Hanaoka–Konno Conjecture 4.2: self-similarity of the Riesz walk

R. Hanaoka and N. Konno, “Return probability and self-similarity of the Riesz walk,” Quantum Information and Computation 21 (2021), 409–422, arXiv:2010.04518v3, DOI 10.26421/QIC21.5-6-5, state Conjecture 4.2:

Let Z_> = {1,2,3,...}. Put T_0(n)={0} and T_k(n)=((k-1)/n,k/n] for 1 <= k <= n. For the Riesz walk with initial state Psi_0=[1,0]^T delta_0, P(X_2t/(2t) in T_k(t)) = P(X_8t/(8t) in T_k(t)) for every t in Z_>.

Issue #15758 preregistered the exact natural-number reading. The delivered declaration is D5/S3/Quantum/Walks/RieszWalkSelfSimilarity.result : claim, and its module state has statement identity sha256:4b9935aefe3ec7033ed45b54d5dec573a17d5471a67ae35f412951ac228f6269.

Statement echo

The source's t >= 1 is the binder t : Nat with 1 <= t; its 0 <= k <= t is k : Nat with k <= t. The source's normalized probability is prob (2t) (Tset t k) and prob (8t) (Tset t k); Tset gives the singleton at k=0 and the left-open, right-closed rational interval for 1 <= k <= t. The conclusion is the equality in claim; result proves exactly that proposition.

The conjecture connects the return law of a quantum walk with the fourth-power self-similarity of the Riesz measure. The settlement identifies the stronger cumulative identity at both scales and supplies a kernel-checked bridge from the CMV representation to the block probabilities.

The paper reports numerical support but no proof for Conjecture 4.2. The preregistered route was the CMV spectral representation, Geronimus identification, and the Riesz decimation identity. The proof uses the five supporting modules frozen in this delivery under escape-witness; the settling module is admitted as open-problem-resolution (#15758; Proved).

The paper's division by z is represented using the pinned RecursiveBoundaryRowSeries.tail owner because PowerSeries.X^{-1}=0. The equations corresponding to the Schur definition and Schur step are proved by schur0_spec and schurStep_spec; this is the convention repair recorded in issue #15758.

Every newly authored public theorem, including the public Szego rho-square owner D5/S3/Weil/Szego/CanonicalTransfer.szego_rho_sq (declaration statement_id: sha256:0c7e849be0a2d5c52755ddb652d3be5c71965267f3356f6b4e69cbbeb8141df6), remains under the unfinished escape audit issue #15854. No declaration is reported as declared_validated.

The Riesz moments stabilize in finite products, giving a positive form on Laurent polynomials. Gram–Schmidt produces the orthonormal Laurent basis and the finite CMV projection representation. Geronimus identifies the Schur parameters with the Verblunsky coefficients. The map z -> z^4 is an isometry after the Riesz density change, and the parity-weighted projection identity equates the two finite projection norms. The cumulative equality F_(2t)(2x)=F_(8t)(8x) then yields the interval identities by subtraction, with the singleton origin block handled separately.

A natural t >= 1 and 0 <= k <= t violating the displayed probability equality would contradict result. At the proof level, failure of the finite projection identity, the Geronimus identification, or the fourth-power Riesz density relation would break the route.

The kernel-checked result is D5.S3.Quantum.Walks.RieszWalkSelfSimilarity.result; its proof uses only the standard axiom closure propext, Classical.choice, and Quot.sound. The five supporting modules and their Scribe projections are frozen in the module-state ledger. The experiment entry https://github.com/the-omega-institute/trureturing-experiments/tree/cc6202d7269efbf26843227e123f34f3652dc67c/docs/reports/hanaoka-konno-2021-riesz-walk-self-similarity was run as python3 check.py, exit 0, and has SHA-256 54eca0560bd45517bdd5010d7540b01b5bf8360a23024434e7fff3b9406cb007. It checks the literal Section 2 construction for t=1,...,32 (times up to 256), with worst discrepancy $4.28 \times 10^{-61}$, and the factor-two control fails in 104 instances. These are finite computed readings.

Tier-1 external conjecture, preregistered in #15758; resolution proved for every natural t >= 1 and 0 <= k <= t. The admission basis is open-problem-resolution (#15758; Proved); utility is none.

What the settlement shows

• Kernel-checked: result proves the stated block equality. The decisive mechanism is CMV.Representation.SpectralReduction.cumulative_projection, CMV.Geronimus.schur_equals_gram, Riesz.Moments.Forms.tau_density, Riesz.Moments.Forms.riesz_residue_orthogonality, Riesz.Decimation.Parity.parity_weighted_projection, and RieszWalkSelfSimilarity.Scaling.cumulative_scale_four. These identify cumulative probabilities with Laurent-window projection norms, express fourth-power decimation with density $1 + \operatorname{Re} w$ and invariance under $z \mapsto iz$, and give $F_{2t}(2x) = F_{8t}(8x)$.
• Kernel-checked: the supporting content witnesses are Riesz.Moments.product_independent, CMV.SzegoRecurrence.OPUCRecurrence.monic_szego, CMV.Geronimus.schur_equals_gram, CMV.Representation.WalkCore.mu_outside, and Riesz.Decimation.Parity.parity_weighted_projection; their conclusions are consumed along result's proof path.
• Computed: the cited experiment checks 32 positive times and the factor-two control; it does not extend the kernel theorem or settle an asymptotic limit.
• Open: Conjectures 4.1 and 4.3, the limit theorem, and general m >= 3 Riesz-type walks remain outside this result. The source's later questions relying on those statements are unchanged.

The bounded literature screen recorded in #15758 did not find a later proof; worldwide novelty, priority, and the absence of an independent proof outside that screen remain unverified. The experiment is finite evidence and does not replace the Lean proof.