bibkey: campbell2025reduced authors: John M. Campbell, James Currie, Narad Rampersad year: 2025 title: Reduced complexities for sequences over finite alphabets doi: 10.48550/arXiv.2509.16034 claim: Section 3 proposes the odd-index recurrence for the reduced abelian complexity of the Thue-Morse word without proving it. strata_touched:

• D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd license: citation-only triage: anchor

Reduced complexities for sequences over finite alphabets

This note anchors only the odd-index equality in Section 3 (Conclusion), page 15 of arXiv:2509.16034v1, for Problems/thue-morse-reduced-abelian-odd.md. The caller-supplied reading, 2026-09-07, transcribes the source sentence as:

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.

The caller reports that the odd equality is not proved anywhere in the paper. The full recursion and the paragraph’s four further open items are outside this anchor: the sign of rho(4n+2) - rho(4n) when nonzero, a recursion for rho(4n), equation (11), and non-k-automaticity of sequence (10).

On page 2, red(w) collapses every maximal constant run to one character. Reduced abelian complexity counts equivalence classes of all length-n factors: two factors are equivalent when their reduced words have equal length and one is a rearrangement of the other. It does not count only prefixes. Equality of reduced Parikh vectors expresses this equivalence, since equal character counts also force equal reduced lengths.

The frozen module D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd uses zero-indexed binary digit parity for thueMorse, List.destutter for runCompress, and R for the number of reduced Parikh vectors over all natural starting positions. Its reducedAbelianComplexity_odd states R (2*n+1) = R (n+1) for every natural n. The companion reducedAbelianComplexity_two_pow_add_one states R (2^k+1) = 3; it is supporting context, not a second problem anchor. The caller’s comparison of R with the paper’s definition is a human reading, not a machine proof of source-to-Lean equivalence.

ASSUMED-UNVERIFIED: the theory volume gives the printed venue as INTEGERS 26 (2026), A34. Neither the caller nor this worker independently verified that venue string. The front matter uses the verified 2025 arXiv metadata and DOI.

Search log

• Caller-supplied reading, 2026-09-07: queried https://export.arxiv.org/api/query?id_list=2509.16034, HTTP 200, totalResults=1. The entry is arXiv:2509.16034v1 with the title and authors above, published 2025-09-19T14:38:51Z, primary category math.CO. No arxiv:doi or arxiv:journal_ref was reported. API response byte count was not supplied.
• Caller-supplied reading, 2026-09-07: HEAD https://doi.org/10.48550/arXiv.2509.16034 returned HTTP 302 to https://arxiv.org/abs/2509.16034; no response byte count was supplied.
• Caller-supplied reading, 2026-09-07: https://arxiv.org/pdf/2509.16034v1 returned HTTP 200, 309503 bytes, 18 pages, SHA-256 b38c8326c8b7d8b0bea8b8f37cc368230f7d9b9c090070177b2b77cfba9a0cbe. The caller located the definition on page 2 and the unproved equality in Section 3 on page 15, and compared the definition against Lean’s R.
• Worker reading, 2026-09-07: read the local all-start factor definitions, public recurrence and corollary, and frozen-state receipt. The HTTP resources above were not fetched again; no venue lookup was performed.

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.

Verified locator

• arXiv: https://arxiv.org/abs/2509.16034v1 (caller-supplied metadata).
• DOI: https://doi.org/10.48550/arXiv.2509.16034 (caller-supplied HTTP 302).
• PDF: https://arxiv.org/pdf/2509.16034v1 (definition, page 2; Section 3, page 15; caller-supplied HTTP 200).