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A Singular-Safe Finite-State Proof for the Dakota Number Wall and Shallit's Three-Value Hankel Problem

Year: 2026
Type: preprint
Abstract: Shallit asked whether a sequence taking only three real values can have every shifted Hankel determinant of positive order nonzero. The construction is based on Fred Lunnon's Dakota sequence. Lunnon's unpublished 2018 manuscript already proves the relevant zero-free Dakota-wall result over nontrivial ternary extensions and its characteristic-zero consequence for irrational parameters; Claudemir de Souza Cavalcante gave the first located public complete account in July 2026 and published a replayable version 2 in September. This paper gives two further finite proofs of the bilateral finite-field theorem. Over F_9 = F_3[u]/(u^2+1), with y_n = 1 + u d_n, the first proof represents the complete Dakota wall by 1,482 exact diamond tiles and uses a reconstruction procedure that remains valid in the presence of zero windows. In the verified F_9 replay every reconstructed cell takes the regular Desnanot-Jacobi branch. The scalar closures used by the second proof exercise the zero-window and singular-frame branches on genuine zeros. The direct determinant half-plane contains 188,992 valid outputs and no zero. The second proof shows that a Hankel determinant of y vanishes exactly when the corresponding determinants of d and d+1 vanish simultaneously, and excludes that simultaneous zero state by a synchronized finite certificate over F_3. For each order and shift the determinant of t+d_n is a nonzero integer polynomial in t of degree at most one. Hence every irrational real t gives a sequence taking exactly the three values t-1, t, t+1 with all shifted Hankel determinants nonzero. The record contains the manuscript PDF, the manuscript and figure source, and a portable exact proof archive. The proof archive distinguishes the complete 4,134-pattern neighbouring language from the 4,126 lower-diamond reconstruction contexts and explicitly verifies the eight input-boundary cases. After extracting the proof archive, run python -B CHECK_PUBLIC_RELEASE.py from its root. The release checker uses Python 3.9 or later and the standard library only. The uploaded proof archive has SHA-256 c0cfdaeebd3c4aef79d361a8ba466378d72d1ff4e45d8faa596ab63879a9dac1. Research context This deposit forms part of a broader investigation into the epistemic consequences of AI-assisted research: how claims generated or substantially developed by generative systems can be assessed, challenged, verified, attributed and, where justified, admitted into a research record. Mathematics provides an unusually clear setting in which to examine these questions. Candidate results can often be stated precisely; counterexamples can decisively reject them; proof obligations can be isolated; symbolic and computational claims can be independently replayed; and the distinction between numerical evidence, conjecture, proof and formal or computational verification can be maintained explicitly. The purpose of this research programme is therefore not to present the author as a conventional mathematician or to treat the production of additional solutions to open problems as an end in itself. The mathematical problems serve as bounded test cases for a wider epistemic question: what processes are required for an AI-generated mathematical claim to become a warranted, inspectable and reproducible piece of knowledge? Accordingly, the released artefacts place particular emphasis on provenance, explicit claim scope, failed approaches and counterexamples where relevant, independent checking, adversarial review, novelty assessment, reproducible computation and the separation of discovery from verification. Each deposit is both a mathematical result and a record of an AI-mediated research process intended to make the basis for accepting or rejecting that result visible. AI-assisted research disclosure This work was developed through a human-directed, AI-assisted mathematical research process conducted principally in OpenAI Codex using GPT-5.6 Sol. The model was used as an active research tool throughout the investigation, including structural exploration, conjecture formation, symbolic derivation, proof development, counterexample search, exact-arithmetic code implementation, construction of finite proof objects, and the design and analysis of verification procedures. The research process was iterative rather than a one-pass generation of a proof. Candidate arguments and computational claims produced during the investigation were tested against exact calculations, explicit reconstruction of the relevant algebra, independently implemented checks, finite-state certificates, replayable verification code, and adversarial review. Approaches that failed under those checks were discarded or retained only as negative results. Universal claims were not inferred from finite numerical agreement alone. For the Dakota number-wall result in particular, the computational proof objects and their verification procedures are released with the work so that the machine-assisted parts of the argument can be inspected and rerun independently. The public package includes exact finite certificates, singular-safe reconstruction checks, boundary verification, and independent replay machinery corresponding to the proof-critical computational claims. Anthropic Claude Fable 5 was also used for implementation review and verification during development. Additional model-assisted adversarial review was used to test the mathematical exposition, certificate boundaries, portability and reproducibility of the release. These review stages were treated as checks on the work rather than as independent mathematical authority. The author directed the research process, selected the problem and scope, determined which candidate results were retained, and set the evidential threshold for release. Responsibility for the mathematical claims, code, citations, proof objects and final text remains with the author. The AI systems are not listed as authors and are not treated as independent sources of mathematical authority.
DOI: 10.5281/zenodo.22347356
Language: English
Open Access status: green
OpenAlex ID: W7208812427
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