slug: campbell-currie-rampersad-eq-11 bibkey: campbell2025reduced doi: 10.48550/arXiv.2509.16034 triage: theorem motivation_gids:

• D5/S1/Words/Complexity/ThueMorseReducedAbelianEven

Equation (11) for the reduced abelian complexity of the Thue-Morse word

Problem

Write rho(n) for the reduced abelian complexity of the Thue-Morse word: the number of classes of length-n factors, where two factors are equivalent when collapsing each maximal constant run to one letter leaves words of equal length that are rearrangements of one another. Campbell, Currie and Rampersad display

|rho(4n+2) - rho(4n)| = 0  if t(n+1) = t(3n+1),  and 1 otherwise,

as equation (11) in the Conclusion of arXiv:2509.16034v1, with the sentence “It appears that (11) holds”, and state a few lines later that they leave proving it as an open problem. The verified journal version retains equation (11) and this question in Section 3, printed page 14; its citation is recorded in Library/Words/campbell2025reduced.md. The paper’s letter indices start at one, so the zero-indexed reading is t(n) = t(3n). Factor lengths do not shift.

Motivation

The odd-index half of the same paragraph is already carried here as Problems/thue-morse-reduced-abelian-odd, together with the frozen module D5/S1/Words/Complexity/ThueMorseReducedAbelianOdd. That module already supplies the all-start factor definitions, the run-collapsed Parikh vector, the code that decides equivalence, and the odd recurrence. Equation (11) is the even-index counterpart and was the nearest unclaimed item in the same paragraph.

Gap

The odd recurrence follows from a bijection between codes at lengths 2n+1 and n+1. No such bijection is available at even lengths: the two lengths 4n and 4n+2 carry genuinely different class counts, and their difference is one of zero and one according to a condition on two Thue-Morse letters. So the even case needs the spectrum of the codes, not a length-to-length correspondence.

Route

Let alternations n s count the letter changes across n edges in the length-n+1 factor at start s, and let minAlternations n and maxAlternations n be its extremes over all starts. These are the paper’s extrema at length n+1. Two steps carry the argument.

First, the parity of the sum of the two extremes is governed by two Thue-Morse letters: (minAlternations n + maxAlternations n) % 2 is zero exactly when thueMorse n = thueMorse (3 * n). The proof is a strong induction that splits n by parity, uses the behaviour of the alternation count under appending a letter, and closes the odd case with a boundary fact about three consecutive Thue-Morse letters.

Second, the reduced Parikh vectors realised at a fixed length form an interval of alternation counts, so the class count is the size of a weighted interval determined by the two extremes. Shifting the length from 4n to 4n+2 moves that interval by a controlled amount, and the resulting difference of the two counts reduces to the difference of the two extremes modulo two. Combining with the first step gives the stated dichotomy.

Falsifier

An n with 0 < n at which the absolute difference is neither zero nor one, or at which it disagrees with the Thue-Morse condition, would contradict the theorem about the defined class count.

Evidence

• Module: D5/S1/Words/Complexity/ThueMorseReducedAbelianEven.lean.
• Theorem: reducedAbelianComplexity_even_difference.
• Supporting public result: alternation_extrema_parity.
• The class count is the frozen R of ThueMorseReducedAbelianOdd, unchanged.
• No finite cutoff appears in either public statement.

Triage

theorem. This dossier’s resolution concerns only the displayed absolute difference. The sign of rho(4n+2) - rho(4n) when nonzero, a recursion for rho(4n), and the full recursion remain outside that result. For the related nonautomaticity question, the Library note and the existing even-complexity Scribe mirror explain how the extrema recurrences and weighted interval count give unbounded R on positive lengths, and hence non-k-automaticity in every integer base k >= 2. That exposition adds no separately kernel-checked endpoint or typed resolution claim and does not enlarge this equation-(11) binding. Unboundedness is compatible with R(2^k+1) = 3.

ASSUMED-UNVERIFIED

The identification of the paper’s rho with this repository’s R rests on a human reading of the printed definition together with term-by-term agreement with the paper’s own printed sequence over a bounded window. It is not a machine proof of that identification. First-publication priority is not established, and no search for a later resolution of equation (11) was performed.