The original ari coproduct preserves every right-factor level filtration

Annika Burmester and Ulf Kühn, On post-Lie structures for free Lie algebras, arXiv:2504.19661v1, Conjecture 4.29, state:

For k ≥ 0 we define the k-level of a word v_{k_1}⋯v_{k_d} by d − #{j | k_j=k}, and extend this to an increasing filtration on Q⟨V⟩. For w∈Q⟨V⟩, write as usual Δ_a(w)=w_(1)⊗w_(2). Then, we always have k-level(w_(2)) ≤ k-level(w).

Let $H=\mathbb Q\langle v_0,v_1,\ldots\rangle$ be the algebra of finite ordinary rational word polynomials, including its unit. For a word $w$, let $\lambda_k(w)$ count letters different from $v_k$ and put

$$ F_r^k=\operatorname{span}_{\mathbb Q} {w:\lambda_k(w)\le r}. $$

The precise filtration assertion is, for every $k,r\in\mathbb N$ and every $P\in F_r^k$,

$$ \Delta_a(P)\in\operatorname{range} \bigl(\operatorname{id}H\otimes\iota_r^k: H\otimes{\mathbb Q}F_r^k\longrightarrow H\otimes_{\mathbb Q}H\bigr), $$

where $\iota_r^k$ is the natural submodule inclusion. All tensor products are ordinary algebraic tensor products. A Sweedler expression is read through this subspace assertion, rather than through a chosen tensor decomposition. The resolution is Proved by D5/S1/Words/PostLie/AriCoproductFiltration.result, with claim : Prop and result : claim.

| Source clause | Formal counterpart | | --- | --- | | Original rational word algebra on all nonnegative indices | AriWordOperations.H = RationalWordPolynomial Nat, with Word = FreeMonoid Nat and the existing ordinary monoid-algebra basis | | Every natural $k$, including zero | The unrestricted k : Nat in claim; the generator $v_0$ and mixed-zero multiplicities retain their source conventions | | 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 $r$ | | Every finite rational polynomial in that span | P : H with membership in filtration k r; no length, weight, homogeneity, coefficient-sign or cancellation restriction | | Original ari product and dual coproduct | letterAction, primitiveUnshuffle, star, transposeProperty, and the proved uniquely characterized delta | | Full ordinary pairing | pairing, tensorPairing, delta_transpose and delta_basis_coefficient, including empty words and both unit tensor terms | | Membership in the actual right-factor tensor image | rightFiltrationImage k r = LinearMap.range (TensorProduct.map id (filtration k r).subtype) and result |

The source operations and locators are documented in the source note. The claim is the entire target preregistered in #23147.

The assertion controls how the original ari coproduct distributes letters away from a chosen generator in its right tensor factor. The source uses this filtration assertion in the context of Theorem 4.30. The present result proves the filtration assertion itself; no independent formalization of that theorem or of every surrounding post-Lie structure is asserted.

The carrier is the existing rational word polynomial algebra. The construction preserves the ordinary coefficient-one word basis and its full tensor pairing, so the result applies to the source algebra without replacing it by a divided-power algebra or a finite truncation.

Conjecture 4.29 asks for a uniform assertion on all words and their rational linear combinations. Checking generator cases or bounded lengths cannot supply that assertion. A coefficient support bound must also give actual membership in the tensor-submodule image, and a proposed dual operation must exist in the ordinary algebraic tensor product.

The finite action presentation is the first equality in the full proof of Proposition 4.19, with generator Lie factors. The source derives it from Definition 4.8 and its enveloping extension, Definition 1.9. Proposition 2.1 supplies the ordered primitive-unshuffle formula for the product. Definition 1.20 and Remark 1.22 supply the full ordinary transpose characterization. These proved source presentations are used directly; no separate magmatic or free-Lie realization, assumed comparison map or assumed coproduct preservation is introduced.

Definition 4.2 has $\beta(0,0)=1$, zero mixed-zero values, and $\beta(a,l)=\binom{a-1}{l-1}$ when $a,l>0$. For a nonempty word $A=v_{a_1}\cdots v_{a_m}$ and $s>0$, the source's finite formula is

$$ A\triangleright_a v_s= \sum_{0\le l_i\le a_i} (-1)^{\sum_i a_i+\sum_i l_i}\prod_i\beta(a_i,l_i) [\cdots[[v_{s+\sum_i(a_i-l_i)},v_{l_1}],v_{l_2}],\ldots,v_{l_m}]. $$

The empty word acts identically; a nonempty word acts by zero on $v_0$. AriWordOperations implements these conventions. It uses only the valid nonzero-multiplicity implication from Remark 4.3, including preservation of zero positions.

The concatenation algebra homomorphism $\pi_k:H\to\mathbb Q[t]$ sends $v_k$ to $t$ and every other generator to zero. For every $d\ge0$, its $t^d$ coefficient is exactly the ordinary coefficient of $v_k^d$. Commutativity of the target annihilates every commutator. Consequently a nonempty left input in a generator action has no pure-$v_k$ output. AriProjection proves this coefficient statement, including degree zero, and the resulting level lower bound.

AriWordProduct assigns each input position independently to one of the ordered slots of the right word and retains the original order inside each slot. Coincident assignments are counted with multiplicity. Each right letter different from $v_k$ contributes at least one such letter to every supported term. Concatenation adds the level, while finite sums and rational cancellation cannot introduce support outside the bound. Thus, for all words $A,B,w$,

$$ [w](A\star_a B)\ne0\quad\Longrightarrow\quad \lambda_k(B)\le\lambda_k(w). $$

Use the original weights $\gamma(0)=1$ and $\gamma(i)=i$ for $i>0$. The action and product preserve total weight. Words of weight at most $N$ have length and every index at most $N$; bounded lists over $\operatorname{Fin}(N+1)$ therefore give an exact finite weight window. AriCoproductFiltration constructs the ordinary tensor transpose in that window and proves its coefficient formula for every pair, including pairs outside the window and empty factors. It proves the full arbitrary-three- polynomial pairing characterization and uniqueness, identifying delta with the source operation.

The support bound places every nonzero tensor basis term over a right word of the required level. Each such term is the image of an actual tensor with a right member of filtration k r. Submodule closure, basis spans and rational linearity prove the stated image membership for every $P\in F_r^k$.

A failure of the defining source correspondence, an incorrect unshuffle multiplicity, a nonzero pure-word coefficient in a nonempty action, or a pair outside the claimed finite window with nonzero transpose coefficient would invalidate the route. A single original $k,r,P$ violating the actual image inclusion would refute the claim. The formal result retains all these quantifiers and proves the actual image membership.

The positive proof is not a finite-instance verification or a designated refutation; it receives no refutation audit exemption. A different Appendix A.7 uri assertion, a single-generator consequence, a bounded length assertion or an assumed filtration-preserving coproduct does not resolve this problem.

The four D5 modules and their corresponding Scribes contain the source operations, the projection/support argument, the finite ordinary transpose and the full filtration result. The terminal declaration is D5/S1/Words/PostLie/AriCoproductFiltration.result; its proposition is D5/S1/Words/PostLie/AriCoproductFiltration.claim.

The module-owned Reg mirrors supply four-slot source-equivalence registrations for all public theorem targets. The current scoped report contains 26 declared_validated records: 11 for AriWordOperations, four for AriProjection, five for AriWordProduct, and six for AriCoproductFiltration, with no registration errors. The reported axiom union is Classical.choice, Quot.sound, and propext. Unknown continuations record residual unknown information; they do not assert undecidability or replace registration evidence.

The typed Scribe resolution binds this one problem to the actual result as Proved. Canonical frozen state records carry the declaration identities. A scoped kernel report establishes the selected mathematical scope, not complete repository admission. Full required CI and independent final review remain separate checks.

This is the whole external named conjecture under admission_basis: open-problem-resolution, with conservative proof_shape: bind-only synthesis. Its positive conclusion is universal; utility: none records that it is not an ordinary positive finite certified instance. The retained auxiliary declarations supply the live mathematical chain to result; no stronger input hypothesis is added.

The bounded literature screen covered relevant passages of arXiv:2508.05024v1, 2603.05038v1, 2609.25168v1 and 2610.37351v1 without identifying an exact settlement. MathDB 368177 concerns the distinct Appendix A.7 uri assertion. These are bounded search conclusions, not a claim of worldwide absence or priority. Other people's exact registered targets remain excluded in every registration state.

Literature completeness and worldwide priority are ASSUMED-UNVERIFIED; rate-limited citation APIs do not establish absence. No complete classification of related post-Lie structures or independent formalization of the paper's magmatic construction is claimed.

Standalone content-check and markdown-check reject the canonical scoped-v1 report at their raw-v3-only schema boundary, before evaluating the candidate. This limitation is tracked in #20682. Those checks have not passed; full current-input admission belongs to the supported required CI route. The draft's mathematical result does not itself certify final review, merge, first dev push, priority or solve credit.