f690fefeb1b538e2f1a919f3615c60cb9c6708fb6412246cf880a2db050e0eba.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A338632"},"id":"literature:url:https://oeis.org/A338632","identifiers":{"url":"https://oeis.org/A338632"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A338632, continued-fraction generating function with odd numerators","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA338632","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A338636"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1ed4bbfefbf67f71329597911616469003c6171cf1078827fa4713008892307b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A338636"},"id":"literature:url:https://oeis.org/A338636","identifiers":{"url":"https://oeis.org/A338636"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A338636, continued-fraction g.f. with odd-square numerators","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA338636","year":2020},{"aliases":["https://oeis.org/A339621"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e77bde0586d54d70b81c20495538a22e35be28635e10b2610f8c8bec50a490b9.json","evidence":{"source_url":"https://oeis.org/A339621"},"id":"literature:url:https://oeis.org/A339621","identifiers":{"url":"https://oeis.org/A339621"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A339621","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA339621"},{"aliases":["https://oeis.org/A339669"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/480a1616bba411ab42afb0adddac053e5d9dc38f890a32eb0de2f7d66dec4b9b.json","evidence":{"source_url":"https://oeis.org/A339669"},"id":"literature:url:https://oeis.org/A339669","identifiers":{"url":"https://oeis.org/A339669"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A339669","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA339669"},{"aliases":["Clark Kimberling","https://oeis.org/A341249"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e64d3711ea683927bcba3e9029a416e7edbe132798386451ee9398659ea72fc3.json","authors":"Clark Kimberling","evidence":{"source_url":"https://oeis.org/A341249"},"id":"literature:url:https://oeis.org/A341249","identifiers":{"url":"https://oeis.org/A341249"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A341249: a(n) = floor(r*floor(s*n)), where r = 2 + sqrt(2) and s = sqrt(2).","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA341249","year":2021},{"aliases":["M. Tebni","https://oeis.org/A341724"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f2f8bb02eeda9ec5362b6352479477bbebfdb3ed512cb8ed872679e6c985dfb.json","authors":"M. Tebni","evidence":{"source_url":"https://oeis.org/A341724"},"id":"literature:url:https://oeis.org/A341724","identifiers":{"url":"https://oeis.org/A341724"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A341724, Column EGF and prime and dyadic row-divisibility conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA341724","year":2023},{"aliases":["C. H. R. Dequatre","https://oeis.org/A343502"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32b6dc3bc54e9d21e4f5e7ad5a0a21c673dba75c354457bed935f24f6a7eb733.json","authors":"C. H. R. Dequatre","evidence":{"source_url":"https://oeis.org/A343502"},"id":"literature:url:https://oeis.org/A343502","identifiers":{"url":"https://oeis.org/A343502"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A343502, Numbers k such that tau(tau(k)) and tau(k+1) are both prime","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA343502","year":2021},{"aliases":["P. Kagey; OEIS Foundation Inc.","https://oeis.org/A343825"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7175a67c0dfdb263b16631914b5cb814b4fa2b3aaeb35f49aa13a5003e405d91.json","authors":"P. Kagey; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A343825"},"id":"literature:url:https://oeis.org/A343825","identifiers":{"url":"https://oeis.org/A343825"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A343825, prime-power sequence minima and A071537","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA343825","year":2021},{"aliases":["Michel Marcus","https://oeis.org/A344083"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b642451f37ffb45d8ac12ffd2073a49fe1961407aeaef5dade0dfd47530b8dd.json","authors":"Michel Marcus","evidence":{"source_url":"https://oeis.org/A344083"},"id":"literature:url:https://oeis.org/A344083","identifiers":{"url":"https://oeis.org/A344083"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A344083, a(n) = f(x)+f(y)+f(z), where (x,y,h) is the n-th Pythagorean triple listed in (A046083, A046084, A009000), and f(m)=A176775(m) is the index of m as k-gonal number for the smallest possible k","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA344083","year":2021},{"aliases":["Peter Bala","https://oeis.org/A344598"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9f11ad4107ebf0934527d01ee77ce65eb4ec2f1303e13deb0de1b3f06c63a66.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A344598"},"id":"literature:url:https://oeis.org/A344598","identifiers":{"url":"https://oeis.org/A344598"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A344598, conjectured alternating gcd-sum formula","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA344598","year":2024},{"aliases":["Gus Wiseman","https://oeis.org/A344611"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d955789018f5b9b33d5b3d3b8e5d55fa1bc4a6b677f3e25a6dd99bb8efaf69f4.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A344611"},"id":"literature:url:https://oeis.org/A344611","identifiers":{"url":"https://oeis.org/A344611"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A344611 — reverse-alternating sums of integer partitions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA344611","year":2021},{"aliases":["Antti Karttunen; Peter Munn","https://oeis.org/A344695"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6da48e686554692a53f131c615e050c3e038e1f903d2615e8a3a9e0f90ffd692.json","authors":"Antti Karttunen; Peter Munn","evidence":{"source_url":"https://oeis.org/A344695"},"id":"literature:url:https://oeis.org/A344695","identifiers":{"url":"https://oeis.org/A344695"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A344695: gcd of the divisor sum and Dedekind psi","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA344695","year":2021},{"aliases":["M. De Vlieger; OEIS Foundation Inc.","https://oeis.org/A347285"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1016b5d2c4cf645cfbf35b38992d42dc4ec5b33d5b253876603a97be8ef8fea2.json","authors":"M. De Vlieger; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A347285"},"id":"literature:url:https://oeis.org/A347285","identifiers":{"url":"https://oeis.org/A347285"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A347285 descending prime-power exponents","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347285","year":2021},{"aliases":["Werner Schulte","https://oeis.org/A347293"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5544fe932267633e5ea87f3c8755cff3f4332ffa17f50663161fcd8fd8164c08.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A347293"},"id":"literature:url:https://oeis.org/A347293","identifiers":{"url":"https://oeis.org/A347293"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A347293, shifted-product gcd triangle and row-sum convolution","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347293","year":2022},{"aliases":["Peter Bala","https://oeis.org/A347854"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/679a86d2eda02e65a9ccdccceb4c742fa8cb5dba7b341a65240068dabfe9af9a.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A347854"},"id":"literature:url:https://oeis.org/A347854","identifiers":{"url":"https://oeis.org/A347854"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A347854–A347858: fractional factorial ratios","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347854","year":2021},{"aliases":["https://oeis.org/A347855"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02041bd164b3defe95481ec17f1609c78d9e155d5803cf46a9aec622d8f60461.json","evidence":{"source_url":"https://oeis.org/A347855"},"id":"literature:url:https://oeis.org/A347855","identifiers":{"url":"https://oeis.org/A347855"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A347855","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347855"},{"aliases":["https://oeis.org/A347856"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a0baaa88d8950199909711cdbad1fa7657dbc20e4867feb61edef3e2c2c74486.json","evidence":{"source_url":"https://oeis.org/A347856"},"id":"literature:url:https://oeis.org/A347856","identifiers":{"url":"https://oeis.org/A347856"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A347856","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347856"},{"aliases":["https://oeis.org/A347857"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8248fab0a0c33dde164d4888cf55319ed5cd492016ab8699618967c524261922.json","evidence":{"source_url":"https://oeis.org/A347857"},"id":"literature:url:https://oeis.org/A347857","identifiers":{"url":"https://oeis.org/A347857"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A347857","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347857"},{"aliases":["https://oeis.org/A347858"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46e36ca65ed2a804b964a703eb64934b684f9ea3aea42efb524e4f15cbd88868.json","evidence":{"source_url":"https://oeis.org/A347858"},"id":"literature:url:https://oeis.org/A347858","identifiers":{"url":"https://oeis.org/A347858"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A347858","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA347858"},{"aliases":["P. Kagey; OEIS Foundation Inc.","https://oeis.org/A350532"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/67c54ff65b3ccf7c4a8094a0b9ff4fed8968c24db71532649046a0016d3724ea.json","authors":"P. Kagey; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A350532"},"id":"literature:url:https://oeis.org/A350532","identifiers":{"url":"https://oeis.org/A350532"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A350532, sparse perfect powers over Z/2Z","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA350532","year":2022},{"aliases":["Mélika Tebni","https://oeis.org/A350759"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3f0221cd757ec865c066d428e080fc9aba70865450cb4065b2545e8a79a09274.json","authors":"Mélika Tebni","evidence":{"source_url":"https://oeis.org/A350759"},"id":"literature:url:https://oeis.org/A350759","identifiers":{"url":"https://oeis.org/A350759"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A350759, COMMENTS prime congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA350759","year":2022},{"aliases":["Werner Schulte","https://oeis.org/A350900"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/11c28fb17979b133bd94a5c9fd92744e1b9b34e2b1420b666255192fb25f22ec.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A350900"},"id":"literature:url:https://oeis.org/A350900","identifiers":{"url":"https://oeis.org/A350900"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A350900, T(n,k) = Sum_{i=1..n} gcd(i,n)/gcd(gcd(i,k),n), with the conjectured row sums (n phi(n)) * (Sum_{d|n} d phi(d))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA350900","year":2022},{"aliases":["Paul D. Hanna","https://oeis.org/A352700"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43c2bd905b906e28eea479f9e4154f8470c03bb2a8812455d86574e31f89fe9b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A352700"},"id":"literature:url:https://oeis.org/A352700","identifiers":{"url":"https://oeis.org/A352700"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A352700: Catalan inverse-power binomial series and modulo 2 and 3 congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA352700","year":2022},{"aliases":["Paul D. Hanna","https://oeis.org/A355872"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f75ea3e73af6cd1e42b3e658f55ec13f7575756bca8e4232e6a5a715be5c2ff8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A355872"},"id":"literature:url:https://oeis.org/A355872","identifiers":{"url":"https://oeis.org/A355872"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A355872, g.f. satisfying x = Sum_{n=-oo..+oo} (-x)^(n^2) * A(x)^((n-1)^2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA355872","year":2022},{"aliases":["Paul D. Hanna","https://oeis.org/A356508"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/43b1dc41e1814382e2b4b010fbdf5be1544ab8eded57af05f6eda2f462fc4ee4.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A356508"},"id":"literature:url:https://oeis.org/A356508","identifiers":{"url":"https://oeis.org/A356508"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A356508, product-partition generating-function congruence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA356508","year":2022},{"aliases":["Eric Angelini, Hans Havermann, and M. F. Hasler","https://oeis.org/A359482"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2a4897f37e0f11f11ba955727318f356e713c4598f7ca2f4e304d32ced68106f.json","authors":"Eric Angelini, Hans Havermann, and M. F. Hasler","evidence":{"source_url":"https://oeis.org/A359482"},"id":"literature:url:https://oeis.org/A359482","identifiers":{"url":"https://oeis.org/A359482"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A359482 and the single-digit conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA359482","year":2023},{"aliases":["Gus Wiseman","https://oeis.org/A360070"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b3d59953bab7aade6bea0cadfefc272b7e9869effb455730775e236c8facae15.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A360070"},"id":"literature:url:https://oeis.org/A360070","identifiers":{"url":"https://oeis.org/A360070"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A360070, Numbers for which there exists an integer partition such that the parts have the same mean as the multiplicities","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA360070","year":2023},{"aliases":["Peter Bala","https://oeis.org/A361034"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3c2dc2464bc372bb4be1452157453ad43862cdbfcb3d4ff8d192d6b86a432769.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A361034"},"id":"literature:url:https://oeis.org/A361034","identifiers":{"url":"https://oeis.org/A361034"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A361034 and A361035: parity of quartic factorial quotients","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA361034","year":2023},{"aliases":["https://oeis.org/A361035"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e0f8f5ca6c241fccaccf5f34c22e33083b61a138bd287b9c85b3672b21dbdf44.json","evidence":{"source_url":"https://oeis.org/A361035"},"id":"literature:url:https://oeis.org/A361035","identifiers":{"url":"https://oeis.org/A361035"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A361035","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA361035"},{"aliases":["Paul D. Hanna","https://oeis.org/A361552"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/23755663daafaf2b5cf657b4191443ba81b9ad4e4c131165e03e016a923ec6b8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A361552"},"id":"literature:url:https://oeis.org/A361552","identifiers":{"url":"https://oeis.org/A361552"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A361552: implicit pentagonal unit series and partition parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA361552","year":2023},{"aliases":["Mohamed Sabba","https://oeis.org/A362534"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ba3f21f332d6fa1e7445f7c5cd6f8fdc106becdd75036e8d5ba568f7f5f276c3.json","authors":"Mohamed Sabba","evidence":{"source_url":"https://oeis.org/A362534"},"id":"literature:url:https://oeis.org/A362534","identifiers":{"url":"https://oeis.org/A362534"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A362534, Numerators of the ratio of the symmetry-constrained bound to the adiabatic bound on polarization transfer in AXn spin-1/2 systems","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA362534","year":2023},{"aliases":["Gus Wiseman","https://oeis.org/A362559"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a857afbb0c4e32a1187cd549b5466d641c41041cb60df3b8dcdfaa449b858fca.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A362559"},"id":"literature:url:https://oeis.org/A362559","identifiers":{"url":"https://oeis.org/A362559"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A362559, Number of integer partitions of n whose weighted sum is divisible by n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA362559","year":2023},{"aliases":["https://oeis.org/A363102"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/20ebc38854db102603515286e4cd2dd3b8edb91f50f7c78a2defdf2b9db0021e.json","evidence":{"source_url":"https://oeis.org/A363102"},"id":"literature:url:https://oeis.org/A363102","identifiers":{"url":"https://oeis.org/A363102"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A363102","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363102"},{"aliases":["Paul D. Hanna","https://oeis.org/A363104"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d4cb82ae537d27dc018dc08ce975c58b9ea22970909f7f47d72b45440646b1c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A363104"},"id":"literature:url:https://oeis.org/A363104","identifiers":{"url":"https://oeis.org/A363104"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363104, g.f. satisfying 4 = Sum_{n in Z} (-x)^n (4 A(x) + x^(n-1))^(n+1)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363104","year":2023},{"aliases":["Paul D. Hanna","https://oeis.org/A363184"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/645b38ba284555aa57e1a22248750ac5c4ab64a850cc8445661b031893981734.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A363184"},"id":"literature:url:https://oeis.org/A363184","identifiers":{"url":"https://oeis.org/A363184"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363184, g.f. satisfying 4 = Sum_{n in Z} (-1)^n x^n (4 A(x) + x^(2n-1))^(n+1)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363184","year":2023},{"aliases":["Paul D. Hanna","https://oeis.org/A363308"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/46fc7794dd553c56f9d414541866e2ee6482fdbbc3cb3f618fbe4a021cc2a661.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A363308"},"id":"literature:url:https://oeis.org/A363308","identifiers":{"url":"https://oeis.org/A363308"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363308, expansion of C(x*C(x)^3) with C the Catalan g.f.","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363308","year":2023},{"aliases":["M. Bouras","https://oeis.org/A363347"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bffa0fe65370c95aa874491568cbc6d8813703439ff4c9b1f86c58d6dbee4fce.json","authors":"M. Bouras","evidence":{"source_url":"https://oeis.org/A363347"},"id":"literature:url:https://oeis.org/A363347","identifiers":{"url":"https://oeis.org/A363347"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363347 and A363102, prime multiplicities of continued-fraction denominators","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363347","year":2023},{"aliases":["P. Bala","https://oeis.org/A363411"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/85d621922b343e98bd81dc8ec62222a825783880e981421e77aab4ec3ef19887.json","authors":"P. Bala","evidence":{"source_url":"https://oeis.org/A363411"},"id":"literature:url:https://oeis.org/A363411","identifiers":{"url":"https://oeis.org/A363411"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363411 and A363412 — quadratic integer product coordinates","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363411","year":2023},{"aliases":["https://oeis.org/A363412"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0f170bea362aa17ca9353ee21cbd6f3fd1fe7370442e466d3119ba9751b58b63.json","evidence":{"source_url":"https://oeis.org/A363412"},"id":"literature:url:https://oeis.org/A363412","identifiers":{"url":"https://oeis.org/A363412"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A363412","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363412"},{"aliases":["Paul D. Hanna","https://oeis.org/A363558"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bee3d09b1cd6ab3b64e4fe8d1ce780523e369977ab97337ab53ab811728c6055.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A363558"},"id":"literature:url:https://oeis.org/A363558","identifiers":{"url":"https://oeis.org/A363558"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363558, A363559, A363561 and A363569: bilateral power sums","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363558","year":2023},{"aliases":["https://oeis.org/A363559"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd39bb6fccea0e75b5e7c17af47cced9ee1a71134412608f49676a50bd0f390d.json","evidence":{"source_url":"https://oeis.org/A363559"},"id":"literature:url:https://oeis.org/A363559","identifiers":{"url":"https://oeis.org/A363559"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A363559","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363559"},{"aliases":["Paul D. Hanna","https://oeis.org/A363560"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cfba78c70b2863f229323bf44edcc40aead1044a3ddd8d39d0fe7efebe730319.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A363560"},"id":"literature:url:https://oeis.org/A363560","identifiers":{"url":"https://oeis.org/A363560"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363560, g.f. satisfying A(x)^3 = 1 + x*(A(x) + A(x)^2 + A(x)^9)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363560","year":2023},{"aliases":["https://oeis.org/A363561"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7da57d71cea6f6b304a49df93fc8e1510fa08adae2db9fbc368e9eb54a6b956c.json","evidence":{"source_url":"https://oeis.org/A363561"},"id":"literature:url:https://oeis.org/A363561","identifiers":{"url":"https://oeis.org/A363561"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A363561","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363561"},{"aliases":["https://oeis.org/A363569"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2aafc845509e4dbca297921fdf1a0d0f9b981d0c1ab84c59b987af8d05120a0b.json","evidence":{"source_url":"https://oeis.org/A363569"},"id":"literature:url:https://oeis.org/A363569","identifiers":{"url":"https://oeis.org/A363569"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A363569","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363569"},{"aliases":["Scott R. Shannon","https://oeis.org/A363956"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e9ddc191ebe00de1c037d828300651ba9ada93d70afb36d917c21544e9867d3.json","authors":"Scott R. Shannon","evidence":{"source_url":"https://oeis.org/A363956"},"id":"literature:url:https://oeis.org/A363956","identifiers":{"url":"https://oeis.org/A363956"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A363956 — least unused multiples indexed by distinct prime factors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA363956","year":2023},{"aliases":["M. Tebni","https://oeis.org/A364822"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/77f8a4b017f424edfd506d60c6afe02b1fbd37275e59d41d954f0cc1df7d6d8d.json","authors":"M. Tebni","evidence":{"source_url":"https://oeis.org/A364822"},"id":"literature:url:https://oeis.org/A364822","identifiers":{"url":"https://oeis.org/A364822"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A364822, Expansion of e.g.f. cosh(x) / (1 - 2*sinh(x))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA364822","year":2023},{"aliases":["Paul D. Hanna","https://oeis.org/A365095"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/eed94530499915b824c06197ab74b800bcc50d2e5b5dea18879844812af12042.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A365095"},"id":"literature:url:https://oeis.org/A365095","identifiers":{"url":"https://oeis.org/A365095"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A365095, expansion of a generating function defined by a vanishing diagonal","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA365095","year":2023},{"aliases":["Paul D. Hanna","https://oeis.org/A365574"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4f8d30047e046e4ebda64c9e5f1b5149e032a49788f2f6c5a43c3ceb0de7bf27.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A365574"},"id":"literature:url:https://oeis.org/A365574","identifiers":{"url":"https://oeis.org/A365574"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A365574: generating function and parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA365574","year":2023},{"aliases":["Paul D. Hanna","https://oeis.org/A365771"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7c043f3695d937acaf2b8de4be18c2c470bbc131f47dc7a668aa8691993fd56.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A365771"},"id":"literature:url:https://oeis.org/A365771","identifiers":{"url":"https://oeis.org/A365771"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A365771: Catalan-binomial residues modulo five","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA365771","year":2023},{"aliases":["Gus Wiseman; Chai Wah Wu","https://oeis.org/A367400"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dc217074b680cb823ee19f3a6fe363b1b0c9628686f6b28cb707345f45c5c61b.json","authors":"Gus Wiseman; Chai Wah Wu","evidence":{"source_url":"https://oeis.org/A367400"},"id":"literature:url:https://oeis.org/A367400","identifiers":{"url":"https://oeis.org/A367400"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A367400: cardinality-sum-avoiding subsets","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA367400","year":2023},{"aliases":["Paul D. Hanna; OEIS Foundation Inc.","https://oeis.org/A368628"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b62d0d581c9a2819d78071743139de35f9ef70be4f98ad3c93ce95868acdae9.json","authors":"Paul D. Hanna; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A368628"},"id":"literature:url:https://oeis.org/A368628","identifiers":{"url":"https://oeis.org/A368628"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A368628 — a piecewise square and fourth-power generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA368628","year":2024},{"aliases":["Václav Kotěšovec","https://oeis.org/A368959"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2a02ecb5ba0fc47311008b86c178ea5d005f286d92768a5932012c1185dee638.json","authors":"Václav Kotěšovec","evidence":{"source_url":"https://oeis.org/A368959"},"id":"literature:url:https://oeis.org/A368959","identifiers":{"url":"https://oeis.org/A368959"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A368959: nonzero complete monomial exponential sums","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA368959","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A369083"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4360a9706952b60efcb909de92a9a1269c02f95d53eb0b9bb945993abdfd3d70.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A369083"},"id":"literature:url:https://oeis.org/A369083","identifiers":{"url":"https://oeis.org/A369083"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A369083, g.f. satisfying A(x) = 1 + x*(5*A(x)^2 - A(-x)^2)/4","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA369083","year":2024},{"aliases":["Paul D. Hanna; N. J. A. Sloane; Reinhard Zumkeller","https://oeis.org/A369673/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/65a63cf9625e5f4fc617cbfcd9978b8ae9cf9cb2799877bfd29a4b4c4d898dee.json","authors":"Paul D. Hanna; N. J. A. Sloane; Reinhard Zumkeller","evidence":{"source_url":"https://oeis.org/A369673/internal"},"id":"literature:url:https://oeis.org/A369673/internal","identifiers":{"url":"https://oeis.org/A369673/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A369673, square classification and five-adic orders, with A127428 and A112765","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA369673%2Finternal","year":2024},{"aliases":["https://oeis.org/A370630/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/445a8872d186723c1749c6fed674925ffe5fc0d81dcf42452c805e8a8aa83492.json","evidence":{"source_url":"https://oeis.org/A370630/internal"},"id":"literature:url:https://oeis.org/A370630/internal","identifiers":{"url":"https://oeis.org/A370630/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A370630/internal","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA370630%2Finternal"},{"aliases":["Seiichi Manyama","https://oeis.org/A371364"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/560eda46a5576bdcddb7353b4d21f44adf84395ec65c192c9c4159708b1b3a90.json","authors":"Seiichi Manyama","evidence":{"source_url":"https://oeis.org/A371364"},"id":"literature:url:https://oeis.org/A371364","identifiers":{"url":"https://oeis.org/A371364"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A371364: normalized reversion of x(1-4x)^2/(1-3x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA371364","year":2024},{"aliases":["Seiichi Manyama; R. J. Mathar","https://oeis.org/A372018"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d0ee6bcbf389d85a587d42e5c631c55a2bbe41c478a7534ed4a1a000fd957b3c.json","authors":"Seiichi Manyama; R. J. Mathar","evidence":{"source_url":"https://oeis.org/A372018"},"id":"literature:url:https://oeis.org/A372018","identifiers":{"url":"https://oeis.org/A372018"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A372018: an algebraic generating function and its odd-index conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA372018","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A372527"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/90e373e94bfbfd0de1a99a0f641ca18565a99c0775f779bd5282933fccc14a4f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A372527"},"id":"literature:url:https://oeis.org/A372527","identifiers":{"url":"https://oeis.org/A372527"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A372527, square-composition generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA372527","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A372577"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d978975c96598017a8ea12b7fbec0887d7574a5482d09d7fb43705e1569237e.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A372577"},"id":"literature:url:https://oeis.org/A372577","identifiers":{"url":"https://oeis.org/A372577"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A372577, expansion of g.f. A(x) satisfying A(x)^2 = A(A(x*A(x) + x*A(x)^2))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA372577","year":2024},{"aliases":["M. Bouras","https://oeis.org/A372763"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5197bd56cb8cbaa32e2c1f899a6d9ee0bec8045062c76ee75e0623d4e1c61777.json","authors":"M. Bouras","evidence":{"source_url":"https://oeis.org/A372763"},"id":"literature:url:https://oeis.org/A372763","identifiers":{"url":"https://oeis.org/A372763"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A372763: denominators of a continued fraction","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA372763","year":2024},{"aliases":["Clark Kimberling; R. J. Mathar","https://oeis.org/A372991"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3cdb4493727cdf9953af681ec5e7dcf9a6dccf0a3ebceef90315bcb7fd5cb8ba.json","authors":"Clark Kimberling; R. J. Mathar","evidence":{"source_url":"https://oeis.org/A372991"},"id":"literature:url:https://oeis.org/A372991","identifiers":{"url":"https://oeis.org/A372991"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A372991, a(n) = (2n)!/(a(n-1)*a(n-2))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA372991","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A373308"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8f607d0c035e19106d2cfa5f906f5ec46642cd5f7edb49c454d7f8bf778759fe.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A373308"},"id":"literature:url:https://oeis.org/A373308","identifiers":{"url":"https://oeis.org/A373308"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A373308, expansion of Product_{n>=0} (1 - x^(2^n))^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA373308","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A373312"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/efd5879a0c6b3542cc57518953fafcfade3604020db46785e28fbc4675596239.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A373312"},"id":"literature:url:https://oeis.org/A373312","identifiers":{"url":"https://oeis.org/A373312"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A373312, g.f. satisfying A(x)^2 = A(x*A(x)/(1-A(x))^2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA373312","year":2024},{"aliases":["Gus Wiseman","https://oeis.org/A373409"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5a6a2b7cefe8868dffe6fd64902c96985a538c441caf0e07025c0fad328657be.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A373409"},"id":"literature:url:https://oeis.org/A373409","identifiers":{"url":"https://oeis.org/A373409"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A373409, Length of the n-th maximal antirun of nonsquarefree numbers differing by more than one","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA373409","year":2024},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A373561"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/625a5af2c98e118396d5ba944f823fcd5b3b80be3dea99cc84d6e4554858291e.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A373561"},"id":"literature:url:https://oeis.org/A373561","identifiers":{"url":"https://oeis.org/A373561"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A373561","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA373561","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A374568"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d3b43e94717ca4ad136ba03dc9a435df98d160713eeff442d48411ffecfe920.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A374568"},"id":"literature:url:https://oeis.org/A374568","identifiers":{"url":"https://oeis.org/A374568"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A374568, g.f. satisfying x = A(x-x^2) - A(x+x^2)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA374568","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A374570"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2e3f59657d8e12a687e69e47fae4f7bb9517ea6f1861a85fb21a305c0a9e26a2.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A374570"},"id":"literature:url:https://oeis.org/A374570","identifiers":{"url":"https://oeis.org/A374570"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A374570, g.f. satisfying A(x)^2 = A(A(x)*C(x)) with C the Catalan function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA374570","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A374571"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6e5099f61d289d55ee7ca76be97dc010c660e2d9fa3b638631145c5e3f293ad8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A374571"},"id":"literature:url:https://oeis.org/A374571","identifiers":{"url":"https://oeis.org/A374571"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A374571, g.f. satisfying A(x) = A(x^2) - x*A(x^2)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA374571","year":2024},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A374911"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5942943268c90d2e7509e1c806ffccbbea42a2f4c5aafde63d8a89764e45c7dd.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A374911"},"id":"literature:url:https://oeis.org/A374911","identifiers":{"url":"https://oeis.org/A374911"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A374911","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA374911","year":2024},{"aliases":["Lechoslaw Ratajczak","https://oeis.org/A375007"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e3fa3cf925292a6b681965e8b418995db75a39999799e069c4ebad072e9db20.json","authors":"Lechoslaw Ratajczak","evidence":{"source_url":"https://oeis.org/A375007"},"id":"literature:url:https://oeis.org/A375007","identifiers":{"url":"https://oeis.org/A375007"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A375007 — isolated quotient-remainder values","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375007","year":2024},{"aliases":["Peter Bala","https://oeis.org/A375178"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4d7d1d6ac4e2da219d35663915ffad9c4855d44eb43ca8d9b62f592581fb99e5.json","authors":"Peter Bala","evidence":{"source_url":"https://oeis.org/A375178"},"id":"literature:url:https://oeis.org/A375178","identifiers":{"url":"https://oeis.org/A375178"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A375178 — binomial cube sums and a fifth-power supercongruence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375178","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A375439"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/59eb79a4476311fdb6222820c3798f0206e1069fe554071481c9debfe68abe02.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A375439"},"id":"literature:url:https://oeis.org/A375439","identifiers":{"url":"https://oeis.org/A375439"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A375439, expansion of A(x) = x + x^2 + (2*A(x)^3 + A(x^3))/3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375439","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A375449"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d915da0d3acc450f6572367dc355cf0b0a6e063641d65b70a75b2959327fc515.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A375449"},"id":"literature:url:https://oeis.org/A375449","identifiers":{"url":"https://oeis.org/A375449"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A375449 and A375452, diagonal derivative parity conjectures","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375449","year":2024},{"aliases":["https://oeis.org/A375452"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/39d8931864e5db5dc8907294d667a3a543f3fb512f7a67927ff90d0c5d850f9e.json","evidence":{"source_url":"https://oeis.org/A375452"},"id":"literature:url:https://oeis.org/A375452","identifiers":{"url":"https://oeis.org/A375452"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A375452","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375452"},{"aliases":["F. Huber","https://oeis.org/A375786"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/03f7abf7f40670c6bdaf60173120a3a130620d0fef340cdf70dbcc531867b62a.json","authors":"F. Huber","evidence":{"source_url":"https://oeis.org/A375786"},"id":"literature:url:https://oeis.org/A375786","identifiers":{"url":"https://oeis.org/A375786"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A375786: minimum volume of a cuboid with the surface of an n-cube","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA375786","year":2024},{"aliases":["Mikhail Kurkov","https://oeis.org/A376317"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50265f38a219440663423704e8a383c21f1511f5ae94dc39f56d347defb45fec.json","authors":"Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A376317"},"id":"literature:url:https://oeis.org/A376317","identifiers":{"url":"https://oeis.org/A376317"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A376317, periodic shifts of a two-index recurrence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA376317","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A376527"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/01ef01b24921fa53d8226f4b4cc40cdce8baafb6a1adfc3ec0db7e493fbc2866.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A376527"},"id":"literature:url:https://oeis.org/A376527","identifiers":{"url":"https://oeis.org/A376527"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A376527, square sums of Carlitz-Riordan q-Catalan rows and parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA376527","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A377100"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fe8970629907524bbe4e15cd88ea8e9ce0ae01263ac1bad709d075c805893d6c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A377100"},"id":"literature:url:https://oeis.org/A377100","identifiers":{"url":"https://oeis.org/A377100"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A377100, a generating-function coefficient conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA377100","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A377106"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32da2bf8ca336ac18e32b533b77a800807f6fd54aaf6046e6cdb80f04c1f985c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A377106"},"id":"literature:url:https://oeis.org/A377106","identifiers":{"url":"https://oeis.org/A377106"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A377106: cubic substitution and power-of-two parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA377106","year":2024},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A378252"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/839930e42b002a34092595bf74103e780ba0d7be29f477b318716aab3d9262a1.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A378252"},"id":"literature:url:https://oeis.org/A378252","identifiers":{"url":"https://oeis.org/A378252"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378252","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378252","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A378258"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f77e2dabf6e36433061775b4764137022aa7408bf7e198e2475b165761dbda24.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378258"},"id":"literature:url:https://oeis.org/A378258","identifiers":{"url":"https://oeis.org/A378258"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378258, a compositional square root and dyadic parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378258","year":2024},{"aliases":["Werner Schulte","https://oeis.org/A378277"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c0f0d5b2aea14ef21209cc1fc5cb1b61682a37a9f64ea838723fba3aa95c0ff3.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A378277"},"id":"literature:url:https://oeis.org/A378277","identifiers":{"url":"https://oeis.org/A378277"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378277, denominators in a harmonic triangle based on products of Fibonacci numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378277","year":2024},{"aliases":["Peter Kagey","https://oeis.org/A378495"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7285d78030059d8694160e230cf3e0f0290c5fcaefea8de79459cc355e43693.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A378495"},"id":"literature:url:https://oeis.org/A378495","identifiers":{"url":"https://oeis.org/A378495"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378495: derangements with no k-cycles","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378495","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A378575"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6c2ea96a19cd7a603a7af4ec240d78d28cfe8e96231ac7e5a107a1bda019f91e.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378575"},"id":"literature:url:https://oeis.org/A378575","identifiers":{"url":"https://oeis.org/A378575"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378575, g.f. satisfying A(x) = x + x*A^5(x) (5-fold compositional iterate)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378575","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A378576"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fccde9b6473a537dc70d57546da8e0c198219c3602daa6193ff45d562304fa47.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378576"},"id":"literature:url:https://oeis.org/A378576","identifiers":{"url":"https://oeis.org/A378576"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378576, g.f. satisfying A(x) = x + x*A^6(x) (6-fold compositional iterate)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378576","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A378578"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/925b7edd012be7bf5447e6fe68affd2d759ba0b8f1a4a3f0a04d1d19c73f4352.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378578"},"id":"literature:url:https://oeis.org/A378578","identifiers":{"url":"https://oeis.org/A378578"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378578, g.f. obtained by removing all factors of 3 from the coefficients of 1 + x*A(x)^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378578","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A378580"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97dcd819803f91d0951294d4561681608ace45949c4679a1e8ce4181f564f8cf.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378580"},"id":"literature:url:https://oeis.org/A378580","identifiers":{"url":"https://oeis.org/A378580"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378580, g.f. A(x) satisfies A(x/A(x)) = theta_3(x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378580","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A378581"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/380a9275689a53af4bd4199f1d6cfa647f372d352ff71ee8a04d5ad9c202875c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A378581"},"id":"literature:url:https://oeis.org/A378581","identifiers":{"url":"https://oeis.org/A378581"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A378581, g.f. satisfying A(x*A(x)) = theta_3(x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA378581","year":2025},{"aliases":["P. D. Hanna","https://oeis.org/A379202"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e38c507bc20f13db9148f7fbb157fb940adcf14d4e01725f0605ada089b51f6f.json","authors":"P. D. Hanna","evidence":{"source_url":"https://oeis.org/A379202"},"id":"literature:url:https://oeis.org/A379202","identifiers":{"url":"https://oeis.org/A379202"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A379202, a bilateral Laurent generating equation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA379202","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A379204"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6103c73b0f77d9327891e7c45f8f5775275c4f8df60fa7d64b6d416509dbc044.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A379204"},"id":"literature:url:https://oeis.org/A379204","identifiers":{"url":"https://oeis.org/A379204"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A379204, g.f. satisfying 1/x = Sum_{n in Z} A(x)^n (A(x)^n + 4)^(n+1)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA379204","year":2024},{"aliases":["Paul D. Hanna","https://oeis.org/A379764"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c44b502074b567d7bc3bc3cbdec8776d7f5962ccda0205fc4664945cbe5adbe8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A379764"},"id":"literature:url:https://oeis.org/A379764","identifiers":{"url":"https://oeis.org/A379764"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A379764, theta-four bilateral generating function and coefficient congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA379764","year":2025},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A380056"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3b705fe47b1f0af450d4a1d58689792a99fe0d8fdf125747712fa886e23ccc64.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A380056"},"id":"literature:url:https://oeis.org/A380056","identifiers":{"url":"https://oeis.org/A380056"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380056","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380056","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380060"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e2ea0acb69b80c79bbfd09c7e5666867d3f3edc2e39b2078be5eed31bf4b323.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380060"},"id":"literature:url:https://oeis.org/A380060","identifiers":{"url":"https://oeis.org/A380060"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380060, bilateral unit-quotient square series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380060","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380066"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7f566009883203bc0288f76209f0e75769c3fe664dd99a55de3c2a7428352a8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380066"},"id":"literature:url:https://oeis.org/A380066","identifiers":{"url":"https://oeis.org/A380066"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380066, bilateral product generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380066","year":2025},{"aliases":["John Tyler Rascoe","https://oeis.org/A380392"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1b2a7b3427619d1b8221443915d36c06fbbe27a4b358f847f41525b00bc738f5.json","authors":"John Tyler Rascoe","evidence":{"source_url":"https://oeis.org/A380392"},"id":"literature:url:https://oeis.org/A380392","identifiers":{"url":"https://oeis.org/A380392"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A380392 — binary matrices counted by their South-East paths of ones","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380392","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380558"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9aebedffcb42812c89ec750c7d840e19390bfa5111b1a50c0ec3e91ced47b671.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380558"},"id":"literature:url:https://oeis.org/A380558","identifiers":{"url":"https://oeis.org/A380558"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380558, g.f. satisfying A(x - A(x)) = x^2/(1 - x^2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380558","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380678"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/191ac9d0fc082e3fbb642115ba62d1f989b230aefde0c2d7d62e6af2a910dd3b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380678"},"id":"literature:url:https://oeis.org/A380678","identifiers":{"url":"https://oeis.org/A380678"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380678, g.f. satisfying A(x - A(x)^2/(1 - A(x)^2)) = x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380678","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380680"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4710d1808e28517415da41c37c0b1426377d2d1d034943fe43a3b615ca6c914c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380680"},"id":"literature:url:https://oeis.org/A380680","identifiers":{"url":"https://oeis.org/A380680"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380680, bilateral rational-quotient square series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380680","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380681"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/907b5956e6822ed6b18412d7af9cc1b8098b5941a7b320ff1c30cdb65338033a.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380681"},"id":"literature:url:https://oeis.org/A380681","identifiers":{"url":"https://oeis.org/A380681"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380681–A380686, prime parameters and integral bilateral series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380681","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380708"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fbbe82fc0bb669a4ea991bcc44bd07c0aa3163a0c0f53a5346b8d3bba00c5f0f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380708"},"id":"literature:url:https://oeis.org/A380708","identifiers":{"url":"https://oeis.org/A380708"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380708, g.f. satisfying A(x) = 1 + x*abs(1/A(x))^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380708","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380709"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ea544d5453384d5067671b7ed7249b48e9489a1b205835834b49d9df0cf9c649.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380709"},"id":"literature:url:https://oeis.org/A380709","identifiers":{"url":"https://oeis.org/A380709"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380709, g.f. satisfying A(x) = 1 + x*abs(1/A(x))^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380709","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380710"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/57297b071e2537c19ad9870ad7dda83a6b4057f91dcfead5470634ae95ec0b60.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380710"},"id":"literature:url:https://oeis.org/A380710","identifiers":{"url":"https://oeis.org/A380710"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380710, g.f. satisfying A(x) = 1 + x*A(x)*abs(1/A(x)^2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380710","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A380711"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7d9d82cb6aa397d52d05733f69743ea844377fff8299bdd970d6ebe549614fae.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A380711"},"id":"literature:url:https://oeis.org/A380711","identifiers":{"url":"https://oeis.org/A380711"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A380711: generating function and parity conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA380711","year":2025},{"aliases":["Eric W. Weisstein; Joerg Arndt; Peter Luschny; Emily Egolf; Veronika Furst","https://oeis.org/A381190"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b8502e44f459dafd47569a6326d84e6c05ac2791cc8f0b2a533b8133ba10a14.json","authors":"Eric W. Weisstein; Joerg Arndt; Peter Luschny; Emily Egolf; Veronika Furst","evidence":{"source_url":"https://oeis.org/A381190"},"id":"literature:url:https://oeis.org/A381190","identifiers":{"url":"https://oeis.org/A381190"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381190, connected minimal dominating sets in the n-trapezohedral graph","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381190","year":2026},{"aliases":["Wouter Meeussen","https://oeis.org/A381265"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd3cd5326558249bfef958fc8a764026973aed2e9268dad2f87573e08b535792.json","authors":"Wouter Meeussen","evidence":{"source_url":"https://oeis.org/A381265"},"id":"literature:url:https://oeis.org/A381265","identifiers":{"url":"https://oeis.org/A381265"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381265, solid partitions with two layers and second layer a plane partition of 3: formula conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381265","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A381362"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/136dbe2d1b6e859fdf287de333628066a5ba9f88f9f08882e1d50aa0b21da18c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A381362"},"id":"literature:url:https://oeis.org/A381362","identifiers":{"url":"https://oeis.org/A381362"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381362, bilateral product generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381362","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A381363"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fcdc7d382b000988b8b68102680aa6493b64629e3e7483b1ae306da50bdeba7d.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A381363"},"id":"literature:url:https://oeis.org/A381363","identifiers":{"url":"https://oeis.org/A381363"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381363, bilateral product generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381363","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A381364"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d05943512402885b28af958227720c06ad7385fe6d07754745f8e81e134b48b1.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A381364"},"id":"literature:url:https://oeis.org/A381364","identifiers":{"url":"https://oeis.org/A381364"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381364, scaled bilateral product generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381364","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A381365"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06e7b672ce4f5ba1a2da55db7929494ca272c8a76ad7717ffeb5c29a9b436395.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A381365"},"id":"literature:url:https://oeis.org/A381365","identifiers":{"url":"https://oeis.org/A381365"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381365, scaled bilateral product generating function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381365","year":2025},{"aliases":["Mike Sheppard","https://oeis.org/A381423/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7b4992fafd71d3558161be8b913acac9903401b772f35556c55f7f0fd9d693f.json","authors":"Mike Sheppard","evidence":{"source_url":"https://oeis.org/A381423/internal"},"id":"literature:url:https://oeis.org/A381423/internal","identifiers":{"url":"https://oeis.org/A381423/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381423, exponent of the maximal signed Hermite coefficient","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381423%2Finternal","year":2025},{"aliases":["Mike Sheppard","https://oeis.org/A381524/internal?rev=18"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a31b363bfc0a0fcc1dfb7e13840289cf88e78b1524a1aebd80013f919495b0fe.json","authors":"Mike Sheppard","evidence":{"source_url":"https://oeis.org/A381524/internal?rev=18"},"id":"literature:url:https://oeis.org/A381524/internal?rev=18","identifiers":{"url":"https://oeis.org/A381524/internal?rev=18"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381524, smallest exponent of a maximal absolute Hermite coefficient","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381524%2Finternal%3Frev%3D18","year":2025},{"aliases":["Thomas Scheuerle","https://oeis.org/A381670"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/faf5eeb539a6448396771d8e61e8de53c24886be797be81d042c09aef5cba923.json","authors":"Thomas Scheuerle","evidence":{"source_url":"https://oeis.org/A381670"},"id":"literature:url:https://oeis.org/A381670","identifiers":{"url":"https://oeis.org/A381670"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A381670, denominators of the compositional-square series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381670","year":2025},{"aliases":["Mike Sheppard","https://oeis.org/A381737/internal?rev=28"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/800aac64a432f0a9a9314f73a0f263e6202c803d2b4ab4a2408442ce0ee00c90.json","authors":"Mike Sheppard","evidence":{"source_url":"https://oeis.org/A381737/internal?rev=28"},"id":"literature:url:https://oeis.org/A381737/internal?rev=28","identifiers":{"url":"https://oeis.org/A381737/internal?rev=28"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"Orders k of Hermite polynomials whose maximal coefficient in absolute value appears twice","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA381737%2Finternal%3Frev%3D28","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A382124"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e49d224cec6b5a2697f308db696aa0a903696b267520d0cbcaa6387bfd1535e5.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A382124"},"id":"literature:url:https://oeis.org/A382124","identifiers":{"url":"https://oeis.org/A382124"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A382124 and A382125, sigma-product exponential series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382124","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A382318"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/28775c160a7170989f57f451218847b5695e6c629b21e23948af9d6ce436ccd8.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A382318"},"id":"literature:url:https://oeis.org/A382318","identifiers":{"url":"https://oeis.org/A382318"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A382318, g.f. A(x) satisfies A(x) = x + ( Sum_{n>=1} A(x^n) )^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382318","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A382319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/325f588749a592d2be3a8d4fac7a7296167c41dd9cf6c889530619e7befdd45b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A382319"},"id":"literature:url:https://oeis.org/A382319","identifiers":{"url":"https://oeis.org/A382319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A382319, g.f. A(x) satisfies A(x) = x/(1-x) + Sum_{n>=1} A(x^n)^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382319","year":2025},{"aliases":["https://oeis.org/A382357"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3413288c57113299873db7ccf4bc5a5e8489cfb336bc99ccdae84a0a63eb8985.json","evidence":{"source_url":"https://oeis.org/A382357"},"id":"literature:url:https://oeis.org/A382357","identifiers":{"url":"https://oeis.org/A382357"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A382357","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382357"},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A382427"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/45f45d2453bc1cb9b8dfc974a69db6a8ac182f7adbcb46302f8a108704bf6a18.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A382427"},"id":"literature:url:https://oeis.org/A382427","identifiers":{"url":"https://oeis.org/A382427"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A382427","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382427","year":2025},{"aliases":["Peter Kagey","https://oeis.org/A382814"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5b7d4d76d807a20e18ba83965ceea2b3e58c4ef8ccc44325e92a9eea50425398.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A382814"},"id":"literature:url:https://oeis.org/A382814","identifiers":{"url":"https://oeis.org/A382814"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A382814: first-player score in the Fibonachos game","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA382814","year":2025},{"aliases":["Gus Wiseman","https://oeis.org/A383093"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4b93334d09bbc8747fe90b45119fb8440a62c98fa8148c19e6377ae6d35ac167.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A383093"},"id":"literature:url:https://oeis.org/A383093","identifiers":{"url":"https://oeis.org/A383093"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A383093 — partitions admitting constant blocks with a common sum","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA383093","year":2025},{"aliases":["Seiichi Manyama; Peter Bala","https://oeis.org/A383165"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1b0c54a233a2e76eb481aac125d29d9e6e3184351bb8a9e76313849896a1e7d.json","authors":"Seiichi Manyama; Peter Bala","evidence":{"source_url":"https://oeis.org/A383165"},"id":"literature:url:https://oeis.org/A383165","identifiers":{"url":"https://oeis.org/A383165"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A383165, Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^2 / 2.","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA383165","year":2026},{"aliases":["Seiichi Manyama; Peter Bala","https://oeis.org/A383166"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dbaca880093f6a857b68c1bfcdab9b3be735f9ba8164cc58e86d8c96b9b3f436.json","authors":"Seiichi Manyama; Peter Bala","evidence":{"source_url":"https://oeis.org/A383166"},"id":"literature:url:https://oeis.org/A383166","identifiers":{"url":"https://oeis.org/A383166"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A383166, Expansion of e.g.f. log(1 + (exp(2*x) - 1)/2)^3 / 6.","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA383166","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A383377"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5554deda50eadea9489fffe198133e8f4d5005b4a5b28e7481800b75a8accc9a.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A383377"},"id":"literature:url:https://oeis.org/A383377","identifiers":{"url":"https://oeis.org/A383377"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A383377, g.f. satisfying A(x) = Sum_{n>=0} x^n * abs(1/A(x)^n)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA383377","year":2025},{"aliases":["Omar E. Pol","https://oeis.org/A384222"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d12fceb38ee936919701c758dcb667f1728705dc0f46df19aed6d151c0189948.json","authors":"Omar E. Pol","evidence":{"source_url":"https://oeis.org/A384222"},"id":"literature:url:https://oeis.org/A384222","identifiers":{"url":"https://oeis.org/A384222"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A384222, lengths of the 2-dense sublists of divisors of n","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384222","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A384267"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cbc427631f12bca145eead69dc417877d294059af045c8562f0c2635804d856f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A384267"},"id":"literature:url:https://oeis.org/A384267","identifiers":{"url":"https://oeis.org/A384267"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A384267, a(n) = abs([x^n] 1 + x/A(x)^2), conjecture C.1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384267","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A384270"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aac2bcf0ad35ba0addc7e8a3ad4c4d2e84ffdb2f483047d907e8cb15b4e9829d.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A384270"},"id":"literature:url:https://oeis.org/A384270","identifiers":{"url":"https://oeis.org/A384270"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A384270, g.f. satisfying A(x) = A(x^4 + 4*x*A(x)^4)/A(x^3 + 3*x*A(x)^3)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384270","year":2025},{"aliases":["David James Sycamore","https://oeis.org/A384309"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c07e9129d37142183d238d2edd5831711788f8c44dc6e9decab71d66a86f3292.json","authors":"David James Sycamore","evidence":{"source_url":"https://oeis.org/A384309"},"id":"literature:url:https://oeis.org/A384309","identifiers":{"url":"https://oeis.org/A384309"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A384309 — the leading decimal digit counter sequence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384309","year":2025},{"aliases":["Gus Wiseman","https://oeis.org/A384350"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8563e5e276ebea8735b98627a8fed87bf1bd4e816654a5841406bcca12340dbc.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A384350"},"id":"literature:url:https://oeis.org/A384350","identifiers":{"url":"https://oeis.org/A384350"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A384350 and A384318 — disjoint strict partition families","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384350","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A384819"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/174e64b6012bba052487f116840fa79cd13ef99f13684858a055434dc9697fb0.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A384819"},"id":"literature:url:https://oeis.org/A384819","identifiers":{"url":"https://oeis.org/A384819"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A384819: bounded integral square-weight corrections","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA384819","year":2025},{"aliases":["Werner Schulte","https://oeis.org/A385590"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/228b39a8a3aa95e5fe211fed0995fb4d7ffc9bba789795e16ad491cf00f6bff4.json","authors":"Werner Schulte","evidence":{"source_url":"https://oeis.org/A385590"},"id":"literature:url:https://oeis.org/A385590","identifiers":{"url":"https://oeis.org/A385590"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A385590, a triangle based on Fibonacci numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA385590","year":2025},{"aliases":["Andrei Zabolotskii","https://oeis.org/A385672"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1fe6b7f9e7dc2bf638be5e1841b96d89e5dbc05af80bf8be999b1725fdb7c3f.json","authors":"Andrei Zabolotskii","evidence":{"source_url":"https://oeis.org/A385672"},"id":"literature:url:https://oeis.org/A385672","identifiers":{"url":"https://oeis.org/A385672"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A385672, Irregular triangle read by rows: T(n, k) is the number of n-step walks on the square lattice having algebraic area k","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA385672","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A386655"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7bbdcf1d0f710ea5503dac48d5bb21492cc32a5a437d11fa5b3d8cb9aae358c2.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A386655"},"id":"literature:url:https://oeis.org/A386655","identifiers":{"url":"https://oeis.org/A386655"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A386655–A386658, Lambert W exponential generating functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA386655","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A386666"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d43913c0583e0ac1d08481eb6daedb6b62320005748510ba58a299d6d824162.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A386666"},"id":"literature:url:https://oeis.org/A386666","identifiers":{"url":"https://oeis.org/A386666"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A386666, g.f. satisfying A(x)^2 = A(x^2 + 4*A(x)^3)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA386666","year":2025},{"aliases":["https://oeis.org/A387118"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/784ba45d1d5e0682bada1b843bce872333abdbd83180597e2949978c7a61f9cb.json","evidence":{"source_url":"https://oeis.org/A387118"},"id":"literature:url:https://oeis.org/A387118","identifiers":{"url":"https://oeis.org/A387118"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A387118","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA387118"},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A387319"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02748133b054c7b51ea956bb94d8d4a9af07d081d23da40e6e230c21815b6583.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A387319"},"id":"literature:url:https://oeis.org/A387319","identifiers":{"url":"https://oeis.org/A387319"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A387319","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA387319","year":2025},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A388711"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ef5f88cc0a428a95138a5c92441483a33ba882219759d5d24622d13e2c2911e.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A388711"},"id":"literature:url:https://oeis.org/A388711","identifiers":{"url":"https://oeis.org/A388711"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A388711","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA388711","year":2025},{"aliases":["Gus Wiseman","https://oeis.org/A388712"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/722c67ac855ca7e13066c17dfbbb02b72f0280e353ed3dbe3fef0146e1ce9dcb.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A388712"},"id":"literature:url:https://oeis.org/A388712","identifiers":{"url":"https://oeis.org/A388712"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A388712, A387118 and A388713","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA388712","year":2025},{"aliases":["https://oeis.org/A388713"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3476fd9a1465b8d6c5e060ee188d0e4ff6682c58c41b7edf2031894820ffedd.json","evidence":{"source_url":"https://oeis.org/A388713"},"id":"literature:url:https://oeis.org/A388713","identifiers":{"url":"https://oeis.org/A388713"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A388713","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA388713"},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A388986"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e43681cfc7dc5f0ce1581664cc31ddce0580597149b18765d3c5905c55f6155.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A388986"},"id":"literature:url:https://oeis.org/A388986","identifiers":{"url":"https://oeis.org/A388986"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A388986","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA388986","year":2026},{"aliases":["Chai Wah Wu","https://oeis.org/A389000"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/38bd184350be27101298ca12c3bfa3e4cb9b8c4890aadff885da5e023da8b2e7.json","authors":"Chai Wah Wu","evidence":{"source_url":"https://oeis.org/A389000"},"id":"literature:url:https://oeis.org/A389000","identifiers":{"url":"https://oeis.org/A389000"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389000, least k with n dividing the digit sums of k and k^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389000","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389472"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b4c1ae12d5c759f3b0f01e9bed10825fbe0df40e43fb13657bd42f378b9fe8b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389472"},"id":"literature:url:https://oeis.org/A389472","identifiers":{"url":"https://oeis.org/A389472"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389472, g.f. satisfying A(x) = A(x^2 + x^3)/x^2 - 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389472","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389476"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/df34d235c741e31006613d905bdc74fc9a827c57d9a6063dd15e759e04fd2a85.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389476"},"id":"literature:url:https://oeis.org/A389476","identifiers":{"url":"https://oeis.org/A389476"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389476, g.f. satisfying A(x - A(x)^2/(1 - A(x))^2) = x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389476","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389536"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0bf4c8baaade659220f17833ff2fdcd9c1d862df0ae039ccde4f7230997b8ab9.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389536"},"id":"literature:url:https://oeis.org/A389536","identifiers":{"url":"https://oeis.org/A389536"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389536, G.f. A(x) = A(x^2 + 2*x^3)/x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389536","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389537"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0c416c57a598c310b6e4edec70b7e91da8a92120f4b7a374e34821d6d2197562.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389537"},"id":"literature:url:https://oeis.org/A389537","identifiers":{"url":"https://oeis.org/A389537"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389537, g.f. satisfying A(A(x) - x) = A(x)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389537","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389540"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2f0c1070d49d111250a8ae62049c0c79841280bd7b885ec953b20d435950c22a.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389540"},"id":"literature:url:https://oeis.org/A389540","identifiers":{"url":"https://oeis.org/A389540"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389540, g.f. satisfying A(x)^2 = A(2*x - 2*A(x))/2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389540","year":2025},{"aliases":["Paul D. Hanna","https://oeis.org/A389542"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2414a701e38a451b4befe07c0cc453cf9430511f269f886ed48738b381d49f03.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A389542"},"id":"literature:url:https://oeis.org/A389542","identifiers":{"url":"https://oeis.org/A389542"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A389542, g.f. satisfying A(A(x)^2 - x^2) = 4*A(x)^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA389542","year":2025},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A390088"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7bfdf8bdfd85eaeabe57bd06b68d0355af89bdb3f17eaa318ba11d8e2ed517c.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A390088"},"id":"literature:url:https://oeis.org/A390088","identifiers":{"url":"https://oeis.org/A390088"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A390088","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA390088","year":2025},{"aliases":["Charles L. Hohn","https://oeis.org/A390148"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d68bd270e8b245b3e268d8b998e3feef206583228ddd944775da3dd9d65c0dd2.json","authors":"Charles L. Hohn","evidence":{"source_url":"https://oeis.org/A390148"},"id":"literature:url:https://oeis.org/A390148","identifiers":{"url":"https://oeis.org/A390148"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A390148 — primitive radii of four mutually tangent spheres and a plane","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA390148","year":2025},{"aliases":["Ctibor O. Zizka","https://oeis.org/A390871"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/55a0f19b448ae780829c323cf728c953a56f4745ac4938fbb414bc229c312253.json","authors":"Ctibor O. Zizka","evidence":{"source_url":"https://oeis.org/A390871"},"id":"literature:url:https://oeis.org/A390871","identifiers":{"url":"https://oeis.org/A390871"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A390871 — square differences of Mersenne form","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA390871","year":2025},{"aliases":["Gus Wiseman","https://oeis.org/A391620"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f9b52151b608636a02fd810e44a21fa93cd7934fa59be5f10d5194af46c4d168.json","authors":"Gus Wiseman","evidence":{"source_url":"https://oeis.org/A391620"},"id":"literature:url:https://oeis.org/A391620","identifiers":{"url":"https://oeis.org/A391620"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A391620, partitions that are not first sums of a composition with all parts > 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA391620","year":2025},{"aliases":["Andrew Howroyd; Eric W. Weisstein","https://oeis.org/A391632/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b89c1ebd097fc019710572aaf381545fda22c137dc9a47ccc3f6de83272e90b.json","authors":"Andrew Howroyd; Eric W. Weisstein","evidence":{"source_url":"https://oeis.org/A391632/internal"},"id":"literature:url:https://oeis.org/A391632/internal","identifiers":{"url":"https://oeis.org/A391632/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A391632: mutual-visibility subsets of Andrasfai graphs","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA391632%2Finternal","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A391810/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2a77645e91433d74f215b4cde8af5ba392e8897bb8b3d8524c6bbac673ce07bf.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A391810/internal"},"id":"literature:url:https://oeis.org/A391810/internal","identifiers":{"url":"https://oeis.org/A391810/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A391810, nonpositive Laurent coefficients and zero row sums","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA391810%2Finternal","year":2025},{"aliases":["Mikhail Kurkov","https://oeis.org/A392095"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5cccb7611827e01b0c70683da9992c30c8c1e35dab7c68691b675288ad01ff5.json","authors":"Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A392095"},"id":"literature:url:https://oeis.org/A392095","identifiers":{"url":"https://oeis.org/A392095"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392095, antidiagonal array and the shifted A088713 column","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392095","year":2025},{"aliases":["Stefano Spezia","https://oeis.org/A392131/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1e336cf2517671490cd913ebf539f668b64a40beb86953fca2621a48a200bad7.json","authors":"Stefano Spezia","evidence":{"source_url":"https://oeis.org/A392131/internal"},"id":"literature:url:https://oeis.org/A392131/internal","identifiers":{"url":"https://oeis.org/A392131/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392131 and A392132: Pell circulant determinant and permanent","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392131%2Finternal","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A392203"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82f6f11d76476c79a530d96f17f96542020ecb14ea83d289aa11838283a92e3a.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A392203"},"id":"literature:url:https://oeis.org/A392203","identifiers":{"url":"https://oeis.org/A392203"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392203, G.f. A(x) satisfies A(x - A(x)) = x^2 + x*A(x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392203","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A392210"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/52c3755558a1b53d7c38d55d0c7bb6c3e73bb9cd9e66ab8c143e58927ccda1da.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A392210"},"id":"literature:url:https://oeis.org/A392210","identifiers":{"url":"https://oeis.org/A392210"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392210, G.f.: x + sq(x/(1-x) + sq(x/(1-2*x) + sq(x/(1-3*x) + ...))), an infinitely nested square","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392210","year":2026},{"aliases":["Thore Posske","https://oeis.org/A392387"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b66a3751480489b9351a45b5e08c40d4a75c08e766757c72d5fc7cc223be3307.json","authors":"Thore Posske","evidence":{"source_url":"https://oeis.org/A392387"},"id":"literature:url:https://oeis.org/A392387","identifiers":{"url":"https://oeis.org/A392387"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392387, Nullspace dimension of the periodic XX Heisenberg chain with n > 1 sites","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392387","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A392525"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/325f8bab7cf568ed11aed65c10b83ebd09ad5c7f22b4eb3115ba9a71e57557f1.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A392525"},"id":"literature:url:https://oeis.org/A392525","identifiers":{"url":"https://oeis.org/A392525"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392525, g.f. satisfying A(x)^3 = A(x^3 + 15*x*A(x)^3)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392525","year":2026},{"aliases":["Xin Chen and Yudian Chen","https://oeis.org/A392664"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/765872a94230509b807fb33d961153eca79b525be490954448003a3326edebb0.json","authors":"Xin Chen and Yudian Chen","evidence":{"source_url":"https://oeis.org/A392664"},"id":"literature:url:https://oeis.org/A392664","identifiers":{"url":"https://oeis.org/A392664"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392664, A392665 and A392666: negative-step Eulerian triangles","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392664","year":2026},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A392694"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/840a70885c534ae35ae1a83b770f93b486226ede89b0c3fd27950a771f4ef758.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A392694"},"id":"literature:url:https://oeis.org/A392694","identifiers":{"url":"https://oeis.org/A392694"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392694","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392694","year":2026},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A392698"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b23a0db57e7b67d7481c1ee37295f8dffa3b67c1ad3443ae2aa62e60709d8424.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A392698"},"id":"literature:url:https://oeis.org/A392698","identifiers":{"url":"https://oeis.org/A392698"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392698","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392698","year":2026},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A392707"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8afed90bb8212dad4abd556a7c0eeb9d389a67a43a2e0f40576f8f99672cd3af.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A392707"},"id":"literature:url:https://oeis.org/A392707","identifiers":{"url":"https://oeis.org/A392707"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A392707","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA392707","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A393170"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f350701138e4de3feb2902a34ae8160dca66bf77059b0ea3d23be3829e839e47.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393170"},"id":"literature:url:https://oeis.org/A393170","identifiers":{"url":"https://oeis.org/A393170"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393170, self-referential inverse series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393170","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A393172"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab7c147d52803d3b133d1c11213fa2fa646156b7d95736c8f281c15032897b71.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393172"},"id":"literature:url:https://oeis.org/A393172","identifiers":{"url":"https://oeis.org/A393172"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393172, g.f. satisfying A(x-x^2) = x + A(x)^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393172","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A393173"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/41afd3fee590c8a50286f452a1ba99ca8f9f282b81a9ad87709cc63deee4ac00.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393173"},"id":"literature:url:https://oeis.org/A393173","identifiers":{"url":"https://oeis.org/A393173"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393173, squared-index inverse-diagonal parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393173","year":2026},{"aliases":["Peter Bala; Sean A. Irvine; Stephen Wolfram","https://oeis.org/A393210"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6328315c14410a1d9a4cbe9bd5fcac73be76e10162866c3f986d2b51bbae4559.json","authors":"Peter Bala; Sean A. Irvine; Stephen Wolfram","evidence":{"source_url":"https://oeis.org/A393210"},"id":"literature:url:https://oeis.org/A393210","identifiers":{"url":"https://oeis.org/A393210"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393210: halting time for Wolfram machine 3626","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393210","year":2026},{"aliases":["https://oeis.org/A393434"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7a4b53d4854a26e661e85d75616923b9152af15b8cdff863dde3fd0cee19d9ce.json","evidence":{"source_url":"https://oeis.org/A393434"},"id":"literature:url:https://oeis.org/A393434","identifiers":{"url":"https://oeis.org/A393434"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A393434","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393434"},{"aliases":["Paul D. Hanna","https://oeis.org/A393856"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d387c876a2f06debfec6e2680355dc6659b672d5503ee48ef0ef3b1f044c3e6c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393856"},"id":"literature:url:https://oeis.org/A393856","identifiers":{"url":"https://oeis.org/A393856"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393856, G.f. satisfies A(x - x*A(4*x)/4) = x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393856","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A393857"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8e55fbc429f644124265272a5f2fef2634f62dd614206e4a79b311f03bbfdee3.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393857"},"id":"literature:url:https://oeis.org/A393857","identifiers":{"url":"https://oeis.org/A393857"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393857, G.f. satisfies A(x - x*A(5*x)/5) = x","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393857","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A393867"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a53750c40b5d52e6ced3db79ab4c3c10a5638e075ef64dffd9336869f697ee8f.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A393867"},"id":"literature:url:https://oeis.org/A393867","identifiers":{"url":"https://oeis.org/A393867"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A393867, logarithmic derivative of A393866 (g.f. with [x^n] A^prime(n) = prime(n) [x^(n-1)] A^prime(n))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA393867","year":2026},{"aliases":["https://oeis.org/A394399"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d34ccf894c3524862f5f81008c63d36ec9730801fa519056a99c23b93aa36451.json","evidence":{"source_url":"https://oeis.org/A394399"},"id":"literature:url:https://oeis.org/A394399","identifiers":{"url":"https://oeis.org/A394399"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A394399","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA394399"},{"aliases":["Felix Huber","https://oeis.org/A394407"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3639b3320f0c0169f2c10a06af6b2cfdb824a6275c06a72116000051fdf3cecd.json","authors":"Felix Huber","evidence":{"source_url":"https://oeis.org/A394407"},"id":"literature:url:https://oeis.org/A394407","identifiers":{"url":"https://oeis.org/A394407"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A394407: first occurrences of the totient-to-divisor-count ratio","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA394407","year":2026},{"aliases":["Jean-Marc Rebert; Sean A. Irvine","https://oeis.org/A394431"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/83ef7c0ffcea3b2dff87e36af117a081842974676d5d045988db4e895d9ce688.json","authors":"Jean-Marc Rebert; Sean A. Irvine","evidence":{"source_url":"https://oeis.org/A394431"},"id":"literature:url:https://oeis.org/A394431","identifiers":{"url":"https://oeis.org/A394431"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A394431, The smallest base b < n where the sum of the digits for the number n in the base b is the largest","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA394431","year":2026},{"aliases":["Clark Kimberling","https://oeis.org/A394578/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7f192ce37bd789179b98d50d06d1fe19a7efcd08db5d161f727429499315ee72.json","authors":"Clark Kimberling","evidence":{"source_url":"https://oeis.org/A394578/internal"},"id":"literature:url:https://oeis.org/A394578/internal","identifiers":{"url":"https://oeis.org/A394578/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A394578: a(n) = 2^(2*n-1) * p(n, n+1/2), where p(n,x) is the Lagrange interpolating polynomial through (1,0) and (k,1) for k=2..n+1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA394578%2Finternal","year":2026},{"aliases":["OEIS Foundation Inc.","https://oeis.org/A394757"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/150e769bc37193847b5e1ca40151fdd22cf65cac7d5a963b66fb3941cbee97b5.json","authors":"OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A394757"},"id":"literature:url:https://oeis.org/A394757","identifiers":{"url":"https://oeis.org/A394757"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A394757","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA394757","year":2026},{"aliases":["Aleksei Udovenko","https://oeis.org/A395531/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a88195db31004b178a4ec0340a1bf4b91c757bc1fafbfe5abf0742e94e2fcede.json","authors":"Aleksei Udovenko","evidence":{"source_url":"https://oeis.org/A395531/internal"},"id":"literature:url:https://oeis.org/A395531/internal","identifiers":{"url":"https://oeis.org/A395531/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A395531, greedy-brick row births","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA395531%2Finternal","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A395833"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/68fab2392b70e8af81adbd1c9c1564a031cea52738d21b6ad1bca3d3c52d2e49.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A395833"},"id":"literature:url:https://oeis.org/A395833","identifiers":{"url":"https://oeis.org/A395833"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A395833, g.f. A(x) satisfies [x^n] A(x/A(x)^(2*n-1)) = 0 for n > 1","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA395833","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A395842"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bdd2aced51edd528496045b693aa45e3541968e17f016606f644c217bf8b7149.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A395842"},"id":"literature:url:https://oeis.org/A395842","identifiers":{"url":"https://oeis.org/A395842"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A395842, diagonal coefficients of the compositional iterates of A177775","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA395842","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396099"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/db0b6e13b4df1bbac3067ca87cd8e77b5db333870193f1c72f99f57c9ea075ee.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396099"},"id":"literature:url:https://oeis.org/A396099","identifiers":{"url":"https://oeis.org/A396099"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396099, g.f. satisfying A(x) = x + A^2(x)*A^3(x) (compositional iterates)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396099","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396102"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e913f417ba5c6fecb930663301838f0cec74ed451307e808724051fae6da47ea.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396102"},"id":"literature:url:https://oeis.org/A396102","identifiers":{"url":"https://oeis.org/A396102"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396102, g.f. satisfying A(A(A(x))) = (1+x)*A(A(x))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396102","year":2026},{"aliases":["Mikhail Kurkov","https://oeis.org/A396563/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e20bd29a0bfb23e74dbd64ef0318844aff76aafedc3891c57246fd605ab1da64.json","authors":"Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/A396563/internal"},"id":"literature:url:https://oeis.org/A396563/internal","identifiers":{"url":"https://oeis.org/A396563/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396563: three-parameter Stirling-minor Vandermonde sum","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396563%2Finternal","year":2026},{"aliases":["https://oeis.org/A396659/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b1936f3a131823a899d424abf10d91d5f45eb9f4bfbb2e37a3e11cef3143e498.json","evidence":{"source_url":"https://oeis.org/A396659/internal"},"id":"literature:url:https://oeis.org/A396659/internal","identifiers":{"url":"https://oeis.org/A396659/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A396659/internal","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396659%2Finternal"},{"aliases":["Paul D. Hanna","https://oeis.org/A396793"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d9cfb10631986ac63c274bc94dd7a1d1fd6a50f6407ff992b4a1d8c4843f7906.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396793"},"id":"literature:url:https://oeis.org/A396793","identifiers":{"url":"https://oeis.org/A396793"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396793, g.f. satisfying A(x) A(A(x)) = x^2 + 9 x^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396793","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396794"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/732c203e331a781bec81f663214525024186d3d946b3fc3e0734360208affa55.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396794"},"id":"literature:url:https://oeis.org/A396794","identifiers":{"url":"https://oeis.org/A396794"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396794, G.f. A(x) satisfies A(x)*A(A(A(x))) = x^2 + 16*x^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396794","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396795"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8ed3f3c095cf80de417511c895299660a7f48d1bb191e4afb37f530af598cb40.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396795"},"id":"literature:url:https://oeis.org/A396795","identifiers":{"url":"https://oeis.org/A396795"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396795, g.f. satisfying A^2(x) A^3(x) = x^2 + 25 x^3","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396795","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396797"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f7f0675ac334fba66140ecd2d5dacadb94831c797f19b03d4ae8527d79bc5e48.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396797"},"id":"literature:url:https://oeis.org/A396797","identifiers":{"url":"https://oeis.org/A396797"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396797, G.f. satisfies A(x) = x + A^3(x)*A^4(x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396797","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396798"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/093b2712380ba64ca4979f8dd9527e7addf4e416ff9fb1dad0ae4956134ba676.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396798"},"id":"literature:url:https://oeis.org/A396798","identifiers":{"url":"https://oeis.org/A396798"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396798: compositional iterates modulo eight","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396798","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396803"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd6825229be9f62d2202e455c0e568464a1360391bf75ea481aa221be6694b91.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396803"},"id":"literature:url:https://oeis.org/A396803","identifiers":{"url":"https://oeis.org/A396803"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396803, A396805 and A396806: iterated-exponential congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396803","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396804"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/168c949ae016fffa9eafeabfee758097df1af09075f826405e95e7ad56c62469.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396804"},"id":"literature:url:https://oeis.org/A396804","identifiers":{"url":"https://oeis.org/A396804"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396804 — E.g.f. satisfies A(x) = x exp(A^4(x))","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396804","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396805"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/599a17cccdfc6937760dfec8ff194f84f0520e36c5910939807c37ed8c90df60.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396805"},"id":"literature:url:https://oeis.org/A396805","identifiers":{"url":"https://oeis.org/A396805"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396805: iterated-exponential congruences at iterate count 5","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396805","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396806"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/596c7e84739de42b4ec1181e00de3898e724ba8913f088941b4ba809f0141639.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396806"},"id":"literature:url:https://oeis.org/A396806","identifiers":{"url":"https://oeis.org/A396806"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396806: iterated-exponential congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396806","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396807"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/746d4820396174614f0ca658efcd93d25e907d62808a906902499f894bef8949.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396807"},"id":"literature:url:https://oeis.org/A396807","identifiers":{"url":"https://oeis.org/A396807"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396807, G.f. satisfies A(x) = x + A^5(x)*A^6(x)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396807","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396808"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2809a214591ee2790f000be4480a666a0ff5ba3d7f9c7f502da219a9def742de.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396808"},"id":"literature:url:https://oeis.org/A396808","identifiers":{"url":"https://oeis.org/A396808"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396808, an implicit power-series coefficient equation","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396808","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396843"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/292616413f01ec2917f8969616ef43db032586a7bbdbbe61458075a0554da511.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396843"},"id":"literature:url:https://oeis.org/A396843","identifiers":{"url":"https://oeis.org/A396843"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396843, g.f. satisfying A(x*A(x) - 3*x*A(x)^2) = x^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396843","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396844"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/04a43d112d4fd17d983f6e08db180a322776b8da1a83e78c179590aed5e5f324.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396844"},"id":"literature:url:https://oeis.org/A396844","identifiers":{"url":"https://oeis.org/A396844"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396844, g.f. satisfying A(x*A(x) - 4*x*A(x)^2) = x^2","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396844","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A396846"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/74eaa60958502188b7baf7e6ed2d81c90e4a9fb3c112bae75b77291286a8c274.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A396846"},"id":"literature:url:https://oeis.org/A396846","identifiers":{"url":"https://oeis.org/A396846"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A396846, l.g.f. log(1 + x + Sum 4n/(4n^2-1) a(n) x^n), mod-8 conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA396846","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397241"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/97e4614b87385eb98dfca9dbced76d65af54b8a2432015834d6c090f8982edc7.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397241"},"id":"literature:url:https://oeis.org/A397241","identifiers":{"url":"https://oeis.org/A397241"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397241, g.f. with n [x^n] A^n = (n-1) [x^n] A^(n+1)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397241","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397242"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/348b49ff18a0aa31bc3e4b0c3d2527e43a489badc392d89fec499cf50357a922.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397242"},"id":"literature:url:https://oeis.org/A397242","identifiers":{"url":"https://oeis.org/A397242"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397242, o.g.f. A = exp(x + Sum (n^2-1) a(n) x^n / n^2)","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397242","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397244"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/95a6f9de73b6b38b1718eb69576739a7e458e3f8f8281f794225c75d95977cee.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397244"},"id":"literature:url:https://oeis.org/A397244","identifiers":{"url":"https://oeis.org/A397244"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397244, quadratic power-diagonal generating series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397244","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397345"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/265eca1d5a5c82c35d0c51f89eec4c23e07c17f115ffc356da2b3fbc9b57de7b.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397345"},"id":"literature:url:https://oeis.org/A397345","identifiers":{"url":"https://oeis.org/A397345"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397345, A397348 and A397346: exponential square-weight congruences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397345","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397346"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/06faf8daf7d47d9a309ccc9a9dbef6581fa980ab8b2ed8503e2bb2008d51f552.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397346"},"id":"literature:url:https://oeis.org/A397346","identifiers":{"url":"https://oeis.org/A397346"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397346: exponential square-weight congruence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397346","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397348"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0ec676de4d1a83d1b2d370f5fa8e85575eb155c5f62fb32da1e15572eb681960.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397348"},"id":"literature:url:https://oeis.org/A397348","identifiers":{"url":"https://oeis.org/A397348"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397348: exponential square-weight congruence","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397348","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397349"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e08b36ba752bb8fb8ba5d55bfe44fb7c3f39980e391132bb8db06d55aa9d8bbc.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397349"},"id":"literature:url:https://oeis.org/A397349","identifiers":{"url":"https://oeis.org/A397349"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397349, logarithmic-weight coefficients: residue and parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397349","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397356"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cd0768cef5bf425c6df646616449697583764a1c229fd737c432d951fc656f99.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397356"},"id":"literature:url:https://oeis.org/A397356","identifiers":{"url":"https://oeis.org/A397356"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397356, reciprocal square-exponent diagonal generating series","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397356","year":2026},{"aliases":["Tanguy Gautier Loic Le Mer","https://oeis.org/A397434"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/58eb3157d8f97173c609cb95724c92fc8aa7bdb826b7a7221fdd64c2ba4a2f3c.json","authors":"Tanguy Gautier Loic Le Mer","evidence":{"source_url":"https://oeis.org/A397434"},"id":"literature:url:https://oeis.org/A397434","identifiers":{"url":"https://oeis.org/A397434"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397434, monomial counts for threshold Boolean functions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397434","year":2026},{"aliases":["Peter Bala; OEIS Foundation Inc.","https://oeis.org/A397436/internal"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/457f1f4cc753fc60e9bf62220139be28236b82d08198e69be76780d9481dfcdf.json","authors":"Peter Bala; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A397436/internal"},"id":"literature:url:https://oeis.org/A397436/internal","identifiers":{"url":"https://oeis.org/A397436/internal"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397436 revision 44, Conjecture 3 on integer-parameter binomial products","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397436%2Finternal","year":2026},{"aliases":["https://oeis.org/A397578"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f1ad9f4d1ebedf8603cc0c2338f167c3aa71c7ecde342d4f9f1f72e3e467edeb.json","evidence":{"source_url":"https://oeis.org/A397578"},"id":"literature:url:https://oeis.org/A397578","identifiers":{"url":"https://oeis.org/A397578"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A397578","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397578"},{"aliases":["Paul D. Hanna","https://oeis.org/A397591"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9e4290edc3204809f29f3e7a379322702a4d5e6d9d297eb69ec803faa1b99dd1.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397591"},"id":"literature:url:https://oeis.org/A397591","identifiers":{"url":"https://oeis.org/A397591"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397591: linear-exponent coefficient equations and dyadic neighbors","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397591","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397592"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9d8d407e47368c3e3076f19945fd5e6a07961a4418aaf46710dd8568a6beef64.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397592"},"id":"literature:url:https://oeis.org/A397592","identifiers":{"url":"https://oeis.org/A397592"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397592: doubled linear rows and dyadic parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397592","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397594"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bbaf16225728c2eefd7d3ff894e7e0d26084b7efd03e3cc05e543fa56b386b5e.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397594"},"id":"literature:url:https://oeis.org/A397594","identifiers":{"url":"https://oeis.org/A397594"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397594: quadrupled linear rows and dyadic parity","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397594","year":2026},{"aliases":["Paul D. Hanna","https://oeis.org/A397902"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa037b3f143e77b3776268cf675fdc320acb2b4ca21e226b3eeec663978c266c.json","authors":"Paul D. Hanna","evidence":{"source_url":"https://oeis.org/A397902"},"id":"literature:url:https://oeis.org/A397902","identifiers":{"url":"https://oeis.org/A397902"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A397902: square-exponent coefficient equations and dyadic parity support","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA397902","year":2026},{"aliases":["David Ulgenes","https://oeis.org/A398064"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a6e35610857090775e61b6089c10d7537ae68c37591c45346bec8def96ed4620.json","authors":"David Ulgenes","evidence":{"source_url":"https://oeis.org/A398064"},"id":"literature:url:https://oeis.org/A398064","identifiers":{"url":"https://oeis.org/A398064"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398064, truncated-binomial limit for Hadamard's Gamma function","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398064","year":2026},{"aliases":["Peter Luschny","https://oeis.org/A398189"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d028dbeb745936ddf8bb31611c5f61f0b04225f688cb0b6581331b0a614a85c4.json","authors":"Peter Luschny","evidence":{"source_url":"https://oeis.org/A398189"},"id":"literature:url:https://oeis.org/A398189","identifiers":{"url":"https://oeis.org/A398189"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398189, binary valuations of generalized Schenker sums","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398189","year":2026},{"aliases":["Charles Cornell Norton","https://oeis.org/A398542"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2cc2554b4dfbec64351a74f856d5b7cd63d6ad1a5f61942e7471536292a169f9.json","authors":"Charles Cornell Norton","evidence":{"source_url":"https://oeis.org/A398542"},"id":"literature:url:https://oeis.org/A398542","identifiers":{"url":"https://oeis.org/A398542"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398542: fixed-bottom polynomial conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398542","year":2026},{"aliases":["David Batista","https://oeis.org/A398581"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3be55ac6170121b471bd72274b0daff5afe310889c014fd33cf52dbc20fec70b.json","authors":"David Batista","evidence":{"source_url":"https://oeis.org/A398581"},"id":"literature:url:https://oeis.org/A398581","identifiers":{"url":"https://oeis.org/A398581"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398581, separation of the lexicographically first and maximum-z Egyptian fraction solutions","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398581","year":2026},{"aliases":["Joshua B. Weinstein","https://oeis.org/A398589"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/35da3139aa38f318f8bbc48212fc4cb0677029d51b76d3d18d1a7441f3509532.json","authors":"Joshua B. Weinstein","evidence":{"source_url":"https://oeis.org/A398589"},"id":"literature:url:https://oeis.org/A398589","identifiers":{"url":"https://oeis.org/A398589"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398589: eventual periodicity of self-banning rows","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398589","year":2026},{"aliases":["F. Chapoton","https://oeis.org/A398690"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1181767a627157a9e6ac0c08a332d2271b08d3d76261819f1ebd35a510a708a1.json","authors":"F. Chapoton","evidence":{"source_url":"https://oeis.org/A398690"},"id":"literature:url:https://oeis.org/A398690","identifiers":{"url":"https://oeis.org/A398690"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398690, triangle of polynomials related to Verlinde numbers","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398690","year":2026},{"aliases":["G. Gordon Thompson","https://oeis.org/A398720"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61d153a0cab209296d985e05795ff1191d400e3ab2bbe6c703eb9a3b8ff1f833.json","authors":"G. Gordon Thompson","evidence":{"source_url":"https://oeis.org/A398720"},"id":"literature:url:https://oeis.org/A398720","identifiers":{"url":"https://oeis.org/A398720"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A398720 — weight distribution of square single parity check product codes","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398720","year":2026},{"aliases":["Vaclav Kotesovec and Alois P. Heinz","https://oeis.org/A398726"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32f9cf67d9810208bdbc8fff3bc3f0b4c5312ffd51e4d8ee978890b5c65089b6.json","authors":"Vaclav Kotesovec and Alois P. Heinz","evidence":{"source_url":"https://oeis.org/A398726"},"id":"literature:url:https://oeis.org/A398726","identifiers":{"url":"https://oeis.org/A398726"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A398726, distinct cycle size asymptotic conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA398726","year":2026},{"aliases":["https://oeis.org/A399155"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fdcd0b837db5b861eb124910b903fc37675091d52f371eee63616bf81ba1510a.json","evidence":{"source_url":"https://oeis.org/A399155"},"id":"literature:url:https://oeis.org/A399155","identifiers":{"url":"https://oeis.org/A399155"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A399155","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA399155"},{"aliases":["Chai Wah Wu","https://oeis.org/A399755"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0e5ef0b8caeee404a06c421160d69360e4c74a0c302a22aa7b0de4ecc985de55.json","authors":"Chai Wah Wu","evidence":{"source_url":"https://oeis.org/A399755"},"id":"literature:url:https://oeis.org/A399755","identifiers":{"url":"https://oeis.org/A399755"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A399755: Numbers k such that the generalized Pell equation x^2-(k^2+1)*(y^2+1) = 0 has more than one fundamental solution in (x, y).","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA399755","year":2026},{"aliases":["Antti Karttunen","https://oeis.org/A400168"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d92bf321850b39af18fa64cea3850bf6e62eb3f8c7fe2ec98de825973dc0d671.json","authors":"Antti Karttunen","evidence":{"source_url":"https://oeis.org/A400168"},"id":"literature:url:https://oeis.org/A400168","identifiers":{"url":"https://oeis.org/A400168"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400168: arithmetic derivative primorial length minus the prime-power-divisor maximum","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400168","year":2026},{"aliases":["Omar E. Pol","https://oeis.org/A400194"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/889933141c4c09276f1a2740748bbed03b9bfdbdaf32c41173a31d2156ffc1dc.json","authors":"Omar E. Pol","evidence":{"source_url":"https://oeis.org/A400194"},"id":"literature:url:https://oeis.org/A400194","identifiers":{"url":"https://oeis.org/A400194"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400194, maximum length of a two-dense divisor block","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400194","year":2026},{"aliases":["Peter Kagey","https://oeis.org/A400216"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bc64cdec2fac3b4b7d36e164e63a74a732e0dfb918555692936147696e0f1cf.json","authors":"Peter Kagey","evidence":{"source_url":"https://oeis.org/A400216"},"id":"literature:url:https://oeis.org/A400216","identifiers":{"url":"https://oeis.org/A400216"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400216, A400217 and A400218, Boerdijk-Coxeter tetrahelix coordinates","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400216","year":2026},{"aliases":["Ctibor O. Zizka; OEIS Foundation Inc.","https://oeis.org/A400361"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/703e4928a18e9029c8a2844cc60f7a91eb18f0629f359914ecd6adb2f2a4d239.json","authors":"Ctibor O. Zizka; OEIS Foundation Inc.","evidence":{"source_url":"https://oeis.org/A400361"},"id":"literature:url:https://oeis.org/A400361","identifiers":{"url":"https://oeis.org/A400361"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400361 revision 17, the second-order Pillai minus floor-gcd conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400361","year":2026},{"aliases":["Hunter Hogan","https://oeis.org/A400429"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f6ea54d21d2ccd37516a57f0c98e2a33bc0bfc8554da44e7045b0a3ba5c37345.json","authors":"Hunter Hogan","evidence":{"source_url":"https://oeis.org/A400429"},"id":"literature:url:https://oeis.org/A400429","identifiers":{"url":"https://oeis.org/A400429"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400429, semi-meanders by crossings and winding number","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400429","year":2026},{"aliases":["J. R. G. Mendonça","https://oeis.org/A400584"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fcb95aa21b2626221961bdb321cc674e17740eeae3f141b1bf235ed889c61e3a.json","authors":"J. R. G. Mendonça","evidence":{"source_url":"https://oeis.org/A400584"},"id":"literature:url:https://oeis.org/A400584","identifiers":{"url":"https://oeis.org/A400584"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"OEIS A400584 and The min-plus configuration triangle: Technical note supporting OEIS A400584","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400584","year":2026},{"aliases":["Jishnu Babu Ranitha; Vaclav Kotesovec","https://oeis.org/A400585"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1256cb5e0f064728c3251aac78ff2873f12b357dac6231e6b5d67384908724af.json","authors":"Jishnu Babu Ranitha; Vaclav Kotesovec","evidence":{"source_url":"https://oeis.org/A400585"},"id":"literature:url:https://oeis.org/A400585","identifiers":{"url":"https://oeis.org/A400585"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"A400585 — odd-cofactor binomial divisor sums and a cubic summatory conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400585","year":2026},{"aliases":["https://oeis.org/A400644"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6573b7c06cf190311bfb06735a40695c4ace34074e7e8b76db398fb749def112.json","evidence":{"source_url":"https://oeis.org/A400644"},"id":"literature:url:https://oeis.org/A400644","identifiers":{"url":"https://oeis.org/A400644"},"kind":"literature","status":"external-literature","summary":"External research source. Attribution is retained; a citation does not establish proof coverage or novelty.","title":"https://oeis.org/A400644","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?record=literature%3Aurl%3Ahttps%3A%2F%2Foeis.org%2FA400644"},{"aliases":["Paul D. Hanna; Mikhail Kurkov","https://oeis.org/history/view?seq=A107667&v=25"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/56166233201ef5d2de82dd2b5a5015e6dc7b85c2838393e3c283cd0486f2ec28.json","authors":"Paul D. Hanna; Mikhail Kurkov","evidence":{"source_url":"https://oeis.org/history/view?seq=A107667&v=25"},"id":"literature:url:https://oeis.org/history/view?seq=A107667&v=25","identifiers":{"url":"https://oeis.org/history/view?seq=A107667&v=25"},"kind":"literature","status":"external-literature","summary":"External research source. 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membership is not a proof-completion claim.","title":"Every original (a,b,d+2,d), 2<=b is not a group} < infinity.\n\nConjecture 5.3 is a separate finite-semigroup assertion and is not a second\nsettlement of this question.\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| every commutative semigroup, including infinite and nonunital ones | `S : Type`, `[AddCommSemigroup S]` |\n| periodic | `Periodic S`: each native `AddSubsemigroup.closure {a}` is finite |\n| positive cyclic generation | `cyclic a`, without adjoining a zeroth multiple |\n| actual cyclic idempotent | `IsCyclicIdempotent` and `cyclicIdempotent`; pointwise existence uses finite cyclic closure |\n| every commutative unitary ring | `R : Type`, `[CommRing R]`; no field hypothesis |\n| ordinary semigroup algebra | upstream `AddMonoidAlgebra R S`, retaining all labels |\n| all independently nonzero positional coefficients | `GoodSequence`: every `c : N -> R`, with `c i != 0` for each occupied position |\n| finite algebraic constant | `algebraicD S R < top`, supremum of finite good lengths |\n| actual principal ideals | `principalIdeal a = {a} union {a+c : c in S}` |\n| Green quotient | native quotient of `greenCon S`, whose relation is equality of those ideals |\n| cyclic nongroup, with its own possible identity | negation of `CyclicIsGroup a`, quantifying identity and inverses inside `cyclic a` |\n| relative constant at the canonical image | `relativeD ((greenCon S).toQuotient a)` |\n| relative irreducibility | no shorter sublist, including empty, has the same sum |\n| one uniform bound over all filtered elements | `nongroupSup S < top` |\n| whole implication and exact negation | closed `claim : Prop`; 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Infinity means unbounded finite lengths; no\ninfinite product or finite truncation substitutes for the source assertion.\nNonempty products require no algebra unit. Length zero contributes only\nzero to the length supremum. The empty sequence sums only to an existing\ntwo-sided identity. In a monoid this is exactly the usual list sum, so the\nempty proper subsequence is included in the counterexample's irreducibility.\nLists enumerate the source's unordered sequences and their submultisets;\ncommutativity makes enumeration irrelevant.","status":"refuted","summary":"Let S be a commutative periodic semigroup, and let R be a commutative unitary ring. 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The algebra is the ordinary semigroup algebra,\nthe scalar quantifier is universal over individually nonzero scalars,\nD is the relative irreducible-length constant at the canonical quotient\nimage, and Gamma consists of the actual stabilizer translation maps.\n\n| Source clause | Lean counterpart |\n| --- | --- |\n| every finite commutative semigroup | `S : Type`, `AddCommSemigroup S`, `Finite S` in `claim` |\n| arbitrary field and both alternatives | `Field K`, `Admissible S K` |\n| ordinary basis and convolution | upstream `AddMonoidAlgebra K S` and its `single` |\n| positive cyclic semigroup and idempotent | `positiveMultiple`, `cyclic`, `IsCyclicIdempotent`, `cyclicIdempotent` |\n| period, exponent, splitting | `period`, `semigroupExponent`, `Splitting` |\n| group property in epsilon | `CyclicIsGroup`, `epsilon` |\n| every independently nonzero coefficient | `GoodSequence` quantifies all nonzero scalar functions of position |\n| supremum, including unboundedness | `algebraicD : ENat`, supremum over every finite list |\n| actual principal ideals and congruence quotient | `Below`, `Green`, `greenCon` and its upstream canonical quotient |\n| relative Davenport constant and empty subsequences | `SequenceSum`, `IrreducibleAt`, `relativeD` |\n| actual stabilizer translations, equality of maps | `HClass`, `Gamma`, composition addition |\n| maximum of the exact summands | `rightSide`, an ENat supremum over the finite carrier |\n| whole universal equality and its negation | `claim : Prop`, `result : Not claim` |\n\nThe source's infinite-length notation is interpreted as unbounded admissible\nfinite lengths, without inventing an infinite product. For a general\nnonunital semigroup algebra only positive products are used; a bottom length\nzero does not affect the supremum. The empty sequence sums to an identity\nonly if the semigroup already has one. The model lies in a monoid over Q,\nwhere the usual nonzero empty product is available. No zero-ring claim is\nneeded for the field-valued conjecture.","status":"refuted","summary":"Wang, arXiv:2509.18724v1, Conjecture 5.3, asks whether every finite commutative semigroup S and field K satisfying either the splitting condition or the characteristic-p / p-power-exponent condition obey","title":"Wang's Conjecture 5.3","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wang-conjecture53"},{"aliases":["D5/S3/Combinatorics/Graph/Saturation/WangTianStrongSaturationRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2394a6c8ff2051957a057e460ab7f85f6d39330fe43d8f96ec79f2393ee17c00.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Graph/Saturation/WangTianStrongSaturationRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2605.01269","statement_id":null},"id":"result:wang-tian-2026-strong-saturation-exact-formula","kind":"result","scope":"Qinglin Wang and Yingzhi Tian, *Extremal Problems for the Family of k-Strongly\nConnected Digraphs*, arXiv:2605.01269v1, Section 4, Conjecture 1, printed PDF\npage 13, state: “For sufficiently large $n$,\n$\\operatorname{ex}(n,\\mathcal D_k)=\\frac32(k-\\frac43)(n-k+1)+\\binom n2$.”\nThe source is [Conjecture 1](https://arxiv.org/html/2605.01269v1#Thmconj1).\n\n### Statement echo\n\nThe closed proposition `claim` quantifies every $k\\in\\mathbb N$ with\n$1\\le k$, then a threshold $N\\in\\mathbb N$ depending on $k$, then every\n$n\\in\\mathbb N$ with $N\\le n$. Its equality is\n\n$$\n(\\operatorname{ex}(n,\\mathcal D_k):\\mathbb Q)\n=\\frac32\\left((k:\\mathbb Q)-\\frac43\\right)\n\\left((n:\\mathbb Q)-(k:\\mathbb Q)+1\\right)\n+(\\operatorname{Nat.choose}(n,2):\\mathbb Q).\n$$\n\nEvery division and subtraction is rational. There is no rounding, parity,\ndivisibility restriction or error term. The local Section 4 reading $k\\ge2$\nis also contradicted by $k=3$.\n\nStrict digraphs have ordered distinct-endpoint arcs, permit reciprocal arcs,\nand have no parallel arcs. A subdigraph chooses arbitrary vertex and arc\nsubsets, retaining isolated vertices. Dipaths are vertex-simple directed\npaths. A separator leaves a nonstrong or singleton remainder. For nonempty\nrecords the connectivity is the attained minimum separator size. Positive\nstrong connectivity excludes empty records; the source leaves null\nconnectivity unspecified. This explicit totalization retains the source's\nLemma 2.2 consequence that a $k$-strong graph has at least $k+1$ vertices.\nSaturation means forbidden-free and that every missing ordered arc creates\na $k$-strong arbitrary subdigraph. The extremum is the actual attained\nmaximum arc count over a proved nonempty finite family of saturated hosts.\nThe Blueprint explains the invertible relabelling to `Fin n`; it does not\nclaim a separate compiled transport theorem.","status":"refuted","summary":"Qinglin Wang and Yingzhi Tian, *Extremal Problems for the Family of k-Strongly Connected Digraphs*, arXiv:2605.01269v1, Section 4, Conjecture 1, printed PDF page 13, state: “For sufficiently large $n$, $\\operatorname{ex}(n,\\mathcal D_k)=\\frac32(k-\\frac43)(n-k+1)+\\binom n2$.” The source is [Conjecture 1](https://arxiv.…","title":"The unrestricted exact Wang–Tian saturation formula is false","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wang-tian-2026-strong-saturation-exact-formula"},{"aliases":["D5/S3/Quantum/Dynamics/BernoulliNoiseWalkGaussianLimit","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/47e7173ccace58d284f6cee87dc258b1e63a28b752039c119c6786e7bfd330ef.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Dynamics/BernoulliNoiseWalkGaussianLimit.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1155/2018/2507265","statement_id":null},"id":"result:wang-wang-tang-ren-2018-bernoulli-noise-walk-gaussian-limit","kind":"result","scope":"C. Wang, C. Wang, Y. Tang and S. Ren, *Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise*, Advances in Mathematical Physics 2018, Article 2507265, DOI 10.1155/2018/2507265, Section 6, “Conclusion Remarks,” published PDF p. 8:\n\n> It is still unclear whether or not the walk has a limit probability distribution for a general localized initial state.\n\n> Thus we conjecture that, for a general localized initial state, the QBN-based walk might still have the same limit probability distribution as the classical random walk.\n\nThe source's coin space $h$ has orthonormal basis $\\{Z_\\sigma:\\sigma\\in\\Gamma\\}$, where $\\Gamma$ consists of the finite subsets of $\\mathbb N$ (Section 2, Eq. (7), p. 2). Its annihilation and creation operators remove or add a coordinate when permitted, and otherwise give zero (Lemmas 1–2, Eqs. (8)–(9)). Lemma 5 and Definition 6, p. 3, specify\n\n$$\nR_k=\\tfrac12(\\partial_k^*+\\partial_k+I),\\qquad\nL_k=\\tfrac12(\\partial_k^*+\\partial_k-I),\\qquad\n\\Phi_{n+1}(x)=R_n\\Phi_n(x-1)+L_n\\Phi_n(x+1).\n$$\n\nA localized initial state is $\\Phi_0(0)=\\varphi$, $\\Phi_0(x)=0$ for $x\\ne0$, with $\\|\\varphi\\|=1$ (Eq. (53), p. 6); the position probabilities are $\\Pr(X_n=x)=\\|\\Phi_n(x)\\|^2$ (Section 3, p. 3).\n\nIssue [#22700](https://github.com/the-omega-institute/trureturing/issues/22700) fixes the reading: for every unit vector $\\varphi\\in h$, the law of $X_n/\\sqrt n$ converges weakly to $N(0,1)$. This is proved in the coefficient-space model below. The source definitions and their locators are also described in [the literature note](../Library/Asymptotics/wangwangtangren2018bernoulliwalk.md).","status":"proved","summary":"It is still unclear whether or not the walk has a limit probability distribution for a general localized initial state.","title":"Every localized quantum Bernoulli noise walk has the standard Gaussian limit","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wang-wang-tang-ren-2018-bernoulli-noise-walk-gaussian-limit"},{"aliases":["D5/S3/Quantum/Entanglement/ConicalDesignConcurrenceComparability","result","arXiv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e4471f5a35fba5f3d9a01887917d832b4263ccce14dd76a381763cacf27bc7a1.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/ConicalDesignConcurrenceComparability.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://arxiv.org/abs/2606.31010v2","statement_id":null},"id":"result:wang-zhou-chen-fei-2026-conical-design-concurrence-comparability","kind":"result","scope":"H.-F. Wang, W. Zhou, L. Chen and S.-M. Fei, *Estimating the concurrence\nfor quantum states via symmetric measurements*, arXiv:2606.31010v2,\nSection IV, after Theorem 2:\n\n> The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.\n\nA finite positive semidefinite family satisfies\n$\\sum_i E_i\\otimes E_i=\\alpha_E I+\\beta_E F$, where\n$\\alpha_E\\geq\\beta_E>0$ and $F$ is the tensor swap. For a positive\nsemidefinite trace-one bipartite state $\\rho$, put\n$P_E(\\rho)_{ij}=\\operatorname{tr}[\\rho(E_i\\otimes E_j)]$ and\n$B_E(\\rho)=c_d(\\|P_E(\\rho)\\|_1-\\alpha_E-\\beta_E)/\\beta_E$,\nwhere $c_d=\\sqrt{2/[d(d-1)]}$ and $d\\geq2$.\nThe bound is the untruncated two-constant expression in Siudzińska,\nJ. Phys. A 58 (2025) 375302, Section 7, Theorem 6, Eqs. (73)–(74),\nwith $S=\\beta_E$ and $\\mathcal C_{\\max}=\\alpha_E+\\beta_E$.","status":"proved","summary":"The lower bounds of concurrence induced by arbitrary two distinct conical 2-designs are comparable.","title":"Comparability of conical-design concurrence bounds","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wang-zhou-chen-fei-2026-conical-design-concurrence-comparability"},{"aliases":["D5/S3/Analytic/ExpRatioCM/Result","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c3b04502ff4a062ac58f436a07d61b3746b18968cc155cdfe232e65689d02dde.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Analytic/ExpRatioCM/Result.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1155/2014/851213","statement_id":null},"id":"result:wei-guo-2014-exponential-derivative-ratio-complete-monotonicity","kind":"result","scope":"Section 5 of the source identified by\n`D5/L/Asymptotics/weiguo2014completemonotonicity` states:\n\n> Conjecture 12. For given i ∈ {0} ∪ ℕ, the functions ℱ_i(t) and 𝒢_i(t) defined in (10) or (27) are completely monotonic on (0, ∞).\n\nEquation (6) defines\n\n$$F_i(t)=(-1)^i\\left(\\frac1{e^t-1}\\right)^{(i)},\\qquad\nG_i(t)=(-1)^i\\left(\\frac1{1-e^{-t}}\\right)^{(i)},$$\n\nand equation (10) defines\n\n$$\\mathcal F_i(t)=\\frac{F_{i+1}(t)}{F_i(t)},\\qquad\n\\mathcal G_i(t)=\\frac{G_{i+1}(t)}{G_i(t)}.$$\n\nThe source verifies $i=0,1$ in (30). Complete monotonicity means\nsmoothness on $(0,\\infty)$ and $(-1)^kh^{(k)}(t)\\ge0$ for every\nnonnegative integer $k$ and every $t>0$.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.ExpRatioCM`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $F_i(t)$ in (6) | `Defs.F` | Exactly the globally defined signed iterated derivative of $1/(e^t-1)$. |\n| $G_i(t)$ in (6) | `Defs.G` | Exactly the globally defined signed iterated derivative of $1/(1-e^{-t})$. |\n| $\\mathcal F_i$ in (10) | `Defs.ratioF` | Exactly $F_{i+1}/F_i$. The denominator is positive for $t>0$. |\n| $\\mathcal G_i$ in (10) | `Defs.ratioG` | Exactly $G_{i+1}/G_i$. The denominator is positive for $t>0$. |\n| Complete monotonicity | `Defs.CompletelyMonotoneOn` | `ContDiffOn` with smoothness index `⊤` on `Set.Ioi 0`, and the sign condition for every natural derivative order at every positive argument. |\n| Conjecture 12 | `Defs.claim` | Both complete-monotonicity assertions for every natural index, including zero. |\n| Exact index sets | `Positive.cm_of_le_two`, `Result.failure_on_every_tail` | The positive theorem covers $i\\le2$, while the failure theorem excludes every $i\\ge3$. |\n| Failure on every right half-line | `Result.failure_on_every_tail` | For every $i\\ge3$ and real $T$, there are a natural $k$ and $t>\\max\\{T,0\\}$ at which both alternating derivatives are negative. |\n| Universal refutation | `Result.result` | Exactly $\\neg\\,\\texttt{Defs.claim}$. |\n\nThe globally defined derivatives agree with derivatives of the smooth\nrestrictions on the open positive half-line by neighborhood equality.\nNo derivative value at zero is used.","status":"refuted","summary":"Conjecture 12. For given i ∈ {0} ∪ ℕ, the functions ℱ_i(t) and 𝒢_i(t) defined in (10) or (27) are completely monotonic on (0, ∞).","title":"Refutation and exact classification of exponential derivative ratios","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wei-guo-2014-exponential-derivative-ratio-complete-monotonicity"},{"aliases":["D5/S3/Quantum/Information/CCZMagicCapacityThreshold","result","arXiv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71befcbb44741bac64f74c5f1afabac15f46e058a3daf5ca3d5065075cb3fa60.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/CCZMagicCapacityThreshold.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://arxiv.org/abs/2410.21215v1","statement_id":null},"id":"result:wei-liu-2024-ccz-magic-capacity-threshold","kind":"result","scope":"Fuchuan Wei and Zi-Wen Liu, *Noise robustness and threshold of many-body quantum magic*, arXiv:2410.21215v1, Section VII:\n\n> We conjecture that the magic capacity threshold for $\\mathrm{CC}Z$ under local depolarizing noise is $1/3$.\n\nThe capacity maximizes robustness of magic over pure six-qubit stabilizer inputs, with $\\mathrm{CC}Z$ and independent single-qubit depolarizing noise acting on the three system qubits $A$ and the three reference qubits $B$ untouched. Robustness is faithful: it equals one exactly on the real convex hull $\\mathrm{STAB}_6$ of pure stabilizer density matrices. The single-qubit channel is $\\rho\\mapsto(1-\\lambda)\\rho+\\lambda\\operatorname{Tr}(\\rho)I_2/2$.\n\nThe literal `claim` has two conjuncts: every pure stabilizer output belongs to `STAB 6` for $\\lambda\\in[1/3,1]$, and a pure stabilizer input with output outside `STAB 6` exists for every $\\lambda\\in[0,1/3)$. Noise follows the gate. The result contradicts the first conjunct at $\\lambda=1/2$.","status":"refuted","summary":"We conjecture that the magic capacity threshold for $\\mathrm{CC}Z$ under local depolarizing noise is $1/3$.","title":"Wei–Liu CCZ magic-capacity threshold","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wei-liu-2024-ccz-magic-capacity-threshold"},{"aliases":["D5/S3/Combinatorics/Hamming/AssociatedMersenneCubePolynomial","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e52f4734efb5da3df5dbaf3d13cbddfad52229cb2256d62e64df999a3e701e81.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Hamming/AssociatedMersenneCubePolynomial.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1007/s00373-025-02936-3","statement_id":null},"id":"result:wei-yang-2024-associated-mersenne-cube-polynomial","kind":"result","scope":"Wei and Yang, *Associated Mersenne graphs*, arXiv:2407.08237v1, Section 6,\nQuestion 6.3 asks:\n\n> What is cube polynomial for Associated Mersenne graph $\\mathcal{M}_n$?\n\nThe associated Mersenne graph has the circular run constrained Boolean words as\nvertices, with Hamming distance one as adjacency. Its cube polynomial is\n$C_n(x)=\\sum_k c_n(k)x^k$, where $c_n(k)$ counts induced copies of $Q_k$.","status":"proved","summary":"What is cube polynomial for Associated Mersenne graph $\\mathcal{M}_n$?","title":"Cube polynomial of the associated Mersenne graphs","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wei-yang-2024-associated-mersenne-cube-polynomial"},{"aliases":["D5/S1/Words/AssociatedMersenne/DegreeGeneratingFunction","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b6ce6abb36e71201fcbcaf7bfec01338ea72d59452c4be435cb7a6fa804ad497.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/AssociatedMersenne/DegreeGeneratingFunction.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2407.08237","statement_id":null},"id":"result:wei-yang-2024-associated-mersenne-degree-series","kind":"result","scope":"Wei and Yang, *Associated Mersenne graphs*, arXiv:2407.08237v1,\nQuestion 6.2: “For given n and k, how many vertices of Associated Mersenne\ngraph ℳₙ have degree k?” The vertices are labelled binary words for which\nevery circular run of ones is followed by a strictly longer run of zeros.\nAdjacent vertices differ at one position. The conventions are\n$\\mathbf M_0=\\{\\lambda\\}$ and $\\mathbf M_1=\\{0\\}$.","status":"proved","summary":"Wei and Yang, *Associated Mersenne graphs*, arXiv:2407.08237v1, Question 6.2: “For given n and k, how many vertices of Associated Mersenne graph ℳₙ have degree k?” The vertices are labelled binary words for which every circular run of ones is followed by a strictly longer run of zeros. Adjacent vertices differ at one…","title":"Degree generating function of Associated Mersenne graphs","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wei-yang-2024-associated-mersenne-degree-series"},{"aliases":["D5/S3/Quantum/Information/PauliRotationCriticalValues","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2bcc5548954390032c9660f8df3db2d7cf52820026ead546f116942438bd8b88.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/PauliRotationCriticalValues.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2511.16736","statement_id":null},"id":"result:wierichs-kottmann-killoran-2025-critical-values","kind":"result","scope":"D. Wierichs, K. Kottmann and N. Killoran, *Unitary synthesis with optimal brick wall circuits*,\narXiv:2511.16736v1, §V (page 9):\n\n> Conjecture 3. The critical values F(C) of a PQC that has full rank almost everywhere in T^d form an at\n> most (d−2)-dimensional subset of G.\n\nA PQC (Appendix F, Definition 1) consists of Clifford gates and $d$ rotations from $\\{R_X,R_Y,R_Z\\}$, each\ninput feeding one rotation without preprocessing, viewed as $F:T^d\\to G\\subset\\mathbb C^{2^n\\times 2^n}$\nwith $d=\\dim G$; $C$ is the set of inputs at which the Jacobian has rank below $d$. The verbatim texts are\nin [the literature note](../Library/QuantumChannels/wierichs2025brickwall.md). Issue\n[#15435](https://github.com/the-omega-institute/trureturing/issues/15435) preregisters the two-qubit,\n$G=SU(4)$, $d=15$ specialization with Hausdorff dimension and the refutation.","status":"refuted","summary":"Conjecture 3. The critical values F(C) of a PQC that has full rank almost everywhere in T^d form an at most (d−2)-dimensional subset of G.","title":"Critical values of parametrized quantum circuits","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wierichs-kottmann-killoran-2025-critical-values"},{"aliases":["D5/S3/Quantum/QuantumChannels/CoherentDiscriminationTriangleRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5ca45cfca5f7e79b5007f6f1b666dd909790dfefb374d3babff6ec585d44921e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/QuantumChannels/CoherentDiscriminationTriangleRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1109/ISIT44484.2020.9174425","statement_id":null},"id":"result:wilde-2020-coherent-discrimination-triangle-refutation","kind":"result","scope":"M. M. Wilde, *Coherent Quantum Channel Discrimination* (IEEE ISIT 2020; arXiv:2001.02668), introduces a coherent version of channel discrimination.\n- The verifier controls the two channels by a qubit R₁ in |+⟩.\n- The prover supplies an input ψ_RA and processes the channel output with a unitary V_RBE′F, returning B and a flag qubit F.\n- The verifier uncomputes the controlled isometric extensions and projects R₁F onto a Bell state.\n\nThe success probability p^coh_s(N⁰, N¹) is the supremum of Eq. (17) over the prover's strategies (Eq. (18)), and 1/2 ≤ p^coh_s ≤ 1 (Proposition 2). Section VI asks: \"An intriguing open question is to determine if p^coh_s − 1/2 is a metric on quantum channels.\" The incoherent analogue p^inc_s − 1/2 is a metric: by Eq. (8) it is half the normalized diamond distance (1/2)‖N⁰ − N¹‖_⋄.\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/wilde2020coherentdiscrimination.md).\n\nIssue [#18289](https://github.com/the-omega-institute/trureturing/issues/18289) fixes the reading: the triangle inequality for d = p^coh_s − 1/2 on qubit channels, which a metric would satisfy. The question is answered in the negative.","status":"refuted","summary":"M. M. Wilde, *Coherent Quantum Channel Discrimination* (IEEE ISIT 2020; arXiv:2001.02668), introduces a coherent version of channel discrimination. - The verifier controls the two channels by a qubit R₁ in |+⟩. - The prover supplies an input ψ_RA and processes the channel output with a unitary V_RBE′F, returning B and…","title":"The coherent discrimination bias is not a metric on quantum channels","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wilde-2020-coherent-discrimination-triangle-refutation"},{"aliases":["D5/S1/Words/Compositions/WisemanA344611OddLengthInjection","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cdcfbfcd4ca1cf875b84af88e75eda25e392176b9309f32b9555f1d40f2b9f35.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Compositions/WisemanA344611OddLengthInjection.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A344611","statement_id":null},"id":"result:wiseman-a344611","kind":"result","scope":"Gus Wiseman's OEIS A344611 revision 27, FORMULA, states: “Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...”. Its NAME is “Number of integer partitions of 2*n with reverse-alternating sum >= 0.” Its COMMENTS define “The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i.” The OFFSET is “0,2”, including the empty partition. A160786 revision 25 gives “a(n) = A027193(2n+1).” A027193 revision 78 counts partitions into an odd number of parts. The electronic sources have no page numbers, DOI or arXiv identifier.\n\nFor a decreasing positive partition $\\lambda=(\\lambda_1,\\ldots,\\lambda_k)$ of weight $m$, set $R(\\lambda)=\\sum_{i=1}^k(-1)^{k-i}\\lambda_i$, with $R(())=0$. The assertion is\n\n$$\n\\forall n\\in\\mathbb N,\\quad\n\\#\\{\\lambda\\vdash2n:R(\\lambda)\\ge0\\}\n\\le\\#\\{\\mu\\vdash2n+1: \\text{the number of parts of }\\mu\\text{ is odd}\\}.\n$$\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive decreasing integer partitions of a specified weight | `Nat.Partition`, with `parts.sort (· ≥ ·)` | Positive multiset parts with the weight constraint; decreasing sorting is inlined. |\n| Reverse signs and the empty sum | `reverseAlternatingSum` | Existing integer recurrence `TrimmedAlternatingPartitions.trimmedSums`, last endpoint with default zero; the update is $z\\mapsto a-z$. |\n| Partitions of $2n$ with $R\\ge0$ | `a344611` | `Nat.card` of the literal nonnegative-endpoint subtype. |\n| Odd number of parts at weight $2n+1$ | `a160786` | `Odd p.parts.card`; no parity restriction on individual parts. |\n| Every $n\\ge0$ and non-strict comparison | `claim`, `result` | The natural-number quantifier includes zero; no added source hypothesis. |","status":"proved","summary":"Gus Wiseman's OEIS A344611 revision 27, FORMULA, states: “Conjecture: a(n) <= A160786(n). The difference is 0, 0, 0, 0, 1, 2, 4, 9, 16, 28, 48, 79, ...”. Its NAME is “Number of integer partitions of 2*n with reverse-alternating sum >= 0.” Its COMMENTS define “The reverse-alternating sum of a partition (y_1,...,y_k) is…","title":"wiseman a344611","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wiseman-a344611"},{"aliases":["D5/S1/Words/Compositions/WisemanA387118","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a9bc4dfa5e56d527cbeca38c068df9f993fd98586c2eb8a8b804339f5f83dfb.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Compositions/WisemanA387118.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A387118","statement_id":null},"id":"result:wiseman-a387118","kind":"result","scope":"OEIS A387118 asks whether the number of integer partitions of $n$ without choosable initial intervals equals the number of non-superdiagonal reversed integer partitions. The conjecture is preregistered in [#18602](https://github.com/the-omega-institute/trureturing/issues/18602).\n\n### Statement echo\n\nThe formal count `a n` ranges over `Nat.Partition n` and filters the descending sort by the negation of `ChoosableInitial`. The comparison count `nonSuperdiagonal n` filters the ascending sort by the negation of `Superdiagonal`. The source predicates and sorting convention are imported from `D5/S1/Words/Compositions/ChoosableInitialIntervals`.","status":"proved","summary":"OEIS A387118 asks whether the number of integer partitions of $n$ without choosable initial intervals equals the number of non-superdiagonal reversed integer partitions. The conjecture is preregistered in [#18602](https://github.com/the-omega-institute/trureturing/issues/18602).","title":"Wiseman A387118 total partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wiseman-a387118"},{"aliases":["D5/S1/Words/Compositions/WisemanA388713","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3d5cab4e0efd158476764411b85f1437b7111ddfc3f265b546453c2c11cdb640.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Compositions/WisemanA388713.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A388713","statement_id":null},"id":"result:wiseman-a388713","kind":"result","scope":"OEIS A388713 asks whether the number of superdiagonal reversed partitions of $n$ with maximum $k$ equals the number of integer partitions of $n$ with maximum $k$ and choosable initial intervals. The conjecture is preregistered in [#18603](https://github.com/the-omega-institute/trureturing/issues/18603).\n\n### Statement echo\n\nThe formal count `T n k` filters `Nat.Partition n` by `p.parts.sup = k` and superdiagonality of the ascending sort. The comparison count `choosable n k` uses the same maximum condition and choosability of the descending sort. The source predicates and sorting convention are imported from `D5/S1/Words/Compositions/ChoosableInitialIntervals`.","status":"proved","summary":"OEIS A388713 asks whether the number of superdiagonal reversed partitions of $n$ with maximum $k$ equals the number of integer partitions of $n$ with maximum $k$ and choosable initial intervals. The conjecture is preregistered in [#18603](https://github.com/the-omega-institute/trureturing/issues/18603).","title":"Wiseman A388713 maximum-refined partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wiseman-a388713"},{"aliases":["D5/S3/Quantum/ResolvedCountWeakValue","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2593411f715e6f134d54d3d82705e653ab5c6c8479110e3d964215ff6d8b589e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/ResolvedCountWeakValue.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1116/5.0137579","statement_id":null},"id":"result:wiseman-steinberg-hallaji-2023-resolved-count-weak-values","kind":"result","scope":"Wiseman, Steinberg and Hallaji, “Obtaining a single-photon weak value from experiments using a strong (many-photon) coherent state”, AVS Quantum Science 5, 024401 (2023), arXiv:2211.15761v2, Section V, page 6:\n\n> This supports earlier approximate results which suggested that if a photon-number resolving detector were used, the weak value post-selected on detecting m photons would exceed the no-click weak value by precisely m times the single-photon number. We leave a rigorous proof of this conjecture for future work.\n\nIssue #17772 fixes the finite-mode Fock conventions. For every $k+1$-mode passive network $A,B$, coherent amplitude $\\alpha\\ne0$, and natural count $m$ of nonzero resolved probability, the conclusion is $W_m-W_0=mw$ when $m>0$. For $m=0$ the difference is zero without requiring a single-photon denominator. The detector acts on mode zero and ignores the remaining modes; the probed mode is also indexed zero, with arbitrary passive evolution between probing and detection.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Arbitrary passive linear network | Every natural $k$ and $A,B:\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin}(k+1))\\simeq_{\\mathbb C}^{\\mathrm{iso}}\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin}(k+1))$ | No fixed dimension, network or intensity |\n| Coherent input in one mode, vacuum in the others | `coherentInput k α`, `EuclideanSpace.single (0 : Fin (k+1)) α` | Literal binding conventions |\n| Weak measurement of number, with other modes ignored | `number`, `numberProjector m`, `generalizedWeakValue` | Unbounded occupation; exact number-domain proofs; projector retains ignored-mode carrier |\n| Detect exactly $m$ photons | Every natural $m$ with `coherentProbability A B α m ≠ 0` | Conditional values retain nonzero-denominator certificates |\n| Compare with no click | Existential nonzero vacuum probability and `coherentWeakValue ... 0` | Vacuum condition is proved, not an added hypothesis |\n| Precisely $m$ times the single-photon value | `claim`, proved by `result` | Same $A,B$ and probed mode; the positive-count branch proves nonzero single-photon probability |\n| Zero count | The explicit `if m = 0` branch of `claim` | Difference zero; no single-photon value assigned when its probability vanishes |\n\nAll public model definitions and the claim retain the mathematical bodies fixed by #17772; its mode-amplitude carrier is written as $\\operatorname{EuclideanSpace}\\,\\mathbb C\\,(\\operatorname{Fin} k)$ without a forwarding declaration. `BeamSplitter.exponentialCoeff` and `BeamSplitter.exponentialVector` supply the one-mode coefficients and vectors. The arbitrary finite-mode completion and totalness are supplied by `PassiveFockModel`.","status":"proved","summary":"This supports earlier approximate results which suggested that if a photon-number resolving detector were used, the weak value post-selected on detecting m photons would exceed the no-click weak value by precisely m times the single-photon number. We leave a rigorous proof of this conjecture for future work.","title":"Resolved-count coherent-state weak values","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wiseman-steinberg-hallaji-2023-resolved-count-weak-values"},{"aliases":["D5/S1/Words/Compositions/WisemanA388712","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a3264330c3c78fef79ab790368c1eeb4c1557ec5ddb01a4455c3faee03dc1c6e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Compositions/WisemanA388712.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A388712","statement_id":null},"id":"result:wiseman388-partitions","kind":"result","scope":"OEIS A388712 states: “Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts without choosable initial intervals.” Its conjecture is: “Also the number of non-superdiagonal reversed partitions of n into k parts, where a sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i = 1..k.” The question is preregistered in [#18601](https://github.com/the-omega-institute/trureturing/issues/18601).\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| Partitions of $n$ into $k$ positive parts | `Nat.Partition n`, filtered by `p.parts.card = k` |\n| Initial intervals admit distinct positive representatives | The imported `ChoosableInitial` predicate on descending parts |\n| Without choosable initial intervals | Negation in `WisemanA388712.T` |\n| Reversed partition, weakly increasing | `p.parts.sort (· ≤ ·)` |\n| $y_i\\ge i$ with one-based indices | The imported `Superdiagonal` predicate uses zero-based index plus one |\n| Non-superdiagonal, with the same number of parts | Negation and the same cardinality filter in `WisemanA388712.nonSuperdiagonal` |\n| Equality for every $n,k$ | `WisemanA388712.claim`, proved by `WisemanA388712.result` |\n\nAll declaration names above have namespace prefix `D5.S1.Words.Compositions.`. The source expressions are retained in the two count definitions. The existing predicates are reused without new aliases. Classical finite cardinalities do not restrict the quantified natural numbers.","status":"proved","summary":"OEIS A388712 states: “Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts without choosable initial intervals.” Its conjecture is: “Also the number of non-superdiagonal reversed partitions of n into k parts, where a sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i =…","title":"Wiseman's nonchoosable partition identity","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wiseman388-partitions"},{"aliases":["D5/S3/Analytic/SeriesInequalities/GurlandLogRemainder","result","arXiv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/78b2f8f49a8569d915844b3b63d9711b9c9bdf2a8ced2c09ce7f6abd187cc739.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Analytic/SeriesInequalities/GurlandLogRemainder.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://arxiv.org/html/2512.07028v1","statement_id":null},"id":"result:wisniewska-2025-gurland-remainder","kind":"result","scope":"Halina Wiśniewska, *Some inequalities for Gurland's ratio of the gamma\nfunctions*, arXiv:2512.07028v1, section 2, Open Problem 1:\n\n> In Corollary 1 we showed that R_n^m(x,y) tends to zero under a sufficient\n> condition. Does R_n^m(x,y) tend to zero hold for every x,y > 0?\n\nFor positive real x,y put a=(x+y)/2 and d=(x-y)/2. In equations (2)–(4),\n\n$$\nG_* = \\frac{\\Gamma(1+x)\\Gamma(1+y)}{\\Gamma(1+a)^2},\\qquad\nH(s,b)=\\sum_{j=0}^{\\infty}\\frac1{(j+b)^s},\n$$\n$$\nS_m=\\sum_{k=1}^{m-1}\\frac{d^{2k}H(2k,1+a)}{k},\\qquad\nR_m=\\log G_*-S_m.\n$$\n\nThe expansion order m tends to infinity through integers, with m >= 2\nin the source. Its n is already summed out. Every positive real pair,\nincluding x=y and positive integer Gamma arguments, is in scope.","status":"proved","summary":"In Corollary 1 we showed that R_n^m(x,y) tends to zero under a sufficient condition. Does R_n^m(x,y) tend to zero hold for every x,y > 0?","title":"Gurland's original logarithmic remainder","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wisniewska-2025-gurland-remainder"},{"aliases":["D5/S1/Recurrence/WitulaSlotaQuasiFibonacciOrderElevenDegree","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c983bfc9af14680238005fdd48e1338edc835cf817dbb60f2b8d1605e6aa60e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Recurrence/WitulaSlotaQuasiFibonacciOrderElevenDegree.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://cs.uwaterloo.ca/journals/JIS/VOL10/Slota2/slota99.pdf","statement_id":null},"id":"result:witula-slota-2007-quasi-fibonacci-order-eleven-degree","kind":"result","scope":"Wituła and Słota define the five polynomial sequences by system (3.12) on\nprinted page 5:\n\n> A_{n+1}(δ) = A_n(δ) + 2δB_n(δ) - δE_n(δ),\n>\n> B_{n+1}(δ) = δA_n(δ) + B_n(δ) + δC_n(δ) - δE_n(δ),\n>\n> C_{n+1}(δ) = δB_n(δ) + C_n(δ) + δD_n(δ) - δE_n(δ),\n>\n> D_{n+1}(δ) = δC_n(δ) + D_n(δ),\n>\n> E_{n+1}(δ) = δD_n(δ) + (1 - δ)E_n(δ),\n\nwith the initial values printed immediately below it:\n\n> A_0(δ) = 1, B_0(δ) = C_0(δ) = D_0(δ) = E_0(δ) = 0.\n\nThe sentence on printed page 19 is:\n\n> Problem. Is it true that deg A_n(∆) = deg B_n(∆) = deg C_n(∆) = deg D_n(∆) = deg E_n(∆) = n for every n = 5, 6, . . .?\n\nSection 6 introduces `∆ := 1/δ` and writes the same polynomial family as\n`A_n(∆),...,E_n(∆)`. The formal module names the indeterminate `X`. It reads\n`degree` in `WithBot ℕ`, rather than `natDegree`, takes the printed threshold\nliterally as `5 ≤ n`, and defines the recurrence over the coefficient ring\n`ℤ`.","status":"proved","summary":"A_{n+1}(δ) = A_n(δ) + 2δB_n(δ) - δE_n(δ), B_{n+1}(δ) = δA_n(δ) + B_n(δ) + δC_n(δ) - δE_n(δ), C_{n+1}(δ) = δB_n(δ) + C_n(δ) + δD_n(δ) - δE_n(δ), D_{n+1}(δ) = δC_n(δ) + D_n(δ), E_{n+1}(δ) = δD_n(δ) + (1 - δ)E_n(δ),","title":"Degrees of the order-eleven quasi-Fibonacci polynomials","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-witula-slota-2007-quasi-fibonacci-order-eleven-degree"},{"aliases":["D5/S3/QuantumBounds/WidthGradientBalancedCutRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7fbc30dcd5a6e6e7bfeb1ca697d6abe582f8629d89988f77ec17ab76c646e823.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/QuantumBounds/WidthGradientBalancedCutRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1088/1367-2630/aa5015","statement_id":null},"id":"result:wolk-guhne-2016-balanced-cut-gradient-bound-refutation","kind":"result","scope":"S. Wölk and O. Gühne, *Characterizing the width of entanglement*, arXiv:1507.07226v2, New J. Phys. 18 (2016) 123024, Appendix D. The conjecture states that every state of N spins with entanglement only within the subgroups {1, …, N/2} and {N/2+1, …, N} satisfies\n\n> $\\langle B^2 \\rangle \\leq \\langle B^2\\rangle_- + \\frac{\\langle B\\rangle_\\textrm {cl}-\\langle B^2\\rangle_-}{2N}\\,\\langle J^2\\rangle$\n\nwith ⟨B²⟩₋ = 6N. The gradient operator and the reference values are quoted in [the literature note](../Library/QuantumContext/wolk2016entanglementwidth.md). The printed numerator has ⟨B⟩_cl; the preceding sentence and the endpoint values fix it as ⟨B²⟩_cl.\n\nIssue [#19442](https://github.com/the-omega-institute/trureturing/issues/19442) fixes the reading:\n- for every even N = 2m ≥ 4 and every density matrix that is separable across the two halves;\n- the coefficient of ⟨J²⟩ is (⟨B²⟩_cl − 6N)/(2N).\n\nThe conjecture is refuted.","status":"refuted","summary":"S. Wölk and O. Gühne, *Characterizing the width of entanglement*, arXiv:1507.07226v2, New J. Phys. 18 (2016) 123024, Appendix D. The conjecture states that every state of N spins with entanglement only within the subgroups {1, …, N/2} and {N/2+1, …, N} satisfies","title":"The balanced-cut gradient bound of Wölk and Gühne fails for eight qubits","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wolk-guhne-2016-balanced-cut-gradient-bound-refutation"},{"aliases":["D5/S0/Certificates/LeeYangPhaseShiftedParityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/07a3b8ad64075624ec9d9ccaf8a69d0f449fa0754f4dc66ed02a7652f430f347.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S0/Certificates/LeeYangPhaseShiftedParityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2602.03605","statement_id":null},"id":"result:wong-bravyi-gosset-liu-2026-conjecture-2-phase-shifted-extension","kind":"result","scope":"Wong, Bravyi, Gosset and Liu, *Lee-Yang tensors and Hamiltonian complexity*, arXiv:2602.03605v1, Section 5.5, Conjecture 2:\n\n> Suppose n ≥ 2, s ∈ [0,1), and the interaction graph is connected. Then the second smallest eigenvalue of H_s coincides with the smallest eigenvalue of H_s restricted to the odd-weight subspace.\n\nSection 5.6 asks:\n\n> We leave as an open question whether Conjectures 1, 2, 3 hold for phase-shifted EPR-like Hamiltonians.\n\nThe question settled here is the extension of Conjecture 2. Preregistration: [#18759](https://github.com/the-omega-institute/trureturing/issues/18759). This is an external published question, tier 1.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| $n\\ge2$ | `claim` quantifies over natural `n` with `2 ≤ n`. |\n| Connected interaction graph | `SimpleGraph (Fin n)`, its adjacency decision procedure, and `G.Connected`. The decision procedure is computational infrastructure. |\n| $0\\le s<1$ | The two inequalities in `claim`. |\n| $h(z)=\\lvert00\\rangle\\langle00\\rvert+z\\lvert11\\rangle\\langle00\\rvert+\\overline z\\lvert00\\rangle\\langle11\\rvert+\\lvert z\\rvert^2\\lvert11\\rangle\\langle11\\rvert$ | `hpq` has exactly these four entries, with the identity on all other bits. |\n| $H_s(\\theta)=-\\sum_{\\{p,q\\}\\in E}h_{p,q}(se^{i\\theta_{p,q}})$ | `H` counts each undirected edge once using `p < q`. |\n| Arbitrary edge angles in $[0,2\\pi)$ | Real representatives in the exponential; the counterexample uses only $0$ and $\\pi$, both in the source interval. |\n| Second smallest eigenvalue | `eigAsc`, sorted with multiplicity, and `List.getD` at index $1$. |\n| Smallest eigenvalue on the odd-weight subspace | Restriction by `Matrix.submatrix` to the odd computational basis and index $0$. The odd weight count is inlined literally in `claim`; the private certificate relates it to Mathlib's `hammingNorm`. |\n| Negative settlement | `result : ¬ claim`. |\n\nThe arbitrary default in `List.getD` and the empty non-Hermitian branch of `eigAsc` totalize the definitions. The witness matrices are Hermitian and the relevant lists have the required entries. The computational basis is `Fin n → Fin 2` directly.","status":"refuted","summary":"Suppose n ≥ 2, s ∈ [0,1), and the interaction graph is connected. Then the second smallest eigenvalue of H_s coincides with the smallest eigenvalue of H_s restricted to the odd-weight subspace.","title":"The phase-shifted extension of Conjecture 2","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wong-bravyi-gosset-liu-2026-conjecture-2-phase-shifted-extension"},{"aliases":["D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dd9f59117ca515ce468914ee25c9af123e996835bead3b483b7f3070acf96007.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/RandomizedGraphNegativityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.89.052335","statement_id":null},"id":"result:wu-2014-randomized-graph-negativity-monotonicity","kind":"result","scope":"J.-Y. Wu, M. Rossi, H. Kampermann, S. Severini, L. C. Kwek, C. Macchiavello,\nD. Bruß, *Randomized Graph States and their Entanglement Properties*, Phys.\nRev. A 89, 052335 (2014), arXiv:1403.3828v3, Section V:\n\n> However, even though this conjecture is supported by numerical evidence,\n> it is an open question whether the monotonic behavior of the negativity in\n> terms of the randomness $p$ is a common feature to all RG states.\n\nThe randomized graph state of a graph $G$ is\n$\\rho^p_G=\\sum_{F\\subseteq E(G)}p^{|F|}(1-p)^{|E(G)\\setminus F|}\n|F\\rangle\\langle F|$, where $|F\\rangle=\\prod_{ab\\in F}\\mathrm{CZ}_{ab}\n|+\\rangle^{\\otimes n}$, and the negativity across a bipartition $A|B$ is\n$N(\\rho)=(\\|\\rho^{\\Gamma_A}\\|_1-1)/2$, evaluated \"with respect to all\npossible bipartitions of the qubits\". The conjecture is that $N$ increases\nmonotonically in $p$; the paper reports it numerically for the complete\ngraphs $K_n$ and the star graphs $S_n$ with $n\\le4$.","status":"refuted","summary":"However, even though this conjecture is supported by numerical evidence, it is an open question whether the monotonic behavior of the negativity in terms of the randomness $p$ is a common feature to all RG states.","title":"Monotonicity of the negativity of randomized graph states","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu-2014-randomized-graph-negativity-monotonicity"},{"aliases":["D5/S1/Digit/Admissibility/WuLouPermutationInvariantNivenDigitSum","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/00572f20fd105d5abd942e293b5f550b4468b90c2ef0ce4293451b2a881a0ca9.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Digit/Admissibility/WuLouPermutationInvariantNivenDigitSum.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.3390/sym18010186","statement_id":null},"id":"result:wu-lou-permutation-invariant-niven-digit-sum-bound","kind":"result","scope":"Wu and Lou, *Permutation-Invariant Niven Numbers*, arXiv:2508.01611v3,\nsection 8.4, state that the relevant nontrivial distinct-digit permutation-invariant\nNiven numbers have digit sums between 3 and 81 and conjecture that no exceptions\nexist beyond the cases excluded there.","status":"proved","summary":"Wu and Lou, *Permutation-Invariant Niven Numbers*, arXiv:2508.01611v3, section 8.4, state that the relevant nontrivial distinct-digit permutation-invariant Niven numbers have digit sums between 3 and 81 and conjecture that no exceptions exist beyond the cases excluded there.","title":"Wu-Lou permutation-invariant Niven digit-sum bound","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu-lou-permutation-invariant-niven-digit-sum-bound"},{"aliases":["D5/S3/Arith/WuPyramidalComplement","wu_conjecture_one","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/7b8d688d81bd6f56c66f3c2050d4f7bfabf92c865471033633a2190b28817dfa.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Arith/WuPyramidalComplement.wu_conjecture_one","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.5281/zenodo.17535229","statement_id":null},"id":"result:wu-pyramidal-complement-conjecture-one","kind":"result","scope":"Chai Wah Wu, *Algorithms for Complementary Sequences*, INTEGERS 25 (2025),\nA95, printed page 12, Conjecture 1, states:\n\n> For k >= 9, (6) holds for all n >= 1.\n\nFor positive `m`, let\n`P_k(m)=m(m+1)(m(k-2)-(k-5))/6`, and let the complementary sequence enumerate\nthe positive integers outside `{P_k(m) | m>=1}` in increasing order. Equation\n(6), on printed page 11, sets `h=floor((6n/(k-2))^(1/3))` and gives `n+h+1`\nwhen `6n` is at least\n`(k-2)h^3+3(k-1)h^2+(2k-1)h+6`; if that test fails, it gives `n+h-1` when\n`6n` is at most `h(h-1)(h(k-2)+k+1)`, and gives `n+h` otherwise.","status":"proved","summary":"For k >= 9, (6) holds for all n >= 1.","title":"Wu's k-gonal-pyramidal complement formula","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu-pyramidal-complement-conjecture-one"},{"aliases":["D5/S3/Quantum/Divergence/UnifiedImaginarityMonotoneComparison","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/de02d76159667dbb47ed4ea85429a00df1c9f44e5535e330bb876aaf10ffd8a7.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Divergence/UnifiedImaginarityMonotoneComparison.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1088/1572-9494/adbdbd","statement_id":null},"id":"result:wu-wu-2025-unified-imaginarity-monotone-comparison","kind":"result","scope":"Chuanfa Wu and Zhaoqi Wu, *Two imaginarity monotones induced by unified $(\\alpha,\\beta)$-relative entropy* (Commun. Theor. Phys. 77, 095101 (2025), DOI 10.1088/1572-9494/adbdbd; arXiv:2506.09799v1), Section 4, Example 1, state:\n\n> We have not found a proof of $M_{\\alpha,\\beta}^H(\\rho)\\geq M_{\\alpha,\\beta}^E(\\rho)$ for any $\\alpha\\in (0,1)$, $\\beta\\in (0,1]$ and any qubit state $\\rho$, however, numerical experiments shows that it might be true, and it seems difficult to find a counterexample. It is also conjectured that $M_{\\alpha,\\beta}^H(\\rho)\\geq M_{\\alpha,\\beta}^E(\\rho)$ for any $\\alpha\\in (0,1)$, $\\beta\\in (0,1]$ and any quantum state $\\rho$.\n\nThis quotation is from lines 573–579 of the downloaded v1 TeX, `20250301WuCF-imaginarity_monotone_R2.tex` (md5 `eb104c03d3638b12044677758c200bae`).\n\nFix the coordinate basis and let $\\bar\\rho$ mean entrywise complex conjugation, denoted $\\rho^*$ in the paper. For density matrices, write\n\n$D_{\\alpha,\\beta}(\\rho\\Vert\\sigma)=\\big[(\\operatorname{Re}\\operatorname{tr}(\\rho^\\alpha\\sigma^{1-\\alpha}))^\\beta-1\\big]/\\big[(\\alpha-1)\\beta\\big]$,\n\n$M^H_{\\alpha,\\beta}(\\rho)=D_{\\alpha,\\beta}(\\rho\\Vert\\bar\\rho)$, and\n\n$M^E_{\\alpha,\\beta}(\\rho)=\\inf_{\\sigma\\in\\mathcal F}D_{\\alpha,\\beta}(\\rho\\Vert\\sigma)$,\n\nwhere $\\mathcal F$ consists of positive semidefinite matrices of trace one with real entries. Matrix powers are spectral powers from continuous functional calculus. These are the module's definitions; the source's Eqs. (7), (13) and (20) use the trace without an explicit real part, and Eq. (20) uses a minimum. The corresponding source definitions and locators are in [the literature note](../Library/Divergence/wuwu2025unifiedimaginarity.md).\n\nIssue [#20971](https://github.com/the-omega-institute/trureturing/issues/20971) fixes the formal reading: for every natural $d\\ge1$, every density matrix $\\rho\\in M_d(\\mathbb C)$, every $0<\\alpha<1$ and every $0<\\beta\\le1$, $M^E_{\\alpha,\\beta}(\\rho)\\le M^H_{\\alpha,\\beta}(\\rho)$. Singular states are included. This infimum comparison is proved.","status":"proved","summary":"We have not found a proof of $M_{\\alpha,\\beta}^H(\\rho)\\geq M_{\\alpha,\\beta}^E(\\rho)$ for any $\\alpha\\in (0,1)$, $\\beta\\in (0,1]$ and any qubit state $\\rho$, however, numerical experiments shows that it might be true, and it seems difficult to find a counterexample. It is also conjectured that $M_{\\alpha,\\beta}^H(\\rho)…","title":"Comparing the Wu–Wu unified imaginarity monotones in every finite dimension","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu-wu-2025-unified-imaginarity-monotone-comparison"},{"aliases":["D5/S3/Quantum/Entanglement/GHZMeasureBiseparableBound","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ab479b271371e08ce2995002fda3502782239d2b3f898e1a9492646ceeb5a27b.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/GHZMeasureBiseparableBound.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2605.02876","statement_id":null},"id":"result:wu-zhong-wu-2026-ghz-measure-biseparable-half","kind":"result","scope":"S. Wu, K. Zhong and J. Wu, “A measure for genuine tripartite entanglement”,\narXiv:2605.02876v3, Section V.C, equation (37), state:\n\n> Numerically maximising E_GHZ over all biseparable A|BC states we find the sharp value sup_{ρ ∈ A|BC} E_GHZ(ρ) = ½, (37) attained e.g. by |0⟩_A ⊗ |Φ⁺⟩_BC and coinciding with the product-state value (34); the analytic proof of the exact constant 1/2 remains open. The same 1/2 holds for the B|AC and C|AB partitions by symmetry.\n\nIssue [#12719](https://github.com/the-omega-institute/trureturing/issues/12719)\npreregisters the Tier 1 A|BC statement: every product of arbitrary density\nmatrices has E_GHZ at most one half, and some such product attains one half.\nThe specified class is exactly ρ_A tensor ρ_BC, including mixed factors.","status":"proved","summary":"Numerically maximising E_GHZ over all biseparable A|BC states we find the sharp value sup_{ρ ∈ A|BC} E_GHZ(ρ) = ½, (37) attained e.g. by |0⟩_A ⊗ |Φ⁺⟩_BC and coinciding with the product-state value (34); the analytic proof of the exact constant 1/2 remains open. The same 1/2 holds for the B|AC and C|AB partitions by sy…","title":"The exact A|BC constant of the GHZ measure","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu-zhong-wu-2026-ghz-measure-biseparable-half"},{"aliases":["D5/S3/Arith/Congruence/WuAdjacentFibonacciResidues","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c116a2f294310ce6ff8c24d5870beabeb9f4a0e384a7f4a480779fb702298d7.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Arith/Congruence/WuAdjacentFibonacciResidues.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A221018/internal","statement_id":null},"id":"result:wu221018-fibdiv","kind":"result","scope":"OEIS A221018, revision 35, NAME:\n\n> Number k such that Fibonacci(k) is divisible by k, k + 1 and k - 1.\n\nIts first FORMULA states:\n\n> a(n) % 10 == 0, 2, or 8 (conjectured). - _Chai Wah Wu_, Aug 06 2026\n\nThe source generator starts at two. Write $F_k$ for the standard Fibonacci sequence, with $F_0=0$, $F_1=1$ and $F_{k+2}=F_k+F_{k+1}$. For every natural $k\\ge2$, the three assumptions $(k-1)\\mid F_k$, $k\\mid F_k$ and $(k+1)\\mid F_k$ imply $k\\bmod10\\in\\{0,2,8\\}$. No converse is asserted.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Fidelity |\n| --- | --- | --- |\n| Generator starts at 2 | Natural index $k$ and $2\\le k$ | Covers the source integers and makes natural subtraction ordinary subtraction. |\n| Fibonacci(k) | Existing `Nat.fib k` | Standard initial values and recurrence; no replacement sequence. |\n| Divisible by k, k+1 and k-1 | The three separate divisibility hypotheses of `claim` | Their implication chain is equivalent to the source conjunction. |\n| Final digit 0, 2 or 8 | The three remainder alternatives in `claim` and `result` | Exact necessary condition; no added hypotheses or converse. |","status":"proved","summary":"Number k such that Fibonacci(k) is divisible by k, k + 1 and k - 1.","title":"Wu’s adjacent-index Fibonacci divisibility conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-wu221018-fibdiv"},{"aliases":["D5/S3/Quantum/Entanglement/EntropicFill/QuditTetrahedronGenuineRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/98147fda13f0153da4f3632878c0f750aa559f564636b76d99617b06e58c58df.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/EntropicFill/QuditTetrahedronGenuineRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.3390/e26030217","statement_id":null},"id":"result:xie-younis-mei-eberly-2024-qudit-tetrahedron-genuine-refutation","kind":"result","scope":"S. Xie, D. Younis, Y. Mei and J. H. Eberly, *Multipartite Entanglement: A Journey Through Geometry*, arXiv:2304.03281v2; Entropy 26(3), 217 (2024), define the genuine requirement in `tetrahedron.tex`, line 57:\n\n> The ``genuine'' requirement requires that a GME measure should only, and always, vanish for biseparable states \\cite{ma2011,xie2021}.\n\nTheir Scalability paragraph, line 220, states:\n\n> Nonetheless, our tetrahedron measure presented in this work maintains its scalability. Specifically, we can consistently partition a large system into four groups and explore genuine quadripartite entanglement among these groups. Essentially, the tetrahedron measure is a genuine four-qudit entanglement measure, where each party possesses more than two dimensions. The extension of entropic fill to qudit systems has been made possible by a recent study \\cite{yang2022}. Additionally, we have conducted numerical verifications, demonstrating the existence of unique nonnegative solutions for Eq.~\\eqref{eqn1} and \\eqref{eqn2} in qudit systems up to $d=50$. Consequently, we conjecture that our tetrahedron measure is also applicable to qudit systems.\n\nThese quotations were checked against the complete v2 TeX source, `tetrahedron.tex` (233 lines, md5 `da0b24bf21c330f90979f63e7679749c`), fetched from https://arxiv.org/src/2304.03281v2. The definitions and further quotations are collected in [the literature note](../Library/QuantumContext/xieyounismeieberly2024journey.md).\n\nIssue [#20861](https://github.com/the-omega-institute/trureturing/issues/20861) fixes the following reading. Each of four parties has Hilbert space $\\mathbb C^d$, with $d\\ge2$, and $\\psi$ is a coefficient function with $\\sum_x\\|\\psi(x)\\|^2=1$. Write $S_X$ for the base-two von Neumann entropy of its reduced density matrix on $X$. Biseparability means factorization across some nonempty proper subset of the four parties.\n\nAn admissible fill consists of six nonnegative real coordinates $\\sigma=(\\sigma_{AB},\\sigma_{AC},\\sigma_{AD},\\sigma_{BC},\\sigma_{BD},\\sigma_{CD})$ and a real parameter $\\lambda$. The face equations are $\\sigma_{ij}+\\sigma_{ik}+\\sigma_{il}=S_i$ for each party $i$. Put $p=\\sqrt{\\sigma_{AB}\\sigma_{CD}}$, $q=\\sqrt{\\sigma_{AC}\\sigma_{BD}}$ and $r=\\sqrt{\\sigma_{AD}\\sigma_{BC}}$. The radical equations are $-p+q+r=\\lambda S_{AB}$, $p-q+r=\\lambda S_{AC}$ and $p+q-r=\\lambda S_{AD}$. These are the source equations labelled `eqn1` and `eqn2`, TeX lines 103–121.\n\nThe source volume, TeX lines 92–101, is $V=(\\sqrt2/3)\\sqrt{2\\sum_{ij}\\sigma_{ij}}\\big((p+q+r)(-p+q+r)(p-q+r)(p+q-r)\\big)^{1/4}$. The formal claim is: for every $d\\ge2$ and every unit $\\psi$, if an admissible fill exists and every admissible fill has $V=0$, then $\\psi$ is biseparable. This implication is refuted.","status":"refuted","summary":"The ``genuine'' requirement requires that a GME measure should only, and always, vanish for biseparable states \\cite{ma2011,xie2021}.","title":"The entropic tetrahedron is not a genuine four-qudit entanglement measure","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xie-younis-mei-eberly-2024-qudit-tetrahedron-genuine-refutation"},{"aliases":["D5/S3/Quantum/Information/ObliqueDiscordMutualInformationRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/32815351a6fcc4e36ab6c2169891b041216e86d270313fcfa69e0546f1d226cc.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/ObliqueDiscordMutualInformationRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1142/S0217979216502568","statement_id":null},"id":"result:xu-2015-oblique-discord-mutual-information-refutation","kind":"result","scope":"Jianwei Xu, *Oblique discord*, arXiv:1506.00404v1, p. 3, Conjecture (22):\n\n> Conjecture: I(ρ^{AB}) ≥ I(Φ_A ρ^{AB}) for any Φ_A and any ρ^{AB}, where I(ρ^{AB}) is the mutual information, Φ_A is defined in Eq.(13).\n\nThe normalized basis immediately before Eq. (13) consists of unit vectors\nv_i and their biorthogonal duals w_i, with ⟨v_i,w_j⟩ = δ_ij. The operation is\nT_b(ρ) = ∑_i (|v_i⟩⟨w_i| ⊗ 1) ρ (|w_i⟩⟨v_i| ⊗ 1),\nN_b(ρ) = tr_B(∑_i ⟨w_i|ρ|w_i⟩), Φ_b(ρ) = T_b(ρ)/N_b(ρ).\nThe Lean normalizer uses partialTraceRight over B on the sum of rectangular\nbra-vector sandwiches, extracting the scalar on the Unit factor. The unit\nnorms prove equality with Re(tr(T_b(ρ))) inside phi, and a local have in\nresult checks the same equality at the witness. Mutual information is\nS(ρ_A) + S(ρ_B) − S(ρ). The conjecture quantifies over all finite dimensions,\nall normalized bases and their duals, and all density states on the domain\nof the normalized operation. Its Lean domain explicitly requires a\npositive normalizing trace.","status":"refuted","summary":"Conjecture: I(ρ^{AB}) ≥ I(Φ_A ρ^{AB}) for any Φ_A and any ρ^{AB}, where I(ρ^{AB}) is the mutual information, Φ_A is defined in Eq.(13).","title":"Xu's oblique-discord mutual-information conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xu-2015-oblique-discord-mutual-information-refutation"},{"aliases":["D5/S3/Quantum/Entanglement/Witness/NoisyGhzWitnessBoundGap","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/be0f3d1541e4e99567ddfb47ec5485ad6664bf844615deb8d4345c7ba26e9bd1.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/Witness/NoisyGhzWitnessBoundGap.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.107.052405","statement_id":null},"id":"result:xu-et-al-2023-noisy-ghz-witness-bound-gap","kind":"result","scope":"X.-Y. Xu, Q. Zhou, S. Zhao, S.-M. Hu, L. Li, N.-L. Liu and K. Chen, *A generic framework for genuine multipartite entanglement detection*, arXiv:2211.05563v1; published as *General framework for genuine multipartite entanglement detection*, Phys. Rev. A 107 (2023) 052405, DOI 10.1103/PhysRevA.107.052405. In the arXiv TeX member `GMEwitness.tex`, subsection “Estimating entanglement measures”, line 325, they ask:\n\n> It remains open whether $\\epsilon_o^{n,d}(p)$ equals $E_G(\\rho_{n,d}(p))$.\n\nThe definitions are at lines 284–311:\n\n- $|GHZ_n^d\\rangle=d^{-1/2}\\sum_{i=0}^{d-1}|i\\rangle^{\\otimes n}$ and $\\rho_{n,d}(p)=pI/d^n+(1-p)|GHZ_n^d\\rangle\\langle GHZ_n^d|$.\n- $E_G(\\psi)=1-\\sup_{\\phi\\text{ biseparable}}|\\langle\\phi,\\psi\\rangle|^2$, where both vectors are unit vectors and biseparability means a product across some nonempty proper bipartition; the mixed-state measure is its convex roof.\n- $\\mathcal W=P/d-|GHZ_n^d\\rangle\\langle GHZ_n^d|$, where $P$ projects onto computational words using at most two distinct symbols. This is the orthogonal sum in lines 297–298, with the distinct arrangements specified by the orthogonal set $S$ at lines 690–692.\n- $w_{n,d}(p)=\\operatorname{Tr}(\\rho_{n,d}(p)\\mathcal W)$, $\\hat E_G(r\\mathcal W)=\\sup_{\\|\\psi\\|=1,\\ \\phi\\text{ unit biseparable}}\\{\\langle\\psi,(r\\mathcal W+|\\phi\\rangle\\langle\\phi|)\\psi\\rangle-1\\}$ over unit vectors, and $\\epsilon_o^{n,d}(p)=\\sup_{r\\in\\mathbb R}\\{r w_{n,d}(p)-\\hat E_G(r\\mathcal W)\\}$.\n\nThe source locators are also collected in [the literature note](../Library/Divergence/xuzhouzhaohu2023gme.md).\n\nIssue [#25044](https://github.com/the-omega-institute/trureturing/issues/25044) fixes the universal reading: for every pair of integers $n,d\\ge3$ and every real $p\\in[0,1]$, $\\epsilon_o^{n,d}(p)=E_G(\\rho_{n,d}(p))$. The formal convex roof takes the real infimum over all finite ensembles with nonnegative weights summing to one, unit pure vectors and exact reconstruction of the density matrix. This universal equality is refuted.","status":"refuted","summary":"It remains open whether $\\epsilon_o^{n,d}(p)$ equals $E_G(\\rho_{n,d}(p))$.","title":"The optimized GHZ witness bound need not equal the geometric GME measure","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xu-et-al-2023-noisy-ghz-witness-bound-gap"},{"aliases":["D5/S3/Quantum/Classicalization/XuImaiGuhneSimultaneousUnboundedness","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16640a7230c702c6ff0fa7ef7cd90bc9ecd6f4224e2775371bcea85d1011fa1e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Classicalization/XuImaiGuhneSimultaneousUnboundedness.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.107.L040401","statement_id":null},"id":"result:xu-imai-guhne-2022-classicalization-simultaneous-unboundedness","kind":"result","scope":"Zhen-Peng Xu, Satoya Imai and Otfried Gühne, *Fate of multiparticle entanglement when one particle becomes classical*, arXiv:2206.12834v2, Phys. Rev. A 107, L040401, Section V.B, page 5:\n\n> Inspired by those two examples, an interesting question arises whether there exist entanglement measures E and states ρ_ABC such that both Δ_E(ρ_ABC) and Δ̃_E(ρ_ABC) − Δ_E(ρ_ABC) can be arbitrarily large in the sense that they are not limited by the size of C, even if C is only a qubit. We leave this question for further research.\n\nThe answer is **Proved** for the fixed tripartite negativity of Equation (4), with no dimension-dependent rescaling. The source and conventions are in [the Library note](../Library/QuantumContext/xuimaiguhne2023classicalization.md); [#19638](https://github.com/the-omega-institute/trureturing/issues/19638) is the preregistration.\n\n### Statement echo\n\n| Source clause | Formal counterpart | Scope |\n| --- | --- | --- |\n| Equation (4), sum of three bipartite negativities | `tripartiteNegativity` | Partial transposes on C, A and B; positive negativity $(\\|\\rho^{T_Y}\\|_1-1)/2$. |\n| Equation (1), $\\sum_i p_i\\sigma_i\\otimes\\tau_i$ | `branch`, `classicalize` | Unnormalized branches $p_i\\sigma_i=\\operatorname{tr}_C(\\rho m_i)$, including zero-probability outcomes. |\n| Equation (2), minimum entanglement loss | `delta` | Infimum over every finite POVM and every finite register dimension; attainment is proved for the constructed family. |\n| Equation (9), discard C and prepare its zero flag | `deltaTilde` | The prepared flag is the zero basis state of a qubit. |\n| Both gaps arbitrarily large on the same state | `claim`, `result` | Every real threshold admits one positive semidefinite trace-one state on $\\operatorname{Fin}(a)\\times\\operatorname{Fin}(b)\\times\\operatorname{Fin}(2)$ exceeding it in both gaps. |\n\nThe density and POVM conditions appear literally in the public proposition and the infimum. The proofs use the frozen `IsDensity` and `IsPOVM` predicates with those same defining conditions. The source prints a sum of negative eigenvalues without an absolute value; its examples use the standard positive negativity adopted here.","status":"proved","summary":"Inspired by those two examples, an interesting question arises whether there exist entanglement measures E and states ρ_ABC such that both Δ_E(ρ_ABC) and Δ̃_E(ρ_ABC) − Δ_E(ρ_ABC) can be arbitrarily large in the sense that they are not limited by the size of C, even if C is only a qubit. We leave this question for furt…","title":"Two unbounded classicalization gaps with a qubit","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xu-imai-guhne-2022-classicalization-simultaneous-unboundedness"},{"aliases":["D5/S3/Analytic/XuYangMillsRemainder/Result","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5e1b5535d11a0f2373a1f9d71540f06719ac3e544a4e218ced1902ecfd65127f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Analytic/XuYangMillsRemainder/Result.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1186/s13660-026-03505-9","statement_id":null},"id":"result:xu-yang-2026-mills-remainder-cm","kind":"result","scope":"H.-Z. Xu and Z.-H. Yang, *Higher order monotonicity involving the Mills\nratio with applications*, Journal of Inequalities and Applications\n2026:98, DOI 10.1186/s13660-026-03505-9.\n\nMills-ratio definition, printed p. 2:\n\n> The Mills ratio [3] is defined by\n>\n> $$R\\equiv R(x)=\\frac{1-\\Phi(x)}{\\phi(x)}\n> =e^{x^2/2}\\int_x^\\infty e^{-t^2/2}\\,dt,\\quad x\\ge0,$$\n>\n> which satisfies the differential equation of first order:\n>\n> $$R'=xR-1.\\qquad(1.1)$$\n\nHere $\\phi(x)=e^{-x^2/2}/\\sqrt{2\\pi}$ and\n$\\Phi(x)=\\int_{-\\infty}^x\\phi(t)\\,dt$ are the standard Gaussian density and\ncumulative distribution function used in the preceding source sentences.\n\nThe bound family in (1.5), printed p. 2, is\n\n$$\\varphi_{p,q,r}(x)=\\frac{p}{\\sqrt{x^2+q}+rx},\n\\qquad p,q>0,\\quad r\\ge0.$$\n\nEquation (1.8), printed p. 3, with its complete surrounding statement:\n\n> Note that all bounds of the form $\\varphi_{p,q,r}(x)$ can be derived from the inequality\n>\n> $$Y_{a,b}(x):=(x^2+a)R^2(x)+bxR(x)<(>)c\\qquad(1.8)$$\n>\n> for all $x>0$, $a,b\\in\\mathbb R$, where\n>\n> $$c=\\lim_{x\\to0}Y_{a,b}(x)=\\frac{a\\pi}{2}\n> \\quad\\text{or}\\quad c=\\lim_{x\\to\\infty}Y_{a,b}(x)=b+1.$$\n\nEquation (3.7), printed p. 6, gives the coefficients following the asymptotic\nexpansion in Theorem 3.1:\n\n> where\n>\n> $$u_n\\equiv u_n(a,b)=(b+1)\\frac{2^n\\Gamma(n+1/2)}{\\Gamma(1/2)}\n> +(n-a)(n-1)!\\sum_{k=0}^{n-1}\n> \\frac{2^k\\Gamma(k+1/2)}{\\Gamma(1/2)k!}.\\qquad(3.7)$$\n>\n> In particular, $u_1=b+2-a$, $u_2=3b+7-2a$, $u_3=15b+36-7a$, and $u_4=105b-36a+249$.\n\nConjecture 7.1, printed p. 27, complete statement:\n\n> Moreover, inspired by Theorem 4.2, we propose the following conjecture.\n>\n> Conjecture 7.1 Let\n>\n> $$D_n(x,a)=Y_{a,a-2}(x)-\\left(a-1+\\sum_{k=1}^{n}\n> \\frac{(-1)^ku_k}{x^{2k}}\\right),$$\n>\n> where $u_k$ is defined by (3.7). Then $x\\mapsto(-1)^nD_n(x,-1)$ is CM on $(0,\\infty)$.\n\n\n### Formal reading\n\nThe namespace is `D5.S3.Analytic.XuYangMillsRemainder.Result`.\nThe definitions `millsRatio`, `Y`, `u` and `D` retain the displayed\nGaussian integral, polynomial expression, Gamma coefficients and finite\nremainder. Here $\\Gamma$ is Euler's Gamma function, $k!$ is the factorial of the\nnonnegative integer $k$, and all integrals use Lebesgue measure.\n`CompletelyMonotoneOn f` means $f$ is $C^\\infty$ on $(0,\\infty)$\nand every signed derivative $(-1)^m f^{(m)}(x)$ is nonnegative there.\n`claim` quantifies over every integer $n\\ge2$; `result` states `claim`.\nThe source does not display a lower bound for $n$ in Conjecture 7.1.\nThe restriction $n\\ge2$ is necessary: $n=1$ fails, as explained in Triage.","status":"proved","summary":"H.-Z. Xu and Z.-H. Yang, *Higher order monotonicity involving the Mills ratio with applications*, Journal of Inequalities and Applications 2026:98, DOI 10.1186/s13660-026-03505-9.","title":"Complete monotonicity of the Xu–Yang Mills remainders","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xu-yang-2026-mills-remainder-cm"},{"aliases":["D5/S3/AnalyticClosure/Polylogarithm/FiniteSlitAdmissibleSmoothing","full_result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/366035959b19cb7702aa36c20249fbc73367e96110ec3be74828e62477896c42.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/AnalyticClosure/Polylogarithm/FiniteSlitAdmissibleSmoothing.full_result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2609.11072","statement_id":null},"id":"result:xu-zhao-trivariate-cauchy-coefficient-signs","kind":"result","scope":"For every nonempty positive composition $k=(k_1,\\ldots,k_d)$ and integer\n$\\ell\\ge1$, put $F_k(z)=z^{-d}\\operatorname{Li}_k(z)$ with its removable\nvalue at zero, and\n$C_n^{j;k;\\ell}=[z^n](1-z)^{-j}F_k(z)^{-\\ell}$ as in source equations\n(7)-(8). Conjecture 1.3 asks for $C_n^{0;k;\\ell}<0$ eventually. For every\nfixed integer $j\\ge1$ it also asks for $C_n^{j;k;\\ell}>0$ eventually and,\nif $k_1>1$, for\n$C_n^{j;k;\\ell}>\\binom{n+j-1}{n}/\\zeta(k)^\\ell$ eventually.\nThe threshold may depend on $k,\\ell,j$; the strict inequality and all\nquantifiers are retained.","status":"proved","summary":"For every nonempty positive composition $k=(k_1,\\ldots,k_d)$ and integer $\\ell\\ge1$, put $F_k(z)=z^{-d}\\operatorname{Li}_k(z)$ with its removable value at zero, and $C_n^{j;k;\\ell}=[z^n](1-z)^{-j}F_k(z)^{-\\ell}$ as in source equations (7)-(8). Conjecture 1.3 asks for $C_n^{0;k;\\ell}<0$ eventually. For every fixed inte…","title":"Xu-Zhao Conjecture 1.3","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-xu-zhao-trivariate-cauchy-coefficient-signs"},{"aliases":["D5/S3/Quantum/Measurement/BiasedTrineJointMeasurabilityThreshold","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16a014da6a0417d6d50927fc00ab005068a54150b2dee01624f8ea54a7f97146.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Measurement/BiasedTrineJointMeasurabilityThreshold.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.110.L060201","statement_id":null},"id":"result:yadavalli-andrejic-kunjwal-2024-biased-trine-joint-measurability","kind":"result","scope":"S. A. Yadavalli, N. Andrejic and R. Kunjwal, *Qualitative equivalence between incompatibility and Bell nonlocality*, arXiv:2008.10100v3; Phys. Rev. A 110, L060201 (2024), Supplemental Material, Section III, Example 1. Its TeX source `qualequiv_supp_arxiv.tex`, lines 151–158, states:\n\n> Based on our numerical evaluation and visual inspection, we conjecture the exact form of $\\eta_1^{(3)}(b)$ to be\n> $$\\eta_1^{(3)}(b)=\\begin{cases}\n> 1+b,\\quad -1\\leq b<-\\frac{1}{3},\\\\\n> \\frac{2}{3},\\quad b\\in\\left[-\\frac{1}{3},\\frac{1}{3}\\right],\\\\\n> 1-b,\\quad \\frac{1}{3}< b\\leq 1.\n> \\end{cases}$$\n\nThe family is defined in lines 112–118, and the threshold is defined in lines 147–150 by triplewise joint measurability if and only if $\\eta\\leq\\eta_1^{(3)}(b)$. For $k\\in\\{1,2,3\\}$ and $x\\in\\{-1,+1\\}$, its effects on $\\mathbb C^2$ are\n\n$$E_k(x)=\\frac12\\big((1+xb)I+\\eta x\\,\\vec n_k\\cdot\\vec\\sigma\\big),\\qquad\n\\vec n_k=\\cos\\frac{2k\\pi}{3}\\vec e_x+\\sin\\frac{2k\\pi}{3}\\vec e_z.$$\n\nThe parameter domain is $-1\\leq b\\leq1$, $0\\leq\\eta\\leq1$ and $|b|\\leq1-\\eta$. Triplewise joint measurability means the existence of eight positive semidefinite complex $2\\times2$ matrices $G_s$, indexed by $s\\in\\{-1,+1\\}^3$, such that $\\sum_sG_s=I$ and $\\sum_{s:s_k=x}G_s=E_k(x)$ for every $k$ and $x$.\n\nIssue [#22692](https://github.com/the-omega-institute/trureturing/issues/22692) fixes the reading: for every real $b,\\eta$ in this domain, these three measurements are triplewise jointly measurable if and only if $\\eta\\leq\\eta_1^{(3)}(b)$ for the displayed piecewise function. This is proved.","status":"proved","summary":"Based on our numerical evaluation and visual inspection, we conjecture the exact form of $\\eta_1^{(3)}(b)$ to be $$\\eta_1^{(3)}(b)=\\begin{cases} 1+b,\\quad -1\\leq b<-\\frac{1}{3},\\\\ \\frac{2}{3},\\quad b\\in\\left[-\\frac{1}{3},\\frac{1}{3}\\right],\\\\ 1-b,\\quad \\frac{1}{3}< b\\leq 1. \\end{cases}$$","title":"The exact triplewise joint-measurability threshold of biased trine qubit measurements","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yadavalli-andrejic-kunjwal-2024-biased-trine-joint-measurability"},{"aliases":["D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c854262af3da0a1d7d60e4f2680986e217e9ee5b6d883974b2ae4ba274a44912.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/QuantumChannels/CoherenceBreakingAmendment.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1007/s10773-018-3890-4","statement_id":null},"id":"result:yang-chen-li-fei-wang-2018-coherence-breaking-amendment","kind":"result","scope":"L.-M. Yang, B. Chen, T. Li, S.-M. Fei and Z.-X. Wang, *Amending Coherence-Breaking Channels via Unitary Operations* (Int. J. Theor. Phys. 57, 3776 (2018); arXiv:1810.09642), study incoherent channels.\n- **Definitions.** An incoherent channel is coherence-breaking if it maps every state to an incoherent state. The coherence-breaking index n(Φ) is the least k with Φ^k coherence-breaking.\n- **Results.** For qubit channels with n(Φ) = 1 and 2 they find unitaries Λα that make Λα ◦ Φ and Φ ◦ Λα ◦ Φ not coherence-breaking.\n- **The conjecture.** Section 5 states: \"for a d-dimensional quantum channel Φ with n(Φ) ≥ 2, one may conjecture that the channel Φ can also be amended by Φ ◦ Λα ◦ Φ ◦ · · · ◦ Φ ◦ Λα ◦ Φ [n Φ] with proper unitary operations Λα.\"\n- **The later preprint.** MiS Preprint 7/2019 asserts the qudit statement in its conclusions without a proof.\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/yang2018coherencebreaking.md).\n\nIssue [#18385](https://github.com/the-omega-institute/trureturing/issues/18385) fixes the reading: for every incoherent trace-preserving Kraus channel with n(Φ) = n ≥ 2, some unitary U makes Φ ◦ (Λ_U ◦ Φ)^{n−1} not coherence-breaking, with the same U in every slot. The conjecture is proved.","status":"proved","summary":"L.-M. Yang, B. Chen, T. Li, S.-M. Fei and Z.-X. Wang, *Amending Coherence-Breaking Channels via Unitary Operations* (Int. J. Theor. Phys. 57, 3776 (2018); arXiv:1810.09642), study incoherent channels. - **Definitions.** An incoherent channel is coherence-breaking if it maps every state to an incoherent state. The cohe…","title":"Every incoherent channel with coherence-breaking index at least two is amended by one unitary","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yang-chen-li-fei-wang-2018-coherence-breaking-amendment"},{"aliases":["D5/S3/Quantum/Measurement/CoherenceFractionIncoherentNonmonotone","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9c6227f72857c23ea2129553ad27be8528fc53332ac2f5fae1f8d4108e9b1513.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Measurement/CoherenceFractionIncoherentNonmonotone.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.100.032324","statement_id":null},"id":"result:yao-li-sun-2019-coherence-fraction-incoherent-nonmonotone","kind":"result","scope":"Y. Yao, D. Li and C. P. Sun, *Quantum coherence fraction*, Phys. Rev. A 100, 032324 (2019), arXiv:1906.09789v1, Section V, `CF.tex`, lines 422–423:\n\n> First, similar to the behavior of fully entangled fraction \\cite{Badziag2000}, we conjecture that $C_{\\mathcal{F}}(\\rho)$ may not satisfy the monotonicity under general incoherent operations.\n\nThe source fixes the computational basis as the incoherent basis. An incoherent operation has a Kraus representation whose individual branches map diagonal density matrices to diagonal matrices; the normalized nonzero branches are again incoherent states (Section II, lines 122–131). Its quantum coherence fraction is $C_{\\mathcal F}(\\rho)=\\max_{|\\phi\\rangle\\in\\mathcal{MCS}}\\langle\\phi|\\rho|\\phi\\rangle$, where the maximally coherent unit vectors have coordinates of modulus $1/\\sqrt d$ (lines 141–150). The definitions and source locator are in [the literature note](../Library/QuadraticForms/yaolisun2019coherencefraction.md).\n\nIssue [#22563](https://github.com/the-omega-institute/trureturing/issues/22563) fixes `claim` as the universal monotonicity assertion: for every $d,m\\in\\mathbb N$, every complex $d\\times d$ matrix $\\rho$ that is positive semidefinite with trace one, and every family $K:\\mathrm{Fin}\\,m\\to M_d(\\mathbb C)$ satisfying $\\sum_\\alpha K_\\alpha^\\dagger K_\\alpha=I$ and preserving diagonality under each sandwich $X\\mapsto K_\\alpha X K_\\alpha^\\dagger$ for every complex diagonal matrix $X$, one has $C_{\\mathcal F}(\\sum_\\alpha K_\\alpha\\rho K_\\alpha^\\dagger)\\le C_{\\mathcal F}(\\rho)$. Here the module defines $C_{\\mathcal F}(A)=\\sup\\{\\operatorname{Re}(z^\\dagger Az)/d: z\\in\\mathbb C^d,\\ |z_j|=1\\text{ for every }j\\}$ using the real `sSup`.\n\nThe formal resolution is `refuted`, because `result : ¬ claim` negates universal monotonicity. This confirms the source's conjecture of non-monotonicity.","status":"refuted","summary":"First, similar to the behavior of fully entangled fraction \\cite{Badziag2000}, we conjecture that $C_{\\mathcal{F}}(\\rho)$ may not satisfy the monotonicity under general incoherent operations.","title":"The quantum coherence fraction can increase under incoherent operations","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yao-li-sun-2019-coherence-fraction-incoherent-nonmonotone"},{"aliases":["D5/S0/Certificates/YoshidaRapidDecreaseRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/50e72aa1559fdda0fb9fb64dd94e52bb7a473352b522eea0ad4c2754e8a62418.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S0/Certificates/YoshidaRapidDecreaseRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.1005.4218","statement_id":null},"id":"result:yoshida-rapid-decrease","kind":"result","scope":"Rintaro Yoshida, *On some questions of Fisk and Brändén*,\n[arXiv:1005.4218v1](https://arxiv.org/abs/1005.4218v1), Open Problem 5,\nprinted page 11, asks:\n\n> Let $f(x)=\\sum_{k=0}^{\\infty}a_kx^k$ where $\\{a_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence. If $S_r[f(x)]=\\sum_{k=0}^{\\infty}(a_k^2-a_{k-r}a_{k+r})x^k:=\\sum_{k=0}^{\\infty}b_kx^k$, is it true that $\\{b_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence?\n\nDefinition 29, printed page 10, requires positive terms and one real $\\alpha$ with\n$\\alpha\\ge\\max\\{2,\\frac{\\sqrt2}{2}(1+\\sqrt{1+a_1})\\}$ such that\n$a_k^2\\ge\\alpha^2a_{k-1}a_{k+1}$ for every $k\\ge1$.\nNegative indices are zero-extended, following Definition 8, printed page 3.\nThe [Library note](../Library/Certificates/yoshida2010fiskbranden.md) carries these locators.\nThe tier-1 preregistration is [#23414](https://github.com/the-omega-institute/trureturing/issues/23414).\n\n### Statement echo\n\nAll declaration names below belong to `D5.S0.Certificates.YoshidaRapidDecreaseRefutation`.\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Positive real sequence indexed by all nonnegative integers | `RapidlyDecreasing` positivity conjunct | Every natural index, including zero; no finite truncation. |\n| A single threshold parameter and all positive-index inequalities | `RapidlyDecreasing` existential and universal clauses | The source threshold is literal, with real square roots and real division. |\n| $a_j=0$ outside the index domain | Integer conditionals in `RapidlyDecreasing` and `sCoeff` | Zero for negative indices; otherwise `Int.toNat` selects the coefficient. No separate forwarding definition. |\n| $b_k=a_k^2-a_{k-r}a_{k+r}$ | `sCoeff` | Subtraction occurs after casting indices to integers. |\n| The source's coefficient object and closure question | `claim` | An explicit existence conjunct plus closure for every real coefficient sequence and every natural $r\\ge1$, as preregistered. The coefficient representation imposes no convergence condition. |\n| Negative answer | `result` | The literal negation of the complete claim; $r=1$ supplies the failure. |","status":"refuted","summary":"Let $f(x)=\\sum_{k=0}^{\\infty}a_kx^k$ where $\\{a_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence. If $S_r[f(x)]=\\sum_{k=0}^{\\infty}(a_k^2-a_{k-r}a_{k+r})x^k:=\\sum_{k=0}^{\\infty}b_kx^k$, is it true that $\\{b_k\\}_{k=0}^{\\infty}$ is a rapidly decreasing sequence?","title":"Yoshida's rapid-decrease closure question","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yoshida-rapid-decrease"},{"aliases":["D5/S3/Quantum/Information/SkewCoherencePolygamyMixedRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d3ac7c9a2a9f098ed0735ab1e58aeee5dcafff12c541cc457ff45f476e100f3b.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/SkewCoherencePolygamyMixedRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.95.042337","statement_id":null},"id":"result:yu-2017-skew-coherence-mixed-state-polygamy","kind":"result","scope":"Chang-shui Yu, *Quantum coherence via skew information and its polygamy*, Phys. Rev. A 95, 042337 (2017), arXiv:1704.04871v1, Discussion and Conclusions, `coherenceLy.tex`, lines 997–999:\n\n> However, we would like to conjecture that it could hold for the bipartite mixed states with the dimension $N \\geq 6$.\n\nAppendix D, “The conjecture,” lines 1396–1397, repeats:\n\n> In this sense, we would like to conjecture that this relation should be satisfied in $(N\\geq 6)$-dimensional systems.\n\nThe quotations are checked against the v1 TeX member `coherenceLy.tex`, md5 `026b6fb07e6d276a7bd743cc480af58c`. Lemma 1, lines 170–178, defines the reference-basis coherence by $C(\\rho)=1-\\sum_k\\langle k|\\sqrt\\rho|k\\rangle^2$, with the positive square root. Write $Q(\\rho)=1-C(\\rho)$. Theorem 2, lines 543–550, states $Q(|\\Psi\\rangle_{AB})\\le Q(\\rho_A)Q(\\rho_B)$ for every bipartite pure state and its two reduced density matrices. The conjecture asks for the same inequality for arbitrary bipartite mixed states of total dimension $N\\ge6$; the $2\\otimes3$, $3\\otimes3$, $3\\otimes4$ and $4\\otimes4$ figure captions identify $N$ as the total dimension.\n\nIssue [#23835](https://github.com/the-omega-institute/trureturing/issues/23835) fixes the reading: for all natural $m,n$ with $mn\\ge6$ and every positive semidefinite complex matrix $\\rho$ of trace one on $\\mathbb C^m\\otimes\\mathbb C^n$, in the product reference basis,\n\n$Q(\\rho)\\le Q(\\operatorname{Tr}_B\\rho)Q(\\operatorname{Tr}_A\\rho)$.\n\nThis is `claim`; `result : ¬ claim` refutes it.","status":"refuted","summary":"However, we would like to conjecture that it could hold for the bipartite mixed states with the dimension $N \\geq 6$.","title":"Yu's mixed-state skew-coherence polygamy conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yu-2017-skew-coherence-mixed-state-polygamy"},{"aliases":["D5/S3/QuantumChannels/SchurChannelMixedUnitaryDistance","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/528f6d0288ac332a2641011882eb4123f4b1445b2f2ef368b0f2b5e7e8c2da3a.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/QuantumChannels/SchurChannelMixedUnitaryDistance.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1109/TIT.2016.2641447","statement_id":null},"id":"result:yu-duan-xu-2017-schur-channel-diamond-equality","kind":"result","scope":"Nengkun Yu, Runyao Duan and Quanhua Xu, *Bounds on the distance between a\nunital quantum channel and the convex hull of unitary channels*, IEEE\nTransactions on Information Theory 63(2), 1299–1310, Section V, Theorem 3,\nEquation (8), ask:\n\n> It seems quite likely that in Eq. (8) the first inequality should be strict and the second one should be an equality. However, this is still an unsettled issue.\n\nThe equality in the second inequality is\n$D(\\Phi,\\operatorname{Conv}(U_d))=D(\\Phi,\\Lambda_d)$ for every Schur channel\n$\\Phi$, where $\\Lambda_d=S_d\\cap\\operatorname{Conv}(U_d)$.\nThe distance is the unhalved diamond distance. The\n[literature note](../Library/QuantumChannels/yuduanxu2017unitarydistance.md)\ncontains the definitions and source quotations. Preregistration\n[#16745](https://github.com/the-omega-institute/trureturing/issues/16745)\nspecifies the full quantified statement.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| Every dimension and every channel | `claim` quantifies natural `d` and `QuantumChannel (Fin d) (Fin d)` | Includes dimension zero; complete positivity and trace preservation are carried by the channel type. |\n| Schur action $\\Phi(X)=C\\circ X$ on every matrix | `IsSchurChannel` | Literal existence of a matrix giving Hadamard action; no replacement by a recovery certificate. |\n| Convex hull of unitary channels | `MixedUnitary` | Finite families, nonnegative real weights of sum one, and unitary conjugation on every complex matrix. |\n| $\\Lambda_d=S_d\\cap\\operatorname{Conv}(U_d)$ | `SchurMixedUnitary` | Literal intersection; diagonal mixtures are obtained in the proof. |\n| Infimum of diamond distances | `distToSet` | Real `sInf` of the image of the unhalved `diamondDistance`; no attained minimum is assumed. |\n| Equality in the second inequality of (8) | `result : claim` | Proved with the Schur hypothesis as the only additional hypothesis. |","status":"proved","summary":"It seems quite likely that in Eq. (8) the first inequality should be strict and the second one should be an equality. However, this is still an unsettled issue.","title":"Schur-channel diamond-distance equality","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yu-duan-xu-2017-schur-channel-diamond-equality"},{"aliases":["D5/S3/Quantum/Entanglement/Witness/AbsolutelyEntangledAllPartitions","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a607646b54d19ce42b5e0cfe6ee66ed340ab324e6d5b367f5581dfe655fce0b2.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/Witness/AbsolutelyEntangledAllPartitions.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.104.032414","statement_id":null},"id":"result:yu-et-al-2021-absolutely-entangled-all-partitions","kind":"result","scope":"B. Yu, P. Jayachandran, A. Burchardt, Y. Cai, N. Brunner and V. Scarani, *Absolutely entangled sets of pure states for bipartitions and multipartitions*, Physical Review A 104, 032414 (2021), DOI 10.1103/PhysRevA.104.032414; arXiv:2106.16216v3, Section III, immediately after the Corollary (`main.tex`, line 509):\n\n> This result guarantees that, given any $U$ and any partition $P$, there exists at least one entangled state. An interesting open question is whether one can find an AES, such that, for every $U$, there exists a state that is entangled according to all partitions. This would arguably be the strongest form of genuine multipartite entanglement.\n\nThe quotation is from the v3 source member `main.tex`, md5 `e11d4dc540598e9ef271fc11bad389a0`. Definition 3 is at lines 310–317, and the preceding Corollary at lines 499–508. The definitions and source locator are also in [the literature note](../Library/Entropy/yuetal2021absolutelyentangled.md).\n\nIssue [#22238](https://github.com/the-omega-institute/trureturing/issues/22238) fixes the reading. A partition $p=(d_1,\\ldots,d_k)$ is an ordered factorization with $k\\ge2$, every $d_i\\ge2$, and $\\prod_i d_i=d$. Its tensor coordinates are fixed by `partitionEquiv`, the mixed-radix identification of local index tuples with the standard basis of $\\mathbb C^d$. The matrix `productState p τ` has entries equal to the products of the corresponding local entries. `Separable p ρ` means membership in the real convex hull of such products with every local matrix positive semidefinite and of trace one.\n\nThe formal assertion is: for every $d,a,b\\in\\mathbb N$ with $a,b\\ge2$ and $d=ab$, there is a finite set $S$ of positive semidefinite $d\\times d$ complex matrices of trace one such that\n\n$\\forall U\\in SU(d),\\ \\exists\\rho\\in S,\\ \\forall p\\in\\mathrm{Partition}(d),\\ \\neg\\mathrm{Separable}(p,U\\rho U^\\dagger).$\n\nThe member may depend on $U$ and must serve every partition simultaneously. This fixed-coordinate assertion is proved. Purity is used in the construction; it is not an additional predicate in the type of `claim`.","status":"proved","summary":"This result guarantees that, given any $U$ and any partition $P$, there exists at least one entangled state. An interesting open question is whether one can find an AES, such that, for every $U$, there exists a state that is entangled according to all partitions. This would arguably be the strongest form of genuine mu…","title":"A finite set supplies one state entangled for all fixed tensor partitions in every frame","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yu-et-al-2021-absolutely-entangled-all-partitions"},{"aliases":["D5/S3/Quantum/Information/SkewInformationLambdaRefutation","result","arXiv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/89fdd56a190980b5b1e4f03be79580ce860fbd1120927df2918a62f7ae265b14.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/SkewInformationLambdaRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://arxiv.org/abs/1303.6404v1","statement_id":null},"id":"result:yu-oh-2013-skew-information-lambda-refutation","kind":"result","scope":"S. Yu and C. H. Oh, *Robertson-Schrödinger uncertainty relation refined by skew information*, arXiv:1303.6404v1, Appendix 2:\n\n> Numerical evidences show that $\\lambda_f=1$ whenever $4f(0)\\le 1$ and we conjecture that $\\lambda_f=\\min\\{1,1/[4f(0)]\\}$ always holds.\n\nHere λ_f is the minimum over x ≥ 0 of F(x) = (1 + x − f(0)(1 − x)²/(2f(x)))/(2f(x)), for f nonnegative on [0, ∞), operator monotone on Hermitian matrices, symmetric (x f(1/x) = f(x)), normalized (f(1) = 1) and regular (f(0) > 0). The definitions are quoted in [the literature note](../Library/QuantumContext/yu2013skewinformation.md).\n\nIssue [#19664](https://github.com/the-omega-institute/trureturing/issues/19664) fixes the reading. For every such f, min{1, 1/(4f(0))} would be the least value of F on [0, ∞). The conjecture is refuted.","status":"refuted","summary":"Numerical evidences show that $\\lambda_f=1$ whenever $4f(0)\\le 1$ and we conjecture that $\\lambda_f=\\min\\{1,1/[4f(0)]\\}$ always holds.","title":"λ_f = min{1, 1/[4f(0)]} fails for a regular symmetric operator monotone function","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yu-oh-2013-skew-information-lambda-refutation"},{"aliases":["D5/S3/Quantum/Information/XStateWeakUnimodalityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/596928d8889b44772f312685dab7dd9ff1c6cff0bf3d8fee2191f379ae13357d.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/XStateWeakUnimodalityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1007/s11128-017-1701-0","statement_id":null},"id":"result:yurischev-2017-xstate-unimodality","kind":"result","scope":"M. A. Yurischev, *Extremal properties of conditional entropy and quantum\ndiscord for XXZ, symmetric quantum states*, Quantum Inf. Process. 16, 249\n(2017), arXiv:1702.03728v3.\n\nIntroduction:\n\n> we suppose the unimodal property for the function $S_{cond}(\\theta)$ (see\n> Appendix). So, if the unimodality hypothesis is valid the only possibility\n> (except the trivial case $S_{cond}(\\theta)=const$) for a single local\n> extremum (minimum or maximum) to appear or disappear inside the open\n> interval by continuous varying the parameters defining the X state is the\n> doubling the extremun at the ends of interval $[0,\\pi/2]$.\n\nThe Appendix writes the conditional entropy of an X state as\n$f_1(x)=-h_2\\bigl(\\frac{1+p_2x}2,\\frac{1-p_2x}2\\bigr)+h_4\\bigl(\\frac{1+p_2x\\pm\\sqrt{r_1}}4,\\frac{1-p_2x\\pm\\sqrt{r_2}}4\\bigr)$\non $x\\in[0,1]$ (Eq. (A1)). Here\n$r_{1,2}=(p_1\\pm p_5x)^2+4w^2(1-x^2)$, $w=(|p_3+p_4|+|p_3-p_4|)/4$, and\n$h_2,h_4$ are Shannon entropies in bits. The Appendix defines weak\nunimodality as follows:\n\n> A function $f(x)$ is a weakly unimodal function in the interval $[a,b]$ if\n> there exists a value $x_m\\in[a,b]$ for which it is weakly monotonically\n> increasing for $x\\le x_m$ and weakly monotonically decreasing for\n> $x\\ge x_m$. […] Analogous definitions are given for the minimum.\n\nIt then states the conjecture:\n\n> The functions $f_1(x)$ and $f_2(x)$ for every choice of parameters\n> $p_1,\\ldots,p_5$ for which all arguments of Shannon functions are\n> non-negative can have at most only one local extremum (minimum or maximum)\n> in the open interval $x\\in(0,1)$. […] It is required to prove or refute\n> this proposition.","status":"refuted","summary":"M. A. Yurischev, *Extremal properties of conditional entropy and quantum discord for XXZ, symmetric quantum states*, Quantum Inf. Process. 16, 249 (2017), arXiv:1702.03728v3.","title":"Unimodality of the X-state conditional entropy","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-yurischev-2017-xstate-unimodality"},{"aliases":["D5/S3/Combinatorics/LatticeWalkNearMaximalArea","result","Integer sequences (OEIS)"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9e61687136b16608319c62b6de31ef96df18ccc79ed1a4b41e13fa3adfab2c8b.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/LatticeWalkNearMaximalArea.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://oeis.org/A385672","statement_id":null},"id":"result:zabolotskii-2025-a385672-near-maximal-area-walks","kind":"result","scope":"OEIS A385672 (Andrei Zabolotskii, 2025) counts the `n`-step walks on the\nsquare lattice by their algebraic area `(Sum_{steps right} y) - (Sum_{steps\nleft} y)`, the integral of `y dx`. Its FORMULA section states\n\n> It appears that T(2*n, n^2 - k) = 2 * A029552(k) for k < n and\n> T(2*n+1, n^2+n - k) = 4 * A098613(k) for k < n.\n\nwhere A029552 and A098613 are the coefficients of `phi(x)/f(-x)` and\n`psi(x^2)/f(-x)`, with generating functions\n`(1 + 2 Σ_{j>0} x^(j^2)) / ∏_{i>0} (1 − x^i)` and\n`Σ_{j>0} x^(j^2 − j) / ∏_{i>0} (1 − x^i)`. Issue #10337 fixes the readings:\nwalks start at the origin, and the coefficients are read through\n`1 / ∏ (1 − x^i) = Σ p(m) x^m` with `p` the partition numbers, so\n`A029552(k) = p(k) + 2 Σ_{j ≥ 1, j² ≤ k} p(k − j²)` and\n`A098613(k) = Σ_{j ≥ 1, j² − j ≤ k} p(k − j² + j)`.","status":"proved","summary":"OEIS A385672 (Andrei Zabolotskii, 2025) counts the `n`-step walks on the square lattice by their algebraic area `(Sum_{steps right} y) - (Sum_{steps left} y)`, the integral of `y dx`. Its FORMULA section states","title":"Walks of near-maximal algebraic area","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zabolotskii-2025-a385672-near-maximal-area-walks"},{"aliases":["D5/S0/Certificates/ZelinskyZhangConjectureTwentyTwoRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/f24342b7c7589fed18c49605278929f996134eef048a1d6d8ce6f58bc478c046.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S0/Certificates/ZelinskyZhangConjectureTwentyTwoRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2501.04209","statement_id":null},"id":"result:zelinsky-zhang-conjecture-22-refutation","kind":"result","scope":"Zelinsky and Zhang, *Kullback-Leibler divergence and primitive non-deficient\nnumbers*, arXiv:2501.04209v2, page 14, Conjecture 22 (verbatim):\n\n> Assume that n is a positive integer and assume that\n> n is not in {1, 12, 24, 30, 36, 48, 60, 72, 120, 180, 240, 360}.\n> Let p be the smallest prime factor of n. Then v(n) >= 1/p^2.\n\nFor a positive integer n, the paper defines\n`v(n) = sum_{d | n, d > 1} (1/d) log((tau(n) - 1)/d)`.\nThe literal claim formalized here quantifies over every positive natural n,\nuses exactly the twelve published exclusions, and takes p to be\n`Nat.minFac n`. No additional restriction on n is imposed.","status":"refuted","summary":"Assume that n is a positive integer and assume that n is not in {1, 12, 24, 30, 36, 48, 60, 72, 120, 180, 240, 360}. Let p be the smallest prime factor of n. Then v(n) >= 1/p^2.","title":"Refutation of Zelinsky and Zhang's Conjecture 22","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zelinsky-zhang-conjecture-22-refutation"},{"aliases":["D5/S1/Words/Patterns/CyclicStackPreimages","zhan_bie_conjectures_3_4","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/2b187139c406d305df00b41813c22dd207d310c0fc1f9488c9883d58bb573a11.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Patterns/CyclicStackPreimages.zhan_bie_conjectures_3_4","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.5281/zenodo.18154216","statement_id":null},"id":"result:zhan-bie-cyclic-stack-preimages","kind":"result","scope":"Alex Zhan and Stella Bie, *Cyclic-Pattern-Avoiding Stacks*, INTEGERS 26\n(2026), A15, Conjectures 3 and 4 on page 12 ask for the complete fibre of the\nconsecutive cyclic `[123]` stack map. Write the stack top-first. For an\nincoming value `x` and top entries `a,b`, pop `a` exactly when\n`x 4213`.\n\nFor every integer `n >= 4`, let `m=floor(n/2)`, let\n`y_n=(1,...,m,n,...,m+1)`, and let `S_n` contain every permutation of\n`1,...,n`. The two claims are\n\n`|{p in S_n : cyclicStackSort(p)=y_n}| = 1` for every even `n >= 4`,\n\n`|{p in S_n : cyclicStackSort(p)=y_n}| = ceil(n/2)` for every odd `n >= 4`.\n\nEquivalently, for every natural `m >= 2`, the cardinalities at `2*m` and\n`2*m+1` are `1` and `m+1`. There is no additional condition on the input\nbeyond being a permutation, and no finite upper bound on `n`.","status":"proved","summary":"Alex Zhan and Stella Bie, *Cyclic-Pattern-Avoiding Stacks*, INTEGERS 26 (2026), A15, Conjectures 3 and 4 on page 12 ask for the complete fibre of the consecutive cyclic `[123]` stack map. Write the stack top-first. For an incoming value `x` and top entries `a,b`, pop `a` exactly when `x For a given $U$, consider two initial states $\\ket{\\psi} = \\sum\\limits_j \\psi_j \\ket{j}$ and $\\ket{\\psi'} = \\sum\\limits_j \\psi'_j \\ket{j}$ such that (i) they are two unequal coefficient-squares, i.e., for some $j$, $|\\psi_j|^2 \\neq |\\psi'_j|^2$, and (ii) they have the same objective function value, i.e., $P(\\ket{\\psi}, U) = P(\\ket{\\psi'}, U)$. We claim that the ``average'' state given by $\\ket{\\psi_{avg}} = \\sum_j \\sqrt{\\frac{|\\psi_j|^2 + |\\psi'_j|^2}{2}} \\ket{j}$ has a lower objective function value, i.e., $P(\\ket{\\psi_{avg}}, U) < P(\\ket{\\psi'}, U)$.\n\nThe quotation is from the v6 source TeX. The [Library note](../Library/QuantumBounds/zhanguptahillery2024detectorsensor.md) records the paper and its encoding. In the source's setting (`problem-related.tex`, lines 17–47), exactly one of $n$ qubit sensors fires, with equal priors $1/n$; its unitary has eigenvalues $e^{\\pm i\\theta}$. The objective is the minimum error of a global POVM discriminating the $n$ possible fired states.\n\nWrite $X_n=\\{0,1\\}^{\\mathrm{Fin}\\,n}$, with $1$ representing the plus eigenvector. A probe is an amplitude function $\\psi:X_n\\to\\mathbb C$, normalized by $\\sum_x|\\psi(x)|^2=1$. For $n\\ge1$, the reused `fire` and `isoSuccess` give $\\phi_i(x)=e^{i\\theta(2x_i-1)}\\psi(x)$ and $S_\\theta(\\psi,M)=n^{-1}\\sum_i\\operatorname{Re}\\langle\\phi_i,M_i\\phi_i\\rangle$. The module defines `optSuccess` as the real indexed supremum of $S_\\theta(\\psi,M)$ over `LocalPOVM (Fin n → Bool) (Fin n)`, `errorProb` as $P_\\theta(\\psi)=1-\\operatorname{optSuccess}_\\theta(\\psi)$, and `average` as $a(\\psi,\\psi')(x)=\\sqrt{(|\\psi(x)|^2+|\\psi'(x)|^2)/2}$, included in $\\mathbb C$ with zero imaginary part.\n\nIssue [#26032](https://github.com/the-omega-institute/trureturing/issues/26032) fixes `claim`: for every natural $n$, every $0<\\theta<\\pi$, and every pair of normalized complex probes, unequal coefficient-squares somewhere and equal $P_\\theta$ imply $P_\\theta(a(\\psi,\\psi')) Let l > 1 and define A_l := {n ∈ lZ⁺ : ‖l^m · n‖_l = m + ‖n‖_l for all m ≥ 0}.\n\n> Can one say anything about the set A_l? For example, is it true that for every n ∈ lZ⁺, there exists m₀ ≥ 0 such that l^{m₀} · n ∈ A_l?\n\nThe every-orbit assertion is proved. The broader request to describe $A_l$\nand to bound the first stable index remains open here. Preregistration: #19862.\n\n### Statement echo\n\n| Source clause | Formal counterpart |\n| --- | --- |\n| $l>1$ | Natural parameter `l` with `1 < l`. |\n| $n\\in l\\mathbb Z^+$ | Natural parameter `n` with `0 < n` and `l ∣ n`. |\n| Minimal number of copies of $l$, with addition, multiplication and parentheses | Public `value` evaluates the existing `KumarPengTranModularComplexity.Expr`; public `complexity` is the natural infimum of attainable `Expr.leaves`. |\n| $A_l$ | Public `A` states positivity, divisibility and the source equality for every natural exponent, including zero. |\n| Some $m_0\\geq0$ with $l^{m_0}n\\in A_l$ | `claim` quantifies over natural `m₀`; public `result : claim` proves it. |\n\nThe natural evaluation and minimum are the literal conventions of #19862.\nThe private `mem_A_iff` identifies them with the frozen evaluator and minimum\nat modulus zero through injectivity of the natural-to-integer cast. The\npublic claim retains the source's natural definitions. No additional premise\nor finite cutoff is present.","status":"proved","summary":"Let l > 1 and define A_l := {n ∈ lZ⁺ : ‖l^m · n‖_l = m + ‖n‖_l for all m ≥ 0}.","title":"Eventual stability of Zhang's l-complexity","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-2024-l-complexity-question-3"},{"aliases":["D5/S3/Quantum/Matrix/ZhangThreeOperatorClarksonRefutation","result","arXiv"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8d90b33121a58af2a1b99866cc2fb13540d197c4718c51e15d02e9144798fa5f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Matrix/ZhangThreeOperatorClarksonRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://arxiv.org/abs/2410.21961v4","statement_id":null},"id":"result:zhang-2024-three-operator-clarkson-mccarthy","kind":"result","scope":"Teng Zhang, *Clarkson–McCarthy type inequalities, part I: $\\ell_p$–$\\ell_p$ and $\\ell_q$–$\\ell_p$ Schatten $p$-estimates* (2024 preprint, arXiv:2410.21961v4), Section 2, subsection “Euler's identity”, states Conjecture `conj:sharp-inequality`:\n\n> Let $A,B,C\\in S_p$. Then for $2\\le p<\\infty$,\n>\n> $\\|A+B+C\\|_p^p+\\|A\\|_p^p+\\|B\\|_p^p+\\|C\\|_p^p\\le\\frac{3^{p-1}+1}{2^p}\\left(\\|A+B\\|_p^p+\\|B+C\\|_p^p+\\|C+A\\|_p^p\\right).$\n>\n> For $00$ by taking $A=B=C\\neq0$.\n\nThis quotation is from lines 574–580 of the downloaded TeX member `v1_Clarkson_McCarthy_type_inequalities_part_I.tex`, MD5 `3ba34aa0913b51484593b3594a9f69d6`; only mathematical display layout and whitespace are normalized. The conjecture is introduced by “numerical evidence suggests the following sharp inequality” (line 571). Zhang restates it in *On five questions of Bourin and Lee* (2026), arXiv:2602.01463v2, TeX member `On_five_questions_of_Bourin_and_Lee.tex`, lines 411–419, MD5 `b3e91436b76494a53687b26bf5b2d06e`.\n\nFor a finite complex matrix and $p>0$, write $\\|X\\|_p^p=\\sum_j s_j(X)^p$. Abbreviate the four-term expression on the left by $L_p(A,B,C)$, the sum of the three pairwise terms by $P_p(A,B,C)$, and the proposed coefficient by $c(p)=(3^{p-1}+1)/2^p$.\n\nIssue [#22304](https://github.com/the-omega-institute/trureturing/issues/22304) fixes the reading settled by `claim`: for every real $p\\ge2$, every natural dimension $n$, and every $A,B,C\\in M_n(\\mathbb C)$, $L_p(A,B,C)\\le c(p)P_p(A,B,C)$. This universal statement is refuted. The reverse clause for $00$ by taking $A=B=C\\neq0$.","title":"Zhang's three-operator Clarkson–McCarthy conjecture is false","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-2024-three-operator-clarkson-mccarthy"},{"aliases":["D5/S3/Quantum/Matrix/TaylorMajorization/Result","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/aa4b7175a3f7c62a46aeee4b6138ef645c5d8bf57571599d5a68bd9e72ed62b7.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Matrix/TaylorMajorization/Result.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2601.07286","statement_id":null},"id":"result:zhang-2026-taylor-coefficient-weak-majorization","kind":"result","scope":"T. Zhang, *Weak majorization inequalities for the cubic and quartic\ncoefficients of e^{(A+B)t} versus e^{At}e^{Bt}*, arXiv:2601.07286v1,\nSection 5, Conjecture 5.5:\n\n> For A, B ∈ Hₙ and every k ≥ 5, λ(H^k) ≺_w λ(R_k).\n\nHere $H_n$ consists of all complex Hermitian $n\\times n$ matrices,\n$H=A+B$,\n\n$$Q_k=\\sum_{p=0}^k\\binom{k}{p}A^pB^{k-p},\\qquad\nR_k=\\operatorname{Re}Q_k=\\frac{Q_k+Q_k^*}{2}.$$\n\nThe list $\\lambda$ is nonincreasing, and $x\\prec_w y$ means\n$\\sum_{i=1}^r x_i\\le\\sum_{i=1}^r y_i$ for every $1\\le r\\le n$.\nThe paper proves the comparisons for $k=3,4$. Its conjecture imposes\nno positivity assumption on $A$ or $B$.\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Quantum.Matrix.TaylorMajorization`;\nthe Pauli matrices are from `D5.S3.Quantum.FiniteDimensional`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $H_n$, $n\\ge1$ | `Matrix (Fin n) (Fin n) ℂ`, `Matrix.IsHermitian` | `Defs.claim` explicitly requires `1 ≤ n` and both Hermitian hypotheses. |\n| $Q_k$ | `Defs.taylorQ` | The ordered binomial sum, with the $A$ powers on the left. |\n| $R_k$ | `Defs.taylorR` | Complex Hermitian symmetrization; `Defs.taylorR_isHermitian` supplies its Hermitian proof. |\n| $\\lambda$, $\\prec_w$ | `Defs.weakMaj` | Mathlib's nonincreasing `eigenvalues₀`, compared by all initial sums up to $n$. The extra $r=0$ sum is automatically zero. |\n| All dimensions, inputs, orders and partial sums | `Defs.claim` | Every natural $n\\ge1$, Hermitian $A,B$, natural $k\\ge5$ and $1\\le r\\le n$. |\n| $\\Delta_k=\\operatorname{Tr}R_k-\\operatorname{Tr}(A+B)^k$ | `Defs.defect` | Real parts of traces; both compared matrices are Hermitian, so their traces are real. |\n| $Z=\\operatorname{diag}(1,-1)$, $X=\\left[\\begin{smallmatrix}0&1\\\\1&0\\end{smallmatrix}\\right]$ | `qubitZ`, `qubitX` | The existing two-dimensional Pauli matrices. |\n| Refutation | `Result.result` | Exactly `¬ Defs.claim`, without additional assumptions. |","status":"refuted","summary":"For A, B ∈ Hₙ and every k ≥ 5, λ(H^k) ≺_w λ(R_k).","title":"Taylor-coefficient weak majorization for Hermitian matrices","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-2026-taylor-coefficient-weak-majorization"},{"aliases":["D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/324ccdb94f6e115ce4192d058a2c839b81a0a420d5d75ac42078c90e99861c0f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.dam.2025.07.017","statement_id":null},"id":"result:zhang-2026-threshold-hypergraph-independence-polynomial","kind":"result","scope":"E. Zhang, *On the Independence Polynomial and Threshold of an Antiregular\nk-Hypergraph*, arXiv:2303.01024v1, Section 5, asks:\n\n> Alternatively, a more general question is, given two T2-threshold\n> k-hypergraphs with the same independence polynomial, are they isomorphic?\n> These are left for future work.\n\nThe universal positive answer is false. The settling declaration is\n`D5/S3/Combinatorics/Hypergraphs/ZhangThresholdIndependenceRefutation.result`.\nThe preregistration is [#19406](https://github.com/the-omega-institute/trureturing/issues/19406).\nThe formal result uses five vertices and $k=3$; no arbitrary-$N$ theorem is delivered.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Scope |\n| --- | --- | --- |\n| A finite $k$-uniform hypergraph | `KHypergraph k V`, its `edges` and `uniform` fields | Every edge has cardinality $k$. |\n| No hyperedge is contained in an independent subset | `IsIndependent` | Every edge and subset are tested. |\n| Equation (8), the sum of $x^{|S|}$ over independent subsets | `indepPoly` | Integer coefficients; includes the empty set. |\n| Definition 4.1(T2), real labels and a strict threshold on every $k$-subset | `IsT2Threshold` | Literal strict inequality, with no positivity restriction on labels. |\n| Isomorphism | `Isomorphic` | A vertex equivalence preserves membership of every edge. |\n| Any two such hypergraphs with equal polynomials are isomorphic | `claim` | Quantifies over $k$, both finite vertex types and both hypergraphs. |\n| Negative answer | `result` | Negates the complete universal implication using $k=3$ and `Fin 5`. |\n\nThe encoding permits $k=0,1,2$, whereas the source question concerns $k\\ge3$;\nthe counterexample lies within the source range and all its edges are nonempty.\nThe finite-type instances enumerate vertices and decide equality; they add no\nmathematical restriction to the finite source objects. The carrier `Polynomial ℤ`\nis the formal integer-coefficient expression of Equation (8).","status":"refuted","summary":"Alternatively, a more general question is, given two T2-threshold k-hypergraphs with the same independence polynomial, are they isomorphic? These are left for future work.","title":"Independence polynomials do not determine T2-threshold hypergraphs","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-2026-threshold-hypergraph-independence-polynomial"},{"aliases":["D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/0a0069721eefd62378f2bc5c626a105272f27dd180dc4ac736a702c65665723f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/ArithSums/ZhangCharacterSumDifferenceRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2506.17235","statement_id":null},"id":"result:zhang-character-sum-difference-question-d-refutation","kind":"result","scope":"Wenpeng Zhang, *Some interesting number theory problems*, arXiv:2506.17235v1,\nprinted page 3, states problem (B) and question (D) verbatim as:\n\n> (B). Whether there are infinitely many pairs of fundamentally different\n> integer coefficients polynomials f(x) and g(x) (That is, (f(x)/p) ≠ (g(x)/p))\n> such that Σ_{x=1}^{p−1} (f(x)/p) − Σ_{x=1}^{p−1} (g(x)/p) = c, (2) where c is\n> a fixed constant.\n\n> (D). Whether the values of c can only be 0 or 2?\n\nIdentity (2) compares the Legendre character sums of two fundamentally\ndifferent integer-coefficient polynomials. Following the preceding corollary,\nthe formal statement uses one constant for every odd prime.","status":"refuted","summary":"(B). Whether there are infinitely many pairs of fundamentally different integer coefficients polynomials f(x) and g(x) (That is, (f(x)/p) ≠ (g(x)/p)) such that Σ_{x=1}^{p−1} (f(x)/p) − Σ_{x=1}^{p−1} (g(x)/p) = c, (2) where c is a fixed constant.","title":"Refutation of Zhang's character-sum constant restriction","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-character-sum-difference-question-d-refutation"},{"aliases":["D5/S3/Quantum/Measurement/QubitIncompatibilityProbability","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e074aacd89e300f7c97ff063a1882967f7f97b274a3deaad9b9f898cf8c092de.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Measurement/QubitIncompatibilityProbability.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevE.100.062139","statement_id":null},"id":"result:zhang-et-al-2019-qubit-incompatibility-probability","kind":"result","scope":"L. Zhang, H. Xiang, X. Li-Jost and S.-M. Fei, *Incompatibility probability of random quantum measurements*, Phys. Rev. E 100, 062139, arXiv:1912.12321, Section III.C, PDF p. 5, states:\n\n> We *conjecture* that the incompatibility probability of a pair of random qubit measurements $\\mathbf A$ and $\\mathbf B$, generated by $(a_0,\\boldsymbol a)$ and $(b_0,\\boldsymbol b)$ in Eq. (4), is given by $\\mathbf{Pr}[\\Theta^{\\mathrm{NJM}}_{2,2}]=\\frac14.$\n\nIssue [#15842](https://github.com/the-omega-institute/trureturing/issues/15842) fixes the literal statement, the squared-form criterion and the source and literature boundaries.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Uniform random qubit observables in $V=\\{(x_0,\\boldsymbol x): \\lvert x_0\\rvert+\\lVert\\boldsymbol x\\rVert\\le1\\}$ | `OrderedRootCoordinates.V` on $\\mathbb R\\times\\mathbb R^3$ | The entire closed body is retained. |\n| Yu–Liu–Li–Oh square-root function | `OrderedRootCoordinates.hYu` | Both $(1+x_0)^2$ and $(1-x_0)^2$ occur under the square roots. |\n| The set of pairs violating Eq. (4) | `OrderedRootCoordinates.NJM` | Both body memberships and the strict violation of the squared criterion are retained. |\n| Product Lebesgue volume divided by $\\operatorname{vol}(V)^2$ | `OrderedRootCoordinates.incompProb` | The measure is actual product Lebesgue measure. |\n| Probability equals $1/4$ | `OrderedRootCoordinates.claim` | Literal equality, without extra hypotheses. |\n| Negation of the conjecture | `QubitIncompatibilityProbability.result` | Unconditional negation of that claim. |\n\n### Definition conventions\n\nThe main-text display of $h$ omits squares on $1\\pm x_0$. The squared form is the expression in Appendix C and in Yu–Liu–Li–Oh, Eq. (5), arXiv:0805.1538. The formal definitions use that preregistered squared form. The strict-interior coordinate maps are smooth bijections; the closed body is not replaced by an interior in the claim. Body boundaries, zero vectors and cosine endpoints are removed only within the integration proof, where their nullity is proved.","status":"refuted","summary":"We *conjecture* that the incompatibility probability of a pair of random qubit measurements $\\mathbf A$ and $\\mathbf B$, generated by $(a_0,\\boldsymbol a)$ and $(b_0,\\boldsymbol b)$ in Eq. (4), is given by $\\mathbf{Pr}[\\Theta^{\\mathrm{NJM}}_{2,2}]=\\frac14.$","title":"The incompatibility probability of random qubit measurements","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-et-al-2019-qubit-incompatibility-probability"},{"aliases":["D5/S3/Quantum/Matrix/ZhangJingPowerCaseSettlement","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/405ddb6e16ea3d94b4fb878fc75441cede190da50de4c03f6756ec0e1e83bdef.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Matrix/ZhangJingPowerCaseSettlement.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.13001/ela.2025.9201","statement_id":null},"id":"result:zhang-jing-2025-block-criterion-power-case-theorem-2-3","kind":"result","scope":"Feng Zhang and Hefang Jing, More inequalities for positive semidefinite matrices, Electronic Journal of Linear Algebra 41 (2025), 595–601, state after Remark 2.13:\n\n> “Due to the limitations of our research method, we cannot solve the problem that Theorem 2.3 holds for $1/2 \\le a \\le 1$.”\n\nTheir Theorem 2.3 states:\n\n> “Let $A \\in M_n$. Then $[[|A|^{2a}, A^*], [A, |A|^{2(1-a)}]] \\ge 0$, for $0 \\le a \\le 1/2$ if and only if $A$ is normal.”\n\nThe notation is “$|A| = (A^*A)^{1/2}$ and $A^*$ is the conjugate transpose of $A$” (p. 595). The delivered claim extends the displayed equivalence to every real $a$ with $1/2 \\le a \\le 1$, every natural dimension, and every complex square matrix, using the existing block construction with $|A|=\\operatorname{CFC.sqrt}(A^*A)$ and real powers.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $|A|=(A^*A)^{1/2}$ and $A^*$ is conjugate transpose | The reused ZhangJingBlockNormalityRefutation.block owner | The source expression is preserved through the frozen owner; no duplicate absolute-value definition is introduced. |\n| $1/2\\le a\\le1$ | claim quantifies a, with both inequalities | Both endpoints are included. |\n| Every n and $A\\in M_n$ | claim quantifies n and A : Matrix (Fin n) (Fin n) Complex | The full finite-dimensional family is retained, including n=0. |\n| The block is positive semidefinite iff A is normal | claim uses Matrix.PosSemidef and $A^*A=A A^*$ | Every block and the normality equality are retained. |\n| $1/2\\le a<1$ survives | The owning Scribe displays the full strict-interval quantified application | Both directions, singular matrices, the half endpoint and dimension zero are retained as an application of canonical suppliers. |\n| The closed interval fails | result : \\lnot claim | The conclusion is the literal refutation. |","status":"refuted","summary":"“Due to the limitations of our research method, we cannot solve the problem that Theorem 2.3 holds for $1/2 \\le a \\le 1$.”","title":"Zhang–Jing Theorem 2.3: the upper power interval","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-jing-2025-block-criterion-power-case-theorem-2-3"},{"aliases":["D5/S3/Quantum/Matrix/ZhangJingBlockNormalityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/704081ebc2b983ed27afe067c18c1a80ee25f32c43b9d71ee1026dc9e0a975f8.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Matrix/ZhangJingBlockNormalityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.13001/ela.2025.9201","statement_id":null},"id":"result:zhang-jing-2025-block-normality-question-2-14","kind":"result","scope":"Feng Zhang and Hefang Jing, *More inequalities for positive semidefinite matrices*, Electronic Journal of Linear Algebra 41 (2025), 595–601, Question 2.14, p. 601:\n\n> Let A ∈ M_n. Then [[f(|A|), A*], [A, g(|A|)]] ≥ 0 if and only if A is normal under the conditions that f(x)g(x) = x² for x ≥ 0, f(x) ≥ g(x) (x ≥ 1) and g(x) ≥ f(x) (0 ≤ x ≤ 1).\n\nThe notation on p. 595 is “|A| = (A*A)^{1/2} and A* is the conjugate transpose of A”. The assertion quantifies over every admissible pair of real functions, every natural dimension, and every complex matrix in that dimension. Preregistration: [#18699](https://github.com/the-omega-institute/trureturing/issues/18699).\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| $|A|=(A^*A)^{1/2}$ | `CFC.sqrt` of the conjugate-transpose product, inlined in `block` | Literal expression; the existing `CFC.abs` construction is not redeclared. |\n| Four displayed blocks, in their given order | `block`, using `Matrix.fromBlocks` and `cfc` | Every block is retained. |\n| $f(x)g(x)=x^2$ for $x\\ge0$ | First quantified premise of `claim` | Same range and equality. |\n| $f(x)\\ge g(x)$ for $x\\ge1$ | Second quantified premise | Same inequality and range. |\n| $g(x)\\ge f(x)$ for $0\\le x\\le1$ | Third quantified premise | Both endpoints included. |\n| Every $A\\in M_n$, block nonnegative iff $A$ normal | Final quantifiers and equivalence in `claim` | `Fin n` labels the indices; positivity is `Matrix.PosSemidef` and normality is $A^*A=AA^*$. |\n| The universal assertion is false | `result` | Its conclusion is the literal negation of `claim`. |\n\nNo continuity, invertibility or spectral injectivity premise is added. The counterexample functions are continuous and satisfy the three required scalar conditions.","status":"refuted","summary":"Let A ∈ M_n. Then [[f(|A|), A*], [A, g(|A|)]] ≥ 0 if and only if A is normal under the conditions that f(x)g(x) = x² for x ≥ 0, f(x) ≥ g(x) (x ≥ 1) and g(x) ≥ f(x) (0 ≤ x ≤ 1).","title":"Zhang–Jing Question 2.14: block positivity and normality","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-jing-2025-block-normality-question-2-14"},{"aliases":["D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/920782ccf82ad3a34f6027d6e4c8581de92554f5bbff24c5f551a998727648d6.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Schatten/ZhangLiRankOneProblemThree.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2609.25552","statement_id":null},"id":"result:zhang-li-2026-rank-one-hadamard-product-variational","kind":"result","scope":"Lei-Hong Zhang and Ren-Cang Li, *On a Conjecture Related to Eigenvalue\nPerturbations*, arXiv:2609.25552v1, §1 Problem 3 and §5:\n\n> Problem 3: Relate min_{W nonsingular} ‖W∘G‖_ui ‖W^{-1}∘G^T‖_ui to min_{P permutation} ‖P∘G‖_ui^2.\n\n> In fact, for rank-one G, Romeo and Tilli [23, Theorem 1.2] proved (1.1) with c1=1 and the unitary restriction (2.3) with c2=1. They explicitly left the corresponding nonsingular version of Problem 3 unresolved; see [23, Remark 3.1].\n\nFor every natural $n$, every complex $n\\times n$ matrix $G$ with\n$\\operatorname{rank}(G)\\le1$, and every unitarily invariant norm $N$, the\nprecise conclusion is\n\n$$\n\\inf_{W\\in\\mathrm{GL}_n(\\mathbb C)}\n N(W\\circ G)N(W^{-1}\\circ G^{\\mathsf T})\n =\\min_{P\\text{ permutation}}N(P\\circ G)^2.\n$$\n\nA permutation minimizing $N(P\\circ G)$ attains the infimum. The transpose\nof $G$ is plain transpose. The zero matrix, zero weights, and $n=0$ are\nincluded.","status":"proved","summary":"Problem 3: Relate min_{W nonsingular} ‖W∘G‖_ui ‖W^{-1}∘G^T‖_ui to min_{P permutation} ‖P∘G‖_ui^2.","title":"Zhang–Li rank-one Hadamard product variational problem","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-li-2026-rank-one-hadamard-product-variational"},{"aliases":["D5/S3/Quantum/QuantumChannels/ProjectiveTestChannelCompatibilityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d4b402eed86e3f4fed9d4afa3fe062397ab012107237f9c2a44f0234538d985f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/QuantumChannels/ProjectiveTestChannelCompatibilityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.3390/e24060805","statement_id":null},"id":"result:zhang-nechita-2022-projective-test-channel-compatibility-refutation","kind":"result","scope":"Qing-Hua Zhang and Ion Nechita, *A Fisher information-based incompatibility criterion for quantum channels*, Entropy 24(6), 805 (2022), DOI 10.3390/e24060805, arXiv:2204.09963v1, Section 4, conjecture immediately following Lemma `lem:channel-POVM-compatible` (`article.tex`, lines 263–265):\n\n> Consider two quantum channels $\\Phi, \\Psi: \\mathcal L(H_d) \\to \\mathcal L(H_d)$ such that, for all orthonormal bases $\\mathbf e = (e_1, \\ldots, e_d)$, $\\mathbf f = (f_1, \\ldots, f_d)$ of $\\mathbb C^d$, the POVMs\n>\n> $\\Big[ \\Phi^*(|e_i\\rangle\\langle e_i|) \\Big]_{i=1}^d \\quad \\text{ and } \\quad \\Big[ \\Psi^*(|f_j\\rangle\\langle f_j|) \\Big]_{j=1}^d$\n>\n> are compatible. Then, $\\Phi$ and $\\Psi$ are compatible channels.\n\nThe source's `\\ketbra` macro is expanded in the displayed quotation. The source archive member `article.tex` has MD5 `614f0961ea15b65fb0b83ac83bd3fffb`.\n\nHere channels are completely positive, trace-preserving complex linear maps. Two POVMs are compatible when a positive normalized family indexed by both outcome sets has them as its marginals. Two channels are compatible when a single channel to the tensor product of their outputs has them as its partial-trace marginals on every input matrix. These definitions are in the source at lines 124–146 and 179–199 and in [the literature note](../Library/QuantumContext/zhangnechita2022fisherchannelincompatibility.md).\n\nIssue [#21608](https://github.com/the-omega-institute/trureturing/issues/21608) fixes the reading: for every positive dimension and every pair of channels on that space, compatibility of the two induced POVMs for every pair of orthonormal bases implies channel compatibility. This is refuted by two copies of the qubit depolarizing channel $\\Phi_{7/10}$.","status":"refuted","summary":"Consider two quantum channels $\\Phi, \\Psi: \\mathcal L(H_d) \\to \\mathcal L(H_d)$ such that, for all orthonormal bases $\\mathbf e = (e_1, \\ldots, e_d)$, $\\mathbf f = (f_1, \\ldots, f_d)$ of $\\mathbb C^d$, the POVMs $\\Big[ \\Phi^*(|e_i\\rangle\\langle e_i|) \\Big]_{i=1}^d \\quad \\text{ and } \\quad \\Big[ \\Psi^*(|f_j\\rangle\\lang…","title":"Compatible projective tests do not imply compatible quantum channels","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-nechita-2022-projective-test-channel-compatibility-refutation"},{"aliases":["D5/S3/Quantum/Subentropy/StrongSubadditivityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/40f0aec53bdc60856cb24997f937df4d033092c075c366d2e62a07a71c28845d.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Subentropy/StrongSubadditivityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.aop.2016.12.024","statement_id":null},"id":"result:zhang-singh-pati-2017-subentropy-strong-subadditivity","kind":"result","scope":"The question attributed to arXiv:1510.08859v2, §7.1 is recorded in\n[zhang2017subentropy](../Library/Entropy/zhang2017subentropy.md).\n\n> A similarly question can be asked for the strong subadditivity of subentropy: Q(ρ ABC) + Q(ρ B) ⩽ Q(ρ AB) + Q(ρ BC).\n\nFor every finite-dimensional tripartite density operator $\\rho_{ABC}$, let\n$\\rho_{AB}=\\operatorname{Tr}_C\\rho_{ABC}$,\n$\\rho_{BC}=\\operatorname{Tr}_A\\rho_{ABC}$ and\n$\\rho_B=\\operatorname{Tr}_{AC}\\rho_{ABC}$. The proposed assertion is\n\n$$\nQ(\\rho_{ABC})+Q(\\rho_B)\n\\le Q(\\rho_{AB})+Q(\\rho_{BC}).\n$$\n\nThe formal statement quantifies over arbitrary finite coordinate\ntypes and the repository's `DensityState`; changing the parentheses of a\nproduct coordinate type only relabels matrix entries. The original assertion\ndoes not impose commutation, full rank, separability or purity.","status":"refuted","summary":"The question attributed to arXiv:1510.08859v2, §7.1 is recorded in [zhang2017subentropy](../Library/Entropy/zhang2017subentropy.md).","title":"Zhang–Singh–Pati: strong subadditivity of subentropy","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-singh-pati-2017-subentropy-strong-subadditivity"},{"aliases":["D5/S3/Combinatorics/Graph/Eccentric/ZhangTianSteinerTreeRadius","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fa0e6d12be3c8718edfab89655b80f6c079496aedbf3ec0e599ebe9518c99e8d.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Graph/Eccentric/ZhangTianSteinerTreeRadius.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2511.22492","statement_id":null},"id":"result:zhang-tian-2025-steiner-diameter-radius-trees","kind":"result","scope":"Qingnan Zhang and Yingzhi Tian, *On the Steiner k-diameter and Steiner\n(k, k′)-radius of trees*, arXiv:2511.22492v1, p. 14, Conjecture 5.1, states:\n\n> Let n, k, k′ be integers with n ≥ k ≥ 3 and k > k′ ≥ 1 and let T be a tree with order n. Then Sd_k(T) ≤ k/(k − k′) Sr_{k,k′}(T) − k′(k′ − 1)/(k − k′).\n\nHere $d_T(S)$ is the minimum edge count among connected subgraphs containing\n$S$; $\\varepsilon_k(v;T)$ maximizes $d_T(S)$ over $k$-sets containing $v$;\n$Sd_k(T)$ maximizes these vertex eccentricities.\nFor an anchor set $B$, $\\varepsilon_{k,r}(B;T)$ maximizes $d_T(S)$ over\n$k$-sets containing $B$, and $Sr_{k,r}(T)$ minimizes this quantity over\n$r$-sets $B$.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| A finite simple tree of order $n$ | `V : Type`, `Fintype V`, `DecidableEq V`, `T : SimpleGraph V`, `T.IsTree`; $n$ is `Fintype.card V` | Exact; decidable equality is classical bookkeeping on a finite carrier. |\n| $n\\ge k\\ge3$ | `k ≤ Fintype.card V`, `3 ≤ k` | Exact. |\n| $k>k'\\ge1$ | `r < k`, `1 ≤ r` | Exact; $r$ denotes $k'$. |\n| Minimum connected-subgraph edge count | `steinerDist` | Literal source definition, using a natural infimum. |\n| Vertex eccentricity and diameter | `steinerEcc`, `steinerDiam` | Literal nested maxima, as natural suprema. |\n| Anchored eccentricity and radius | `steinerSetEcc`, `steinerRadius` | Literal extrema over terminal sets of the specified cardinality. |\n| The rational inequality | `claim`, proved by `result` | All distances and parameters are cast into $\\mathbb Q$ before subtraction and division. |\n| Auxiliary cut representation | `Separates`, `cutEdges`, `steinerDist_cut` | Repository-derived objects characterize the source's literal distance; they do not replace its definition. |\n| Arbitrary terminal $k$-set $A$ and anchor $r$-set $B$ | `pointwise_bound` | Strengthening; no disjointness, hull containment or diameter-maximization hypothesis. |","status":"proved","summary":"Let n, k, k′ be integers with n ≥ k ≥ 3 and k > k′ ≥ 1 and let T be a tree with order n. Then Sd_k(T) ≤ k/(k − k′) Sr_{k,k′}(T) − k′(k′ − 1)/(k − k′).","title":"Zhang–Tian Conjecture 5.1: Steiner diameter and anchored radius of trees","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-tian-2025-steiner-diameter-radius-trees"},{"aliases":["D5/S3/Quantum/Information/ZhangWuFeiSuperadditivityRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e5c9ca38ea045c42acefe71825bb292f764deee899fdfa335f35778d94076e7d.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/ZhangWuFeiSuperadditivityRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.physleta.2013.05.047","statement_id":null},"id":"result:zhang-wu-fei-2012-relative-entropy-superadditivity-conjecture-4-2","kind":"result","scope":"Zhang, Wu and Fei, arXiv:1210.4720v4, p. 6, §4, conjecture (4.2), state:\n\n> Based on the above result, we propose the following conjecture: There exist unitary operators U_A ∈ U(H_A), U_B ∈ U(H_B) and U_AB ∈ U(H_A ⊗ H_B) such that S(U_AB ρ_AB U_AB† ‖ σ_AB) ⩾ S(U_A ρ_A U_A† ‖ σ_A) + S(U_B ρ_B U_B† ‖ σ_B), (4.2) where the reference state σ_AB is required to be full-ranked state. Indeed, our numerical calculations show that inequality (4.2) is true.\n\nThe source is recorded in [the Library note](../Library/Divergence/zhangwufei2013superadditivity.md).\nThe assertion ranges over every pair of finite subsystem dimensions and every joint density\n$\\rho_{AB}$ and full-rank joint reference $\\sigma_{AB}$; all four marginals are actual partial traces.\nThe relative entropy is $S(\\rho\\Vert\\sigma)=\\operatorname{Re}\\operatorname{Tr}(\\rho(\\log\\rho-\\log\\sigma))$\nwith natural logarithms. The independent existential joint and local unitaries are unrestricted.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Finite $H_A,H_B$ | Natural dimensions `a`, `b`, with coordinates `Fin a`, `Fin b` | All finite dimensions; a density state supplies trace one |\n| Joint density matrices $\\rho_{AB},\\sigma_{AB}$ | `DensityState (Fin a × Fin b)` | Positive semidefinite complex matrices of trace one |\n| Full-ranked $\\sigma_{AB}$ | `(CStarMatrix.ofMatrix.symm σ.1).PosDef` | Positive definiteness of the reference matrix |\n| $\\rho_A,\\sigma_A$ | `marginalRight ρ`, `marginalRight σ` | Keeps the first product factor |\n| $\\rho_B,\\sigma_B$ | `marginalLeft ρ`, `marginalLeft σ` | Keeps the second product factor |\n| Three existential unitaries | `UAB`, `UA`, `UB` and `Matrix.unitaryGroup` membership proofs | The membership proofs express unitarity, with no extra hypotheses |\n| Inequality (4.2) | `claim` | Sum of the two marginal divergences is at most the joint divergence |\n| Refutation | `result` | The negation of the complete universally quantified `claim` |","status":"refuted","summary":"Based on the above result, we propose the following conjecture: There exist unitary operators U_A ∈ U(H_A), U_B ∈ U(H_B) and U_AB ∈ U(H_A ⊗ H_B) such that S(U_AB ρ_AB U_AB† ‖ σ_AB) ⩾ S(U_A ρ_A U_A† ‖ σ_A) + S(U_B ρ_B U_B† ‖ σ_B), (4.2) where the reference state σ_AB is required to be full-ranked state. Indeed, our num…","title":"Zhang–Wu–Fei: relative-entropy superadditivity after unitary conjugation","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-wu-fei-2012-relative-entropy-superadditivity-conjecture-4-2"},{"aliases":["D5/S3/Quantum/Information/BargmannImaginaryMaximizers","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b97f15234158737d7124f01bd19428be118e3cce180c9f76bcd844c0580cf291.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/BargmannImaginaryMaximizers.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2601.01858","statement_id":null},"id":"result:zhang-xie-2026-bargmann-imaginary-maximizers","kind":"result","scope":"Lin Zhang and Bing Xie, *A Survey of Bargmann Invariants: Geometric Foundations and Applications*, arXiv:2601.01858v2, p. 26, ask:\n\n> The quantity τ_n predicts the upper bound for the imaginary part of any nth-order Bargmann invariant. This immediately raises the question: what is the complete set of n-tuples Ψ = (|ψ_1⟩, …, |ψ_n⟩) in ℂ² that achieve Im Tr(ψ_1 … ψ_n) = τ_n? We leave this characterization, along with the operational interpretation of the condition z ∈ 𝓑_{n+1} ∖ 𝓑_n, as interesting open problems for future work.\n\nThe first question is **Proved** with the exact conventions in [preregistration #19224](https://github.com/the-omega-institute/trureturing/issues/19224). For every $n\\ge3$, a tuple of unit vectors in $\\mathbb C^2$ attains\n\n$$\\tau_n=\\frac{\\cos^n(\\pi/n)}{\\cos^{n-1}(\\pi/[2(n-1)])}$$\n\nif and only if $\\psi_j=\\eta_jUv_j$ for one common $2\\times2$ unitary $U$, independent phases $|\\eta_j|=1$, and\n\n$$v_j=(\\sqrt{1-t_n},\\sqrt{t_n}\\,e^{2\\pi ij/n}),\\qquad t_n=\\frac{1-\\tan(\\pi/[2(n-1)])/\\tan(\\pi/n)}2.$$\n\nThe [Library note](../Library/QuantumContext/zhangxie2026bargmann.md) gives Definition 3.1, Corollary 5.5, Eq. (5.15), and the source question.\n\n### Statement echo\n\n| Source clause | Lean counterpart | Fidelity |\n| --- | --- | --- |\n| Every $n\\ge3$ | `claim` universally quantifies `n` with `3 ≤ n` | Exact |\n| Unit vectors in $\\mathbb C^2$ | `Ψ : Fin n → Fin 2 → ℂ`, with `star (Ψ j) ⬝ᵥ Ψ j = 1` for every `j` | Exact; the wave-function condition is inlined |\n| Pure state $|\\psi_j\\rangle\\langle\\psi_j|$ and ordered trace | `bargmann` uses `Matrix.vecMulVec`, `List.ofFn`, `List.prod`, and `Matrix.trace` | Exact; the projector expression is inlined |\n| Maximal imaginary value | `tau` is the source's cosine-power quotient | Exact; natural subtraction occurs in the exponent, real subtraction in the angle |\n| Complete set | `result : claim` proves both directions of the equality/orbit equivalence | Exact |\n| Explicit candidate answer | `maximizer` is the displayed latitude vector; one `U ∈ Matrix.unitaryGroup (Fin 2) ℂ` and phases `η` | The answer proposed in #19224, with no added hypothesis |\n\nNo clause is dropped or weakened. The source's second question is outside this equivalence.","status":"proved","summary":"The quantity τ_n predicts the upper bound for the imaginary part of any nth-order Bargmann invariant. This immediately raises the question: what is the complete set of n-tuples Ψ = (|ψ_1⟩, …, |ψ_n⟩) in ℂ² that achieve Im Tr(ψ_1 … ψ_n) = τ_n? We leave this characterization, along with the operational interpretation of…","title":"All qubit tuples maximizing the imaginary Bargmann invariant","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-xie-2026-bargmann-imaginary-maximizers"},{"aliases":["D5/S3/Combinatorics/Matroids/SpeyerUniformNormality","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/cc0a268b6453b25e17da11b6a9e08a014ca11ce028d2d05efb65bc25f8fbd094.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Matroids/SpeyerUniformNormality.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1007/s10114-026-4502-5","statement_id":null},"id":"result:zhang-zhao-2024-speyer-g-uniform-normality","kind":"result","scope":"Rong Zhang and James Jing Yu Zhao, *Analytic properties of Speyer’s g-polynomial of uniform matroids*, arXiv:2408.15506v1, Conjecture 4.8:\n\n> If $d=d(n)$ is a real function of $n$ such that $d(n)\\to+\\infty$ as $n\\to+\\infty$, then the coefficient of Speyer’s g-polynomial $g_{n,d(n)}(t)$, namely $S_i(n,d(n))$, is asymptotically normal by local and central limit theorems.\n\nThe source defines $g_{n,d}$ for integer ranks and uses the standing restriction $1\\le d\\le\\lfloor n/2\\rfloor$ after its rank-complement symmetry. The formal claim quantifies $d:\\mathbb N\\to\\mathbb N$, imposes that restriction eventually, and assumes $d(n)\\to\\infty$. It does not extend the polynomial to arbitrary real-valued rank functions. Both limits are uniform in every real threshold and use the source's derivative-ratio mean and variance.\n\n### Statement echo\n\n| Source clause or convention | Lean counterpart | Scope |\n| --- | --- | --- |\n| Proposition 1.1 factorial sum | `SpeyerUniformNormality.g` | Natural subtraction occurs before casting, on the prescribed finite summation range. |\n| Integer matroid rank under the standing half-rank restriction | `d : ℕ → ℕ` and eventual `1 ≤ d n ∧ d n ≤ n / 2` | The source's wording “real function” is interpreted within its integer rank domain; eventual validity allows arbitrary finitely many initial values. |\n| Rank tends to positive infinity | `Tendsto d atTop atTop` | No monotonicity or growth-rate hypothesis is added. |\n| Theorem 4.2 mean and variance | `BernoulliTriangularLimits.mu`, `sigmaSq`, `sigma` | The same derivative ratios and nonnegative square root. |\n| Coefficient probability | `BernoulliTriangularLimits.prob` | Coefficient divided by evaluation at one, and zero at negative integer indices. |\n| Standard normal distribution and density | `StatLean.HypothesisTesting.stdNormalCDF` and `stdNormalPDF` | Reused Gaussian owners inside `CLT` and `LLT`; `claim` is their conjunction for the Speyer polynomial sequence. Their integral and density correspondences are established inside the consuming proofs. |\n| Central limit theorem, uniform over real thresholds | `BernoulliTriangularLimits.CLT` | The epsilon form uses the finite coefficient support. |\n| Local limit theorem, with integer floor | `BernoulliTriangularLimits.LLT` | The epsilon form retains the literal integer floor. |\n| Both limits | `SpeyerUniformNormality.result : claim` | The conclusion is their conjunction for every admissible divergent rank sequence. |","status":"proved","summary":"If $d=d(n)$ is a real function of $n$ such that $d(n)\\to+\\infty$ as $n\\to+\\infty$, then the coefficient of Speyer’s g-polynomial $g_{n,d(n)}(t)$, namely $S_i(n,d(n))$, is asymptotically normal by local and central limit theorems.","title":"Normal limits for Speyer g-polynomials of growing uniform-matroid rank","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-zhao-2024-speyer-g-uniform-normality"},{"aliases":["D5/S3/Combinatorics/Graph/DomiRankEntropyRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/479baebe9e43f7038e709f03600397c1d87c920262acb05e85f3728349af6212.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/Graph/DomiRankEntropyRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2610.00107","statement_id":null},"id":"result:zhang-zhao-2026-domirank-entropy-monotonicity","kind":"result","scope":"Yingying Zhang and Chengye Zhao, *Theoretical Analysis of DomiRank\nCentrality: Automorphism, Entropy, and Graph Transformations*,\narXiv:2610.00107v1, Section 4.1, Remark 4.6, identify the unconditional\nassertion\n\n> the DomiRank entropy of a connected non-regular graph decreases\n> monotonically with σ\n\nas an open problem. The claim quantifies over every finite connected\nnonregular unweighted undirected simple graph and every pair\n$0<\\sigma<\\tau<-1/\\lambda_{\\min}(A)$. It asserts\n$H_G(\\tau)\\le H_G(\\sigma)$, with the actual inverse equilibrium\n$\\Gamma(s)=s(I+sA)^{-1}A\\mathbf1$, positive clipping, normalization,\nand Shannon entropy in bits. There is no raw-positivity or extra\nnonzero-normalizer premise. Conjecture 4.7 is excluded.","status":"refuted","summary":"Yingying Zhang and Chengye Zhao, *Theoretical Analysis of DomiRank Centrality: Automorphism, Entropy, and Graph Transformations*, arXiv:2610.00107v1, Section 4.1, Remark 4.6, identify the unconditional assertion","title":"DomiRank entropy need not decrease with competition","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhang-zhao-2026-domirank-entropy-monotonicity"},{"aliases":["D5/S3/Quantum/Information/CoherenceAssistanceRoofEquality","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d12329d597e408ca69d7f6f61ec541b0edaae81dd7990c7ed69fff73740a193d.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/CoherenceAssistanceRoofEquality.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1038/s41598-021-85273-8","statement_id":null},"id":"result:zhao-pereira-ma-fei-2021-coherence-assistance-roof-equality","kind":"result","scope":"M.-J. Zhao, R. Pereira, T. Ma and S.-M. Fei, *Coherence of assistance and assisted maximally coherent states*, Scientific Reports 11, 5935 (2021), arXiv:2103.08818v1, ask in the paragraph “Relation between the convex roof extension and the least concave majorant extension” (`ZhaoMJ-AMC4.tex`, line 303):\n\n> The strict relation between the coherence of convex roof and the coherence of assistance, that is, whether $C_c(\\rho)< C_a(\\rho)$ holds for all mixed quantum states is an interesting topic.\n\nThe source's standing assumption is $f\\in\\mathfrak F\\setminus\\{0\\}$ (lines 189–190): $f$ is real, symmetric and concave on the probability simplex and vanishes at its vertices. The pure-state coherence is $C_f(|\\psi\\rangle)=f(\\mu(|\\psi\\rangle))$, where $\\mu(|\\psi\\rangle)_i=|\\psi_i|^2$ (lines 207–211).\n\nIn the formal reading, for every natural dimension $n$, `IsCoherenceFunction f` requires concavity on $\\Omega_n=\\{p\\in\\mathbb R^n:p_i\\ge0,\\ \\sum_i p_i=1\\}$, invariance under every coordinate permutation on $\\Omega_n$, and $f(e_i)=0$ for every vertex. Nonzeroness means $\\exists p\\in\\Omega_n,\\ f(p)\\ne0$.\n\nA finite pure-state decomposition has nonnegative real weights $p_j$ summing to one, unit vectors $\\psi_j\\in\\mathbb C^n$ for every index, and $\\rho=\\sum_jp_j|\\psi_j\\rangle\\langle\\psi_j|$. Let $D_f(\\rho)$ be the set of all averages $\\sum_jp_jf(\\mu(\\psi_j))$ over these decompositions. The module defines $C_c(\\rho)=\\inf D_f(\\rho)$ and $C_a(\\rho)=\\sup D_f(\\rho)$, using real `sInf` and `sSup`; the source uses minimum and maximum (lines 212–216 and 232–236). `IsMixed ρ` means $\\rho\\succeq0$, $\\operatorname{tr}\\rho=1$ and $\\operatorname{tr}(\\rho^2)\\ne1$.\n\nIssue [#22770](https://github.com/the-omega-institute/trureturing/issues/22770) fixes the claim: for every $n$, every such nonzero $f$, and every mixed $\\rho$, $C_c(\\rho) For n ≥ 2, it holds that max over π in 𝔖ₙ of |SC₁₂̲₃̲⁻¹(π)| equals max over π in 𝔖ₙ of |SC₃₂̲₁̲⁻¹(π)| equals 2ⁿ⁻².\n\nThe formal claim uses the paper's one-based permutation entries and the right-greedy stack convention fixed by its four worked figures. The flag `false` denotes 1-underline(23), and `true` denotes 3-underline(21).","status":"refuted","summary":"For n ≥ 2, it holds that max over π in 𝔖ₙ of |SC₁₂̲₃̲⁻¹(π)| equals max over π in 𝔖ₙ of |SC₃₂̲₁̲⁻¹(π)| equals 2ⁿ⁻².","title":"Refutation of Zhao's maximum preimage conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhao-vincular-stack-maximum-preimages-refutation"},{"aliases":["D5/S1/Words/Patterns/ZhaoVincularPreimageRefutations","resultSecondLargest","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/71ac80ef47a3411e72248b5b953ca9ddf4a3afc6d3015c1c16d6f42324abe602.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S1/Words/Patterns/ZhaoVincularPreimageRefutations.resultSecondLargest","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.disc.2025.114834","statement_id":null},"id":"result:zhao-vincular-stack-second-largest-preimages-refutation","kind":"result","scope":"Zhao, arXiv:2410.17057v1, Conjecture 5.2 (Section 5, printed page 20), states:\n\n> The second-largest number of preimages under SC₁₂̲₃̲ that a permutation in 𝔖ₙ can have is 2ⁿ⁻³, for n ≥ 3. Furthermore, the number of permutations π in 𝔖ₙ satisfying |SC₁₂̲₃̲⁻¹(π)| = 2ⁿ⁻³ is 2n − 2.\n\nThe formal claim retains both clauses under the same universal quantifier. The flag `false` denotes 1-underline(23), with one-based permutation entries and the right-greedy convention fixed by the paper's figures.","status":"refuted","summary":"The second-largest number of preimages under SC₁₂̲₃̲ that a permutation in 𝔖ₙ can have is 2ⁿ⁻³, for n ≥ 3. Furthermore, the number of permutations π in 𝔖ₙ satisfying |SC₁₂̲₃̲⁻¹(π)| = 2ⁿ⁻³ is 2n − 2.","title":"Refutation of Zhao's second-largest preimage conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhao-vincular-stack-second-largest-preimages-refutation"},{"aliases":["D5/S3/Combinatorics/VincularStack/VincularStackSort","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3442808f911bd3e64278fa3b23e176dcdc18f01d117e460ce77d9a3fb19ded8b.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/VincularStack/VincularStackSort.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.disc.2025.114834","statement_id":null},"id":"result:zhao-vincular-stack-sorting-schroeder","kind":"result","scope":"Zhao, arXiv:2410.17057v1, Conjecture 3.30 (Section 3.2, printed page 11):\n\n> The sorting class of SC_{23̲1} is enumerated by |Sort_n(SC_{23̲1})| = S_{n−1}.\n\nSC_{23̲1} is the right-greedy stack map whose stack, read from top to bottom, must avoid the vincular pattern\n23̲-1 (the entries playing 2 and 3 adjacent); Sort_n(SC_{23̲1}) is the set of permutations of length n whose\nimage avoids 231, and S_m is the large Schröder number.","status":"proved","summary":"The sorting class of SC_{23̲1} is enumerated by |Sort_n(SC_{23̲1})| = S_{n−1}.","title":"Schröder Enumeration of the Sorting Class of the Stack Avoiding 23-1","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhao-vincular-stack-sorting-schroeder"},{"aliases":["D5/S3/Combinatorics/VincularStack/VincularStackThree","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dcca8567b614a296b15ad985b7cb2ab4edd6ee12292e0f559747ebbf5f6f8b56.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/VincularStack/VincularStackThree.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.disc.2025.114834","statement_id":null},"id":"result:zhao-vincular-stack-three-sorting-classes","kind":"result","scope":"Zhao, arXiv:2410.17057v1, Conjecture 5.1 (Section 5, printed page 20):\n\n> The sorting classes of SC_{312} and SC_{31̲2} are identical. That is, Sort_n(SC_{312}) = Sort_n(SC_{31̲2}) =\n> Sort_n(SC_{31̲2̲}). Furthermore, for τ ∈ Sort_n(SC_{312}), we have SC_{312}(τ) = SC_{31̲2}(τ) = SC_{31̲2̲}(τ).\n\nSC_σ is the right-greedy stack map whose stack, read from top to bottom, must avoid σ; σ is the classical\npattern 312, the vincular pattern in which the entries playing 3 and 1 are adjacent, or the vincular pattern in\nwhich the entries playing 1 and 2 are adjacent. Sort_n(SC_σ) is the set of permutations of length n whose image\navoids 231.","status":"proved","summary":"The sorting classes of SC_{312} and SC_{31̲2} are identical. That is, Sort_n(SC_{312}) = Sort_n(SC_{31̲2}) = Sort_n(SC_{31̲2̲}). Furthermore, for τ ∈ Sort_n(SC_{312}), we have SC_{312}(τ) = SC_{31̲2}(τ) = SC_{31̲2̲}(τ).","title":"Equal Sorting Classes and Outputs of the Stacks Avoiding 312, 31-2 and 3-12","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhao-vincular-stack-three-sorting-classes"},{"aliases":["D5/S3/Quantum/Information/StabilizerLogicalZSupportRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/82ec47b8b48c07fa21bc3da3f1d735ecd7526ee77fe38fa9a4d045faba207594.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/StabilizerLogicalZSupportRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.22331/q-2025-09-15-1858","statement_id":null},"id":"result:zheng-liu-2024-logical-z-support-refutation","kind":"result","scope":"Y. Zheng and D. E. Liu, *From Magic State Distillation to Dynamical Systems*, Quantum 9, 1858 (2025), arXiv:2412.04402, model magic state distillation by a stabilizer code as a map of the Bloch ball.\n\nProposition 3 gives a sufficient condition for z = 0 to be a fixed plane of the map: for every logical operator Z̄_i and every stabilizer element s_j, the product s_j Z̄_i contains a Pauli Z tensor factor. The authors write after Eq. (23): \"This condition is satisfied by most known protocols that we have examined and we conjecture it to be generally true for any stabilizer codes by properly choosing the logical Z operators without modifying the stabilizer groups.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/zhengliu2024magicdistillationdynamics.md).\n\nIssue [#17951](https://github.com/the-omega-institute/trureturing/issues/17951) fixes the reading. For every stabilizer code with at least one logical qubit, some nontrivial logical Pauli word L has a literal Z factor in every product s·L with s in the stabilizer group. This conjecture is refuted.","status":"refuted","summary":"Y. Zheng and D. E. Liu, *From Magic State Distillation to Dynamical Systems*, Quantum 9, 1858 (2025), arXiv:2412.04402, model magic state distillation by a stabilizer code as a map of the Bloch ball.","title":"Logical Z operators with Z support on every stabilizer coset need not exist","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zheng-liu-2024-logical-z-support-refutation"},{"aliases":["D5/S3/Estimation/TransmissivityBetaPriorProbeRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/81c2fb900791cb12c8a6b0eab6661bb604f62a5e6f9c30abd82d6feefe9db9c8.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Estimation/TransmissivityBetaPriorProbeRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevResearch.5.043033","statement_id":null},"id":"result:zhou-bash-guha-gagatsos-2023-transmissivity-beta-prior-refutation","kind":"result","scope":"B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean\nsquare error for transmissivity sensing”, arXiv:2304.05539v1, Phys. Rev.\nResearch 5, 043033 (2023), Section VI (p. 6):\n\n> The numerical results indicate that the optimal input states with non-integer\n> mean photon number, have the form of Eq. (27), which means that for\n> $\\bar{n} \\in \\mathbb{N}$ the optimal states are Fock states.\n\nand Section VII (p. 7):\n\n> We note that for the beta prior PDF, we were not able to prove analytically\n> that the Fock states or the in-between states of Eq. (27) states are optimal,\n> even though our numerical computations support such conjecture.\n\nThe beta prior is eq. (16), $P(\\tau)=\\tau^{\\alpha-1}(1-\\tau)^{\\beta-1}/\\mathcal B(\\alpha,\\beta)$\nwith $\\alpha,\\beta>0$. The verbatim definitions are in\n[the literature note](../Library/QuantumStates/zhou2023transmissivity.md).\nIssue [#14698](https://github.com/the-omega-institute/trureturing/issues/14698)\npreregisters the reading with the parameters free, the witness and the exact\ncertificates.","status":"refuted","summary":"The numerical results indicate that the optimal input states with non-integer mean photon number, have the form of Eq. (27), which means that for $\\bar{n} \\in \\mathbb{N}$ the optimal states are Fock states.","title":"In-between-state optimality for transmissivity sensing with a beta prior","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhou-bash-guha-gagatsos-2023-transmissivity-beta-prior-refutation"},{"aliases":["D5/S3/Estimation/TransmissivityTwoPointProbeRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/6bdb215f6ed5a2fa3bad2418cacb27a9dbe097caeebf4119868f05d060c591d2.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Estimation/TransmissivityTwoPointProbeRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevResearch.5.043033","statement_id":null},"id":"result:zhou-bash-guha-gagatsos-2023-transmissivity-in-between-probe-refutation","kind":"result","scope":"B. Zhou, B. A. Bash, S. Guha and C. N. Gagatsos, “Bayesian minimum mean\nsquare error for transmissivity sensing”, arXiv:2304.05539v1, Section V,\np. 4, state:\n\n> For the two-point prior PDF (15), we verify that for integer n̄ the optimal\n> state is the Fock state with the same photon number. For real n̄, our numerical\n> results support the conjecture that the optimal state has the form of the\n> state (27), up to a phase, i.e., |Φ′ₙ̄⟩ = e^{iφn̂}|Φₙ̄⟩.\n\nIssue [#11653](https://github.com/the-omega-institute/trureturing/issues/11653)\npreregisters this Tier 1 conjecture, the quantified reading and the witness.\nThe two-point prior is\n$P(\\tau)=q\\delta(\\tau-\\tau_0)+(1-q)\\delta(\\tau-\\tau_1)$,\nwith $q,\\tau_0,\\tau_1\\in[0,1]$.","status":"refuted","summary":"For the two-point prior PDF (15), we verify that for integer n̄ the optimal state is the Fock state with the same photon number. For real n̄, our numerical results support the conjecture that the optimal state has the form of the state (27), up to a phase, i.e., |Φ′ₙ̄⟩ = e^{iφn̂}|Φₙ̄⟩.","title":"In-between-state optimality for Bayesian transmissivity sensing","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhou-bash-guha-gagatsos-2023-transmissivity-in-between-probe-refutation"},{"aliases":["D5/S3/Quantum/Measurement/ErrorDisturbanceEndpointRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c01d9c9d6d20f4e6b11905daa36a49c31cda1dba0faaa3678bf2b0fc69a96852.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Measurement/ErrorDisturbanceEndpointRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1016/j.physleta.2016.03.046","statement_id":null},"id":"result:zhou-wu-chau-2016-error-disturbance-endpoint-refutation","kind":"result","scope":"S. S. Zhou, S. Wu and H. F. Chau, *State-Independent Error-Disturbance Tradeoff For Measurement Operators*, arXiv:1509.03812v1 (2015), published as *State-independent error-disturbance trade-off for measurement operators*, Physics Letters A 380(22–23), 1918–1924 (2016), DOI 10.1016/j.physleta.2016.03.046, state Conjecture 3 in Section IV. The v1 source `new_912arxiv.tex`, lines 625–638, reads:\n\n> For all $d \\geq 2$, the following trade-off between error and disturbance holds: $\\varepsilon(\\mathcal{A},\\mathcal{A}') + \\eta(\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$ and $\\Delta(\\mathcal{A},\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$, where $f(\\mathcal{A},\\mathcal{B}) = \\min \\{ \\varepsilon(\\mathcal{A},\\mathcal{B}),\\eta(\\mathcal{A},\\mathcal{B}) \\}$. Moreover, the equalities are both attained either when for all $i$, $\\ket{a'_i} = \\ket{a_i}$, or when for all $i$, $\\ket{a'_i} = \\ket{b_i}$.\n\nThe source uses rank-one projective measurements associated with orthonormal bases of $\\mathbb C^d$. Section II defines the outcome-distance by $D(x,y)=\\max_i|x_i-y_i|$ and compares the $\\mathcal B$ statistics before and after the nonselective $\\mathcal A'$ measurement. Its overall quantity is explicitly\n\n> $\\Delta(\\mathcal{A}, \\mathcal{A}', \\mathcal{B}) = \\max_\\rho \\left( \\varepsilon_{\\rho}( \\mathcal{A}, \\mathcal{A}') + \\eta_{\\rho}( \\mathcal{A}', \\mathcal{B})\\right).$\n\nThis is line 221 of the source. Both terms use one common density matrix. Immediately before the conjecture, lines 616–620 relax the endpoint error by\n\n> $\\varepsilon(\\mathcal{A},\\mathcal{B}) = \\min_{\\tilde{\\mathcal B}} \\varepsilon(\\mathcal{A},\\tilde{\\mathcal{B}})$,\n\nwith the explanation\n\n> This relaxed definition of $\\varepsilon$ means that we only distinguish between the classical probability distributions resulting from measurements $\\mathcal{A}$ and $\\mathcal{B}$ with no regard to the labeling of different measurement outcomes.\n\nThe definitions and source locator are collected in [the literature note](../Library/Estimation/zhouwuchau2016errordisturbance.md).\n\nIssue [#23208](https://github.com/the-omega-institute/trureturing/issues/23208) fixes the statement: for every natural $d\\ge2$ and all three unitary measurement bases $A,A',B$, the minimum of the endpoint error, minimized over outcome permutations, and the disturbance caused by $A$ is at most $\\Delta(A,A',B)$. This second inequality is refuted. The first inequality, concerning the sum of two separately maximized quantities, is not settled here.","status":"refuted","summary":"For all $d \\geq 2$, the following trade-off between error and disturbance holds: $\\varepsilon(\\mathcal{A},\\mathcal{A}') + \\eta(\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$ and $\\Delta(\\mathcal{A},\\mathcal{A}',\\mathcal{B}) \\geq f(\\mathcal{A},\\mathcal{B})$, where $f(\\mathcal{A},\\mathcal{B}) = \\min \\{ \\vare…","title":"The Zhou–Wu–Chau endpoint lower bound on overall error is false","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhou-wu-chau-2016-error-disturbance-endpoint-refutation"},{"aliases":["D5/S3/Combinatorics/ArrowThirtyTwoOneThree","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1c0b369f92aa16fbfae9118f2d7de57e76f049d893cddff9044ee212a6496ecd.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/ArrowThirtyTwoOneThree.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2609.29392","statement_id":null},"id":"result:zhou-yu-arrow-32-13-enumeration","kind":"result","scope":"Robin D.P. Zhou and Xinyang Yu, *Arrow-Wilf equivalences and enumerative results for short arrow\npatterns*, arXiv:2609.29392v1, abstract:\n\n> Together with earlier work of Archer and Laudone, this leaves only (32;1→3) unresolved for |ν|≤2, which we pose as an open problem.\n\nHere `S_n(32; 1 → 3)` consists of permutations of `1, …, n` avoiding the arrow pattern in the\npaper's sense: the selected values have the required order in the one-line word, and the prescribed\narrow holds in `π̂ = θ⁻¹(π)`.","status":"proved","summary":"Together with earlier work of Archer and Laudone, this leaves only (32;1→3) unresolved for |ν|≤2, which we pose as an open problem.","title":"Enumeration of the Arrow Pattern (32; 1 → 3)","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhou-yu-arrow-32-13-enumeration"},{"aliases":["D5/S3/Combinatorics/ArrowWilfEquivalence","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dfc58ef27e22076ea5ea74c0db90be437ec5419cd16f087a05912dcd90db3153.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Combinatorics/ArrowWilfEquivalence.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2609.29392","statement_id":null},"id":"result:zhou-yu-arrow-wilf-equivalence","kind":"result","scope":"Robin D.P. Zhou and Xinyang Yu, *Arrow-Wilf equivalences and enumerative results for short arrow\npatterns*, arXiv:2609.29392v1, Section 7:\n\n> We have computed the avoidance counts for the arrow patterns (12; 3 → 3) and (23; 1 → 1) and found that\n> the resulting sequences agree for all values of n tested. This suggests that these two patterns may be\n> arrow-Wilf-equivalent. However, the explicit formulas we obtained for the two sequences are quite\n> different in form, and we have not been able to establish the equivalence by a direct bijection or by\n> any other method. We leave this as an open problem.\n\nArrow-Wilf equivalence means `|S_n(12; 3 → 3)| = |S_n(23; 1 → 1)|` for every `n ≥ 1`, with containment of\nan arrow pattern as defined in Section 1 of the paper through `π̂ = θ⁻¹(π)`.","status":"proved","summary":"We have computed the avoidance counts for the arrow patterns (12; 3 → 3) and (23; 1 → 1) and found that the resulting sequences agree for all values of n tested. This suggests that these two patterns may be arrow-Wilf-equivalent. However, the explicit formulas we obtained for the two sequences are quite different in f…","title":"The Arrow Patterns (12; 3 → 3) and (23; 1 → 1) Are Arrow-Wilf-Equivalent","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhou-yu-arrow-wilf-equivalence"},{"aliases":["D5/S3/Analytic/CothBernoulliDegree/Result","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/18cf6154d4e12c76a4531bfbd46dde7600e06eb20965a50f500d041deca0a352.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Analytic/CothBernoulliDegree/Result.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.3934/math.2020224","statement_id":null},"id":"result:zhu-2020-coth-bernoulli-remainder-derivative-signs","kind":"result","scope":"Zhu, *Completely monotonic integer degrees for a class of special functions*,\nConjecture 6.4:\n\n> Let m ≥ 1, n ≥ 2, and pₙ(t) be defined as (2.2). Then ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−2)} > 0 holds for all t ∈ (0, ∞) and ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−1)} > 0 is not true for all t ∈ (0, ∞).\n\nEquation (2.2) is\n\n$$p_n(t)=\\frac{\\coth t}{t}\n-\\sum_{k=0}^{n}\\frac{2^{2k}B_{2k}}{(2k)!}t^{2k-2}.$$\n\nPut $P(t)=(-1)^nt^mp_n(t)$ and $M=m+2n$. The first clause is (6.12),\nand the failure of universal strict positivity in the second clause is (6.13).\n\n### Formal reading\n\nAll Lean names below have prefix `D5.S3.Analytic.CothBernoulliDegree`.\n\n| Source object | Lean name | Correspondence |\n| --- | --- | --- |\n| $p_n(t)$ in (2.2) | `Defs.p` | The finite Bernoulli sum includes $k=0$ and $k=n$; its powers of $t$ are integer powers. |\n| $P(t)=(-1)^nt^mp_n(t)$ | `Defs.P` | The sign and the natural power $t^m$ are retained. |\n| Integers $m\\ge1$, $n\\ge2$ | Natural `m`, `n` with `1 ≤ m`, `2 ≤ n` | The domains coincide on the stated positive integer ranges. |\n| $P^{(M-2)}(t)>0$ for all $t>0$ | First conjunct of `Defs.claim` | The derivative is `iteratedDeriv (m + 2 * n - 2) (P m n)`. |\n| $P^{(M-1)}(t)>0$ is not true for all $t>0$ | Second conjunct of `Defs.claim` | The universal strict positivity assertion is negated, with the same real argument domain. |\n| Both clauses | `Result.result` | Exactly `Defs.claim`, with no additional hypothesis. |\n\nThe value assigned by real division and integer powers at zero is immaterial:\n`PartialFraction.hasSum_iteratedDeriv_P` uses neighborhood agreement at each\npositive argument to transfer all iterated derivatives.","status":"proved","summary":"Let m ≥ 1, n ≥ 2, and pₙ(t) be defined as (2.2). Then ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−2)} > 0 holds for all t ∈ (0, ∞) and ((−1)ⁿ tᵐ pₙ(t))^{(m+2n−1)} > 0 is not true for all t ∈ (0, ∞).","title":"Derivative signs of the power-weighted hyperbolic cotangent remainder","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhu-2020-coth-bernoulli-remainder-derivative-signs"},{"aliases":["D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fde656aa50d580e7e955f4cf11dc25f678a6808af37e824321c6f3c8aadf6b0f.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Magic/CliffordThirdMomentAggregateRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2410.13575","statement_id":null},"id":"result:zhu-2024-clifford-third-moment-aggregate-lower-bound","kind":"result","scope":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and\n$|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on\n$\\Sigma(d)$ together with the aggregate bounds $6\\le\\kappa(\\Psi,\\Sigma(d))\\le2d+2$,\n$0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})\\le2d-4$ and $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$. Here\n$\\kappa(\\Psi,\\mathscr T)=\\sum_{\\mathcal T\\in\\mathscr T}\\kappa(\\Psi,\\mathcal T)$ and\n$\\mathscr T_{\\rm iso}=\\{\\mathcal T_O : O\\in O_3(d)\\}$. The verbatim statements are in\n[the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\n\nThe pointwise clause was refuted earlier with one qudit at $d=11$\n([dossier](zhu-2024-clifford-third-moment-kappa-negativity.md)); for that state the aggregate bounds\nhold. Issue [#14561](https://github.com/the-omega-institute/trureturing/issues/14561) targets the\nremaining aggregate lower bound in its printed form $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$.","status":"refuted","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic orbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and $|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on $\\Sigma(d)$ together with the agg…","title":"The isotropic aggregate of a two-qudit state in dimension five is below six","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhu-2024-clifford-third-moment-aggregate-lower-bound"},{"aliases":["D5/S3/Quantum/Magic/CliffordThirdMomentNegativity","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bbe2fb0471e6f2562adb97069ac0a86b558c66c3a87b5c548ccba18dc788ed48.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Magic/CliffordThirdMomentNegativity.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2410.13575","statement_id":null},"id":"result:zhu-2024-clifford-third-moment-kappa-negativity","kind":"result","scope":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1 (Commun. Math. Phys. 407, 242 (2026)), Conjecture 2: for an odd\nprime $d$ and $|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$, $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ for every\nstochastic Lagrangian subspace $\\mathcal T\\in\\Sigma(d)$, together with three aggregate\ninequalities that the authors derive from it. Here\n$\\kappa(\\Psi,\\mathcal T)=\\operatorname{tr}[R(\\mathcal T)(|\\Psi\\rangle\\langle\\Psi|)^{\\otimes3}]$,\n$R(\\mathcal T)=r(\\mathcal T)^{\\otimes n}$ and $r(\\mathcal T)=\\sum_{(x;y)\\in\\mathcal T}|x\\rangle\\langle y|$.\nThe verbatim statements are in [the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\n\nIssue [#14367](https://github.com/the-omega-institute/trureturing/issues/14367) targets the\npointwise clause for every odd prime $d$, every $n$, every normalized $\\Psi$ and every\n$\\mathcal T\\in\\Sigma(d)$.","status":"refuted","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic orbits*, arXiv:2410.13575v1 (Commun. Math. Phys. 407, 242 (2026)), Conjecture 2: for an odd prime $d$ and $|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$, $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ for every stochastic Lagrangian subspace…","title":"A negative Clifford third-moment expectation at dimension eleven","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhu-2024-clifford-third-moment-kappa-negativity"},{"aliases":["D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5cd928ee8dfc877d6889062c7ee7826e26a687c9c164837f33af12fa5416f2d7.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Magic/CliffordThirdMomentSigmaRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.48550/arXiv.2410.13575","statement_id":null},"id":"result:zhu-2024-clifford-third-moment-sigma-lower-bounds","kind":"result","scope":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic\norbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and\n$|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on\n$\\Sigma(d)$ together with the aggregate bounds $6\\le\\kappa(\\Psi,\\Sigma(d))\\le2d+2$,\n$0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})\\le2d-4$ and $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$. Here\n$\\Sigma(d)$ is the set of stochastic Lagrangian subspaces of $\\mathbb F_d^6$,\n$\\mathscr T_{\\rm sym}=\\{\\mathcal T_O\\mid O\\in S_3\\}$, $\\mathscr T_{\\rm ns}=\\Sigma(d)\\setminus\\mathscr T_{\\rm sym}$\nand $\\kappa(\\Psi,\\mathscr T)=\\sum_{\\mathcal T\\in\\mathscr T}\\kappa(\\Psi,\\mathcal T)$. The verbatim statements are in\n[the literature note](../Library/QuantumStates/zhu2024thirdmoments.md).\nThe pointwise clause and the clause $\\kappa(\\Psi,\\mathscr T_{\\rm iso})\\ge6$ are refuted\n([pointwise](zhu-2024-clifford-third-moment-kappa-negativity.md),\n[isotropic aggregate](zhu-2024-clifford-third-moment-aggregate-lower-bound.md)). Issue\n[#14620](https://github.com/the-omega-institute/trureturing/issues/14620) targets the two remaining lower\nbounds, $6\\le\\kappa(\\Psi,\\Sigma(d))$ and $0\\le\\kappa(\\Psi,\\mathscr T_{\\rm ns})$.","status":"refuted","summary":"H. Zhu, C. Mao and C. Yi, *Third moments of qudit Clifford orbits and 3-designs based on magic orbits*, arXiv:2410.13575v1, Section VII.1, Conjecture 2, asserts for an odd prime $d$ and $|\\Psi\\rangle\\in\\mathcal H_d^{\\otimes n}$ the pointwise bounds $0\\le\\kappa(\\Psi,\\mathcal T)\\le1$ on $\\Sigma(d)$ together with the agg…","title":"At dimension five the full and non-symmetric aggregates of a two-qudit state fall below their conjectured lower bounds","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhu-2024-clifford-third-moment-sigma-lower-bounds"},{"aliases":["D5/S3/Quantum/Entanglement/GeometricMeasure/SmolinNoisyHilbertSchmidtDistance","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/deb9f4c68415e29afe890bcf5a27563a96117ca329f2acb7fa97a59bb4c36a19.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/GeometricMeasure/SmolinNoisyHilbertSchmidtDistance.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/82gn-78lv","statement_id":null},"id":"result:zhu-huang-zhang-nie-zhao-lu-2026-smolin-hilbert-schmidt-distance","kind":"result","scope":"X. Zhu, K. Huang, C. Zhang, X. Nie, Q. Zhao and D. Lu, *Universal entanglement detection with geometrically optimal witnesses*, arXiv:2610.17592v1 [quant-ph]; Phys. Rev. A 114, 042409 (2026), Appendix, “Four-qubit Smolin state”. The source TeX is `manuscript.tex` (1053 lines, MD5 `db9c376c24127cc118b3e270cf74af42`). Lines 672–687 state:\n\n> A natural conjecture is that\n>\n> $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\|\\mathcal{N}_p(\\rho_S) - \\mathcal{N}_{\\frac{1}{3}}(\\rho_S)\\|, & p \\in (\\frac{1}{3}, 1], \\end{cases}$\n>\n> which simplifies to\n>\n> $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\frac{\\sqrt{3}}{4} \\left( p - \\frac{1}{3} \\right), & p \\in (\\frac{1}{3}, 1]. \\end{cases}$\n\nHere the norm is Hilbert–Schmidt, $\\|X\\|_{\\mathrm{HS}}=\\sqrt{\\operatorname{Re}\\operatorname{tr}(X^\\dagger X)}$. The appendix defines $\\rho_S=(I+\\sigma_x^{\\otimes4}+\\sigma_y^{\\otimes4}+\\sigma_z^{\\otimes4})/16$ at lines 659–661, $D(\\rho)=\\min_{\\sigma\\in\\mathrm{SEP}(\\mathcal P)}\\|\\rho-\\sigma\\|$ at lines 662–665, and $\\mathcal N_p(\\rho_S)=p\\rho_S+(1-p)I/16$ at lines 669–671. Its four representative partitions are $\\{1\\}:\\{2,3,4\\}$, $\\{1\\}:\\{2\\}:\\{3,4\\}$, $\\{1\\}:\\{2\\}:\\{3\\}:\\{4\\}$ and $\\{1,2\\}:\\{3,4\\}$ (line 666).\n\nIssue [#20844](https://github.com/the-omega-institute/trureturing/issues/20844) fixes the reading: for every $p\\in[0,1]$, the distance is the displayed function $f(p)$ for the first three partitions, and zero for the two-pair partition, which the source identifies as separable at line 668. Every minimum is attained. This is proved. The source and definitions are also recorded in [the literature note](../Library/QuantumContext/zhuhuangzhangniezhaolu2026universal.md).","status":"proved","summary":"A natural conjecture is that $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\|\\mathcal{N}_p(\\rho_S) - \\mathcal{N}_{\\frac{1}{3}}(\\rho_S)\\|, & p \\in (\\frac{1}{3}, 1], \\end{cases}$ which simplifies to $D(\\mathcal{N}_p(\\rho_S)) = \\begin{cases} 0, & p \\in [0, \\frac{1}{3}], \\\\ \\frac{\\sqrt{3}}{4} \\…","title":"Exact Hilbert–Schmidt distances for noisy four-qubit Smolin states","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zhu-huang-zhang-nie-zhao-lu-2026-smolin-hilbert-schmidt-distance"},{"aliases":["D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/16faaeaee4086919e61c211a1e6bc0be5aef6e0de432585c1cd6976058f85e0e.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/NormCompression/StrictSpectralPolyaRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.4064/bc112-0-16","statement_id":null},"id":"result:zietak-2017-strict-spectral-polya-conjecture","kind":"result","scope":"K. Ziętak, “From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix,” Banach Center Publications 112 (2017), Conjecture 1.1, asks whether Schatten-p approximant residuals converge to the residual of a strict spectral approximant for every complex matrix subspace.\n\nThe source statement is:\n\n> Let M be a linear subspace of Cᵐˣⁿ and let Bₚ ∈ M be the cₚ-approximant to A ∈ Cᵐˣⁿ. As p tends to ∞, the matrix Rₚ = A − Bₚ converges to A − B⁽ˢᵗ⁾, where B⁽ˢᵗ⁾ is the strict spectral approximant to A.\n\n### Statement echo\n\nThe Lean `claim` quantifies over all natural dimensions, complex matrix subspaces, targets, strict spectral approximants, and real-parameter functions of approximants. Its hypotheses require strict spectral minimality and a Schatten-$p$ minimizing approximant for every $p>1$; its conclusion is convergence of residual matrices. The module's `result` is the literal negation `¬ claim`.","status":"refuted","summary":"K. Ziętak, “From the strict Chebyshev approximant of a vector to the strict spectral approximant of a matrix,” Banach Center Publications 112 (2017), Conjecture 1.1, asks whether Schatten-p approximant residuals converge to the residual of a strict spectral approximant for every complex matrix subspace.","title":"Ziętak's strict spectral approximant convergence conjecture","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zietak-2017-strict-spectral-polya-conjecture"},{"aliases":["D5/S3/Quantum/QuantumChannels/ChannelAssemblageLOSRRefutation","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/ebaea8bbcc385d5f4dcd44b5545b081b32021f0c766d80109218810d152dbeef.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/QuantumChannels/ChannelAssemblageLOSRRefutation.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.22331/q-2023-10-10-1134","statement_id":null},"id":"result:zjawin-schmid-hoban-sainz-2023-channel-assemblage-losr-refutation","kind":"result","scope":"B. Zjawin, D. Schmid, M. J. Hoban and A. B. Sainz, *The resource theory of nonclassicality of channel assemblages*, Quantum 7, 1134 (2023), arXiv:2209.10177v3, Conjecture 3:\n\n> Let the two resources $\\In^{\\,\\theta_1}_{\\A|\\X}$ and $\\In^{\\,\\theta_2}_{\\A|\\X}$ belong to the infinite family of assemblages $\\FA$. Then, the conversion $\\In^{\\,\\theta_1}_{\\A|\\X} \\overset{\\text{LOSR}}{\\longrightarrow} \\In^{\\,\\theta_2}_{\\A|\\X}$ is possible if and only if $\\theta_1 \\geq \\theta_2$.\n\nThis is the verbatim text of `channel-quantum.tex`, line 354, in the [v3 source](https://arxiv.org/src/2209.10177v3), md5 `1478712c3cb6058c040039d9ae9978e6`. The source macros `\\FA`, `\\In`, `\\A` and `\\X` denote $\\mathsf R$, $\\mathbf I$, $\\mathbb A$ and $\\mathbb X$. The definitions and locators are also in [the literature note](../Library/QuantumContext/zjawinschmidhobansainz2023channel.md).\n\nThe family has binary outcomes and settings, $a,x\\in\\{0,1\\}$, and angles $\\theta\\in(0,\\pi/2]$. Alice and Bob share $|\\phi\\rangle=(|00\\rangle+|11\\rangle)/\\sqrt2$. Bob applies the controlled gate $|0\\rangle\\langle0|_B\\otimes I+|1\\rangle\\langle1|_B\\otimes U_\\theta$, where $U_\\theta$ has rows $(\\cos(\\theta/2),-\\sin(\\theta/2))$ and $(\\sin(\\theta/2),\\cos(\\theta/2))$. The construction traces out $B$, applies Alice's effect $M_{a|0}=(I+(-1)^a\\sigma_z)/2$ or $M_{a|1}=(I+(-1)^a\\sigma_x)/2$, and traces out $A$ (Eqs. `eq:CR` and `eq:CRfam`, TeX lines 316–333).\n\nIssue [#21490](https://github.com/the-omega-institute/trureturing/issues/21490) fixes the reading: for every pair of angles in $(0,\\pi/2]$, LOSR conversion between these literal assemblages is possible exactly when the source angle is at least the target angle. The conjecture is refuted.","status":"refuted","summary":"Let the two resources $\\In^{\\,\\theta_1}_{\\A|\\X}$ and $\\In^{\\,\\theta_2}_{\\A|\\X}$ belong to the infinite family of assemblages $\\FA$. Then, the conversion $\\In^{\\,\\theta_1}_{\\A|\\X} \\overset{\\text{LOSR}}{\\longrightarrow} \\In^{\\,\\theta_2}_{\\A|\\X}$ is possible if and only if $\\theta_1 \\geq \\theta_2$.","title":"The controlled-rotation channel assemblages do not obey the proposed LOSR angle ordering","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zjawin-schmid-hoban-sainz-2023-channel-assemblage-losr-refutation"},{"aliases":["D5/S3/Quantum/Information/QutritOverlapBoundNonOptimality","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fb7ba4fdea025c1be3742c603ca909db17bf4b753e811b9495eca3e1f3c64e7c.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Information/QutritOverlapBoundNonOptimality.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1088/1751-8113/47/49/495302","statement_id":null},"id":"result:zozor-bosyk-portesi-2014-qutrit-overlap-bound-non-optimality","kind":"result","scope":"S. Zozor, G. M. Bosyk and M. Portesi, *General entropy-like uncertainty relations in finite dimensions*, arXiv:1311.5602v3, J. Phys. A 47 (2014) 495302, Section 3, after Corollary 2:\n\n> We suspect that this corollary is also valid when $N=3$, but we have not been\n> able to prove it yet.\n\nCorollary 2 states that for c > 1/√2 and N = 2 or N ≥ 4 the bound (22), B(c) = min over θ ∈ [0, arccos c] of D(θ) + D(arccos c − θ), is c-optimal. That is, it equals the minimum of the entropy sum over all states and all unitary T ∈ U(N) with c(T) = max |T_ij| = c. The entropies, the overlap and the function D are quoted in [the literature note](../Library/QuantumContext/zozor2014entropylikeuncertainty.md).\n\nIssue [#19610](https://github.com/the-omega-institute/trureturing/issues/19610) fixes the reading for the Tsallis pair of order 2, S₂(p) = 1 − Σ p_i², an admissible pair of the corollary. The suspicion then says that, for every c with 1/√2 < c ≤ 1, some T ∈ U(3) with c(T) = c and some state attain B(c). The suspicion is refuted.","status":"refuted","summary":"We suspect that this corollary is also valid when $N=3$, but we have not been able to prove it yet.","title":"The overlap entropy bound is not c-optimal for qutrits","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zozor-bosyk-portesi-2014-qutrit-overlap-bound-non-optimality"},{"aliases":["D5/S3/Quantum/Entanglement/NonselectiveNegativityGainOptimum","result","Open problem"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dcbd21ad5acb19c9c4814d583593f60ba058fcba86ee05a94f378bb3e40fcf55.json","evidence":{"assessment":"kernel-verified","declaration":"D5/S3/Quantum/Entanglement/NonselectiveNegativityGainOptimum.result","release_membership":"kernel-verified","source_commit":"b52262a023c4db9bb30a128ebfd80ea4ef4c1a53","source_url":"https://doi.org/10.1103/PhysRevA.99.042319","statement_id":null},"id":"result:zuppardo-ganardi-miller-bandyopadhyay-paterek-2018-nonselective-negativity-optimum","kind":"result","scope":"M. Zuppardo, R. Ganardi, M. Miller, S. Bandyopadhyay and T. Paterek, *Entanglement gain in measurements with unknown results* (Phys. Rev. A 99, 042319 (2019); arXiv:1806.05465), study how a non-selective measurement in an orthonormal basis changes entanglement. It maps a pure state |φ⟩ to Σ_j |⟨ψ_j|φ⟩|² |ψ_j⟩⟨ψ_j|.\n- **The derived optimum.** For two qubits and the measurement bases (28), they derive the largest final negativity: N^max_f − N_i = 1/4 − N_i/2 (Eq. (33)).\n- **The open point.** \"Although we were not able to provide a proof that this is indeed the optimal choice among all possible measurement bases, we have conducted numerical check which supports optimality.\"\n\nThe verbatim statements are in [the literature note](../Library/QuantumChannels/zuppardo2019entanglementgain.md).\n\nIssue [#18399](https://github.com/the-omega-institute/trureturing/issues/18399) fixes the reading: for every unit two-qubit ψ and every orthonormal basis, the measured state has negativity at most N(ψψ†)/2 + 1/4, and some basis attains it. This is proved.","status":"proved","summary":"M. Zuppardo, R. Ganardi, M. Miller, S. Bandyopadhyay and T. Paterek, *Entanglement gain in measurements with unknown results* (Phys. Rev. A 99, 042319 (2019); arXiv:1806.05465), study how a non-selective measurement in an orthonormal basis changes entanglement. It maps a pure state |φ⟩ to Σ_j |⟨ψ_j|φ⟩|² |ψ_j⟩⟨ψ_j|. -…","title":"The two-qubit non-selective measurement optimum N/2 + 1/4 is global","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#resolved-zuppardo-ganardi-miller-bandyopadhyay-paterek-2018-nonselective-negativity-optimum"},{"aliases":["Fourier and optimization"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/61adc3a9db3534abf051f543cff5353f6256c0bbfeae91278f38da1fb10301da.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:circulant-fourier-reduction","kind":"target","scope":"The SDP uses trace one while a scaled first-row LP can use x_0=1. Confusing these conventions changes the objective by n.","status":"proposed-target","summary":"Can cyclic averaging identify the theta SDP with a finite Fourier positivity problem without changing its optimum?","title":"A normalization-safe circulant theta reduction","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=circulant-fourier-reduction"},{"aliases":["Fourier and optimization"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d8ef93974fe91d596cba5c765faf6f028fe0aff4ea45f3ef2bff849279f5c81d.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:circulant-primal-dual-certificate","kind":"target","scope":"Solver tolerance and approximate Fourier eigenvalues can falsely certify feasibility. Real algebraic or interval enclosures need a soundness theorem.","status":"proposed-target","summary":"Can explicit primal and dual witnesses give rigorous theta bounds for a supplied circulant graph?","title":"Certified bounds for a concrete circulant graph","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=circulant-primal-dual-certificate"},{"aliases":["Automata"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d5e4f505494c123660b270717ad514779ac50774d6f1b19379bb594f0d925136.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2405.02727"},"id":"target:dfao-finite-unsat","kind":"target","scope":"A solver verdict needs a proof-checking and encoding-soundness bridge. A SAT candidate still requires all-powers verification.","status":"proposed-target","summary":"Can a finite set of exact power-input digit constraints already exclude all DFAOs with at most 21 states under the chosen conventions?","title":"A finite, replayable 21-state UNSAT obstruction","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=dfao-finite-unsat"},{"aliases":["Automata"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/66308e9ab9e126a1ba7a5c25c91f9b72977158a1ecebb5da4418c83a5ba2e9d6.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2405.02727"},"id":"target:dfao-sparse-distinguishability","kind":"target","scope":"Each separation witness must constrain both words inside the sparse domain. Full-language Myhill-Nerode witnesses may be irrelevant here.","status":"proposed-target","summary":"Can 22 prefixes be found such that every pair has a common suffix making both completed words legal power inputs with different required outputs?","title":"A sparse-language distinguishability certificate","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=dfao-sparse-distinguishability"},{"aliases":["Probability"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fed31dff7c42c5ef8f6b5698b04b289e074a49f69a1ecc8bbdf6dca52b90c9fd.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2210.11038"},"id":"target:game-path-measures","kind":"target","scope":"Move multiplicities and suffix-path counts affect transition weights. Treating the measures as interchangeable changes the target.","status":"proposed-target","summary":"Can exact finite recursions distinguish uniform terminal-path sampling from locally uniform successor sampling on the game DAG?","title":"Two exact game-length generating functions","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=game-path-measures"},{"aliases":["Probability"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/179b6da4830434aaaa143af9f401627f33b2528e9559bbbbb7744ab3c4cb2813.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2210.11038"},"id":"target:game-regenerative-barrier","kind":"target","scope":"Finite carry locality need not survive conditioning on entire random histories. The relevant mixing-CLT summability and variance hypotheses must be checked explicitly.","status":"proposed-target","summary":"Can a canonical digit barrier bound the dependence of separated split indicators uniformly over initial input size?","title":"A uniform regenerative-barrier estimate","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=game-regenerative-barrier"},{"aliases":["Additive combinatorics"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b77089376428af5b7f99bf0cd1a2c49998390b9cab2bf909746a0b4a1ccb897c.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2503.04122"},"id":"target:greedy-g-plus-one-blocks","kind":"target","scope":"The single-interval periodic decomposition from Conjecture 17 cannot be reused without handling both new intervals and all cross-sums.","status":"proposed-target","summary":"Can the candidate set with exceptions 1,2g+1,6g+1 and residue intervals [g,2g] and [6g+2,7g+1] modulo 10g+3 be characterized by restricted triple sums?","title":"Accepted blocks and forbidden sums for g+1","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=greedy-g-plus-one-blocks"},{"aliases":["Additive combinatorics"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/3fd1c79fc45caaf98945a61be4af176a435e6707d7ba51b98a7d355666d7311f.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2503.04122"},"id":"target:greedy-g-plus-one-literal","kind":"target","scope":"A sumfree candidate by itself does not prove that the greedy algorithm generates it.","status":"proposed-target","summary":"Does the new candidate describe every term selected by the literal greedy process from 1,g,g+1?","title":"Connect the candidate to the least-next-entry sequence","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=greedy-g-plus-one-literal"},{"aliases":["Rewriting games"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/94164b82f27ba4699ef009130b6f1bf60956ba70ac2e0aa87a66116d2066587a.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2508.20222"},"id":"target:lgs-bellman-exchange","kind":"target","scope":"The game DAG needs its own termination proof. Local exchange lemmas may fail even when global optimality survives.","status":"proposed-target","summary":"Can each LGS priority choice be shown to attain an argmax successor of the longest-remaining-path Bellman recursion?","title":"Height-preserving local exchanges for LGS","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=lgs-bellman-exchange"},{"aliases":["Rewriting games"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/b0a9051f232fb35f7f7d90b440c24eab3bcd95b9ceb462d5781576eeb5c1b064.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2508.20222"},"id":"target:lgs-switch-order","kind":"target","scope":"The universal LGS statement depends on this interpretation; normal-form uniqueness does not imply equal run lengths.","status":"proposed-target","summary":"Do all allowed switch tie choices lead to the same LGS terminal length from the specified initial states?","title":"Does switch-order freedom preserve LGS length?","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=lgs-switch-order"},{"aliases":["Quantum geometry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/bd4358c79b6ddd55ad092d6d683ff9ceb73b533e399f1927d9025bb1cf0d2670.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2410.23997"},"id":"target:mub-basis-context","kind":"target","scope":"Rank-one, completeness, orthogonality, labelling and phase equivalence all need explicit hypotheses.","status":"proposed-target","summary":"Can orthonormal bases modulo individual phases be represented faithfully as labelled complete rank-one projector contexts, preserving the MUB overlap condition?","title":"An exact basis-to-projector bridge","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=mub-basis-context"},{"aliases":["Quantum geometry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5f27f547569e2db6861e58cdb72543fbd7a9cc1695bdf94343477d0b53551c38.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2410.23997"},"id":"target:mub-joint-compatibility","kind":"target","scope":"A classification claim alone leaves pairwise and joint compatibility to prove. Numerical near-solutions do not satisfy exact equations.","status":"proposed-target","summary":"Can an independently verified order-six Hadamard parameterization support a complete exact exclusion or construction for three Hadamards relative to the identity basis?","title":"A certificate for joint Hadamard compatibility","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=mub-joint-compatibility"},{"aliases":["Quantum geometry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/fff19f31e3cc5c8fcb0ec29412e346cf348c7dc753370516abb76c4922ee3fb7.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2410.23997"},"id":"target:mub-projector-sos-scope","kind":"target","scope":"The existing elementary projector bound permits four, five, six and seven bases at d=6. No stronger exclusion is supplied by that bound.","status":"proposed-target","summary":"After checking the vector-coordinate obstruction, can a faithful projector formulation impose a dimension-six constraint stronger than m <= d+1?","title":"Which projector-coordinate constraint could exclude four bases?","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=mub-projector-sos-scope"},{"aliases":["Golden numeration"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9efa7afe99336e0a53737ea5c445715c185f48f2ee82fafcab18a718359c3d65.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2305.08349"},"id":"target:negative-prefix-return-itinerary","kind":"target","scope":"Existing golden-word return theorems apply only after the cylinder/subshift identification is established.","status":"proposed-target","summary":"Can each negative-prefix cylinder be identified with the paper's F, G or H itinerary, or its allowed three-component interleaving?","title":"From negative cylinders to exact return itineraries","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=negative-prefix-return-itinerary"},{"aliases":["Golden numeration"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/992bb8ce20597d09446c741fd14b653c4d0151114af810275e41bbcbebd43966.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2305.08349"},"id":"target:negative-prefix-transducer","kind":"target","scope":"Contractive intuition is insufficient: carry residues and a uniform reachable-state bound for fixed m need proof.","status":"proposed-target","summary":"For each fixed prefix length m, can a finite-state transducer read canonical Zeckendorf input and output the first m negative-position base-phi digits?","title":"A bounded-state negative-prefix transducer","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=negative-prefix-transducer"},{"aliases":["Phase retrieval"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/62f53a46390d54e041ed3961d46e4b19ae19c254a6872ea914b49379ebb60545.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:omega-deterministic-decay","kind":"target","scope":"Average-case small-ball estimates and the existence of a badly conditioned example do not imply a worst-case upper bound.","status":"proposed-target","summary":"Can row-subset geometry force an exponentially small least singular value uniformly over every normalized full-spark frame?","title":"A deterministic obstruction stronger than a Gaussian example","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=omega-deterministic-decay"},{"aliases":["Phase retrieval"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/5c1b1b0d11a42177d2b1c64090eab47491319088e7c2073eb94a73534e8535d5.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:omega-full-spark-reduction","kind":"target","scope":"The source definition also allows larger row sets; its equivalence needs a rank and Loewner-order argument.","status":"proposed-target","summary":"Under full spark at N=2M-1, is omega(A) exactly the minimum of sigma_min(A_T) over M-row sets T?","title":"An exact square-submatrix formulation of omega","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=omega-full-spark-reduction"},{"aliases":["Phase retrieval"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/dafcd653516f8f35d6c7a7166d96b25f0cac1321407983602bd4c7aec3543244.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:phase-bad-event-probability","kind":"target","scope":"A single real-algebraic solution may lie in a measure-zero stratum. 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Its formalization needs the same canonical digit and symmetry conventions and does not settle the trident conjecture.","status":"proposed-target","summary":"Can the exponent-doubling criterion in Shallit and Vukusic Theorem 5 be transported faithfully to the repository's exact two-sided phi representation?","title":"Use the proved Kimberling criterion as a conversion check","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=phi-kimberling-conversion-check"},{"aliases":["Arithmetic"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/e77f61ef38529aebc3b156d0330706fe163b26d8a03f2a703f418aa23c26a8a4.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.25343"},"id":"target:pisano-first-lift-defect","kind":"target","scope":"The argument requires the reduction-order relation and a justified calculation of the order of I+pB, including small primes.","status":"proposed-target","summary":"With r = pi(p) and A^r = I + pB modulo p², is pi(p²) equal to r exactly when B vanishes modulo p, and otherwise equal to pr?","title":"The first-lift defect criterion modulo p²","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=pisano-first-lift-defect"},{"aliases":["Arithmetic"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/1f042e99c19b3b700c3f685fd92617492ff3b9dae7dd98aaf99dcaafd06bddea.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.25343"},"id":"target:pisano-matrix-order","kind":"target","scope":"Matrix order concerns the whole matrix, so equality with the orbit period needs proof; invertibility and indexing must be explicit.","status":"proposed-target","summary":"For m >= 2, can the pair-recurrence period be proved equal to the multiplicative order of A = [[0,1],[1,1]] over Z/mZ and of its golden-unit representation?","title":"Pisano period as matrix and golden-unit order","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=pisano-matrix-order"},{"aliases":["Real-rooted polynomials"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/deea50382e310400ba1154d6ce6243355b202c7407d150f545ff80342148d3db.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2608.03723"},"id":"target:pochhammer-degree-monotonicity","kind":"target","scope":"Monotonicity was bundled with a false strict bound in the source conjecture. Its validity has to be established separately.","status":"proposed-target","summary":"With c_(2k)(a) defined independently from the admissible set, does c2(a) <= c4(a) hold for every a > 0?","title":"Compare the degree-two and degree-four radii","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=pochhammer-degree-monotonicity"},{"aliases":["Real-rooted polynomials"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a676026a456a94483e44b282fdadf0910586aa83b1464a847fd578574c13fe3e.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2608.03723"},"id":"target:pochhammer-quartic-interval","kind":"target","scope":"The quadratic discriminant argument does not classify all four complex roots of the quartic.","status":"proposed-target","summary":"For each positive a, is the set of t for which every root of L_a((X+t)^4) lies in [-1,0] a single interval?","title":"A certified degree-four admissible interval","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=pochhammer-quartic-interval"},{"aliases":["Sequence complexity"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/a1083927efe16fd783a13184c30c0a17c4a06b0bb8558a914396efd29adca7e6.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2106.09959"},"id":"target:polynomial-carry-window","kind":"target","scope":"Unbounded carry propagation and noninjective history-to-state maps are possible obstructions.","status":"proposed-target","summary":"For square inputs, can a history of length O(N^(1/4)) determine enough boundary carry information to determine the next Zeckendorf digit-sum parity?","title":"Can bounded carry state force the next parity bit?","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=polynomial-carry-window"},{"aliases":["Sequence complexity"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/243e5fbacccf8114be8495352a62605ade1a30430f3fffcdd22d4b5a4f25ede7.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2106.09959"},"id":"target:polynomial-successor-certificate","kind":"target","scope":"Off-by-one conventions and special zero-memory cases must match the source definition.","status":"proposed-target","summary":"Can finite maximum order complexity be certified by equal-length block collisions and their following bits, including degenerate constant-prefix cases?","title":"An exact successor-collision certificate","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=polynomial-successor-certificate"},{"aliases":["Quantum geometry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/70ead040b44007ac47617f058cd3c6fe902e1c3cc8e2ce02d9f6351490f315a8.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:sic-exact-fiducial-certificate","kind":"target","scope":"A small floating residual does not establish equality or existence. An imposed order-three symmetry narrows the search class.","status":"proposed-target","summary":"Can a proposed algebraic fiducial be certified against all required overlap and normalization equations?","title":"An exact certificate behind a numerical SIC candidate","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=sic-exact-fiducial-certificate"},{"aliases":["Quantum geometry"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/9a4035bd01f21be5c3f8892f3eba6a2667d29fe5cdd50463d5550ab42be980cb.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2603.29571"},"id":"target:sic-fiducial-orbit","kind":"target","scope":"Trace orthogonality is available but the orbit normalization, distinctness and identity resolution must be derived.","status":"proposed-target","summary":"Can exact displacement-overlap equations for a unit fiducial imply the d²-vector equiangular tight-frame and POVM identities?","title":"From a verified fiducial to a SIC orbit","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=sic-fiducial-orbit"},{"aliases":["Weil spectra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/34c89bdad6e1ac7bebd0bce71b033e3fd2ce63365d6d72ad9d1c670eba9505c2.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2606.09096"},"id":"target:suzuki-boundary-identification","kind":"target","scope":"A finite atomic measure chosen to look like a Clark measure does not establish this identification.","status":"proposed-target","summary":"Can the paper's deficiency-space boundary characteristic be identified with a resolvent or Fredholm construction using the genuine screw kernel?","title":"Identify the actual finite-window characteristic","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=suzuki-boundary-identification"},{"aliases":["Weil spectra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/02564c915bbb472f5a1afa078444c8c76de5bbe737b0955433372eb1ef5b70b5.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2606.09096"},"id":"target:suzuki-unconditional-shift","kind":"target","scope":"The source's global Hilbert-space heuristic assumes RH. Finiteness of prime terms at each fixed a gives no uniform estimate.","status":"proposed-target","summary":"Can the shift needed to define the window Hilbert spaces be controlled as a grows, without assuming positivity of every Weil window?","title":"Control the spectral shift without assuming RH","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=suzuki-unconditional-shift"},{"aliases":["Sequence complexity"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/8655a1cd5e9dfa5e16b19783a9a4d86e8f2b85a747eec174f1f8fd92c7c4f7a8.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2509.16034"},"id":"target:thue-morse-even-difference","kind":"target","scope":"The released absolute-difference theorem fixes the magnitude only; it leaves the sign and R(4n) outside its stated scope.","status":"proposed-target","summary":"Can the difference R(4n+2)-R(4n) be characterized for every n >= 1, including its sign when nonzero?","title":"Determine the sign and the remaining even recursion","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=thue-morse-even-difference"},{"aliases":["Sequence complexity"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/d735b962e9a43f8f69fdd0cdfe1941448f2a48041e5dfe94a68677916664a4f1.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2509.16034"},"id":"target:thue-morse-nonautomaticity","kind":"target","scope":"The dossier points to a written unboundedness argument, but does not bind a separate kernel-checked nonautomaticity endpoint.","status":"proposed-target","summary":"Can one prove that R(n), for n >= 1, is not k-automatic for any k >= 2?","title":"Formalize the nonautomaticity argument","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=thue-morse-nonautomaticity"},{"aliases":["Weil spectra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/4a866191cf22e81601cdd9ff83d4129370083a06376c4c513cbddc6a568c7e87.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2511.22755"},"id":"target:weil-moving-window-rate","kind":"target","scope":"Increasing numerical precision or an exterior cutoff does not reduce the physical error by itself; gaps may deteriorate with the window.","status":"proposed-target","summary":"Can arithmetic cancellation and complement coercivity control the actual-mode Fourier error along an unbounded sequence of support windows?","title":"A physical-window rate for the true mode error","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=weil-moving-window-rate"},{"aliases":["Weil spectra"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c58b40792cf2e14b4a2babf48d8912cdfb7629b516ce80c0c3e1ef74dc59c5a2.json","evidence":{"source_url":"https://doi.org/10.48550/arXiv.2511.22755"},"id":"target:weil-same-family-residual","kind":"target","scope":"A retained matrix residual leaves an exterior term. Domain error and candidate normalization must be paid separately.","status":"proposed-target","summary":"Can a full-space residual and complement-coercivity certificate bound the Fourier error for the same physical Weil operator and prolate trial?","title":"Compare one genuine mode and its actual candidate","url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=weil-same-family-residual"},{"aliases":["OEIS","sequences","数列","整数序列"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/c7347d543774d481f90fe79ac3b387461c29ecf8a2ef00540ce666949f9e5ee9.json","id":"topic:integer-sequences","kind":"topic","status":"curated-topic","summary":"Formal results about integer sequences, their structure and remaining questions.","title":"Integer sequences","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?q=Integer%20sequences"},{"aliases":["algebraic logic","equational logic","equational implication","逻辑代数","逻辑与代数","等式推理"],"api_url":"https://the-omega-institute.github.io/trureturing-pages/api/v1/records/995c2536a307137eacc1cb6249fa570b4cf951a2c664466da29655635525e53f.json","id":"topic:logic-algebra","kind":"topic","status":"curated-topic","summary":"Equational implication, proof certificates and countermodels over algebraic structures.","title":"Logic and algebra","url":"https://the-omega-institute.github.io/trureturing-pages/discover.html?q=Logic%20and%20algebra"}],"relation_types":{"acknowledges":"Source-authored acknowledgement; no theorem-use or novelty claim.","addresses":"Reviewed result addresses exactly the stated public question and scope.","builds_on":"Proposed follow-up based on a reviewed result, not a proved implication.","cites":"Source-authored attribution to prior literature; not an extracted proof dependency.","explains":"A worked explanation is available; independent human understanding has not been assessed.","formalized_in":"Pinned result belongs to this module; current release certification is not inferred.","paper_context":"Related in the cited paper; no proof coverage is implied.","proposed_target":"Proposed subproblem of a research family.","proposed_transfer":"Candidate use on a further question; no successful transfer or benefit claimed.","question_source":"The publication supplies the question or its research context.","representation":"A named equality or characterization in the reviewed pinned Lean source.","source_anchor":"Source-authored motivation link to a released module, not theorem-use evidence.","studied_object":"Result studies this sequence; the sequence itself is not marked solved.","topic_member":"Curated subject association, not a proof dependency."},"relations":[{"evidence_url":"https://doi.org/10.48550/arXiv.2509.16034","kind":"question_source","scope":"Section 3, Campbell-Currie-Rampersad, arXiv:2509.16034v1","source":"claim:thue-morse-reduced-abelian-odd","target":"literature:arxiv:2509.16034"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/","kind":"addresses","scope":"This dossier deliberately anchors only the equality\n`rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1)` for every nonnegative integer\n`n`, proposed in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1 and\nprinted page 14 of the verified journal version. The shared Library note\nrecords the journal citation. The source sentence is:\n\n> Although it appears that rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for\n> nonnegative integers n, the problem of determining a full recursion for\n> rho^{ab,red}_t(n) seems to be challenging.\n\n`red(w)` collapses every maximal constant run; reduced abelian complexity\ncounts all length-`n` factors up to rearrangement of their reduced words\nwith equal reduced length. 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current open/solved literature status is not inferred.","source":"dossier:oeis-a008590-gcd-sum-parity-characterization","target":"literature:url:https://oeis.org/A008590"},{"evidence_url":"https://oeis.org/A018800","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a018800-least-prime-prefix-suffix-refutation","target":"literature:url:https://oeis.org/A018800"},{"evidence_url":"https://oeis.org/A023887","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a023887-yanev-sigma-radical-identity","target":"literature:url:https://oeis.org/A023887"},{"evidence_url":"https://oeis.org/A030101","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a030101-yanev-binary-reversal-position-identity","target":"literature:url:https://oeis.org/A030101"},{"evidence_url":"https://oeis.org/A035513","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a035513-wythoff-column-residue-pairs","target":"literature:url:https://oeis.org/A035513"},{"evidence_url":"https://oeis.org/A038867","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a038867-cube-gap-five-no-square","target":"literature:url:https://oeis.org/A038867"},{"evidence_url":"https://oeis.org/A046528","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization","target":"literature:url:https://oeis.org/A046528"},{"evidence_url":"https://oeis.org/A049473","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference","target":"literature:url:https://oeis.org/A049473"},{"evidence_url":"https://oeis.org/A049591","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a049591-prime-gap-divisor-characterization-refutation","target":"literature:url:https://oeis.org/A049591"},{"evidence_url":"https://oeis.org/A051903","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence","target":"literature:url:https://oeis.org/A051903"},{"evidence_url":"https://oeis.org/A054096","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a054096-stephan-triangle-second-column","target":"literature:url:https://oeis.org/A054096"},{"evidence_url":"https://oeis.org/A059894","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a059894-stephan-complement-reverse-recurrence","target":"literature:url:https://oeis.org/A059894"},{"evidence_url":"https://oeis.org/A060693","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a060693-schroeder-peak-alternating-quadratic-moment","target":"literature:url:https://oeis.org/A060693"},{"evidence_url":"https://oeis.org/A064618","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a064618-stirling-transform-totient-period","target":"literature:url:https://oeis.org/A064618"},{"evidence_url":"https://oeis.org/A068012","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a068012-correction-recurrence","target":"literature:url:https://oeis.org/A068012"},{"evidence_url":"https://oeis.org/A068012","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a068012-subset-sum-doubling","target":"literature:url:https://oeis.org/A068012"},{"evidence_url":"https://oeis.org/A072872","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a072872-power-minus-index-prime-bound-refutation","target":"literature:url:https://oeis.org/A072872"},{"evidence_url":"https://oeis.org/A074724","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma","target":"literature:url:https://oeis.org/A074724"},{"evidence_url":"https://oeis.org/A076502","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a076502-nested-recurrence-floor-refutation","target":"literature:url:https://oeis.org/A076502"},{"evidence_url":"https://oeis.org/A077864","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series","target":"literature:url:https://oeis.org/A077864"},{"evidence_url":"https://oeis.org/A077866","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a077866-least-two-subset-count","target":"literature:url:https://oeis.org/A077866"},{"evidence_url":"https://oeis.org/A079063","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a079063-sqrt-prime-gap-lower-bound-refutation","target":"literature:url:https://oeis.org/A079063"},{"evidence_url":"https://oeis.org/A079278","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence","target":"literature:url:https://oeis.org/A079278"},{"evidence_url":"https://oeis.org/A084068","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a084068-pell-companion-gcd","target":"literature:url:https://oeis.org/A084068"},{"evidence_url":"https://oeis.org/A089610","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a089610-square-interval-prime-count-eventual-increase-refutation","target":"literature:url:https://oeis.org/A089610"},{"evidence_url":"https://oeis.org/A090079","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a090079-yanev-run-compression-closed-form","target":"literature:url:https://oeis.org/A090079"},{"evidence_url":"https://oeis.org/A090825","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a090825-bernoulli-integrality-refutation","target":"literature:url:https://oeis.org/A090825"},{"evidence_url":"https://oeis.org/A091190","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a091190-somos-prime-integrality","target":"literature:url:https://oeis.org/A091190"},{"evidence_url":"https://oeis.org/A091259","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a096127-factorial-square-divisibility-prime-power","target":"literature:url:https://oeis.org/A096127"},{"evidence_url":"https://oeis.org/A097602","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a097602-square-counting-recurrence-square-values","target":"literature:url:https://oeis.org/A097602"},{"evidence_url":"https://oeis.org/A100952","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a100952-log-bounded-semiprime-partition-refutation","target":"literature:url:https://oeis.org/A100952"},{"evidence_url":"https://oeis.org/A103585","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a103585-sloping-binary-period-refutation","target":"literature:url:https://oeis.org/A103585"},{"evidence_url":"https://oeis.org/A104863","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a104863-floor-sqrt-recurrence-refutation","target":"literature:url:https://oeis.org/A104863"},{"evidence_url":"https://oeis.org/A105595","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a105595-barry-triangle-row-sums-odd","target":"literature:url:https://oeis.org/A105595"},{"evidence_url":"https://oeis.org/A107928","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a107928-harmonic-mean-numerator-closed-form","target":"literature:url:https://oeis.org/A107928"},{"evidence_url":"https://oeis.org/A110454","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a110454-distinct-partition-concatenation-zero-indices","target":"literature:url:https://oeis.org/A110454"},{"evidence_url":"https://oeis.org/A110491","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a115039-prime-square-perfect-power-representation","target":"literature:url:https://oeis.org/A115039"},{"evidence_url":"https://oeis.org/A115770","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a115770-carryless-seven-five-equivalence","target":"literature:url:https://oeis.org/A115770"},{"evidence_url":"https://oeis.org/A115801","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a115801-carryless-multiplier-equivalence","target":"literature:url:https://oeis.org/A115801"},{"evidence_url":"https://oeis.org/A115803","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a115803-carryless-thirteen-eleven-containment","target":"literature:url:https://oeis.org/A115803"},{"evidence_url":"https://oeis.org/A115809","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a115809-carryless-seventeen-nine-containment","target":"literature:url:https://oeis.org/A115809"},{"evidence_url":"https://oeis.org/A116184","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a116184-generalized-harmonic-thirty-seven-cube-progression","target":"literature:url:https://oeis.org/A116184"},{"evidence_url":"https://oeis.org/A119623","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a119623-divisor-pair-product-prime-preimages","target":"literature:url:https://oeis.org/A119623"},{"evidence_url":"https://oeis.org/A119690","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a119690-layman-odd-power-factorial-residue","target":"literature:url:https://oeis.org/A119690"},{"evidence_url":"https://oeis.org/A122399","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a122399-stirling-power-factorial-prime-period","target":"literature:url:https://oeis.org/A122399"},{"evidence_url":"https://oeis.org/A124418","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a124418-mixed-parity-block-partition-product","target":"literature:url:https://oeis.org/A124418"},{"evidence_url":"https://oeis.org/A126347","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a126347-kurkov-reversed-qbell-row-polynomial","target":"literature:url:https://oeis.org/A126347"},{"evidence_url":"https://oeis.org/A126762","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a126762-least-witness-exponent-shift-refutation","target":"literature:url:https://oeis.org/A126762"},{"evidence_url":"https://oeis.org/A129598","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a129598-gcd-totient-position-refutation","target":"literature:url:https://oeis.org/A129598"},{"evidence_url":"https://oeis.org/A131853","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five","target":"literature:url:https://oeis.org/A131853"},{"evidence_url":"https://oeis.org/A132393","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a132393-kurkov-nested-stirling-identity","target":"literature:url:https://oeis.org/A132393"},{"evidence_url":"https://oeis.org/A133901","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a133901-palindrome-square-reversal-infinitude","target":"literature:url:https://oeis.org/A133901"},{"evidence_url":"https://oeis.org/A133907","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime","target":"literature:url:https://oeis.org/A133907"},{"evidence_url":"https://oeis.org/A136162","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a136162-krizek-next-prime-product-quadruplet","target":"literature:url:https://oeis.org/A136162"},{"evidence_url":"https://oeis.org/A140869","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a140869-sundaram-triangle-complement-prime","target":"literature:url:https://oeis.org/A140869"},{"evidence_url":"https://oeis.org/A144837","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a144837-bala-lucas-five-adic-congruence","target":"literature:url:https://oeis.org/A144837"},{"evidence_url":"https://oeis.org/A145110","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a145110-granvik-redheffer-support-refutation","target":"literature:url:https://oeis.org/A145110"},{"evidence_url":"https://oeis.org/A146557","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a146557-collinear-triple-divisibility","target":"literature:url:https://oeis.org/A146557"},{"evidence_url":"https://oeis.org/A152020","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a152020-rational-denominator-cube-shift","target":"literature:url:https://oeis.org/A152020"},{"evidence_url":"https://oeis.org/A152661","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a152661-mathar-first-recurrence","target":"literature:url:https://oeis.org/A152661"},{"evidence_url":"https://oeis.org/A160627","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a160627-israel-laguerre-four-parity","target":"literature:url:https://oeis.org/A160627"},{"evidence_url":"https://oeis.org/A163617","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form","target":"literature:url:https://oeis.org/A163617"},{"evidence_url":"https://oeis.org/A165719","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a165719-eighth-quadratic-integrality-and-sole-prime","target":"literature:url:https://oeis.org/A165719"},{"evidence_url":"https://oeis.org/A167138","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a167138-logarithmic-power-closure","target":"literature:url:https://oeis.org/A167138"},{"evidence_url":"https://oeis.org/A167540","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a167540-logarithmic-affine-closure","target":"literature:url:https://oeis.org/A167540"},{"evidence_url":"https://oeis.org/A167857","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a167857-sorted-divisor-polynomial-three-primes","target":"literature:url:https://oeis.org/A167857"},{"evidence_url":"https://oeis.org/A175406","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a175406-log-two-floor-refutation","target":"literature:url:https://oeis.org/A175406"},{"evidence_url":"https://oeis.org/A177680","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a177680-twin-prime-average-multiple-of-five","target":"literature:url:https://oeis.org/A177680"},{"evidence_url":"https://oeis.org/A178294","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a178294-four-grid-collinear-triples","target":"literature:url:https://oeis.org/A178294"},{"evidence_url":"https://oeis.org/A181278","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation","target":"literature:url:https://oeis.org/A181278"},{"evidence_url":"https://oeis.org/A189733","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a189733-curtz-denominator-conjecture","target":"literature:url:https://oeis.org/A189733"},{"evidence_url":"https://oeis.org/A192023","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a192023-comb-wiener-determinant-refutation","target":"literature:url:https://oeis.org/A192023"},{"evidence_url":"https://oeis.org/A192763","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a192763-granvik-mertens-array","target":"literature:url:https://oeis.org/A192763"},{"evidence_url":"https://oeis.org/A193020","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a193020-minus-four-eventual-identity","target":"literature:url:https://oeis.org/A193020"},{"evidence_url":"https://oeis.org/A193020","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a193020-plus-one-eventual-identity","target":"literature:url:https://oeis.org/A193020"},{"evidence_url":"https://oeis.org/A193351","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a193351-alternating-divisor-sum-prime-square","target":"literature:url:https://oeis.org/A193351"},{"evidence_url":"https://oeis.org/A196226","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation","target":"literature:url:https://oeis.org/A196226"},{"evidence_url":"https://oeis.org/A204061","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a204061-companion-pell-exp-base-five-residue","target":"literature:url:https://oeis.org/A204061"},{"evidence_url":"https://oeis.org/A208342","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a208342-joint-polynomial-pair-binomial-coefficients","target":"literature:url:https://oeis.org/A208342"},{"evidence_url":"https://oeis.org/A208509","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients","target":"literature:url:https://oeis.org/A208509"},{"evidence_url":"https://oeis.org/A219431","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a219431-hanna-overpartition-log-derivative-mod-four","target":"literature:url:https://oeis.org/A219431"},{"evidence_url":"https://oeis.org/A220181","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a220181-alternating-weight-stirling-prime-period","target":"literature:url:https://oeis.org/A220181"},{"evidence_url":"https://oeis.org/A221077","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a221077-tanh-power-prime-period","target":"literature:url:https://oeis.org/A221077"},{"evidence_url":"https://oeis.org/A222001","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a222001-descent-lex-arrays","target":"literature:url:https://oeis.org/A222001"},{"evidence_url":"https://oeis.org/A222013","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a222013-factorial-product-sum-catalan-parity","target":"literature:url:https://oeis.org/A222013"},{"evidence_url":"https://oeis.org/A222014","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a222014-factorial-square-exponent-sum-parity","target":"literature:url:https://oeis.org/A222014"},{"evidence_url":"https://oeis.org/A224899","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a224899-sinh-power-prime-period","target":"literature:url:https://oeis.org/A224899"},{"evidence_url":"https://oeis.org/A226839","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a226839-triangular-egf-mod-three","target":"literature:url:https://oeis.org/A226839"},{"evidence_url":"https://oeis.org/A226857","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a226857-fibonacci-sum-product-recurrence","target":"literature:url:https://oeis.org/A226857"},{"evidence_url":"https://oeis.org/A228558","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a228558-price-seventeen-four-prime-exponent","target":"literature:url:https://oeis.org/A228558"},{"evidence_url":"https://oeis.org/A229140","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation","target":"literature:url:https://oeis.org/A229140"},{"evidence_url":"https://oeis.org/A231329","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a231329-price-nineteen-four-prime-exponent","target":"literature:url:https://oeis.org/A231329"},{"evidence_url":"https://oeis.org/A231548","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a231548-antisigma-decrease-refutation","target":"literature:url:https://oeis.org/A231548"},{"evidence_url":"https://oeis.org/A239293","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a239293-ordowski-immediate-successor-weak-pseudoprime","target":"literature:url:https://oeis.org/A239293"},{"evidence_url":"https://oeis.org/A239907","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a239907-kurkov-run-alternating-count","target":"literature:url:https://oeis.org/A239907"},{"evidence_url":"https://oeis.org/A245786","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a245786-integrality-gcd-record-refutation","target":"literature:url:https://oeis.org/A245786"},{"evidence_url":"https://oeis.org/A249621","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a249621-square-prepend-one-append-twenty-five","target":"literature:url:https://oeis.org/A249621"},{"evidence_url":"https://oeis.org/A249759","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne","target":"literature:url:https://oeis.org/A249759"},{"evidence_url":"https://oeis.org/A254074","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a254074-wu-digit-range-square-not-multiple-of-five","target":"literature:url:https://oeis.org/A254074"},{"evidence_url":"https://oeis.org/A254748","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor","target":"literature:url:https://oeis.org/A254748"},{"evidence_url":"https://oeis.org/A258409","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a258409-divisor-difference-gcd-heinz","target":"literature:url:https://oeis.org/A258409"},{"evidence_url":"https://oeis.org/A260310","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation","target":"literature:url:https://oeis.org/A260310"},{"evidence_url":"https://oeis.org/A260407","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a260407-krizek-all-multiples-of-the-order","target":"literature:url:https://oeis.org/A260407"},{"evidence_url":"https://oeis.org/history/view?seq=A265714&v=23","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a265714-positive-threshold-common-minimum","target":"literature:url:https://oeis.org/history/view?seq=A265714&v=23"},{"evidence_url":"https://oeis.org/A266285","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a266285-rule13-barker-zizka","target":"literature:url:https://oeis.org/A266285"},{"evidence_url":"https://oeis.org/A266489","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a266489-diagonal-index-divisibility","target":"literature:url:https://oeis.org/A266489"},{"evidence_url":"https://oeis.org/A266489","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a266489-negative-power-diagonal-mod-two","target":"literature:url:https://oeis.org/A266489"},{"evidence_url":"https://oeis.org/A266614","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a266614-rule41-barker","target":"literature:url:https://oeis.org/A266614"},{"evidence_url":"https://oeis.org/A267681","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a267681-rule201-barker-hasler","target":"literature:url:https://oeis.org/A267681"},{"evidence_url":"https://oeis.org/A270236","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a270236-rgf-antepenultimate-label-count","target":"literature:url:https://oeis.org/A270236"},{"evidence_url":"https://oeis.org/A270236","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a270236-rgf-penultimate-label-count","target":"literature:url:https://oeis.org/A270236"},{"evidence_url":"https://oeis.org/A271931","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a271931-hanna-prime-substitution-integrality","target":"literature:url:https://oeis.org/A271931"},{"evidence_url":"https://oeis.org/A274010","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a274010-stechkin-divisor-count","target":"literature:url:https://oeis.org/A274010"},{"evidence_url":"https://oeis.org/A277829","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a277829-a278770-catalan-square-hankel-limits","target":"literature:url:https://oeis.org/A277829"},{"evidence_url":"https://oeis.org/A280258","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity","target":"literature:url:https://oeis.org/A280258"},{"evidence_url":"https://oeis.org/A284005","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a284005-kurkov-lower-bit-recurrence","target":"literature:url:https://oeis.org/A284005"},{"evidence_url":"https://oeis.org/A284005","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a284005-kurkov-weight-product-binomial","target":"literature:url:https://oeis.org/A284005"},{"evidence_url":"https://oeis.org/A290322","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a290322-cyclotomic-five-residue-sum","target":"literature:url:https://oeis.org/A290322"},{"evidence_url":"https://oeis.org/A292394","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a292394-diagonal-index-divisibility","target":"literature:url:https://oeis.org/A292394"},{"evidence_url":"https://oeis.org/A299406","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a299406-schulte-liouville-cube-mobius","target":"literature:url:https://oeis.org/A299406"},{"evidence_url":"https://oeis.org/A300657","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite","target":"literature:url:https://oeis.org/A300657"},{"evidence_url":"https://oeis.org/A300732","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a300732-diagonal-index-divisibility","target":"literature:url:https://oeis.org/A300732"},{"evidence_url":"https://oeis.org/A300733","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a300733-diagonal-index-divisibility","target":"literature:url:https://oeis.org/A300733"},{"evidence_url":"https://oeis.org/A300734","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a300734-diagonal-index-divisibility","target":"literature:url:https://oeis.org/A300734"},{"evidence_url":"https://oeis.org/A301420","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a301420-positive-newton-sums","target":"literature:url:https://oeis.org/A301420"},{"evidence_url":"https://oeis.org/A301976","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a301976-grid-no-leaf-mod-ten","target":"literature:url:https://oeis.org/A301976"},{"evidence_url":"https://oeis.org/A301976","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a301976-grid-no-leaf-recurrence","target":"literature:url:https://oeis.org/A301976"},{"evidence_url":"https://oeis.org/A302975","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a302975-divisor-count-power-ratio-denominator-square","target":"literature:url:https://oeis.org/A302975"},{"evidence_url":"https://oeis.org/A305550","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a305550-distinct-part-egf-totient-period","target":"literature:url:https://oeis.org/A305550"},{"evidence_url":"https://oeis.org/A305550","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a305550-integral-egf-substitution-totient-period","target":"literature:url:https://oeis.org/A305550"},{"evidence_url":"https://oeis.org/A306270","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification","target":"literature:url:https://oeis.org/A306270"},{"evidence_url":"https://oeis.org/A308090","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion","target":"literature:url:https://oeis.org/A308090"},{"evidence_url":"https://oeis.org/A308815","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a308815-dfinite-recurrence","target":"literature:url:https://oeis.org/A308815"},{"evidence_url":"https://oeis.org/A319197","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility","target":"literature:url:https://oeis.org/A319197"},{"evidence_url":"https://oeis.org/A319927","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation","target":"literature:url:https://oeis.org/A319927"},{"evidence_url":"https://oeis.org/A322327","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative","target":"literature:url:https://oeis.org/A322327"},{"evidence_url":"https://oeis.org/A326244","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a326244-noncrossing-nonnesting-graph-recurrence","target":"literature:url:https://oeis.org/A326244"},{"evidence_url":"https://oeis.org/A328959","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a328959-divisor-count-prime-factor-bound","target":"literature:url:https://oeis.org/A328959"},{"evidence_url":"https://oeis.org/A329278","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a329278-triangular-residue-descent-count","target":"literature:url:https://oeis.org/A329278"},{"evidence_url":"https://oeis.org/A329369","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a329369-printed-row-sum-range","target":"literature:url:https://oeis.org/A329369"},{"evidence_url":"https://oeis.org/A331474","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a331474-hankel-generating-function","target":"literature:url:https://oeis.org/A331474"},{"evidence_url":"https://oeis.org/A334595","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a334595-value-one","target":"literature:url:https://oeis.org/A334595"},{"evidence_url":"https://oeis.org/A335901","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a335901-self-referential-doubling-first-occurrence","target":"literature:url:https://oeis.org/A335901"},{"evidence_url":"https://oeis.org/A335925","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a335925-self-referential-quotient-first-occurrence","target":"literature:url:https://oeis.org/A335925"},{"evidence_url":"https://oeis.org/A338040","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a338040-weighted-stirling-power-prime-period","target":"literature:url:https://oeis.org/A338040"},{"evidence_url":"https://oeis.org/A338193","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a338193-egf-coefficients","target":"literature:url:https://oeis.org/A338193"},{"evidence_url":"https://oeis.org/A338636","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a338636-continued-fraction-odd-square-mod-eight","target":"literature:url:https://oeis.org/A338636"},{"evidence_url":"https://oeis.org/A339621","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index","target":"literature:url:https://oeis.org/A339621"},{"evidence_url":"https://oeis.org/A339669","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a363569-hanna-bilateral-prime-values","target":"literature:url:https://oeis.org/A363569"},{"evidence_url":"https://oeis.org/A363956","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a363956-positive-permutation","target":"literature:url:https://oeis.org/A363956"},{"evidence_url":"https://oeis.org/A365095","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a365095-perturbed-diagonal-square-divisibility","target":"literature:url:https://oeis.org/A365095"},{"evidence_url":"https://oeis.org/A365574","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1","target":"literature:url:https://oeis.org/A365574"},{"evidence_url":"https://oeis.org/A367400","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a372991-factorial-quotient-linear-recurrence","target":"literature:url:https://oeis.org/A372991"},{"evidence_url":"https://oeis.org/A373308","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a373308-cubed-binary-product-mod-three","target":"literature:url:https://oeis.org/A373308"},{"evidence_url":"https://oeis.org/A373312","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a373312-unit-reversion-square-parity","target":"literature:url:https://oeis.org/A373312"},{"evidence_url":"https://oeis.org/A373561","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a373561-gcd-bucketed-quadruple-sum","target":"literature:url:https://oeis.org/A373561"},{"evidence_url":"https://oeis.org/A374568","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a374568-catalan-double-composition-powers-of-four","target":"literature:url:https://oeis.org/A374568"},{"evidence_url":"https://oeis.org/A374570","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a374570-catalan-composition-square-parity","target":"literature:url:https://oeis.org/A374570"},{"evidence_url":"https://oeis.org/A374571","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a374571-fibbinary-square-substitution-parity","target":"literature:url:https://oeis.org/A374571"},{"evidence_url":"https://oeis.org/A375439","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a375439-cubic-ternary-substitution-parity","target":"literature:url:https://oeis.org/A375439"},{"evidence_url":"https://oeis.org/A375449","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a376527-q-catalan-square-sum-parity","target":"literature:url:https://oeis.org/A376527"},{"evidence_url":"https://oeis.org/A377106","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a377106-hanna-parity-reversion","target":"literature:url:https://oeis.org/A377106"},{"evidence_url":"https://oeis.org/A378252","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a378252-running-sum-residue-zero","target":"literature:url:https://oeis.org/A378252"},{"evidence_url":"https://oeis.org/A378258","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a378258-hanna-mod-six","target":"literature:url:https://oeis.org/A378258"},{"evidence_url":"https://oeis.org/A378258","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine","target":"literature:url:https://oeis.org/A381364"},{"evidence_url":"https://oeis.org/A381365","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine","target":"literature:url:https://oeis.org/A381365"},{"evidence_url":"https://oeis.org/A381423/internal","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381423-hermite-differences","target":"literature:url:https://oeis.org/A381423/internal"},{"evidence_url":"https://oeis.org/A381524/internal?rev=18","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381524-hermite-absolute-differences","target":"literature:url:https://oeis.org/A381524/internal?rev=18"},{"evidence_url":"https://oeis.org/A381670","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381670-compositional-square-dyadic-denominators","target":"literature:url:https://oeis.org/A381670"},{"evidence_url":"https://oeis.org/A381737/internal?rev=28","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381737-hermite-absolute-ties","target":"literature:url:https://oeis.org/A381737/internal?rev=28"},{"evidence_url":"https://oeis.org/A381737/internal?rev=28","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381737-hermite-tie-generating-function","target":"literature:url:https://oeis.org/A381737/internal?rev=28"},{"evidence_url":"https://oeis.org/A381737/internal?rev=28","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a381737-hermite-tie-recurrence","target":"literature:url:https://oeis.org/A381737/internal?rev=28"},{"evidence_url":"https://oeis.org/A382124","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a384222-two-dense-divisor-blocks-palindrome","target":"literature:url:https://oeis.org/A384222"},{"evidence_url":"https://oeis.org/A384267","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a384267-absolute-reciprocal-square-shift-parity","target":"literature:url:https://oeis.org/A384267"},{"evidence_url":"https://oeis.org/A384270","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a384270-quartic-cubic-catalan-quotient-parity","target":"literature:url:https://oeis.org/A384270"},{"evidence_url":"https://oeis.org/A385590","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a385590-alternating-binomial","target":"literature:url:https://oeis.org/A385590"},{"evidence_url":"https://oeis.org/A386655","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six","target":"literature:url:https://oeis.org/A386655"},{"evidence_url":"https://oeis.org/A386666","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a386666-signed-catalan-cubic-substitution-mod-three","target":"literature:url:https://oeis.org/A386666"},{"evidence_url":"https://oeis.org/A387319","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a387319-multiple-of-three-coincidence","target":"literature:url:https://oeis.org/A387319"},{"evidence_url":"https://oeis.org/A388711","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a388711-choosable-initial-intervals","target":"literature:url:https://oeis.org/A388711"},{"evidence_url":"https://oeis.org/A388986","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a388986-euler-form-divisor-sum","target":"literature:url:https://oeis.org/A388986"},{"evidence_url":"https://oeis.org/A389000","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a389000-digit-sum-square-least-witness","target":"literature:url:https://oeis.org/A389000"},{"evidence_url":"https://oeis.org/A389472","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a389472-square-cube-substitution-odd-index-parity","target":"literature:url:https://oeis.org/A389472"},{"evidence_url":"https://oeis.org/A389476","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a389476-quadratic-reversion-dyadic-support-parity","target":"literature:url:https://oeis.org/A389476"},{"evidence_url":"https://oeis.org/A389536","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a394407-totient-tau-primes-huber","target":"literature:url:https://oeis.org/A394407"},{"evidence_url":"https://oeis.org/A394578/internal","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a394578-interpolation-ratio-limit","target":"literature:url:https://oeis.org/A394578/internal"},{"evidence_url":"https://oeis.org/A394757","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a394757-sigma-gcd-two-or-three","target":"literature:url:https://oeis.org/A394757"},{"evidence_url":"https://oeis.org/A394757","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a394757-twin-prime-sigma-gcd","target":"literature:url:https://oeis.org/A394757"},{"evidence_url":"https://oeis.org/A395531/internal","kind":"question_source","scope":"Release-bound dossier source; 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current open/solved literature status is not inferred.","source":"dossier:oeis-a396102-triple-iterate-shift-mod-three","target":"literature:url:https://oeis.org/A396102"},{"evidence_url":"https://oeis.org/search?q=id:A396596&fmt=text","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396596-divisor-power-union","target":"literature:url:https://oeis.org/search?q=id:A396596&fmt=text"},{"evidence_url":"https://oeis.org/A396659/internal","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396659-edge-collision-ratio","target":"literature:url:https://oeis.org/A396659/internal"},{"evidence_url":"https://oeis.org/A396793","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396793-iterate-product-nine-mod-three","target":"literature:url:https://oeis.org/A396793"},{"evidence_url":"https://oeis.org/A396794","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396794-triple-iterate-product-mod-four","target":"literature:url:https://oeis.org/A396794"},{"evidence_url":"https://oeis.org/A396795","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five","target":"literature:url:https://oeis.org/A396795"},{"evidence_url":"https://oeis.org/A396797","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396797-three-four-iterate-product-mod-six","target":"literature:url:https://oeis.org/A396797"},{"evidence_url":"https://oeis.org/A396798","kind":"question_source","scope":"Release-bound dossier source; current open/solved literature status is not inferred.","source":"dossier:oeis-a396798-eighth-iterate-mod-eight","target":"literature:url:https://oeis.org/A396798"},{"evidence_url":"https://oeis.org/A396798","kind":"question_source","scope":"Release-bound dossier source; 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000005","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384222-two-dense-divisor-blocks-palindrome/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000005","target":"dossier:oeis-a384222-two-dense-divisor-blocks-palindrome"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000007","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000009","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000009","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000009","target":"dossier:oeis-a392694-zero-prepended-first-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000009","target":"dossier:oeis-a392694-zero-prepended-first-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392698-trimmed-alternating-partitions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000009","target":"dossier:oeis-a392698-trimmed-alternating-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:bala-machine3626-exact-halting-time"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala126156/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:bala126156"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a004123-generalized-weak-orders-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:oeis-a004123-generalized-weak-orders-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a064618-stirling-transform-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:oeis-a064618-stirling-transform-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000010","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000040","target":"dossier:oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000040","target":"dossier:oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000041","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000041","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a361552-literal-partition-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000041","target":"dossier:oeis-a361552-literal-partition-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a104863-floor-sqrt-recurrence-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000045","target":"dossier:oeis-a104863-floor-sqrt-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000045","target":"dossier:oeis-a382814-fibonachos-score"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000058","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000071-bala-strong-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000071","target":"dossier:oeis-a000071-bala-strong-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-cyclic-4123-1324-tetranacci/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000078","target":"dossier:archer-cyclic-4123-1324-tetranacci"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000108","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000108","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363308-catalan-cubic-composition-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000108","target":"dossier:oeis-a363308-catalan-cubic-composition-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374570-catalan-composition-square-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000108","target":"dossier:oeis-a374570-catalan-composition-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000110","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126347-kurkov-reversed-qbell-row-polynomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000110","target":"dossier:oeis-a126347-kurkov-reversed-qbell-row-polynomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a059894-stephan-complement-reverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a059894-stephan-complement-reverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a284005-kurkov-lower-bit-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000120","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000149","target":"dossier:kok-jaco-exponential-domination-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000149","target":"dossier:kok-jaco-exponential-domination-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000149","target":"dossier:kok-jaco-exponential-domination-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000149","target":"dossier:kok-jaco-exponential-domination-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-exponential-domination-refutation/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000149","target":"dossier:kok-jaco-exponential-domination-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/richman-2023-a053871-signed-congruence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000166","target":"dossier:richman-2023-a053871-signed-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000188","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000203","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000203","target":"dossier:oeis-a023887-yanev-sigma-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000203","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000203","target":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000219","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000219","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000219","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000290","target":"dossier:oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000293","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000293","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/coprime-divisor-tuples/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000312","target":"dossier:coprime-divisor-tuples"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2004-solid-partition-first-column-tau4/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000334","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000364-secant-number-pure-period-refutation/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000364","target":"dossier:oeis-a000364-secant-number-pure-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000364-secant-number-pure-period-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000364","target":"dossier:oeis-a000364-secant-number-pure-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335925-self-referential-quotient-first-occurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000522","target":"dossier:oeis-a335925-self-referential-quotient-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000523","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000557","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000557","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000557","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000557","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000668","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000680-schulte-half-factorial-prime-criterion/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000680","target":"dossier:oeis-a000680-schulte-half-factorial-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000680-schulte-half-factorial-prime-criterion/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000680","target":"dossier:oeis-a000680-schulte-half-factorial-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000680-schulte-half-factorial-prime-criterion/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000680","target":"dossier:oeis-a000680-schulte-half-factorial-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006472-schulte-factorial-square-prime-criterion/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000680","target":"dossier:oeis-a006472-schulte-factorial-square-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308090-caceres-gcd-factorial-power-prime-criterion/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000680","target":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000792","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000792","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000792","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000792","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000792","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-cyclic-4132-1324-padovan/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000931","target":"dossier:archer-cyclic-4132-1324-padovan"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/archer-cyclic-4132-1324-padovan/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000931","target":"dossier:archer-cyclic-4132-1324-padovan"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000990","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A000990","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-alternating-jacobsthal/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001045","target":"dossier:oeis-a391810-alternating-jacobsthal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-alternating-jacobsthal/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001045","target":"dossier:oeis-a391810-alternating-jacobsthal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001045","target":"dossier:oeis-a391810-laurent-first-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001105","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001105","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001108","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001108","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a084068-pell-companion-gcd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001108","target":"dossier:oeis-a084068-pell-companion-gcd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a084068-pell-companion-gcd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001109","target":"dossier:oeis-a084068-pell-companion-gcd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001157","target":"dossier:oeis-a023887-yanev-sigma-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001157","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001221","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001222","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001333","target":"dossier:oeis-a204061-companion-pell-exp-base-five-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001333","target":"dossier:oeis-a204061-companion-pell-exp-base-five-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a113409-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001405","target":"dossier:mathar-2012-a113409-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001481","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001511","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001511","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001511","target":"dossier:oeis-a400194-two-dense-divisor-block-maximum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001542","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001615","target":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001615","target":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001694","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-monomial-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001694","target":"dossier:kotesovec-2024-a368959-monomial-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001764","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001783","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001783","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001783","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001906","target":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001906","target":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001953","target":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A001954","target":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002000-bala-cubic-three-adic-congruence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002000","target":"dossier:oeis-a002000-bala-cubic-three-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002000-bala-cubic-three-adic-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002000","target":"dossier:oeis-a002000-bala-cubic-three-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002034","target":"dossier:oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396596-divisor-power-union/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002182","target":"dossier:oeis-a396596-divisor-power-union"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002203","target":"dossier:oeis-a204061-companion-pell-exp-base-five-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-xz-plane-path-count-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002212","target":"dossier:florez-junes-ramirez-cubic-lattice-xz-plane-path-count-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-xz-plane-path-count-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002212","target":"dossier:florez-junes-ramirez-cubic-lattice-xz-plane-path-count-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002212","target":"dossier:florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002212","target":"dossier:florez-junes-ramirez-cubic-lattice-yz-plane-path-count-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002315","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192763-granvik-mertens-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002321","target":"dossier:oeis-a192763-granvik-mertens-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002322","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002448","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002487","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002487","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002618","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002620-forberg-exponential-floor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A002620","target":"dossier:oeis-a002620-forberg-exponential-floor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a163617-gray-code-binary-recurrence-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003188","target":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003319","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003319","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003319","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003415","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/lillo-rosas-2026-priority-lattice-principal-filters/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003422","target":"dossier:lillo-rosas-2026-priority-lattice-principal-filters"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003557","target":"dossier:b164660-chebfloor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003557","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003558-marcus-tent-map-suborder/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003558","target":"dossier:oeis-a003558-marcus-tent-map-suborder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-unique-maximum-values/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003622","target":"dossier:kok-jaco-unique-maximum-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-unique-maximum-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003622","target":"dossier:kok-jaco-unique-maximum-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-jaco-unique-maximum-values/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003622","target":"dossier:kok-jaco-unique-maximum-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a005590-zero-set-fibbinary-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a005590-zero-set-fibbinary-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374571-fibbinary-square-substitution-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a374571-fibbinary-square-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374571-fibbinary-square-substitution-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a374571-fibbinary-square-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374571-fibbinary-square-substitution-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a374571-fibbinary-square-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A003714","target":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a004123-generalized-weak-orders-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a004123-generalized-weak-orders-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a004123-generalized-weak-orders-totient-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a004123-generalized-weak-orders-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a004123-generalized-weak-orders-totient-period/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a004123-generalized-weak-orders-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a064618-stirling-transform-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a064618-stirling-transform-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a064618-stirling-transform-totient-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a064618-stirling-transform-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a064618-stirling-transform-totient-period/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a064618-stirling-transform-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004123","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004760","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A004760","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005361","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005361","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005361","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005590","target":"dossier:oeis-a005590-zero-set-fibbinary-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005590","target":"dossier:oeis-a005590-zero-set-fibbinary-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a005590-zero-set-fibbinary-characterization/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005590","target":"dossier:oeis-a005590-zero-set-fibbinary-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090079-yanev-run-compression-closed-form/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005811","target":"dossier:oeis-a090079-yanev-run-compression-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090079-yanev-run-compression-closed-form/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005811","target":"dossier:oeis-a090079-yanev-run-compression-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090079-yanev-run-compression-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005811","target":"dossier:oeis-a090079-yanev-run-compression-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246056-a246423-legendre-central-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A005836","target":"dossier:hanna-a246056-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384267-absolute-reciprocal-square-shift-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006013","target":"dossier:oeis-a384267-absolute-reciprocal-square-shift-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006068","target":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006068","target":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006183","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006183","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006183","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006183","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006183","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006318","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A006318","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/elizalde-luo-nonnesting-1132-2213/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007305","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007305","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007306","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a002487-adamson-stern-farey-inverse/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007306","target":"dossier:oeis-a002487-adamson-stern-farey-inverse"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-13/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007502","target":"dossier:fried-fibonacci-shift-non-fibonacci"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/cigler-signed-strip-cycle-walks/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007582","target":"dossier:cigler-signed-strip-cycle-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007917","target":"dossier:oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007947","target":"dossier:oeis-a023887-yanev-sigma-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A007947","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008287-schulte-quadrinomial-alternating-binomial/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008287","target":"dossier:oeis-a008287-schulte-quadrinomial-alternating-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008287-schulte-quadrinomial-alternating-binomial/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008287","target":"dossier:oeis-a008287-schulte-quadrinomial-alternating-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008287-schulte-quadrinomial-alternating-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008287","target":"dossier:oeis-a008287-schulte-quadrinomial-alternating-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008365-detlefs-rough-residue-characterization-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008365","target":"dossier:oeis-a008365-detlefs-rough-residue-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008365-detlefs-rough-residue-characterization-refutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008365","target":"dossier:oeis-a008365-detlefs-rough-residue-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008472","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008472","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008472","target":"dossier:oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008474","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008474-prime-exponent-sum-iteration-reaches-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008474","target":"dossier:oeis-a008474-prime-exponent-sum-iteration-reaches-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300657-krizek-divisor-sigma-mod-noncomposite/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008578","target":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008590-gcd-sum-parity-characterization/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008590","target":"dossier:oeis-a008590-gcd-sum-parity-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a008590-gcd-sum-parity-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008590","target":"dossier:oeis-a008590-gcd-sum-parity-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/granvik-2024-a292273-real-part/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008683","target":"dossier:granvik-2024-a292273-real-part"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008683","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A008977","target":"dossier:bala361035-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A009195","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://arxiv.org/html/2509.16034v1#S3","kind":"studied_object","scope":"The proved odd recurrence concerns reduced abelian complexity of the Thue-Morse word. A010060 is the underlying binary sequence, not the derived complexity sequence. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A023855","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A023855","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A023855","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A023887","target":"dossier:oeis-a023887-yanev-sigma-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a023887-yanev-sigma-radical-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A023887","target":"dossier:oeis-a023887-yanev-sigma-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kag329479-binarynull/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A024493","target":"dossier:kag329479-binarynull"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kag329479-binarynull/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A024495","target":"dossier:kag329479-binarynull"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231548-antisigma-decrease-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A024816","target":"dossier:oeis-a231548-antisigma-decrease-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A026009","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A026010","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A026010","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a026023-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A026023","target":"dossier:mathar-2012-a026023-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a026023-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A026023","target":"dossier:mathar-2012-a026023-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027170","target":"dossier:mathar-a027182"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027170","target":"dossier:mathar-a027182"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027175","target":"dossier:mathar-a027182"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027182","target":"dossier:mathar-a027182"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a027182/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027182","target":"dossier:mathar-a027182"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027193","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380678-quadratic-square-reversion-dyadic-support-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027383","target":"dossier:oeis-a380678-quadratic-square-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380678-quadratic-square-reversion-dyadic-support-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027383","target":"dossier:oeis-a380678-quadratic-square-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389476-quadratic-reversion-dyadic-support-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027383","target":"dossier:oeis-a389476-quadratic-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389476-quadratic-reversion-dyadic-support-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027383","target":"dossier:oeis-a389476-quadratic-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027907","target":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027907","target":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A027907","target":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-bilateral-reversion-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A028309","target":"dossier:hanna-bilateral-reversion-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-bilateral-reversion-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A028309","target":"dossier:hanna-bilateral-reversion-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A029552","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A029552","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A029552","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A030101","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A030101","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A030101","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A030109","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A033042","target":"dossier:hanna365771-mod5"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-two/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A033042","target":"dossier:hanna365771-mod5-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-zero/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A033042","target":"dossier:hanna365771-mod5-zero"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-13/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A033321","target":"dossier:egge-fishburn-conjecture-10-13"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A034444","target":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A034444","target":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A034444","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a096127-factorial-square-divisibility-prime-power/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A034841","target":"dossier:oeis-a096127-factorial-square-divisibility-prime-power"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A035263","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A035263","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A035263","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a035513-wythoff-column-residue-pairs/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A035513","target":"dossier:oeis-a035513-wythoff-column-residue-pairs"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a035513-wythoff-column-residue-pairs/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A035513","target":"dossier:oeis-a035513-wythoff-column-residue-pairs"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A037952","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-powers/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A038464","target":"dossier:oeis-a397242-mod-three-powers"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A038464","target":"dossier:oeis-a397242-mod-three-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-sums/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A038464","target":"dossier:oeis-a397242-mod-three-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375439-cubic-ternary-substitution-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A038754","target":"dossier:oeis-a375439-cubic-ternary-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386666-signed-catalan-cubic-substitution-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A039966","target":"dossier:oeis-a386666-signed-catalan-cubic-substitution-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394407-totient-tau-primes-huber/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A045344","target":"dossier:oeis-a394407-totient-tau-primes-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046528","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046528","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046528","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046660","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046660","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046660","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046666","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046666","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046667","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A046667","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A047999","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048105","target":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048105","target":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala295462/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048718","target":"dossier:bala295462"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala295462/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048718","target":"dossier:bala295462"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115801-carryless-multiplier-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115801-carryless-multiplier-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A048720","target":"dossier:oeis-a115809-carryless-seventeen-nine-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049020","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049020","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049020","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049020","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049473","target":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049473","target":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049473-kimberling-rounded-half-root-two-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049473","target":"dossier:oeis-a049473-kimberling-rounded-half-root-two-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a049591-prime-gap-divisor-characterization-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049591","target":"dossier:oeis-a049591-prime-gap-divisor-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A049660","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A050168","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A050399","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A050399","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A050399","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051064","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051064","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051281","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a046528-sigma-tau-rational-power-mersenne-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051281","target":"dossier:oeis-a046528-sigma-tau-rational-power-mersenne-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051903","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a051903-yanev-max-exponent-radical-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051903","target":"dossier:oeis-a051903-yanev-max-exponent-radical-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A051924","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a356508-hanna-product-partition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A052002","target":"dossier:oeis-a356508-hanna-product-partition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090825-bernoulli-integrality-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053176","target":"dossier:oeis-a090825-bernoulli-integrality-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090825-bernoulli-integrality-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053176","target":"dossier:oeis-a090825-bernoulli-integrality-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053251","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053251","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053251","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053251","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053251","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077866-least-two-subset-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053599","target":"dossier:oeis-a077866-least-two-subset-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053644","target":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053645","target":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053645","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053645","target":"dossier:oeis-a284005-kurkov-lower-bit-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/richman-2023-a053871-signed-congruence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053871","target":"dossier:richman-2023-a053871-signed-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/richman-2023-a053871-signed-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A053871","target":"dossier:richman-2023-a053871-signed-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a196226-layman-divisor-power-congruence-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054024","target":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300657-krizek-divisor-sigma-mod-noncomposite/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054024","target":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054090","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054090","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054091","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054091","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054091","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054096","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a054096-stephan-triangle-second-column/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054096","target":"dossier:oeis-a054096-stephan-triangle-second-column"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a059894-stephan-complement-reverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A054429","target":"dossier:oeis-a059894-stephan-complement-reverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A055235","target":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A055235","target":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192763-granvik-mertens-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A055615","target":"dossier:oeis-a192763-granvik-mertens-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a096127-factorial-square-divisibility-prime-power/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A057599","target":"dossier:oeis-a096127-factorial-square-divisibility-prime-power"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A057660","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A057660","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A057660","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A057660","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400644-rounded-three-term-norm/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A058265","target":"dossier:oeis-a400644-rounded-three-term-norm"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a059894-stephan-complement-reverse-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A059894","target":"dossier:oeis-a059894-stephan-complement-reverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a060693-schroeder-peak-alternating-quadratic-moment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A060693","target":"dossier:oeis-a060693-schroeder-peak-alternating-quadratic-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/coprime-divisor-tuples/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A062319","target":"dossier:coprime-divisor-tuples"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/coprime-divisor-tuples/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A062319","target":"dossier:coprime-divisor-tuples"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/witula-slota-2007-quasi-fibonacci-order-eleven-degree/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A064875","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A064875","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A071324","target":"dossier:oeis-a193351-alternating-divisor-sum-prime-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193351-alternating-divisor-sum-prime-square/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A071324","target":"dossier:oeis-a193351-alternating-divisor-sum-prime-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A071537","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A071537","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A071537","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a072872-power-minus-index-prime-bound-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A072872","target":"dossier:oeis-a072872-power-minus-index-prime-bound-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a072872-power-minus-index-prime-bound-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A072872","target":"dossier:oeis-a072872-power-minus-index-prime-bound-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167857-sorted-divisor-polynomial-three-primes/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A073103","target":"dossier:oeis-a167857-sorted-divisor-polynomial-three-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A073759","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-oeis-a073759-least-prime-factor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A073759","target":"dossier:mathar-oeis-a073759-least-prime-factor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001108-krizek-triangular-square-sigma-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A074285","target":"dossier:oeis-a001108-krizek-triangular-square-sigma-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A074724","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A074724","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A074724","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a076502-nested-recurrence-floor-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A076502","target":"dossier:oeis-a076502-nested-recurrence-floor-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152661-mathar-first-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077613","target":"dossier:oeis-a152661-mathar-first-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152661-mathar-first-recurrence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077613","target":"dossier:oeis-a152661-mathar-first-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077864","target":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077864-trinomial-odd-diagonal-rational-series/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077864","target":"dossier:oeis-a077864-trinomial-odd-diagonal-rational-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077866-least-two-subset-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077866","target":"dossier:oeis-a077866-least-two-subset-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077866-least-two-subset-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A077866","target":"dossier:oeis-a077866-least-two-subset-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/egge-fishburn-conjecture-10-13/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A078482","target":"dossier:egge-fishburn-conjecture-10-13"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079063-sqrt-prime-gap-lower-bound-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079063","target":"dossier:oeis-a079063-sqrt-prime-gap-lower-bound-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079063-sqrt-prime-gap-lower-bound-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079063","target":"dossier:oeis-a079063-sqrt-prime-gap-lower-bound-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079269","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079269","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079278","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079278","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-valuations/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079998","target":"dossier:bala3634-valuations"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala3634-valuations/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A079998","target":"dossier:bala3634-valuations"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/condon-dugan-goldman-williams-2026-l-polyomino-skipped-sequence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A080353","target":"dossier:condon-dugan-goldman-williams-2026-l-polyomino-skipped-sequence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A080581","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A080582","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a084068-pell-companion-gcd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A084068","target":"dossier:oeis-a084068-pell-companion-gcd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a084068-pell-companion-gcd/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A085282","target":"dossier:cigler-signed-strip-cycle-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A085582","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A085582","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a018800-least-prime-prefix-suffix-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A085608","target":"dossier:oeis-a018800-least-prime-prefix-suffix-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a018800-least-prime-prefix-suffix-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A085608","target":"dossier:oeis-a018800-least-prime-prefix-suffix-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392095-antidiagonal-source-series/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A088713","target":"dossier:oeis-a392095-antidiagonal-source-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392095-antidiagonal-source-series/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A088713","target":"dossier:oeis-a392095-antidiagonal-source-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A088838","target":"dossier:oeis-a074724-cloitre-fib-four-three-adic-valuation-sigma"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a089610-square-interval-prime-count-eventual-increase-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A091200","target":"dossier:oeis-a091190-somos-prime-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091259-reduced-numerator-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A091259","target":"dossier:oeis-a091259-reduced-numerator-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A091915","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A091915","target":"dossier:oeis-a091915-stephan-even-product-partition-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a091915-stephan-even-product-partition-recurrence/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A093178","target":"dossier:oeis-a110491-ratajczak-t-transform-determinant"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A093369","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A093369","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A093372","target":"dossier:oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A094401","target":"dossier:oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A094802","target":"dossier:oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a094802-perry-least-factorial-divisible-by-prefix-lcm/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A096126","target":"dossier:oeis-a096127-factorial-square-divisibility-prime-power"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a096127-factorial-square-divisibility-prime-power/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A096126","target":"dossier:oeis-a096127-factorial-square-divisibility-prime-power"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a096127-factorial-square-divisibility-prime-power/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A098530","target":"dossier:meeussen-2004-solid-partition-first-column-tau4"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A098613","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A098613","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A098613","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a100952-log-bounded-semiprime-partition-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A100952","target":"dossier:oeis-a100952-log-bounded-semiprime-partition-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a100952-log-bounded-semiprime-partition-refutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A100952","target":"dossier:oeis-a100952-log-bounded-semiprime-partition-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A102370","target":"dossier:oeis-a103585-sloping-binary-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A102370","target":"dossier:oeis-a103585-sloping-binary-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A103447","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A103585","target":"dossier:oeis-a103585-sloping-binary-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A103585","target":"dossier:oeis-a103585-sloping-binary-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a103585-sloping-binary-period-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A103585","target":"dossier:oeis-a103585-sloping-binary-period-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a104863-floor-sqrt-recurrence-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A104863","target":"dossier:oeis-a104863-floor-sqrt-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A105594","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A105594","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A105594","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A105595","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a105595-barry-triangle-row-sums-odd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A105595","target":"dossier:oeis-a105595-barry-triangle-row-sums-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a107099-half-iterate-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A107099","target":"dossier:hanna-a107099-half-iterate-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kur107667-ballot/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A107667","target":"dossier:kur107667-ballot"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398690-palindromic/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A107735","target":"dossier:oeis-a398690-palindromic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398690-palindromic/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A107735","target":"dossier:oeis-a398690-palindromic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a107928-harmonic-mean-numerator-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A107928","target":"dossier:oeis-a107928-harmonic-mean-numerator-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a108958-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A108958","target":"dossier:mathar-2012-a108958-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a108958-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A108958","target":"dossier:mathar-2012-a108958-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110454-distinct-partition-concatenation-zero-indices/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A110454","target":"dossier:oeis-a110454-distinct-partition-concatenation-zero-indices"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110491-ratajczak-t-transform-determinant/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A110491","target":"dossier:oeis-a110491-ratajczak-t-transform-determinant"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110491-ratajczak-t-transform-determinant/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A110491","target":"dossier:oeis-a110491-ratajczak-t-transform-determinant"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a110491-ratajczak-t-transform-determinant/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A110491","target":"dossier:oeis-a110491-ratajczak-t-transform-determinant"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A111089","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112041-dale-triple-scaling-prime-multiples-of-ten/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112041","target":"dossier:oeis-a112041-dale-triple-scaling-prime-multiples-of-ten"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112041-dale-triple-scaling-prime-multiples-of-ten/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112041","target":"dossier:oeis-a112041-dale-triple-scaling-prime-multiples-of-ten"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112302-lerch-somos-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112302","target":"dossier:oeis-a112302-lerch-somos-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112302-lerch-somos-limit/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112302","target":"dossier:oeis-a112302-lerch-somos-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112399","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112399","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112575","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112575","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A112765","target":"dossier:h369673"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/dement-2005-a113249-odd-index-squares/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A113249","target":"dossier:dement-2005-a113249-odd-index-squares"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/dement-2005-a113249-odd-index-squares/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A113249","target":"dossier:dement-2005-a113249-odd-index-squares"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/dement-2005-a113249-odd-index-squares/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A113254","target":"dossier:dement-2005-a113249-odd-index-squares"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a113409-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A113409","target":"dossier:mathar-2012-a113409-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-2012-a113409-recurrence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A113409","target":"dossier:mathar-2012-a113409-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/barry-2013-riordan-pascal-kernel-determinant/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A114188","target":"dossier:barry-2013-riordan-pascal-kernel-determinant"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115039-prime-square-perfect-power-representation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115038","target":"dossier:oeis-a115039-prime-square-perfect-power-representation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115039-prime-square-perfect-power-representation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115039","target":"dossier:oeis-a115039-prime-square-perfect-power-representation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115770","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115770","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115801-carryless-multiplier-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115801-carryless-multiplier-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115801-carryless-multiplier-equivalence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115801-carryless-multiplier-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115809-carryless-seventeen-nine-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115809-carryless-seventeen-nine-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115801","target":"dossier:oeis-a115809-carryless-seventeen-nine-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115803","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115803","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115805","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115805","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115805","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115803-carryless-thirteen-eleven-containment/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115805","target":"dossier:oeis-a115803-carryless-thirteen-eleven-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115809-carryless-seventeen-nine-containment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115809","target":"dossier:oeis-a115809-carryless-seventeen-nine-containment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115872","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a115770-carryless-seven-five-equivalence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A115872","target":"dossier:oeis-a115770-carryless-seven-five-equivalence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a116184-generalized-harmonic-thirty-seven-cube-progression/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A116184","target":"dossier:oeis-a116184-generalized-harmonic-thirty-seven-cube-progression"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/benyi-mansour-ramirez-weak-ascent-210/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A117106","target":"dossier:benyi-mansour-ramirez-weak-ascent-210"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/benyi-mansour-ramirez-weak-ascent-210/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A117106","target":"dossier:benyi-mansour-ramirez-weak-ascent-210"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335901-self-referential-doubling-first-occurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A117261","target":"dossier:oeis-a335901-self-referential-doubling-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335901-self-referential-doubling-first-occurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A117261","target":"dossier:oeis-a335901-self-referential-doubling-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119616","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119616","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119616","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119616","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119623","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119623-divisor-pair-product-prime-preimages/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119623","target":"dossier:oeis-a119623-divisor-pair-product-prime-preimages"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119690","target":"dossier:oeis-a119690-layman-odd-power-factorial-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A119690","target":"dossier:oeis-a119690-layman-odd-power-factorial-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A121854","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A121854","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a077866-least-two-subset-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122196","target":"dossier:oeis-a077866-least-two-subset-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a122399-stirling-power-factorial-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122399","target":"dossier:oeis-a122399-stirling-power-factorial-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122399","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122399","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122974","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A122974","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A124418","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A124419","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a124418-mixed-parity-block-partition-product/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A124526","target":"dossier:oeis-a124418-mixed-parity-block-partition-product"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala126156/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126156","target":"dossier:bala126156"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala126156/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126156","target":"dossier:bala126156"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126347-kurkov-reversed-qbell-row-polynomial/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126347","target":"dossier:oeis-a126347-kurkov-reversed-qbell-row-polynomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126347-kurkov-reversed-qbell-row-polynomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126347","target":"dossier:oeis-a126347-kurkov-reversed-qbell-row-polynomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126762-least-witness-exponent-shift-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126762","target":"dossier:oeis-a126762-least-witness-exponent-shift-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a126762-least-witness-exponent-shift-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126762","target":"dossier:oeis-a126762-least-witness-exponent-shift-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A126762","target":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A127428","target":"dossier:h369673"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A127428","target":"dossier:h369673"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A127473","target":"dossier:oeis-a347293-schulte-shifted-product-gcd-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kimberling-array-first-occurrence-l4v5/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A128440","target":"dossier:kimberling-array-first-occurrence-l4v5"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A128921","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A129598","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A129598","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A129598","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a129598-gcd-totient-position-refutation/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A129598","target":"dossier:oeis-a129598-gcd-totient-position-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131851","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131852","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131853","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131853","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131854","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A131855","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kurk396563/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A132393","target":"dossier:kurk396563"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A132961","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A132961","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133901","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133901","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133901","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133901","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133901-palindrome-square-reversal-infinitude/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133901","target":"dossier:oeis-a133901-palindrome-square-reversal-infinitude"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133907","target":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A133907","target":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-pythagorean-perimeter-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A134492","target":"dossier:fibonacci-pythagorean-perimeter-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112302-lerch-somos-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A135002","target":"dossier:oeis-a112302-lerch-somos-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a136162-krizek-next-prime-product-quadruplet/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A136162","target":"dossier:oeis-a136162-krizek-next-prime-product-quadruplet"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a136162-krizek-next-prime-product-quadruplet/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A136162","target":"dossier:oeis-a136162-krizek-next-prime-product-quadruplet"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/irwin-a138099/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A138099","target":"dossier:irwin-a138099"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/irwin-a138099/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A138099","target":"dossier:irwin-a138099"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a140869-sundaram-triangle-complement-prime/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A140869","target":"dossier:oeis-a140869-sundaram-triangle-complement-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chu-2026-schreier-multiset-recurrence-q-two/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A141015","target":"dossier:chu-2026-schreier-multiset-recurrence-q-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A143104","target":"dossier:oeis-a145110-granvik-redheffer-support-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A143104","target":"dossier:oeis-a145110-granvik-redheffer-support-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A143104","target":"dossier:oeis-a145110-granvik-redheffer-support-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a140869-sundaram-triangle-complement-prime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A144652","target":"dossier:oeis-a140869-sundaram-triangle-complement-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A144837","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A144837","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A145110","target":"dossier:oeis-a145110-granvik-redheffer-support-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a145110-granvik-redheffer-support-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A145110","target":"dossier:oeis-a145110-granvik-redheffer-support-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/shah-kiselev-2026-late-growing-ratio-limit/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A147681","target":"dossier:shah-kiselev-2026-late-growing-ratio-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152020-rational-denominator-cube-shift/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A152020","target":"dossier:oeis-a152020-rational-denominator-cube-shift"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a152661-mathar-first-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A152661","target":"dossier:oeis-a152661-mathar-first-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a140869-sundaram-triangle-complement-prime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A153085","target":"dossier:oeis-a140869-sundaram-triangle-complement-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/a156216-diagonalghost/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A156216","target":"dossier:a156216-diagonalghost"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/a156216-diagonalghost/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A156216","target":"dossier:a156216-diagonalghost"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a160627-israel-laguerre-four-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A160627","target":"dossier:oeis-a160627-israel-laguerre-four-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a160627-israel-laguerre-four-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A160627","target":"dossier:oeis-a160627-israel-laguerre-four-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a160627-israel-laguerre-four-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A160628","target":"dossier:oeis-a160627-israel-laguerre-four-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a160627-israel-laguerre-four-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A160628","target":"dossier:oeis-a160627-israel-laguerre-four-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A160786","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a006068-kurkov-gray-inverse-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A163617","target":"dossier:oeis-a006068-kurkov-gray-inverse-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a090079-yanev-run-compression-closed-form/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A163617","target":"dossier:oeis-a090079-yanev-run-compression-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a163617-gray-code-binary-recurrence-closed-form/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A163617","target":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a163617-gray-code-binary-recurrence-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A163617","target":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a163617-gray-code-binary-recurrence-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A163618","target":"dossier:oeis-a163617-gray-code-binary-recurrence-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A164658","target":"dossier:b164660-chebfloor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A164659","target":"dossier:b164660-chebfloor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A164660","target":"dossier:b164660-chebfloor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/b164660-chebfloor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A164660","target":"dossier:b164660-chebfloor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a165719-eighth-quadratic-integrality-and-sole-prime/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A165719","target":"dossier:oeis-a165719-eighth-quadratic-integrality-and-sole-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167138","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167138","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167139","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167139","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167141","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167138-logarithmic-power-closure/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167141","target":"dossier:oeis-a167138-logarithmic-power-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167540-logarithmic-affine-closure/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167540","target":"dossier:oeis-a167540-logarithmic-affine-closure"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a167857-sorted-divisor-polynomial-three-primes/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A167857","target":"dossier:oeis-a167857-sorted-divisor-polynomial-three-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/knight-dialler-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A169696","target":"dossier:knight-dialler-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a174193-order-four-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A174193","target":"dossier:mathar-a174193-order-four-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mathar-a174193-order-four-recurrence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A174193","target":"dossier:mathar-a174193-order-four-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a174470/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A174470","target":"dossier:hanna-a174470"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a174470/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A174470","target":"dossier:hanna-a174470"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399155-value-finiteness/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A175126","target":"dossier:oeis-a399155-value-finiteness"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a175406-log-two-floor-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A175406","target":"dossier:oeis-a175406-log-two-floor-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A175567","target":"dossier:oeis-a119690-layman-odd-power-factorial-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A175567","target":"dossier:oeis-a119690-layman-odd-power-factorial-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a119690-layman-odd-power-factorial-residue/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A175567","target":"dossier:oeis-a119690-layman-odd-power-factorial-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni350759/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A176118","target":"dossier:tebni350759"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-1989-mahler-binary-digit-square-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A176189","target":"dossier:erdos-1989-mahler-binary-digit-square-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a177680-twin-prime-average-multiple-of-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A177680","target":"dossier:oeis-a177680-twin-prime-average-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a177680-twin-prime-average-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A177680","target":"dossier:oeis-a177680-twin-prime-average-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a178294-four-grid-collinear-triples/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A178294","target":"dossier:oeis-a178294-four-grid-collinear-triples"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a181278-barker-binary-gram-recurrence-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A181274","target":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a181278-barker-binary-gram-recurrence-refutation/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A181278","target":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a181278-barker-binary-gram-recurrence-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A181278","target":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a181278-barker-binary-gram-recurrence-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A181278","target":"dossier:oeis-a181278-barker-binary-gram-recurrence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A183300","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A183300","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A185897","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a338632-conjecture/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A186776","target":"dossier:hanna-a338632-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338636-continued-fraction-odd-square-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A186776","target":"dossier:oeis-a338636-continued-fraction-odd-square-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/witula-slota-2007-quasi-fibonacci-order-eleven-degree/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A189235","target":"dossier:witula-slota-2007-quasi-fibonacci-order-eleven-degree"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a189733-curtz-denominator-conjecture/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A189731","target":"dossier:oeis-a189733-curtz-denominator-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a189733-curtz-denominator-conjecture/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A189733","target":"dossier:oeis-a189733-curtz-denominator-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a189733-curtz-denominator-conjecture/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A189733","target":"dossier:oeis-a189733-curtz-denominator-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A190275","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A190275","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A190275","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a338632-conjecture/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A191107","target":"dossier:hanna-a338632-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338636-continued-fraction-odd-square-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A191107","target":"dossier:oeis-a338636-continued-fraction-odd-square-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192023-comb-wiener-determinant-refutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A192023","target":"dossier:oeis-a192023-comb-wiener-determinant-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192023-comb-wiener-determinant-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A192023","target":"dossier:oeis-a192023-comb-wiener-determinant-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192023-comb-wiener-determinant-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A192023","target":"dossier:oeis-a192023-comb-wiener-determinant-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192023-comb-wiener-determinant-refutation/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A192023","target":"dossier:oeis-a192023-comb-wiener-determinant-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a192763-granvik-mertens-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A192763","target":"dossier:oeis-a192763-granvik-mertens-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-minus-four-eventual-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193020","target":"dossier:oeis-a193020-minus-four-eventual-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-minus-four-eventual-identity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193020","target":"dossier:oeis-a193020-minus-four-eventual-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-plus-one-eventual-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193020","target":"dossier:oeis-a193020-plus-one-eventual-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193020-plus-one-eventual-identity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193020","target":"dossier:oeis-a193020-plus-one-eventual-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193236/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193236","target":"dossier:bala193236"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193236/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193236","target":"dossier:bala193236"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193236/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193236","target":"dossier:bala193236"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193237/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193236","target":"dossier:bala193237"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193236/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193237","target":"dossier:bala193236"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193237/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193237","target":"dossier:bala193237"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193237/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193237","target":"dossier:bala193237"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala193237/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193237","target":"dossier:bala193237"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a193351-alternating-divisor-sum-prime-square/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A193351","target":"dossier:oeis-a193351-alternating-divisor-sum-prime-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A195415","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a221077-tanh-power-prime-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A195415","target":"dossier:oeis-a221077-tanh-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a196226-layman-divisor-power-congruence-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A196226","target":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a196226-layman-divisor-power-congruence-refutation/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A196226","target":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a196226-layman-divisor-power-congruence-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A196226","target":"dossier:oeis-a196226-layman-divisor-power-congruence-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zhao-vincular-stack-three-sorting-classes/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A202062","target":"dossier:zhao-vincular-stack-three-sorting-classes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A202854","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A204061","target":"dossier:oeis-a204061-companion-pell-exp-base-five-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a204061-companion-pell-exp-base-five-residue/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A204061","target":"dossier:oeis-a204061-companion-pell-exp-base-five-residue"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A206715","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A206716","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208342-joint-polynomial-pair-binomial-coefficients/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208342","target":"dossier:oeis-a208342-joint-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208342-joint-polynomial-pair-binomial-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208342","target":"dossier:oeis-a208342-joint-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208342","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208342-joint-polynomial-pair-binomial-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208343","target":"dossier:oeis-a208342-joint-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208508","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208508","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208508","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208509","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208509","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a208509-affine-polynomial-pair-binomial-coefficients/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A208509","target":"dossier:oeis-a208509-affine-polynomial-pair-binomial-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hardin-barker-a209723-edge-differences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A209723","target":"dossier:hardin-barker-a209723-edge-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A210826","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A210826","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A210826","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A212424","target":"dossier:oeis-a000040-detlefs-fibonacci-fermat-prime-characterization-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A213591","target":"dossier:oeis-a396798-eighth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A213591","target":"dossier:oeis-a396798-fifth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A213591","target":"dossier:oeis-a396798-fourth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A213591","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A213591","target":"dossier:oeis-a396798-third-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A215004","target":"dossier:oeis-a382814-fibonachos-score"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A216793","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A216793","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A216955","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A216955","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A217209","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chaffin-sloane-gijswijt-merge-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A217209","target":"dossier:chaffin-sloane-gijswijt-merge-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a219431-hanna-overpartition-log-derivative-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A219430","target":"dossier:oeis-a219431-hanna-overpartition-log-derivative-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a219431-hanna-overpartition-log-derivative-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A219431","target":"dossier:oeis-a219431-hanna-overpartition-log-derivative-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a219431-hanna-overpartition-log-derivative-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A219431","target":"dossier:oeis-a219431-hanna-overpartition-log-derivative-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A220181","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A220181","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A220181","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wu221018-fibdiv/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A221018","target":"dossier:wu221018-fibdiv"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A221077","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a221077-tanh-power-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A221077","target":"dossier:oeis-a221077-tanh-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222001-descent-lex-arrays/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222001","target":"dossier:oeis-a222001-descent-lex-arrays"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222001-descent-lex-arrays/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222001","target":"dossier:oeis-a222001-descent-lex-arrays"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222013-factorial-product-sum-catalan-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222013","target":"dossier:oeis-a222013-factorial-product-sum-catalan-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222014-factorial-square-exponent-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222013","target":"dossier:oeis-a222014-factorial-square-exponent-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222014-factorial-square-exponent-sum-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222013","target":"dossier:oeis-a222014-factorial-square-exponent-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222014-factorial-square-exponent-sum-parity/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222014","target":"dossier:oeis-a222014-factorial-square-exponent-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a222014-factorial-square-exponent-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222760","target":"dossier:noe-2013-a222759-prime-cube-growth"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/noe-2013-a222759-prime-cube-growth/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222760","target":"dossier:noe-2013-a222759-prime-cube-growth"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222955","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222955","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A222955","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A224899","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a224899-sinh-power-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A224899","target":"dossier:oeis-a224899-sinh-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/divisor-doubling-multiple-divides-square/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A225004","target":"dossier:divisor-doubling-multiple-divides-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/divisor-doubling-multiple-divides-square/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A225004","target":"dossier:divisor-doubling-multiple-divides-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wu221018-fibdiv/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A225219","target":"dossier:wu221018-fibdiv"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226077","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226602","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a226839-triangular-egf-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226838","target":"dossier:oeis-a226839-triangular-egf-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a226839-triangular-egf-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226839","target":"dossier:oeis-a226839-triangular-egf-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a226857-fibonacci-sum-product-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226857","target":"dossier:oeis-a226857-fibonacci-sum-product-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a226857-fibonacci-sum-product-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A226857","target":"dossier:oeis-a226857-fibonacci-sum-product-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h227051-cossinhmod10/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A227051","target":"dossier:h227051-cossinhmod10"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376527-q-catalan-square-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A227543","target":"dossier:oeis-a376527-q-catalan-square-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376527-q-catalan-square-sum-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228058","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228058","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228558","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228558","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228558","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228558","target":"dossier:oeis-a231329-price-nineteen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228884","target":"dossier:sun228884"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sun228884/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A228884","target":"dossier:sun228884"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A229140","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A229140","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sun-2013-circular-primitive-permutation-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A229232","target":"dossier:sun-2013-circular-primitive-permutation-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a228558-price-seventeen-four-prime-exponent/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a228558-price-seventeen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a231329-price-nineteen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a231329-price-nineteen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231329-price-nineteen-four-prime-exponent/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231329","target":"dossier:oeis-a231329-price-nineteen-four-prime-exponent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231548-antisigma-decrease-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231548","target":"dossier:oeis-a231548-antisigma-decrease-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a231548-antisigma-decrease-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A231548","target":"dossier:oeis-a231548-antisigma-decrease-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232686/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232686","target":"dossier:hefg-a232686"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232686/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232686","target":"dossier:hefg-a232686"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232686/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232694","target":"dossier:hefg-a232694"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232694/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232694","target":"dossier:hefg-a232694"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232694/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232694","target":"dossier:hefg-a232694"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232694/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232694","target":"dossier:hefg-a232694"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232694/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232694","target":"dossier:hefg-a232694"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232695/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232695","target":"dossier:hefg-a232695"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232695/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232696","target":"dossier:hefg-a232696"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232696/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232696","target":"dossier:hefg-a232696"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hefg-a232696/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A232696","target":"dossier:hefg-a232696"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A234387","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A235224","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-adjacent-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A238474","target":"dossier:oeis-a378495-adjacent-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A239293","target":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239293-ordowski-immediate-successor-weak-pseudoprime/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A239293","target":"dossier:oeis-a239293-ordowski-immediate-successor-weak-pseudoprime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239293-ordowski-immediate-successor-weak-pseudoprime/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A239293","target":"dossier:oeis-a239293-ordowski-immediate-successor-weak-pseudoprime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A239907","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A239907","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a189733-curtz-denominator-conjecture/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A242926","target":"dossier:oeis-a189733-curtz-denominator-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a224899-sinh-power-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A245322","target":"dossier:oeis-a224899-sinh-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A245778","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A245786","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A245786","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a245786-integrality-gcd-record-refutation/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A245786","target":"dossier:oeis-a245786-integrality-gcd-record-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246056-a246423-legendre-central-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246056","target":"dossier:hanna-a246056-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246056-a246423-legendre-central-binomial/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246056","target":"dossier:hanna-a246056-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246423-legendre-central-binomial/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246056","target":"dossier:hanna-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246056-a246423-legendre-central-binomial/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246423","target":"dossier:hanna-a246056-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246423-legendre-central-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246423","target":"dossier:hanna-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a246423-legendre-central-binomial/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A246423","target":"dossier:hanna-a246423-legendre-central-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a072872-power-minus-index-prime-bound-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247248","target":"dossier:oeis-a072872-power-minus-index-prime-bound-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247477","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247477","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247477","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247477","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bradshaw-arithmetic-derivative-collatz-squarefree-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A247583","target":"dossier:bradshaw-arithmetic-derivative-collatz-squarefree-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fried-fibonacci-shift-non-fibonacci/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A248982","target":"dossier:fried-fibonacci-shift-non-fibonacci"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fried-fibonacci-shift-non-fibonacci/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A248982","target":"dossier:fried-fibonacci-shift-non-fibonacci"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/greedy-fibonacci-average-closed-form/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A248982","target":"dossier:greedy-fibonacci-average-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/greedy-fibonacci-average-closed-form/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A248982","target":"dossier:greedy-fibonacci-average-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/greedy-fibonacci-average-closed-form/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A248982","target":"dossier:greedy-fibonacci-average-closed-form"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249621-square-prepend-one-append-twenty-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249621","target":"dossier:oeis-a249621-square-prepend-one-append-twenty-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249759","target":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249759","target":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249759","target":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249759","target":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a249759-krizek-sigma-prime-fermat-mersenne/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249761","target":"dossier:oeis-a249759-krizek-sigma-prime-fermat-mersenne"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249872","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249872","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A249872","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A253368","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254074-wu-digit-range-square-not-multiple-of-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A254074","target":"dossier:oeis-a254074-wu-digit-range-square-not-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A254748","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a254748-twin-prime-mean-no-superdivisor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A254748","target":"dossier:oeis-a254748-twin-prime-mean-no-superdivisor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258409","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258409","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-a258899-totient-period/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258899","target":"dossier:bala-a258899-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-a258899-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258899","target":"dossier:bala-a258899-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-a258899-totient-period/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258903","target":"dossier:bala-a258899-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-a258899-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A258903","target":"dossier:bala-a258899-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260072","target":"dossier:oeis-a260407-krizek-all-multiples-of-the-order"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-a363569-hanna-bilateral-power-and-prime-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363558-a363569-hanna-bilateral-power-and-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-a363569-hanna-bilateral-power-and-prime-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363558-a363569-hanna-bilateral-power-and-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-hanna-bilateral-prime-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363558-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363558-hanna-bilateral-prime-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363558-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-power-of-two-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363559-hanna-bilateral-power-of-two-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-power-of-two-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363559-hanna-bilateral-power-of-two-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-prime-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363559-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363559-hanna-bilateral-prime-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363559-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-power-of-two-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363561-hanna-bilateral-power-of-two-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-power-of-two-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363561-hanna-bilateral-power-of-two-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-prime-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363561-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363561-hanna-bilateral-prime-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363561-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363569-hanna-bilateral-power-of-two-values/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363569-hanna-bilateral-power-of-two-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363569-hanna-bilateral-power-of-two-values/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260147","target":"dossier:oeis-a363569-hanna-bilateral-prime-values"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260310","target":"dossier:oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260310","target":"dossier:oeis-a260310-balanced-prime-divisor-pair-mod-six-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260406","target":"dossier:oeis-a260407-krizek-all-multiples-of-the-order"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260407","target":"dossier:oeis-a260407-krizek-all-multiples-of-the-order"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260407","target":"dossier:oeis-a260407-krizek-all-multiples-of-the-order"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a260407-krizek-all-multiples-of-the-order/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A260407","target":"dossier:oeis-a260407-krizek-all-multiples-of-the-order"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380709-absolute-reciprocal-cube-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263132","target":"dossier:oeis-a380709-absolute-reciprocal-cube-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a369083-reflected-quadratic-quarter-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263133","target":"dossier:oeis-a369083-reflected-quadratic-quarter-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263968","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263968","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263968","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/parisse-stirling-fibonacci-alternating-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A263968","target":"dossier:parisse-stirling-fibonacci-alternating-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/partition-weighted-sum-reversal/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A264034","target":"dossier:partition-weighted-sum-reversal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A264596","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a030101-yanev-binary-reversal-position-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A264596","target":"dossier:oeis-a030101-yanev-binary-reversal-position-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a265714-positive-threshold-common-minimum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A265714","target":"dossier:oeis-a265714-positive-threshold-common-minimum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotdp-weighteddistinct/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A265840","target":"dossier:kotdp-weighteddistinct"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotdp-weighteddistinct/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A265841","target":"dossier:kotdp-weighteddistinct"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotdp-weighteddistinct/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A265842","target":"dossier:kotdp-weighteddistinct"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266285-rule13-barker-zizka/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266285","target":"dossier:oeis-a266285-rule13-barker-zizka"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266285-rule13-barker-zizka/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266285","target":"dossier:oeis-a266285-rule13-barker-zizka"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-diagonal-index-divisibility/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266489","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266614-rule41-barker/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266614","target":"dossier:oeis-a266614-rule41-barker"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266614-rule41-barker/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A266614","target":"dossier:oeis-a266614-rule41-barker"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a267681-rule201-barker-hasler/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A267581","target":"dossier:oeis-a267681-rule201-barker-hasler"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a267681-rule201-barker-hasler/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A267680","target":"dossier:oeis-a267681-rule201-barker-hasler"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a267681-rule201-barker-hasler/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A267681","target":"dossier:oeis-a267681-rule201-barker-hasler"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a267681-rule201-barker-hasler/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A267681","target":"dossier:oeis-a267681-rule201-barker-hasler"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A268293","target":"dossier:oeis-a266489-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A268922","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A268922","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A269591","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a144837-bala-lucas-five-adic-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A269591","target":"dossier:oeis-a144837-bala-lucas-five-adic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-antepenultimate-label-count/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-antepenultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-antepenultimate-label-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-antepenultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-antepenultimate-label-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-antepenultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-penultimate-label-count/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-penultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-penultimate-label-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-penultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a270236-rgf-penultimate-label-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A270236","target":"dossier:oeis-a270236-rgf-penultimate-label-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-second-bit-run-length/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271591","target":"dossier:fibonacci-second-bit-run-length"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271931","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271931","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271932","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271932","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271933","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271933","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271959","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A271960","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-second-bit-run-length/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A272170","target":"dossier:fibonacci-second-bit-run-length"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/fibonacci-second-bit-run-length/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A272170","target":"dossier:fibonacci-second-bit-run-length"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/manetti-ricciardi-2015-higher-antibracket-coefficients/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A272688","target":"dossier:manetti-ricciardi-2015-higher-antibracket-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/manetti-ricciardi-2015-higher-antibracket-coefficients/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A272688","target":"dossier:manetti-ricciardi-2015-higher-antibracket-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/manetti-ricciardi-2015-higher-antibracket-coefficients/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A272688","target":"dossier:manetti-ricciardi-2015-higher-antibracket-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a274010-stechkin-divisor-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274010","target":"dossier:oeis-a274010-stechkin-divisor-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112302-lerch-somos-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274181","target":"dossier:oeis-a112302-lerch-somos-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274190","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274190","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274190","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274190","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274199","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274199","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/gabric-shallit-2023-block-palindrome-expected-width/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274199","target":"dossier:gabric-shallit-2023-block-palindrome-expected-width"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274478","target":"dossier:hanna-a274479-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274478","target":"dossier:hanna-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274479","target":"dossier:hanna-a274479-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274479","target":"dossier:hanna-a274479-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274479","target":"dossier:hanna-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274484","target":"dossier:hanna-a274479-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A274484","target":"dossier:hanna-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-shift-square-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A276370","target":"dossier:hanna-shift-square-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h277403-div10/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A277403","target":"dossier:h277403-div10"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h277403-div10/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A277403","target":"dossier:h277403-div10"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/saied-marshall-anand-rieffel-2025-heralding-rate-monotonicity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A277458","target":"dossier:saied-marshall-anand-rieffel-2025-heralding-rate-monotonicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a277829-a278770-catalan-square-hankel-limits/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A277829","target":"dossier:oeis-a277829-a278770-catalan-square-hankel-limits"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a277829-a278770-catalan-square-hankel-limits/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A278770","target":"dossier:oeis-a277829-a278770-catalan-square-hankel-limits"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280258","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280258","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280258","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a280258-krizek-phitorial-divisor-sum-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280258","target":"dossier:oeis-a280258-krizek-phitorial-divisor-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280523","target":"dossier:oeis-a382814-fibonachos-score"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/knight-dialler-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A280594","target":"dossier:knight-dialler-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283303","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283303","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283304","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283304","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283477","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A283477","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A284005","target":"dossier:oeis-a284005-kurkov-lower-bit-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A284005","target":"dossier:oeis-a284005-kurkov-lower-bit-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A284005","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A284005","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A289508","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A289508","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A289508","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A289832","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A289832","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a290322-cyclotomic-five-residue-sum/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A290309","target":"dossier:oeis-a290322-cyclotomic-five-residue-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/granvik-2024-a292273-real-part/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A292273","target":"dossier:granvik-2024-a292273-real-part"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a292394-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A292394","target":"dossier:oeis-a292394-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a292394-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A292395","target":"dossier:oeis-a292394-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A293211","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A293211","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A293452","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A293452","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A293452","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347854-bala-half-index-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347854-bala-half-index-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-aux-b-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347855-bala-aux-b-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-aux-c-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347855-bala-aux-c-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347855-bala-third-index-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347855-bala-third-index-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347856-bala-half-index-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347856-bala-half-index-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347857-bala-half-index-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347857-bala-half-index-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347858-bala-half-index-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295432","target":"dossier:oeis-a347858-bala-half-index-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala295462/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295462","target":"dossier:bala295462"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala295462/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A295462","target":"dossier:bala295462"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A296306","target":"dossier:bala-machine3626-exact-halting-time"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A296306","target":"dossier:bala-machine3626-exact-halting-time"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a001222-schulte-omega-binomial-convolution/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A299406","target":"dossier:oeis-a001222-schulte-omega-binomial-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A299406","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A299406","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A299406","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a299406-schulte-liouville-cube-mobius/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A299406","target":"dossier:oeis-a299406-schulte-liouville-cube-mobius"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a359482-positive-permutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300000","target":"dossier:oeis-a359482-positive-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300657-krizek-divisor-sigma-mod-noncomposite/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300657","target":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300657-krizek-divisor-sigma-mod-noncomposite/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300657","target":"dossier:oeis-a300657-krizek-divisor-sigma-mod-noncomposite"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300732-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300732","target":"dossier:oeis-a300732-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300733-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300733","target":"dossier:oeis-a300733-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a300734-diagonal-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300734","target":"dossier:oeis-a300734-diagonal-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300787","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300788","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/balanced-composition-paths/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A300789","target":"dossier:balanced-composition-paths"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301417","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301420","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301420","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301420","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301421","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-mod-ten/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301976","target":"dossier:oeis-a301976-grid-no-leaf-mod-ten"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-mod-ten/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301976","target":"dossier:oeis-a301976-grid-no-leaf-mod-ten"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301976","target":"dossier:oeis-a301976-grid-no-leaf-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301976-grid-no-leaf-recurrence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A301976","target":"dossier:oeis-a301976-grid-no-leaf-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A302764","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a301420-positive-newton-sums/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A302764","target":"dossier:oeis-a301420-positive-newton-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a302975-divisor-count-power-ratio-denominator-square/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A302975","target":"dossier:oeis-a302975-divisor-count-power-ratio-denominator-square"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/partition-weighted-sum-reversal/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A304818","target":"dossier:partition-weighted-sum-reversal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-distinct-part-egf-totient-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-distinct-part-egf-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a305550-integral-egf-substitution-totient-period/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A305550","target":"dossier:oeis-a305550-integral-egf-substitution-totient-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A306270","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A306270","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A306270","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a306270-ordowski-semiprime-carmichael-square-classification/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A306270","target":"dossier:oeis-a306270-ordowski-semiprime-carmichael-square-classification"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A307408","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A307409","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308090-caceres-gcd-factorial-power-prime-criterion/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A308090","target":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308090-caceres-gcd-factorial-power-prime-criterion/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A308090","target":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308090-caceres-gcd-factorial-power-prime-criterion/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A308090","target":"dossier:oeis-a308090-caceres-gcd-factorial-power-prime-criterion"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a308815-dfinite-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A308815","target":"dossier:oeis-a308815-dfinite-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A309221","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A309221","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399155-value-finiteness/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A309892","target":"dossier:oeis-a399155-value-finiteness"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a133907-ordowski-binomial-power-sum-least-prime/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A317358","target":"dossier:oeis-a133907-ordowski-binomial-power-sum-least-prime"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp322844-serpentine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A317614","target":"dossier:sp322844-serpentine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp322844-serpentine-higher/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A317614","target":"dossier:sp322844-serpentine-higher"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/partition-weighted-sum-reversal/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A318283","target":"dossier:partition-weighted-sum-reversal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319197","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319197","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319197-lang-fib-dyadic-index-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319197","target":"dossier:oeis-a319197-lang-fib-dyadic-index-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319227","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-linear-jaco-maximum-degree/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319433","target":"dossier:kok-linear-jaco-maximum-degree"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-linear-jaco-maximum-degree/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319433","target":"dossier:kok-linear-jaco-maximum-degree"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kok-linear-jaco-maximum-degree/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319433","target":"dossier:kok-linear-jaco-maximum-degree"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a319927-odd-nonunitary-power-sum-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A319927","target":"dossier:oeis-a319927-odd-nonunitary-power-sum-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/a320043-square-values-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A320043","target":"dossier:a320043-square-values-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A320632","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322327","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322327","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322327","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322327","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a322327-schulte-exponent-product-omega-power-multiplicative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322328","target":"dossier:oeis-a322327-schulte-exponent-product-omega-power-multiplicative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322437","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp322844-serpentine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322844","target":"dossier:sp322844-serpentine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp322844-serpentine-higher/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A322844","target":"dossier:sp322844-serpentine-higher"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A323689","target":"dossier:hanna323-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A323689","target":"dossier:hanna323-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A323689","target":"dossier:hanna323-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A323689","target":"dossier:hanna324-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A323689","target":"dossier:hanna324-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A324300","target":"dossier:hanna323-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna323-gaussparity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A324300","target":"dossier:hanna323-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A324300","target":"dossier:hanna324-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A324300","target":"dossier:hanna324-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna324-gaussparity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A324300","target":"dossier:hanna324-gaussparity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a326244-noncrossing-nonnesting-graph-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326244","target":"dossier:oeis-a326244-noncrossing-nonnesting-graph-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a326244-noncrossing-nonnesting-graph-recurrence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326244","target":"dossier:oeis-a326244-noncrossing-nonnesting-graph-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326954","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326954","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326954","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326955","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326955","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sloane-2019-a309221-knight-walk-range-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A326955","target":"dossier:sloane-2019-a309221-knight-walk-range-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/knight-dialler-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A327692","target":"dossier:knight-dialler-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/knight-dialler-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A327692","target":"dossier:knight-dialler-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328023","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328023","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328023","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328163","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a258409-divisor-difference-gcd-heinz/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328164","target":"dossier:oeis-a258409-divisor-difference-gcd-heinz"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328803","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328803","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a328959-divisor-count-prime-factor-bound/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A328959","target":"dossier:oeis-a328959-divisor-count-prime-factor-bound"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329278-triangular-residue-descent-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329278","target":"dossier:oeis-a329278-triangular-residue-descent-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329320","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329320","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a239907-kurkov-run-alternating-count/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329320","target":"dossier:oeis-a239907-kurkov-run-alternating-count"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-lower-bit-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329369","target":"dossier:oeis-a284005-kurkov-lower-bit-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a284005-kurkov-weight-product-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329369","target":"dossier:oeis-a284005-kurkov-weight-product-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329369","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kag329479-binarynull/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329479","target":"dossier:kag329479-binarynull"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kag329479-binarynull/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A329479","target":"dossier:kag329479-binarynull"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp330700/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A330287","target":"dossier:sp330700"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp330700/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A330700","target":"dossier:sp330700"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A330714","target":"dossier:oeis-a131853-binary-gaussian-evaluation-zero-multiple-of-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331473","target":"dossier:oeis-a331474-hankel-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331473","target":"dossier:oeis-a331474-hankel-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331473","target":"dossier:oeis-a331474-hankel-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331474","target":"dossier:oeis-a331474-hankel-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a331474-hankel-generating-function/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331474","target":"dossier:oeis-a331474-hankel-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331859","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331859","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331903","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331903","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331903","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A331904","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kagey-2020-a331859-colliding-blocks-records/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A332045","target":"dossier:kagey-2020-a331859-colliding-blocks-records"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A332565","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a334595-value-one/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A334595","target":"dossier:oeis-a334595-value-one"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a334595-value-one/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A334595","target":"dossier:oeis-a334595-value-one"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a334595-value-one/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A334595","target":"dossier:oeis-a334595-value-one"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335901-self-referential-doubling-first-occurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A335901","target":"dossier:oeis-a335901-self-referential-doubling-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335901-self-referential-doubling-first-occurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A335901","target":"dossier:oeis-a335901-self-referential-doubling-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335925-self-referential-quotient-first-occurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A335925","target":"dossier:oeis-a335925-self-referential-quotient-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a335925-self-referential-quotient-first-occurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A335925","target":"dossier:oeis-a335925-self-referential-quotient-first-occurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a220181-alternating-weight-stirling-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338040","target":"dossier:oeis-a220181-alternating-weight-stirling-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338040-weighted-stirling-power-prime-period/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338040","target":"dossier:oeis-a338040-weighted-stirling-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338040-weighted-stirling-power-prime-period/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338040","target":"dossier:oeis-a338040-weighted-stirling-power-prime-period"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338193-egf-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338193","target":"dossier:oeis-a338193-egf-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338325","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338325","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338325","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-monomial-sums/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338325","target":"dossier:kotesovec-2024-a368959-monomial-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-monomial-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338325","target":"dossier:kotesovec-2024-a368959-monomial-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a338632-conjecture/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338632","target":"dossier:hanna-a338632-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a338632-conjecture/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338632","target":"dossier:hanna-a338632-conjecture"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a338636-continued-fraction-odd-square-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A338636","target":"dossier:oeis-a338636-continued-fraction-odd-square-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A339621","target":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A339621","target":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339621-fibonacci-divisor-sum-even-index/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A339621","target":"dossier:oeis-a339621-fibonacci-divisor-sum-even-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339669-odd-lucas-square-fibonacci-divisors/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A339669","target":"dossier:oeis-a339669-odd-lucas-square-fibonacci-divisors"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a339669-odd-lucas-square-fibonacci-divisors/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A339669","target":"dossier:oeis-a339669-odd-lucas-square-fibonacci-divisors"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kimb341249-bound/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A341724","target":"dossier:tebni341724-row-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A341862","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/erdos-1989-mahler-binary-digit-square-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A342546","target":"dossier:erdos-1989-mahler-binary-digit-square-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/joshi-rust-2025-thue-morse-first-longest-start/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A342827","target":"dossier:joshi-rust-2025-thue-morse-first-longest-start"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a387319-multiple-of-three-coincidence/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343311","target":"dossier:oeis-a387319-multiple-of-three-coincidence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343502-divisor-count-primes-dequatre/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343502","target":"dossier:oeis-a343502-divisor-count-primes-dequatre"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343502-divisor-count-primes-dequatre/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343656","target":"dossier:coprime-divisor-tuples"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343825","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343825","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a343825-kagey-power-sequences/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A343881","target":"dossier:oeis-a343825-kagey-power-sequences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344083-marcus-pythagorean-nonpolygonal-triple/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344083","target":"dossier:oeis-a344083-marcus-pythagorean-nonpolygonal-triple"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344083-marcus-pythagorean-nonpolygonal-triple/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344083","target":"dossier:oeis-a344083-marcus-pythagorean-nonpolygonal-triple"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344598-alternating-gcd-sum-pillai/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344372","target":"dossier:oeis-a344598-alternating-gcd-sum-pillai"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344598-alternating-gcd-sum-pillai/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344598","target":"dossier:oeis-a344598-alternating-gcd-sum-pillai"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344611","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344611","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344611","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344695","target":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a344695-gcd-sigma-psi-odd-primes/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344695","target":"dossier:oeis-a344695-gcd-sigma-psi-odd-primes"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344743","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344743","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a344611/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A344743","target":"dossier:wiseman-a344611"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni350759/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A345652","target":"dossier:tebni350759"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/tebni350759/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A345652","target":"dossier:tebni350759"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347285-de-vlieger-prime-power-descent/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A347285","target":"dossier:oeis-a347285-de-vlieger-prime-power-descent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347285-de-vlieger-prime-power-descent/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A347285","target":"dossier:oeis-a347285-de-vlieger-prime-power-descent"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a347285-de-vlieger-prime-power-descent/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A350759","target":"dossier:tebni350759"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A350900","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A350900","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/onnis-2022-square-three-step-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A351327","target":"dossier:onnis-2022-square-three-step-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363956-positive-permutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A351495","target":"dossier:oeis-a363956-positive-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352698","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352698","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352699","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352700","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352700","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352700","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352701","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352702","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a352700-hanna-catalan-congruences/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A352702","target":"dossier:oeis-a352700-hanna-catalan-congruences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/dekking-keane-2022-two-block-thue-morse-frequency/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A354896","target":"dossier:dekking-keane-2022-two-block-thue-morse-frequency"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A355615","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a079278-quet-rational-iteration-denominator-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A355615","target":"dossier:oeis-a079278-quet-rational-iteration-denominator-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a355872-bilateral-theta-reversion-mod-sixteen/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A355872","target":"dossier:oeis-a355872-bilateral-theta-reversion-mod-sixteen"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A356247","target":"dossier:bouras"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras-a363102/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A356247","target":"dossier:bouras-a363102"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a356508-hanna-product-partition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A356508","target":"dossier:oeis-a356508-hanna-product-partition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A357127","target":"dossier:bouras"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras-a363102/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A357127","target":"dossier:bouras-a363102"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kimberling-array-first-occurrence-l4v5/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A358359","target":"dossier:kimberling-array-first-occurrence-l4v5"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A359402","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A359402","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/least-squares-zero-slope-binary-words/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A359402","target":"dossier:least-squares-zero-slope-binary-words"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a359482-positive-permutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A359482","target":"dossier:oeis-a359482-positive-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a361552-literal-partition-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A359914","target":"dossier:oeis-a361552-literal-partition-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a361552-literal-partition-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A363184","target":"dossier:oeis-a363184-bilateral-quarter-double-shift-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a363308-catalan-cubic-composition-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A363308","target":"dossier:oeis-a363308-catalan-cubic-composition-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365095","target":"dossier:oeis-a365095-perturbed-diagonal-square-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365574-hanna-lagrange-parity-conjecture-1/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365574","target":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365574-hanna-lagrange-parity-conjecture-1/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365574","target":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365574-hanna-lagrange-parity-conjecture-1/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365574","target":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380711-hanna-absolute-cubic-congruence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365574","target":"dossier:oeis-a380711-hanna-absolute-cubic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365771","target":"dossier:hanna365771-mod5"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-two/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365771","target":"dossier:hanna365771-mod5-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-two/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365771","target":"dossier:hanna365771-mod5-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-zero/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365771","target":"dossier:hanna365771-mod5-zero"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna365771-mod5-zero/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A365771","target":"dossier:hanna365771-mod5-zero"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A367396","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A367396","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A367400","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a367400-cardinality-sum-avoiding-subsets/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A367400","target":"dossier:oeis-a367400-cardinality-sum-avoiding-subsets"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A368959","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-family/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A368959","target":"dossier:kotesovec-2024-a368959-family"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-monomial-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A368959","target":"dossier:kotesovec-2024-a368959-monomial-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kotesovec-2024-a368959-monomial-sums/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A368959","target":"dossier:kotesovec-2024-a368959-monomial-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a369083-reflected-quadratic-quarter-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A369083","target":"dossier:oeis-a369083-reflected-quadratic-quarter-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A369673","target":"dossier:h369673"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h369673/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A369673","target":"dossier:h369673"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370089","target":"dossier:oeis-a381190-trapezohedral-domination-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370380","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370380","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370381","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a003319-kurkov-double-array/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370381","target":"dossier:oeis-a003319-kurkov-double-array"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370630","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370630","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370630","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a370630-zeckendorf-greedy-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A370631","target":"dossier:oeis-a370630-zeckendorf-greedy-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a372527-square-composition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372527","target":"dossier:hanna-a372527-square-composition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a372577-double-composition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372577","target":"dossier:oeis-a372577-double-composition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372706","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372706","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372712","target":"dossier:oeis-a381190-trapezohedral-domination-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bouras372763-once/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372763","target":"dossier:bouras372763-once"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a372991-factorial-quotient-linear-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A372991","target":"dossier:oeis-a372991-factorial-quotient-linear-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a350900-schulte-gcd-quotient-row-sum-convolution/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A373059","target":"dossier:oeis-a350900-schulte-gcd-quotient-row-sum-convolution"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a329369-printed-row-sum-range/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A373183","target":"dossier:oeis-a329369-printed-row-sum-range"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373308-cubed-binary-product-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A373308","target":"dossier:oeis-a373308-cubed-binary-product-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373312-unit-reversion-square-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A373312","target":"dossier:oeis-a373312-unit-reversion-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a373561-gcd-bucketed-quadruple-sum/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A373561","target":"dossier:oeis-a373561-gcd-bucketed-quadruple-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374568-catalan-double-composition-powers-of-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A374568","target":"dossier:oeis-a374568-catalan-double-composition-powers-of-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374570-catalan-composition-square-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A374570","target":"dossier:oeis-a374570-catalan-composition-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a374571-fibbinary-square-substitution-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A374571","target":"dossier:oeis-a374571-fibbinary-square-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397436-general-supercongruence/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375178","target":"dossier:oeis-a397436-general-supercongruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375439-cubic-ternary-substitution-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375439","target":"dossier:oeis-a375439-cubic-ternary-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375449","target":"dossier:oeis-a375449-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375449","target":"dossier:oeis-a375449-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375449","target":"dossier:oeis-a375449-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375449","target":"dossier:oeis-a375452-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375449","target":"dossier:oeis-a375452-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375452","target":"dossier:oeis-a375449-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375449-hanna-diagonal-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375452","target":"dossier:oeis-a375449-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375452","target":"dossier:oeis-a375452-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375452","target":"dossier:oeis-a375452-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375452-hanna-diagonal-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375452","target":"dossier:oeis-a375452-hanna-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375786-cuboid-minimum-volume-huber/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375785","target":"dossier:oeis-a375786-cuboid-minimum-volume-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375786-cuboid-minimum-volume-huber/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375785","target":"dossier:oeis-a375786-cuboid-minimum-volume-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375786-cuboid-minimum-volume-huber/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375786","target":"dossier:oeis-a375786-cuboid-minimum-volume-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a375786-cuboid-minimum-volume-huber/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A375786","target":"dossier:oeis-a375786-cuboid-minimum-volume-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376224","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376317","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376317","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376388","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376388","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376317-kurkov-periodic-boundary-schroder/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376388","target":"dossier:oeis-a376317-kurkov-periodic-boundary-schroder"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376527-q-catalan-square-sum-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376527","target":"dossier:oeis-a376527-q-catalan-square-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a376527-q-catalan-square-sum-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A376527","target":"dossier:oeis-a376527-q-catalan-square-sum-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a274479-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A377100","target":"dossier:hanna-a274479-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A377100","target":"dossier:hanna-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a377100-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A377100","target":"dossier:hanna-a377100-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a271931-hanna-prime-substitution-integrality/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A377106","target":"dossier:oeis-a271931-hanna-prime-substitution-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378247","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378252-running-sum-residue-zero/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378252","target":"dossier:oeis-a378252-running-sum-residue-zero"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378258-hanna-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378258","target":"dossier:oeis-a378258-hanna-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378258-hanna-square-root-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378258","target":"dossier:oeis-a378258-hanna-square-root-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378258-hanna-square-root-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378258","target":"dossier:oeis-a378258-hanna-square-root-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378277-harmonic-triangle-alternating-row-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378277","target":"dossier:oeis-a378277-harmonic-triangle-alternating-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-adjacent-difference/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378495","target":"dossier:oeis-a378495-adjacent-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-adjacent-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378495","target":"dossier:oeis-a378495-adjacent-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378495-terminal-difference/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378495","target":"dossier:oeis-a378495-terminal-difference"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378575-shifted-iterate-fixed-point-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378575","target":"dossier:oeis-a378575-shifted-iterate-fixed-point-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378575-shifted-iterate-fixed-point-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378575","target":"dossier:oeis-a378575-shifted-iterate-fixed-point-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378576-shifted-iterate-fixed-point-mod-five/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378575","target":"dossier:oeis-a378576-shifted-iterate-fixed-point-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378575-shifted-iterate-fixed-point-mod-four/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378576","target":"dossier:oeis-a378575-shifted-iterate-fixed-point-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378576-shifted-iterate-fixed-point-mod-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378576","target":"dossier:oeis-a378576-shifted-iterate-fixed-point-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378576-shifted-iterate-fixed-point-mod-five/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378576","target":"dossier:oeis-a378576-shifted-iterate-fixed-point-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378578-strip-three-ternary-catalan-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378578","target":"dossier:oeis-a378578-strip-three-ternary-catalan-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378578-strip-three-ternary-catalan-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378578","target":"dossier:oeis-a378578-strip-three-ternary-catalan-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378580-quotient-theta-composition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378580","target":"dossier:oeis-a378580-quotient-theta-composition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a378581-theta-self-composition-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378581","target":"dossier:oeis-a378581-theta-self-composition-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378582","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A378582","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-bilateral-reversion-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379202","target":"dossier:hanna-bilateral-reversion-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379204-bilateral-quartic-power-theta-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379204","target":"dossier:oeis-a379204-bilateral-quartic-power-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393434-factorial-heraclitus/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379719","target":"dossier:oeis-a393434-factorial-heraclitus"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379764","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379764","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a379764-hanna-theta-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A379764","target":"dossier:oeis-a379764-hanna-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380056-fourth-index-divisible-by-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380056","target":"dossier:oeis-a380056-fourth-index-divisible-by-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a380060-bilateral-theta-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380060","target":"dossier:hanna-a380060-bilateral-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a380060-bilateral-theta-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380060","target":"dossier:hanna-a380060-bilateral-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380060","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380060","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380060","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380066-hanna-bilateral-product-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380066","target":"dossier:oeis-a380066-hanna-bilateral-product-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380066-hanna-bilateral-product-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380066","target":"dossier:oeis-a380066-hanna-bilateral-product-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380066-hanna-bilateral-product-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380066","target":"dossier:oeis-a380066-hanna-bilateral-product-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380558","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380558","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380558-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380558","target":"dossier:oeis-a380558-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380678-quadratic-square-reversion-dyadic-support-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380678","target":"dossier:oeis-a380678-quadratic-square-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a380060-bilateral-theta-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380680","target":"dossier:hanna-a380060-bilateral-theta-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380680","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380680","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380680-hanna-bilateral-mod-four/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380680","target":"dossier:oeis-a380680-hanna-bilateral-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380681","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380681","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380681","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380682","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380683","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380684","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380685","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380686","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380686","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380681-hanna-prime-bilateral-integrality/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380686","target":"dossier:oeis-a380681-hanna-prime-bilateral-integrality"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380708","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380708","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380709","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380709-absolute-reciprocal-cube-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380709","target":"dossier:oeis-a380709-absolute-reciprocal-cube-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380708-absolute-reciprocal-square-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380710","target":"dossier:oeis-a380708-absolute-reciprocal-square-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380709-absolute-reciprocal-cube-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380710","target":"dossier:oeis-a380709-absolute-reciprocal-cube-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380710-absolute-reciprocal-square-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380710","target":"dossier:oeis-a380710-absolute-reciprocal-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a365574-hanna-lagrange-parity-conjecture-1/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380711","target":"dossier:oeis-a365574-hanna-lagrange-parity-conjecture-1"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380711-hanna-absolute-cubic-congruence/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380711","target":"dossier:oeis-a380711-hanna-absolute-cubic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380711-hanna-absolute-cubic-congruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380711","target":"dossier:oeis-a380711-hanna-absolute-cubic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380711-hanna-absolute-cubic-congruence/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A380711","target":"dossier:oeis-a380711-hanna-absolute-cubic-congruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381190","target":"dossier:oeis-a381190-trapezohedral-domination-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381190","target":"dossier:oeis-a381190-trapezohedral-domination-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/meeussen-2025-a381265-two-layer-solid/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381265","target":"dossier:meeussen-2025-a381265-two-layer-solid"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380066-hanna-bilateral-product-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a380066-hanna-bilateral-product-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a381363-bilateral-product-cubic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381362","target":"dossier:oeis-a381363-bilateral-product-cubic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381363","target":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381362-bilateral-product-quadratic-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381363","target":"dossier:oeis-a381362-bilateral-product-quadratic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381363","target":"dossier:oeis-a381363-bilateral-product-cubic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381363","target":"dossier:oeis-a381363-bilateral-product-cubic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381363-bilateral-product-cubic-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381363","target":"dossier:oeis-a381363-bilateral-product-cubic-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381364","target":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381364","target":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381364","target":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381364","target":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381364","target":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381365","target":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381364-scaled-bilateral-product-linear-mod-nine/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381365","target":"dossier:oeis-a381364-scaled-bilateral-product-linear-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381365","target":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381365","target":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381365-scaled-bilateral-product-quadratic-mod-nine/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381365","target":"dossier:oeis-a381365-scaled-bilateral-product-quadratic-mod-nine"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381423-hermite-differences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381423","target":"dossier:oeis-a381423-hermite-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381423-hermite-differences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381423","target":"dossier:oeis-a381423-hermite-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381524-hermite-absolute-differences/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381423","target":"dossier:oeis-a381524-hermite-absolute-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381524-hermite-absolute-differences/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381524","target":"dossier:oeis-a381524-hermite-absolute-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381524-hermite-absolute-differences/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381524","target":"dossier:oeis-a381524-hermite-absolute-differences"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381670-compositional-square-dyadic-denominators/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381669","target":"dossier:oeis-a381670-compositional-square-dyadic-denominators"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381670-compositional-square-dyadic-denominators/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381670","target":"dossier:oeis-a381670-compositional-square-dyadic-denominators"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382427-constant-blocks-distinct-run-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381717","target":"dossier:oeis-a382427-constant-blocks-distinct-run-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382427-constant-blocks-distinct-run-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381717","target":"dossier:oeis-a382427-constant-blocks-distinct-run-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381737-hermite-absolute-ties/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381737","target":"dossier:oeis-a381737-hermite-absolute-ties"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381737-hermite-tie-generating-function/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381737","target":"dossier:oeis-a381737-hermite-tie-generating-function"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381737-hermite-tie-recurrence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381737","target":"dossier:oeis-a381737-hermite-tie-recurrence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a381190-trapezohedral-domination-series/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A381794","target":"dossier:oeis-a381190-trapezohedral-domination-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382122","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382122","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382124-hanna-sigma-exponential-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382124","target":"dossier:oeis-a382124-hanna-sigma-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382125-hanna-sigma-exponential-mod-three-cube/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382124","target":"dossier:oeis-a382125-hanna-sigma-exponential-mod-three-cube"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382124-hanna-sigma-exponential-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382125","target":"dossier:oeis-a382124-hanna-sigma-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382125-hanna-sigma-exponential-mod-three-cube/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382125","target":"dossier:oeis-a382125-hanna-sigma-exponential-mod-three-cube"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382318","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382318-a382319-lambert-cube-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382318-a382319-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a382319-a382318-lambert-cube-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382319","target":"dossier:hanna-a382319-a382318-lambert-cube-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382357-valuation-step-permutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382357","target":"dossier:oeis-a382357-valuation-step-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382357-valuation-step-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382357","target":"dossier:oeis-a382357-valuation-step-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382357-valuation-step-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382374","target":"dossier:oeis-a382357-valuation-step-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382357-valuation-step-permutation/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382376","target":"dossier:oeis-a382357-valuation-step-permutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382427-constant-blocks-distinct-run-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382427","target":"dossier:oeis-a382427-constant-blocks-distinct-run-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382814","target":"dossier:oeis-a382814-fibonachos-score"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382814-fibonachos-score/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382814","target":"dossier:oeis-a382814-fibonachos-score"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a382427-constant-blocks-distinct-run-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A382876","target":"dossier:oeis-a382427-constant-blocks-distinct-run-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383165","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383165","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383165","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383165","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383166","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383165-scaled-log-column-periodicity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383166","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383166","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383166","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380709-absolute-reciprocal-cube-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383377","target":"dossier:oeis-a380709-absolute-reciprocal-cube-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380710-absolute-reciprocal-square-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383377","target":"dossier:oeis-a380710-absolute-reciprocal-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a380710-absolute-reciprocal-square-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383377","target":"dossier:oeis-a380710-absolute-reciprocal-square-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383377-absolute-reciprocal-geometric-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383377","target":"dossier:oeis-a383377-absolute-reciprocal-geometric-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bradshaw-arithmetic-derivative-collatz-squarefree-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A383619","target":"dossier:bradshaw-arithmetic-derivative-collatz-squarefree-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384222-two-dense-divisor-blocks-palindrome/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384222","target":"dossier:oeis-a384222-two-dense-divisor-blocks-palindrome"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384222","target":"dossier:oeis-a400194-two-dense-divisor-block-maximum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384222","target":"dossier:oeis-a400194-two-dense-divisor-block-maximum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384267-absolute-reciprocal-square-shift-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384267","target":"dossier:oeis-a384267-absolute-reciprocal-square-shift-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a384270-quartic-cubic-catalan-quotient-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384270","target":"dossier:oeis-a384270-quartic-cubic-catalan-quotient-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h384819-primepower/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384819","target":"dossier:h384819-primepower"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h384819-primepower/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384819","target":"dossier:h384819-primepower"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h384819-primepower/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384820","target":"dossier:h384819-primepower"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/h384819-primepower/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A384820","target":"dossier:h384819-primepower"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385236","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a229140-stephan-least-coordinate-distinctness-refutation/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385236","target":"dossier:oeis-a229140-stephan-least-coordinate-distinctness-refutation"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a385590-alternating-binomial/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385590","target":"dossier:oeis-a385590-alternating-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a385590-alternating-binomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385590","target":"dossier:oeis-a385590-alternating-binomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385672","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/zabolotskii-2025-a385672-near-maximal-area-walks/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A385672","target":"dossier:zabolotskii-2025-a385672-near-maximal-area-walks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386655-hanna-lambert-w-egf-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386655","target":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386655-hanna-lambert-w-egf-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386656","target":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386655-hanna-lambert-w-egf-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386657","target":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386655-hanna-lambert-w-egf-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386658","target":"dossier:oeis-a386655-hanna-lambert-w-egf-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a386666-signed-catalan-cubic-substitution-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386666","target":"dossier:oeis-a386666-signed-catalan-cubic-substitution-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386667","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386667","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-a271960-dyadic-vanishing/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A386667","target":"dossier:hanna-a271960-dyadic-vanishing"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a387118/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387118","target":"dossier:wiseman-a387118"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a387118/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387118","target":"dossier:wiseman-a387118"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387118","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387118","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a387319-multiple-of-three-coincidence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387319","target":"dossier:oeis-a387319-multiple-of-three-coincidence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A387880","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388711-choosable-initial-intervals/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388711","target":"dossier:oeis-a388711-choosable-initial-intervals"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388712","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388712","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388712","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a388713/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388713","target":"dossier:wiseman-a388713"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman-a388713/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388713","target":"dossier:wiseman-a388713"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388713","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388713","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/wiseman388-partitions/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388724","target":"dossier:wiseman388-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388986","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A388986","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389079","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a388986-euler-form-divisor-sum/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389219","target":"dossier:oeis-a388986-euler-form-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389472-square-cube-substitution-odd-index-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389472","target":"dossier:oeis-a389472-square-cube-substitution-odd-index-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389476-quadratic-reversion-dyadic-support-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389476","target":"dossier:oeis-a389476-quadratic-reversion-dyadic-support-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389536-substitution-square-cube-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389536","target":"dossier:oeis-a389536-substitution-square-cube-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389537-alternating-dyadic-reversion-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389537","target":"dossier:oeis-a389537-alternating-dyadic-reversion-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a389542-quadratic-square-reversion-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389542","target":"dossier:oeis-a389542-quadratic-square-reversion-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/arndt-2017-a293452-torus-column-sandpile/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A389565","target":"dossier:arndt-2017-a293452-torus-column-sandpile"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391620-first-sums-partition-characterization/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391620","target":"dossier:oeis-a391620-first-sums-partition-characterization"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391632-andrasfai-mutual-visibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391632","target":"dossier:oeis-a391632-andrasfai-mutual-visibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-alternating-jacobsthal/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-alternating-jacobsthal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-full-laurent-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-full-laurent-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-laurent-first-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-laurent-first-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-laurent-first-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-second-moment/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391810","target":"dossier:oeis-a391810-second-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391811","target":"dossier:oeis-a391810-full-laurent-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391811","target":"dossier:oeis-a391810-full-laurent-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-full-laurent-row-sums/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391811","target":"dossier:oeis-a391810-full-laurent-row-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a391810-laurent-first-moment/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A391811","target":"dossier:oeis-a391810-laurent-first-moment"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392095-antidiagonal-source-series/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392095","target":"dossier:oeis-a392095-antidiagonal-source-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392095-antidiagonal-source-series/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392095","target":"dossier:oeis-a392095-antidiagonal-source-series"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp3921/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392131","target":"dossier:sp3921"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp3921/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392131","target":"dossier:sp3921"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/sp3921/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392203","target":"dossier:hanna-shift-square-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna-nested-square-odd-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392210","target":"dossier:hanna-nested-square-odd-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/posske-2026-a392387-xx-chain-nullspace-2p/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392387","target":"dossier:posske-2026-a392387-xx-chain-nullspace-2p"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/posske-2026-a392387-xx-chain-nullspace-2p/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392387","target":"dossier:posske-2026-a392387-xx-chain-nullspace-2p"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392525-cubic-fifteen-substitution-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392525","target":"dossier:oeis-a392525-cubic-fifteen-substitution-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chen-step-m-eulerian-parity-positivity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392664","target":"dossier:chen-chen-step-m-eulerian-parity-positivity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chen-step-m-eulerian-parity-positivity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392664","target":"dossier:chen-chen-step-m-eulerian-parity-positivity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chen-step-m-eulerian-parity-positivity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392665","target":"dossier:chen-chen-step-m-eulerian-parity-positivity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/chen-chen-step-m-eulerian-parity-positivity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392666","target":"dossier:chen-chen-step-m-eulerian-parity-positivity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392694","target":"dossier:oeis-a392694-zero-prepended-first-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392698","target":"dossier:oeis-a392694-zero-prepended-first-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392698-trimmed-alternating-partitions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392698","target":"dossier:oeis-a392698-trimmed-alternating-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392698-trimmed-alternating-partitions/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392703","target":"dossier:oeis-a392698-trimmed-alternating-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392698-trimmed-alternating-partitions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392703","target":"dossier:oeis-a392698-trimmed-alternating-partitions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392703","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392694-zero-prepended-first-sums/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392707","target":"dossier:oeis-a392694-zero-prepended-first-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a392707-gapfree-odd-combinatorial-bridge/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A392707","target":"dossier:oeis-a392707-gapfree-odd-combinatorial-bridge"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393170","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393170","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393172-catalan-shift-square-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393172","target":"dossier:oeis-a393172-catalan-shift-square-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393173","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393173","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393173","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393210","target":"dossier:bala-machine3626-exact-halting-time"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/bala-machine3626-exact-halting-time/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393210","target":"dossier:bala-machine3626-exact-halting-time"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393434-factorial-heraclitus/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393434","target":"dossier:oeis-a393434-factorial-heraclitus"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393850","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393173-hanna-power-of-two-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393850","target":"dossier:oeis-a393173-hanna-power-of-two-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393856-scaled-reversion-mod-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393856","target":"dossier:oeis-a393856-scaled-reversion-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393856-scaled-reversion-mod-five/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393856","target":"dossier:oeis-a393856-scaled-reversion-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393857-scaled-reversion-mod-six/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393856","target":"dossier:oeis-a393857-scaled-reversion-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393857-scaled-reversion-mod-six/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393856","target":"dossier:oeis-a393857-scaled-reversion-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393857-scaled-reversion-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393857","target":"dossier:oeis-a393857-scaled-reversion-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-all-terms-odd/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393866","target":"dossier:oeis-a393867-all-terms-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-all-terms-odd/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393867","target":"dossier:oeis-a393867-all-terms-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-prime-power-shift-log-derivative/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393867","target":"dossier:oeis-a393867-prime-power-shift-log-derivative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-prime-power-shift-log-derivative/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393867","target":"dossier:oeis-a393867-prime-power-shift-log-derivative"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-all-terms-odd/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A393868","target":"dossier:oeis-a393867-all-terms-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394399-twin-prime-sigma-gcd-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394399","target":"dossier:oeis-a394399-twin-prime-sigma-gcd-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394407-totient-tau-primes-huber/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394407","target":"dossier:oeis-a394407-totient-tau-primes-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394407-totient-tau-primes-huber/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394407","target":"dossier:oeis-a394407-totient-tau-primes-huber"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/irvine-least-maximal-digit-sum-base/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394431","target":"dossier:irvine-least-maximal-digit-sum-base"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394757-sigma-gcd-two-or-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394757","target":"dossier:oeis-a394757-sigma-gcd-two-or-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a394757-twin-prime-sigma-gcd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A394757","target":"dossier:oeis-a394757-twin-prime-sigma-gcd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395531-greedy-brick-all-index/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395531","target":"dossier:oeis-a395531-greedy-brick-all-index"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a266489-negative-power-diagonal-mod-two"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a266489-negative-power-diagonal-mod-two/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a383165-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a383166-scaled-log-column-periodicity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a383166-scaled-log-column-periodicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-diagonal-exponent-self-divisibility/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-diagonal-exponent-self-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-diagonal-exponent-self-divisibility/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-diagonal-exponent-self-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-diagonal-exponent-self-divisibility/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-diagonal-exponent-self-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-diagonal-exponent-self-divisibility/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-diagonal-exponent-self-divisibility"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#falsifier","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Falsifier section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a395833-negative-power-diagonal-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A395833","target":"dossier:oeis-a395833-negative-power-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396099-iterate-product-two-three-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396099-iterate-product-two-three-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396798-eighth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396798-fifth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396798-fourth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396099","target":"dossier:oeis-a396798-third-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396102-triple-iterate-shift-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396102","target":"dossier:oeis-a396102-triple-iterate-shift-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/kurk396563/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396563","target":"dossier:kurk396563"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396596-divisor-power-union/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396596","target":"dossier:oeis-a396596-divisor-power-union"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396596-divisor-power-union/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396596","target":"dossier:oeis-a396596-divisor-power-union"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396596-divisor-power-union/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396596","target":"dossier:oeis-a396596-divisor-power-union"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396659-edge-collision-ratio/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396659","target":"dossier:oeis-a396659-edge-collision-ratio"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396793-iterate-product-nine-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396793","target":"dossier:oeis-a396793-iterate-product-nine-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396795-iterate-product-twenty-five-mod-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396793","target":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396794-triple-iterate-product-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396794","target":"dossier:oeis-a396794-triple-iterate-product-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396795-iterate-product-twenty-five-mod-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396794","target":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396795-iterate-product-twenty-five-mod-five/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396795","target":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396795-iterate-product-twenty-five-mod-five/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396795","target":"dossier:oeis-a396795-iterate-product-twenty-five-mod-five"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396797-three-four-iterate-product-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396797-three-four-iterate-product-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396798-eighth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396798-fifth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396798-fourth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396797","target":"dossier:oeis-a396798-third-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-eighth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-fifth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-fourth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396798","target":"dossier:oeis-a396798-third-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396803-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396803-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396803-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396803-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-mod-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396805-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396805-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-mod-six/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396806-iterate-exponential-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396803","target":"dossier:oeis-a396806-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396804","target":"dossier:oeis-a396803-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396804-quartic-egf-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396804","target":"dossier:oeis-a396804-quartic-egf-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396804","target":"dossier:oeis-a396805-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396804","target":"dossier:oeis-a396806-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. 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This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396805","target":"dossier:oeis-a396806-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396803-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-mod-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396803-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396803-iterate-exponential-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396803-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-mod-three/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396805-iterate-exponential-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396805-iterate-exponential-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396805-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-mod-six/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396806-iterate-exponential-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-mod-six/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396806-iterate-exponential-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-mod-six/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396806-iterate-exponential-mod-six"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396806-iterate-exponential-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396806","target":"dossier:oeis-a396806-iterate-exponential-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-eighth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396807","target":"dossier:oeis-a396798-eighth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fifth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396807","target":"dossier:oeis-a396798-fifth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-fourth-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396807","target":"dossier:oeis-a396798-fourth-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-seventh-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396807","target":"dossier:oeis-a396798-seventh-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396798-third-iterate-mod-eight/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396807","target":"dossier:oeis-a396798-third-iterate-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a393867-all-terms-odd/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396843","target":"dossier:oeis-a393867-all-terms-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396843-compositional-square-all-odd-coefficients/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396843","target":"dossier:oeis-a396843-compositional-square-all-odd-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396843-compositional-square-all-odd-coefficients/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396843","target":"dossier:oeis-a396843-compositional-square-all-odd-coefficients"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396844-scaled-quadratic-reversion-mod-sixteen/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396844","target":"dossier:oeis-a396844-scaled-quadratic-reversion-mod-sixteen"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a396846-logarithmic-quarter-normalization-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A396846","target":"dossier:oeis-a396846-logarithmic-quarter-normalization-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-diagonal-power-ratio-all-odd/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397241","target":"dossier:oeis-a397241-diagonal-power-ratio-all-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-mod-four/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397241","target":"dossier:oeis-a397241-mod-four"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397241-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397241","target":"dossier:oeis-a397241-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-exponential-square-weight-catalan-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397242-exponential-square-weight-catalan-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-powers/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397242-mod-three-powers"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397242-mod-three-sums/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397242-mod-three-sums"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397345-square-weight-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397345-square-weight-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397346-square-weight-mod-eight/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397346-square-weight-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397348-square-weight-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397242","target":"dossier:oeis-a397348-square-weight-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397244","target":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397244","target":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397244","target":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397244-quadratic-power-diagonal-fibbinary-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397244","target":"dossier:oeis-a397244-quadratic-power-diagonal-fibbinary-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397315","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397345-square-weight-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397345","target":"dossier:oeis-a397345-square-weight-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397346-square-weight-mod-eight/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397346","target":"dossier:oeis-a397346-square-weight-mod-eight"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397348-square-weight-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397348","target":"dossier:oeis-a397348-square-weight-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397349-logarithmic-weight-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397349","target":"dossier:oeis-a397349-logarithmic-weight-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397349-logarithmic-weight-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397349","target":"dossier:oeis-a397349-logarithmic-weight-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-mod-three/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397356","target":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-mod-three/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397356","target":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-mod-three"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397356-reciprocal-square-exponent-diagonal-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397356","target":"dossier:oeis-a397356-reciprocal-square-exponent-diagonal-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397436-general-supercongruence/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397436","target":"dossier:oeis-a397436-general-supercongruence"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397578-sigma-neighbour-peaks/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397578","target":"dossier:oeis-a397578-sigma-neighbour-peaks"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397590","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397590","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397591","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397591","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397591","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397592","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397592","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397592","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#motivation","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Motivation section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397592","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397592","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/hanna397594/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397594","target":"dossier:hanna397594"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#route","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Route section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397594","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397592-dyadic-parity/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A397594","target":"dossier:oeis-a397592-dyadic-parity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/ulg398064/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398064","target":"dossier:ulg398064"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/ulg398064/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398064","target":"dossier:ulg398064"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398542-fixed-bottom-polynomial/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398446","target":"dossier:oeis-a398542-fixed-bottom-polynomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398542-fixed-bottom-polynomial/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398542","target":"dossier:oeis-a398542-fixed-bottom-polynomial"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398690-palindromic/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398690","target":"dossier:oeis-a398690-palindromic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398690-palindromic/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398690","target":"dossier:oeis-a398690-palindromic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398726","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a398726-distinct-cycle-size-limit/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A398726","target":"dossier:oeis-a398726-distinct-cycle-size-limit"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399155-value-finiteness/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399155","target":"dossier:oeis-a399155-value-finiteness"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399155-value-finiteness/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399155","target":"dossier:oeis-a399155-value-finiteness"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399232-radical-identity/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399232","target":"dossier:oeis-a399232-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399232-radical-identity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399232","target":"dossier:oeis-a399232-radical-identity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399491","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399491","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399491","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a397315-three-row-blackout/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399580","target":"dossier:oeis-a397315-three-row-blackout"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#evidence","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Evidence section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399755","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399755","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a399755-pell-class-successor/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A399755","target":"dossier:oeis-a399755-pell-class-successor"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400164","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400168","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400168","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400171","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400168-primorial-gap/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400171","target":"dossier:oeis-a400168-primorial-gap"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400194","target":"dossier:oeis-a400194-two-dense-divisor-block-maximum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400194-two-dense-divisor-block-maximum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400195","target":"dossier:oeis-a400194-two-dense-divisor-block-maximum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400203","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a112399-morken-first-divisor-sum/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400203","target":"dossier:oeis-a112399-morken-first-divisor-sum"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400216-a400218-tetrahelix-generating-functions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400216","target":"dossier:oeis-a400216-a400218-tetrahelix-generating-functions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400216-a400218-tetrahelix-generating-functions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400217","target":"dossier:oeis-a400216-a400218-tetrahelix-generating-functions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400216-a400218-tetrahelix-generating-functions/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400218","target":"dossier:oeis-a400216-a400218-tetrahelix-generating-functions"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400429-semi-meander-second-diagonal/#assumed-unverified","kind":"paper_context","scope":"Sequence mentioned in the dossier’s ASSUMED-UNVERIFIED section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400429","target":"dossier:oeis-a400429-semi-meander-second-diagonal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400429-semi-meander-second-diagonal/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400429","target":"dossier:oeis-a400429-semi-meander-second-diagonal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400429-semi-meander-second-diagonal/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400429","target":"dossier:oeis-a400429-semi-meander-second-diagonal"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/mendonca-oeis-a400584-strict-monotonicity/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400584","target":"dossier:mendonca-oeis-a400584-strict-monotonicity"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400585-odd-cofactor-asymptotic/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400585","target":"dossier:oeis-a400585-odd-cofactor-asymptotic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400585-odd-cofactor-asymptotic/#triage","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Triage section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400585","target":"dossier:oeis-a400585-odd-cofactor-asymptotic"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400644-rounded-three-term-norm/#gap","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Gap section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400644","target":"dossier:oeis-a400644-rounded-three-term-norm"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research/oeis-a400644-rounded-three-term-norm/#problem","kind":"paper_context","scope":"Sequence mentioned in the dossier’s Problem section. This does not certify a resolution of any OEIS conjecture.","source":"oeis:A400644","target":"dossier:oeis-a400644-rounded-three-term-norm"},{"evidence_url":"https://openreview.net/forum?id=xOAZvprLXO","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:cross-engine-admission","target":"literature:url:https://openreview.net/forum?id=xOAZvprLXO"},{"evidence_url":"https://arxiv.org/abs/2610.18816","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:eof-coincidence","target":"literature:arxiv:2610.18816"},{"evidence_url":"https://arxiv.org/abs/2609.00706","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:eqt2","target":"literature:arxiv:2609.00706"},{"evidence_url":"https://arxiv.org/abs/2609.09753","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:generalized-fidelity","target":"literature:arxiv:2609.09753"},{"evidence_url":"https://arxiv.org/abs/2610.39941","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:insertion-degree-descent","target":"literature:arxiv:2610.39941"},{"evidence_url":"https://arxiv.org/abs/2610.20329","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:klein-four-ca","target":"literature:arxiv:2610.20329"},{"evidence_url":"https://arxiv.org/abs/2609.33421","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:padovan","target":"literature:arxiv:2609.33421"},{"evidence_url":"https://arxiv.org/abs/2610.05623","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:paper-cotfas-gaussian-discrete-approximation","target":"literature:arxiv:2610.05623"},{"evidence_url":"https://arxiv.org/abs/2611.01064","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:paper-singhvi-deletion-channel-input-entropy","target":"literature:arxiv:2611.01064"},{"evidence_url":"https://arxiv.org/abs/2611.04568","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:paper-two-qutrit-time-invariant-negativity","target":"literature:arxiv:2611.04568"},{"evidence_url":"https://arxiv.org/abs/2610.24059","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:petersen-zero-forcing","target":"literature:arxiv:2610.24059"},{"evidence_url":"https://arxiv.org/abs/2610.29608","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:residue-sums","target":"literature:arxiv:2610.29608"},{"evidence_url":"https://arxiv.org/abs/2607.04534","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:routing","target":"literature:arxiv:2607.04534"},{"evidence_url":"https://arxiv.org/abs/2609.25128","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:subquorum","target":"literature:arxiv:2609.25128"},{"evidence_url":"https://www.rairo-ita.org/articles/ita/abs/2026/01/ita20260032/ita20260032.html","kind":"cites","scope":"Publication record linked to its bibliographic identity.","source":"publication:zeckendorf","target":"literature:doi:10.1051/ita/2026032"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"question_source","scope":"The Gaussian asymptotic reported in July is a random-matrix result. The target quantifies over every admissible deterministic matrix.","source":"question:balan-wang-universal-stability","target":"literature:arxiv:2603.29571"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Compare extremal deterministic families using certified spectral intervals; seek a packing, determinant or partition argument with dimension-independent constants.","source":"question:balan-wang-universal-stability","target":"target:omega-deterministic-decay"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Formalize rectangular row restriction and the variational least singular value. Preserve M>1 and the source's complement-rank convention.","source":"question:balan-wang-universal-stability","target":"target:omega-full-spark-reduction"},{"evidence_url":"https://doi.org/10.48550/arXiv.2305.08349","kind":"question_source","scope":"Negative-position base-phi digits need a value-preserving conversion, followed by an identification with the specific subshifts in the paper.","source":"question:base-phi-negative-prefix-trident","target":"literature:arxiv:2305.08349"},{"evidence_url":"https://doi.org/10.48550/arXiv.2305.08349","kind":"proposed_target","scope":"Start with one admissible single-component prefix; derive its gap word and Lucas parameters by exact desubstitution.","source":"question:base-phi-negative-prefix-trident","target":"target:negative-prefix-return-itinerary"},{"evidence_url":"https://doi.org/10.48550/arXiv.2305.08349","kind":"proposed_target","scope":"Compute exact Z[phi] residue traces, propose a state invariant, and prove transition closure and output correctness.","source":"question:base-phi-negative-prefix-trident","target":"target:negative-prefix-transducer"},{"evidence_url":"https://doi.org/10.48550/arXiv.2305.08349","kind":"proposed_target","scope":"Port the symmetry and integer-value predicates, replay the exact algebraic criterion, and use it as a regression case for two-sided digit conversion.","source":"question:base-phi-negative-prefix-trident","target":"target:phi-kimberling-conversion-check"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"question_source","scope":"Generic injectivity, existence of one exceptional injective frame, and injectivity probability are distinct. The old universal 4M-4 necessity statement is already false.","source":"question:complex-phase-retrieval-probability","target":"literature:arxiv:2603.29571"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Analyze the rank-two difference incidence set on a concrete Grassmannian chart and verify a local persistence or measure argument.","source":"question:complex-phase-retrieval-probability","target":"target:phase-bad-event-probability"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Prove the trace identity and represent the difference cone exactly; relate normalized nonzero signals to the rank-one projector predicate.","source":"question:complex-phase-retrieval-probability","target":"target:phase-intensity-lift"},{"evidence_url":"https://doi.org/10.48550/arXiv.2405.02727","kind":"question_source","scope":"Minimizing the full valid-input machine does not establish minimality on the sparse powers-of-four language.","source":"question:golden-ratio-base4-dfao-minimality","target":"literature:arxiv:2405.02727"},{"evidence_url":"https://doi.org/10.48550/arXiv.2405.02727","kind":"proposed_target","scope":"Encode a bounded labelled sample, export an independently checkable UNSAT certificate, and account for state renaming and leading-zero invariance.","source":"question:golden-ratio-base4-dfao-minimality","target":"target:dfao-finite-unsat"},{"evidence_url":"https://doi.org/10.48550/arXiv.2405.02727","kind":"proposed_target","scope":"Build exact labelled prefix trees; search for a 22-clique of pairwise incompatible prefixes and retain both completed power inputs for every edge.","source":"question:golden-ratio-base4-dfao-minimality","target":"target:dfao-sparse-distinguishability"},{"evidence_url":"https://doi.org/10.48550/arXiv.2503.04122","kind":"question_source","scope":"Putting d=1 into Conjecture 17 violates its hypothesis d>=2. The g+1 family has different exceptional values, modulus and two residue intervals; its universal claim needs a new proof.","source":"question:greedy-three-sumfree-third-seed-one","target":"literature:arxiv:2503.04122"},{"evidence_url":"https://doi.org/10.48550/arXiv.2503.04122","kind":"proposed_target","scope":"Prove three-sumfreeness and explicit representations of every excluded later value, separating the initial segment from periodic translates.","source":"question:greedy-three-sumfree-third-seed-one","target":"target:greedy-g-plus-one-blocks"},{"evidence_url":"https://doi.org/10.48550/arXiv.2503.04122","kind":"proposed_target","scope":"Adapt the prefix invariant and uniqueness argument to the new accepted/forbidden partition, including the seed boundary.","source":"question:greedy-three-sumfree-third-seed-one","target":"target:greedy-g-plus-one-literal"},{"evidence_url":"https://doi.org/10.48550/arXiv.2410.23997","kind":"question_source","scope":"Conditional tomography identities assume the family. A complete seven-basis design identity does not automatically constrain a four-basis family.","source":"question:mub-six-fourth-basis","target":"literature:arxiv:2410.23997"},{"evidence_url":"https://doi.org/10.48550/arXiv.2410.23997","kind":"proposed_target","scope":"Formalize both directions and prove trace(P_i Q_j) = |inner(v_i,w_j)|^2, using the repository context definitions.","source":"question:mub-six-fourth-basis","target":"target:mub-basis-context"},{"evidence_url":"https://doi.org/10.48550/arXiv.2410.23997","kind":"proposed_target","scope":"First check parameterization coverage and exceptional branches, then express every required cross-Gram modulus equation and seek exact certificates.","source":"question:mub-six-fourth-basis","target":"target:mub-joint-compatibility"},{"evidence_url":"https://doi.org/10.48550/arXiv.2410.23997","kind":"proposed_target","scope":"Track polynomial degree before and after P=vv*. First reproduce the stated separation, then test exact certificates for a genuinely stronger constraint.","source":"question:mub-six-fourth-basis","target":"target:mub-projector-sos-scope"},{"evidence_url":"https://doi.org/10.48550/arXiv.2508.20222","kind":"question_source","scope":"Ordered states, the exact game orientation, path length and the meaning of arbitrary switch order require separate treatment.","source":"question:ordered-zeckendorf-long-game-strategy","target":"literature:arxiv:2508.20222"},{"evidence_url":"https://doi.org/10.48550/arXiv.2508.20222","kind":"proposed_target","scope":"Test swap-first, leftmost-one merge and rightmost-split exchanges against exact state heights; save both suffix witnesses at the first failure.","source":"question:ordered-zeckendorf-long-game-strategy","target":"target:lgs-bellman-exchange"},{"evidence_url":"https://doi.org/10.48550/arXiv.2508.20222","kind":"proposed_target","scope":"Enumerate exact ordered states and all switch tie choices for bounded n; retain differing lengths and full move traces.","source":"question:ordered-zeckendorf-long-game-strategy","target":"target:lgs-switch-order"},{"evidence_url":"https://doi.org/10.48550/arXiv.2608.03723","kind":"question_source","scope":"The original conjunction is false. Its interval and monotonicity clauses must be tested independently, without importing the disproved bound or assuming that a > 1/24 repairs every degree.","source":"question:pochhammer-higher-even-intervals","target":"literature:arxiv:2608.03723"},{"evidence_url":"https://doi.org/10.48550/arXiv.2608.03723","kind":"proposed_target","scope":"Use the proved c2 closed form as a baseline for the exact quartic endpoint, preserving infimum existence and boundedness obligations.","source":"question:pochhammer-higher-even-intervals","target":"target:pochhammer-degree-monotonicity"},{"evidence_url":"https://doi.org/10.48550/arXiv.2608.03723","kind":"proposed_target","scope":"Construct L_a((X+t)^4) from the same basis action and use exact root isolation or a Sturm certificate, including repeated roots and endpoints.","source":"question:pochhammer-higher-even-intervals","target":"target:pochhammer-quartic-interval"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"question_source","scope":"The finite Fourier reduction does not remove the remaining probabilistic loss. A theta SDP, normalization bridge and random-graph model must be connected explicitly.","source":"question:random-circulant-lovasz-theta","target":"literature:arxiv:2603.29571"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Prove averaging preserves PSD, edge constraints and objective, then derive eigenvalues using the existing ZMod characters, with sign and scale recorded.","source":"question:random-circulant-lovasz-theta","target":"target:circulant-fourier-reduction"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Construct exact small-graph witnesses and certified residual margins; test graph-complement and scaling conventions independently.","source":"question:random-circulant-lovasz-theta","target":"target:circulant-primal-dual-certificate"},{"evidence_url":"https://doi.org/10.48550/arXiv.2210.11038","kind":"question_source","scope":"Uniform terminal-path sampling and locally uniform move sampling give different measures. A limit theorem needs dependence control and nondegenerate variance.","source":"question:random-zeckendorf-game-gaussianity","target":"literature:arxiv:2210.11038"},{"evidence_url":"https://doi.org/10.48550/arXiv.2210.11038","kind":"proposed_target","scope":"Compute both rational generating functions independently on small value-preserving state DAGs, with the terminal state contributing 1.","source":"question:random-zeckendorf-game-gaussianity","target":"target:game-path-measures"},{"evidence_url":"https://doi.org/10.48550/arXiv.2210.11038","kind":"proposed_target","scope":"Search exact history pairs that share a proposed barrier, then measure and bound influence-crossing events under the chosen measure.","source":"question:random-zeckendorf-game-gaussianity","target":"target:game-regenerative-barrier"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"question_source","scope":"For d>1 the target requires squared overlaps 1/(d+1) and sum of rank-one projectors dI. This is distinct from the MUB version of Zauner's conjecture.","source":"question:sic-povm-all-dimensions","target":"literature:arxiv:2603.29571"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Use explicit number-field embeddings and exact polynomial reductions; list which equations and embedding choices are certified.","source":"question:sic-povm-all-dimensions","target":"target:sic-exact-fiducial-certificate"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.29571","kind":"proposed_target","scope":"Use the existing displacement composition and trace owners; derive twirling and the orbit Gram entries rather than introducing another Weyl convention.","source":"question:sic-povm-all-dimensions","target":"target:sic-fiducial-orbit"},{"evidence_url":"https://doi.org/10.48550/arXiv.2606.09096","kind":"question_source","scope":"The spectral shift lambda < lambda_a, the physical characteristic function and the treatment of poles in compact-uniform convergence need explicit control.","source":"question:suzuki-boundary-characteristic-limit","target":"literature:arxiv:2606.09096"},{"evidence_url":"https://doi.org/10.48550/arXiv.2606.09096","kind":"proposed_target","scope":"Fix the domain, boundary convention and inner product; derive the relevant boundary function and measure rather than assume their desired expansion.","source":"question:suzuki-boundary-characteristic-limit","target":"target:suzuki-boundary-identification"},{"evidence_url":"https://doi.org/10.48550/arXiv.2606.09096","kind":"proposed_target","scope":"Use explicit lower bounds and compare shifted constructions on common test-function cores; track any effect on characteristic zeros and normalization.","source":"question:suzuki-boundary-characteristic-limit","target":"target:suzuki-unconditional-shift"},{"evidence_url":"https://doi.org/10.48550/arXiv.2509.16034","kind":"question_source","scope":"Equation (11) fixes the absolute difference, but its published resolution does not cover the sign, R(4n), or a full recursion. Nonautomaticity has a written unboundedness argument; a separate kernel-checked endpoint remains to be identified or formalized.","source":"question:thue-morse-reduced-abelian-even","target":"literature:arxiv:2509.16034"},{"evidence_url":"https://doi.org/10.48550/arXiv.2509.16034","kind":"proposed_target","scope":"Read the released even-complexity proof and its alternation-extrema recurrences, then isolate the additional information needed to recover the signed difference.","source":"question:thue-morse-reduced-abelian-even","target":"target:thue-morse-even-difference"},{"evidence_url":"https://doi.org/10.48550/arXiv.2509.16034","kind":"proposed_target","scope":"Locate any newer formal endpoint first. Otherwise formalize the written unboundedness argument for positive lengths and connect it to the finite-output definition of k-automaticity.","source":"question:thue-morse-reduced-abelian-even","target":"target:thue-morse-nonautomaticity"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.25343","kind":"question_source","scope":"The exceptional lift from p to p² is missing. A finite search with no examples does not refute existence.","source":"question:wall-sun-sun-golden-unit-lift","target":"literature:arxiv:2603.25343"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.25343","kind":"proposed_target","scope":"Define B as a residue class, prove the kernel-order dichotomy, and cross-check recurrence and matrix computations independently.","source":"question:wall-sun-sun-golden-unit-lift","target":"target:pisano-first-lift-defect"},{"evidence_url":"https://doi.org/10.48550/arXiv.2603.25343","kind":"proposed_target","scope":"Prove the Fibonacci formula for A^n and compare the return condition (F_n,F_(n+1)) = (0,1) modulo m.","source":"question:wall-sun-sun-golden-unit-lift","target":"target:pisano-matrix-order"},{"evidence_url":"https://doi.org/10.48550/arXiv.2511.22755","kind":"question_source","scope":"An explicit prolate model and a finite-window certificate leave actual-mode identification, full-space coercivity and a growing-window comparison to establish.","source":"question:weil-ground-mode-xi-limit","target":"literature:arxiv:2511.22755"},{"evidence_url":"https://doi.org/10.48550/arXiv.2511.22755","kind":"proposed_target","scope":"Separate the physical window, finite matrix dimension and numerical tail cutoff. Seek bounds uniform over each target compact set with an explicit vanishing budget.","source":"question:weil-ground-mode-xi-limit","target":"target:weil-moving-window-rate"},{"evidence_url":"https://doi.org/10.48550/arXiv.2511.22755","kind":"proposed_target","scope":"Check the existing numerical and proof-script outputs before creating another certificate; assemble prime, pole, Gamma and tail terms with one set of parameters.","source":"question:weil-ground-mode-xi-limit","target":"target:weil-same-family-residual"},{"evidence_url":"https://doi.org/10.48550/arXiv.2106.09959","kind":"question_source","scope":"Factor complexity is a different invariant. The upper bound needs a uniform successor-determinism or carry-state argument.","source":"question:zeckendorf-polynomial-maximum-order-complexity","target":"literature:arxiv:2106.09959"},{"evidence_url":"https://doi.org/10.48550/arXiv.2106.09959","kind":"proposed_target","scope":"Track exact normalization of 2nh+h² at Fibonacci/Lucas scales, then search for equal observed histories with conflicting next bits.","source":"question:zeckendorf-polynomial-maximum-order-complexity","target":"target:polynomial-carry-window"},{"evidence_url":"https://doi.org/10.48550/arXiv.2106.09959","kind":"proposed_target","scope":"Cross-check a naive implementation against a suffix-based method; emit equal blocks with unequal successors for every claimed lower bound.","source":"question:zeckendorf-polynomial-maximum-order-complexity","target":"target:polynomial-successor-certificate"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/conjectures.html#rp=thue-morse-reduced-abelian-even","kind":"proposed_transfer","scope":"A next question from this result. Transfer success and improvement over a baseline have not been measured.","source":"representation:thue-morse-reduced-abelian-odd","target":"question:thue-morse-reduced-abelian-even"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/","kind":"explains","scope":"A worked example and proof walkthrough. Independent reader assessment is still needed.","source":"result:thue-morse-reduced-abelian-odd","target":"explanation:thue-morse-reduced-abelian-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/assets/proofs/thue-morse-reduced-abelian-odd.lean","kind":"formalized_in","scope":"Pinned result belongs to this module; current release certification is not inferred.","source":"result:thue-morse-reduced-abelian-odd","target":"module:D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd"},{"evidence_url":"https://doi.org/10.48550/arXiv.2509.16034","kind":"builds_on","scope":"The odd-index recurrence and the even absolute-difference result study the same run-compressed class count and its alternation extrema.","source":"result:thue-morse-reduced-abelian-odd","target":"question:thue-morse-reduced-abelian-even"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/assets/proofs/thue-morse-reduced-abelian-odd.lean","kind":"representation","scope":"For every factor length and start, the class encoding equals character counts after literal run compression. This preserves the reduced abelian class, not the original word.","source":"result:thue-morse-reduced-abelian-odd","target":"representation:thue-morse-reduced-abelian-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/results/thue-morse-reduced-abelian-odd/","kind":"topic_member","scope":"Formal results about integer sequences, their structure and remaining questions.","source":"topic:integer-sequences","target":"result:thue-morse-reduced-abelian-odd"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research.html#eqt2","kind":"topic_member","scope":"Equational implication, proof certificates and countermodels over algebraic structures.","source":"topic:logic-algebra","target":"publication:eqt2"},{"evidence_url":"https://the-omega-institute.github.io/trureturing-pages/research.html#routing","kind":"topic_member","scope":"Equational implication, proof certificates and countermodels over algebraic structures.","source":"topic:logic-algebra","target":"publication:routing"}],"research_paths":[{"benefit_claim":"not-established","id":"thue-morse-reduced-abelian-odd","result":"result:thue-morse-reduced-abelian-odd","source_sha256":"83c07d9a519f88c99485f090e7f988bf5774f32061de9961748a619a22642b18","stages":[{"kind":"public_problem","records":["claim:thue-morse-reduced-abelian-odd","literature:arxiv:2509.16034"],"status":"source-located"},{"kind":"general_theorem","records":["result:thue-morse-reduced-abelian-odd"],"status":"reviewed-pinned-result"},{"kind":"representation","records":["representation:thue-morse-reduced-abelian-odd"],"status":"reviewed-pinned-statement"},{"kind":"new_problem","records":["question:thue-morse-reduced-abelian-even"],"status":"proposed","transfer_benefit":"not-measured"},{"independent_assessment":"not-assessed","kind":"human_understanding","records":["explanation:thue-morse-reduced-abelian-odd"],"status":"explanation-available"}],"title":"Thue-Morse reduced abelian complexity: the odd-length recurrence"}],"reviewed":"2026-09-09","schema_version":"pages-discovery.v1","truth_release_digest":"sha256:76a8db6a0f78609b1b9d4a959c8392f2752f2daaddc1930e046e34ce07d0f608"}