# AI provenance - The minimum number of distinct abelian squares in a binary circular word (v2)

Human author and submitting contributor: Deep Bhattacharjee, ORCID 0000-0003-0466-750X (sole author since release v2.1.0; the only contributor on hexagon:2610.00473v1).

Text classification: to be chosen by the submitting author on the form (version 1 was submitted as "a mix of human-written and AI-generated text").
Human-understanding declaration: to be chosen by the submitting author on the form (version 1 carries "some parts").
Basis: the submitting author's own declaration on the Hexagon form, not a claim verified by the assistant.

Claude Code (Anthropic) assisted, under the author's direction, with the proofs of Sections 1 to 4 and their Lean 4 formalisation (versions 1 and 2), the checking programs in C, C++, Python, Julia and shell, the figures, the LaTeX text, and in this version the mathematics and text of Sections 5 and 6 (the balance identities, the jump and descent lemmas, the two-run formula, Conjecture 5.5 and its verification, the literature status of the open problems, and Theorem 6.1 on occurrences), the three C programs of those sections, and the submission package. OpenAI Codex assisted with the source audit and the amsart conversion of v2.2.0, as the paper's own declaration states.

Documented tools: Anthropic Claude / Claude Code; OpenAI Codex (v2.2.0 conversion); Lean 4.34.1 with Mathlib; C, C++, Python 3, Julia 1.10 and Bash; LaTeX / pdfLaTeX (amsart). Exact historical model versions are not inferred from tool names.

Verification scope: Theorem 1.1 and its corollary are proved in the text and formalised in Lean 4 (six main theorems, axioms propext, Classical.choice and Quot.sound only, no sorry). Theorem 6.1 and Lemmas and Propositions 5.1 to 5.4 are proved in the text by hand. Conjecture 5.5 is not proved; it is verified to length 27 by a program, as is the linear-word minimum to length 24. The programs are in release v2.3.0 of the repository; no proved statement rests on them alone.

The author retains responsibility for the final mathematics, citations and approval of the manuscript. AI systems are not added as human contributors. The package creation step prepared the source archive and the submission metadata; it did not submit the manuscript to Hexagon. No private prompts, chat history or hidden reasoning traces are included or represented as an exported transcript.

Earlier Hexagon version: https://hexagonmath.org/2610.00473v1
Research manuscript/source DOI: https://doi.org/10.5281/zenodo.23692500 (release v2.3.0; all versions https://doi.org/10.5281/zenodo.23313258)
Repository: https://github.com/creelie/literate-octo-lamp