slug: thue-morse-reduced-abelian-odd bibkey: campbell2025reduced doi: 10.48550/arXiv.2509.16034 triage: theorem motivation_gids:

• D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd

Odd-index reduced abelian complexity of the Thue-Morse word

Problem

This dossier deliberately anchors only the equality rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for every nonnegative integer n, proposed in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1. The caller-supplied reading, 2026-09-07, quotes its source sentence:

Although it appears that rho^{ab,red}_t(2n+1) = rho^{ab,red}_t(n+1) for nonnegative integers n, the problem of determining a full recursion for rho^{ab,red}_t(n) seems to be challenging.

red(w) collapses every maximal constant run; reduced abelian complexity counts all length-n factors up to rearrangement of their reduced words with equal reduced length. The full recursion and the four further open items in the paragraph, namely the nonzero sign of rho(4n+2)-rho(4n), a recursion for rho(4n), equation (11), and non-k-automaticity of sequence (10), are deliberately out of scope.

Motivation

The frozen motivation module defines factors at every natural start and counts their reduced Parikh vectors. It provides the exact odd recurrence needed for this external proposition, making the all-start interpretation explicit rather than depending on a finite sampled prefix.

Gap

The caller reports that the paper does not prove this odd equality. Its formal counterpart is already frozen; the missing item addressed here is the literature-backed pool entry. No assertion about a full recursion or any of the other four open items follows from this dossier.

Route

Use reducedAbelianComplexity_odd (n : Nat). The formal thueMorse is zero-indexed binary digit parity; factor length start ranges over every natural start, runCompress uses List.destutter, and R length counts the resulting reduced Parikh classes. Equal Parikh vectors force equal reduced length and identical character multiplicities, matching the paper’s equivalence relation according to the caller’s reading.

The frozen proof transfers a bijection on reduced class codes back to these all-start Parikh classes. No prefix-only count is substituted. The R (2^k+1) = 3 corollary is supporting evidence only; the sole problem anchor remains the odd recurrence. Claim binding is a later Scribe layer.

Falsifier

A nonnegative n for which the two exact all-start reduced complexity counts differ would refute the proposition. A discrepancy in a finite sample of starting positions is insufficient without a proof that the sample exhausts every reduced class at both lengths. No fresh numerical search or exhaustive factor computation was performed here.

Evidence

• Frozen module: D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd.lean.
• Public theorem: reducedAbelianComplexity_odd, stating R (2*n+1) = R (n+1) for every natural n.
• Companion public theorem: reducedAbelianComplexity_two_pow_add_one, stating R (2^k+1) = 3; it is not another anchored open problem.
• Machine-checkable frozen-state receipt: Golden/Frozen/state/D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd.lean.json. The worker’s test -f exited 0 on 2026-09-07.
• Literature reading and locators: Library/Words/campbell2025reduced.md. Theory candidates 6.223 and 6.224 supply provenance context only.

Triage

theorem. The exact odd recurrence for the all-start definition has a frozen kernel-verified proof, subject to the source correspondence limitation below. This classification neither covers the paper’s other open items nor binds a resolution claim.

ASSUMED-UNVERIFIED

• No repository machine verifies that the Lean statement is equivalent to the paper’s natural-language proposition. The caller-supplied comparison of the all-start definitions, 2026-09-07, is a human reading, not a proof of equivalence between the source text and Lean.
• The theory volume’s printed venue string, INTEGERS 26 (2026), A34, was not independently verified by the caller or this worker. The Library note binds the verified arXiv DOI and metadata.
• The API, DOI redirect, PDF, and report that the equality is unproved in the paper are caller-supplied. This worker did not fetch those sources again or rebuild Lean; frozen-state existence was checked locally.
• No literature search for a later resolution of the conjecture was performed; the open status recorded in the problem candidate is the status stated in this arXiv version, not an assessment of the subsequent literature.