slug: joshi-rust-2025-thue-morse-infinite-odd-fibers
bibkey: joshirust2025monochromatic
doi: 10.1016/j.tcs.2025.115391
url: https://arxiv.org/html/2501.05830v2
triage: theorem
motivation_gids:
  - D5/S1/Words/ThueMorseMapFirstStart

Infinite odd fibers of Thue-Morse global maxima

Problem

Joshi and Rust, arXiv:2501.05830v2, Section 3.2.1, Question 3.7, second clause asks: “Are there infinitely many values of n for which O_max(n)=∞?” The first and third clauses are outside this target.

Exact question locator: https://arxiv.org/html/2501.05830v2#S3.Thmtheorem7.

The staged claim is that the set of natural lengths n for which the set of positive odd differences d satisfying ExactMax(d,n) is infinite is itself infinite. ExactMax requires a start attaining length n and a universal bound on every monochromatic AP length at every natural start. MAP and the actual zero-indexed thueMorse are reused from the existing Words sources. This covers both colors and the entire infinite word.

Motivation

This question concerns the distribution of actual progression maxima, rather than one explicit family of long progressions. The existing first-start result supplies the actual word, MAP and general dyadic identities. The latter are shared through ThueMorseDyadic.

Gap

Unbounded attained lengths alone do not give infinite exact-max fibers. For each lower bound, one needs infinitely many distinct odd differences, a uniform finite upper bound over all starts, and actual maximum attainment.

Route

Fix odd m>=3 and put M=2^m, c=M^2+M+1, B=2^(4m+1), a=c(M^2-1), U=2B+2. Base-M digits give a true triple at a,a+c,a+2c inside one B block, and t(cj)=t(j) for j<M. Coprimality of B and c transports the triple into every U-letter c-spaced window. No such window is a color translate of consecutive Thue-Morse letters, since those contain no monochromatic triple.

For R=2^k>=max(2U+1,M) and d=cR+1, the single-carry split bounds every monochromatic progression at every start by 2U. Start zero attains M. The finite nonempty set of attained lengths has an actual maximum in [M,2U]. Infinite pigeonhole across distinct positive odd differences gives an infinite exact-max fiber at some n>=M. Unbounded odd exponents yield infinitely many such lengths.

Falsifier

Only finitely many natural lengths with infinite positive odd exact-max fibers would refute the claim. A missing attaining start, a longer progression at any start or color, or use of a different word convention would invalidate the source correspondence.

Evidence

• Formal source: D5/S1/Words/ThueMorseMapInfiniteFibers.lean.
• Public declarations: ExactMax, claim, result.
• ExactMax(d,n) requires an attaining start and bounds every AP length at every natural start. Both Boolean colors are unrestricted.
• The public import check verifies the expanded statement, explicit color quantifiers and the Scribe unbounded-quantifier formulation.
• The result’s axiom closure is propext, Classical.choice, Quot.sound.
• Exact source: https://arxiv.org/html/2501.05830v2#S3.Thmtheorem7.

Triage

Theorem: the Lean result establishes the exact second clause. Its mechanism combines replicated triples, residue transport, dyadic carry control, finite attainment and infinite pigeonhole. The selected exact maxima are not given an explicit formula. The first and third clauses, a classification of all infinite fibers, and an explicit maximum formula remain unproved by this result. No other paper result is used as a consequence of this clause in the formal proof.

ASSUMED-UNVERIFIED

Compilation status and exact source hashes belong to the staging receipts. Integration, canonical gates and external acceptance are controller-owned. The published question and library suppliers are source checked; no exhaustive later-literature audit is claimed. Worldwide novelty, priority and official acceptance are unverified.