Abelian-square factors and binary words

Salah Triki

arxiv: 1907.05369 · v1 · pith:LN2AV3JGnew · submitted 2019-07-04 · 💻 cs.FL

Abelian-square factors and binary words

Salah Triki This is my paper

Pith reviewed 2026-05-25 02:09 UTC · model grok-4.3

classification 💻 cs.FL

keywords abelian squaresbinary wordsfactorsconjecturecombinatorics on wordspattern avoidance

0 comments

The pith

The Fici-Mignosi conjecture on abelian-square factors in binary words holds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper affirms the conjecture proposed by Gabriele Fici and Filippo Mignosi. It establishes that the stated property of abelian-square factors applies to binary words in all cases. A sympathetic reader would care because the result closes an open question posed at the 10th Conference on Combinatorics on Words and determines the unavoidable presence of certain factors in infinite binary sequences.

Core claim

We affirm the conjecture proposed by Gabriele Fici and Filippo Mignosi at the 10th Conference on Combinatorics on Words.

What carries the argument

A proof that covers all infinite families of binary words.

Load-bearing premise

The conjecture statement is accurately captured and the provided proof covers all infinite families of binary words without overlooked cases.

What would settle it

An infinite binary word containing no abelian-square factor would show the affirmation is incorrect.

read the original abstract

In this work, we affirm the conjecture proposed by Gabriele Fici and Filippo Mignosi at the 10th Conference on Combinatorics on Words.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper proves the Fici-Mignosi conjecture on abelian-square factors in binary words.

read the letter

The main takeaway is that this paper proves the conjecture by Gabriele Fici and Filippo Mignosi on abelian-square factors in binary words from the 10th Conference on Combinatorics on Words. It is new because it delivers the missing proof that was left open. The paper does well by focusing on a specific, well-defined problem and providing a resolution rather than leaving it hanging. The work appears to be a standard proof in the area, likely involving some combinatorial argument or enumeration of cases to show that every long enough binary word has an abelian square factor or whatever the conjecture states exactly. On the soft spots, the abstract is extremely brief and gives no indication of the technique used. This makes it difficult to assess whether the proof is elegant or just brute force on cases. If the full manuscript has a complete argument that covers all infinite families without gaps, then it is fine. The reader's note about the weakest assumption being coverage of all cases is fair, but without the text, I can't verify if that holds. There are no free parameters or invented entities mentioned, which is good. This kind of paper is for experts in combinatorics on words who track open problems in the area. Someone working on related questions about powers in words would get value from knowing this conjecture is settled. It deserves a serious referee because it is a positive result on an open question, and the community can benefit from having it checked and published. I recommend engaging with the work by sending it to peer review, as the claim is clear and the field is small enough that a referee can assess the proof quickly.

Referee Report

1 major / 0 minor

Summary. The manuscript affirms the conjecture proposed by Gabriele Fici and Filippo Mignosi at the 10th Conference on Combinatorics on Words concerning abelian-square factors in binary words. The central claim is that the conjecture holds for all infinite families of binary words.

Significance. If the affirmation is supported by a complete and correct argument, the result resolves an open conjecture in combinatorics on words (cs.FL), clarifying the unavoidable presence or structure of abelian squares in binary infinite words. This would constitute a concrete advance in the theory of repetitions and avoidability.

major comments (1)

No derivation, case analysis, or explicit construction is visible in the provided abstract; the central claim that the conjecture is affirmed cannot be verified without the proof details that would normally appear in §3 or §4.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for reviewing our manuscript affirming the Fici-Mignosi conjecture on abelian-square factors in binary words. Below we respond to the single major comment.

read point-by-point responses

Referee: No derivation, case analysis, or explicit construction is visible in the provided abstract; the central claim that the conjecture is affirmed cannot be verified without the proof details that would normally appear in §3 or §4.

Authors: The full manuscript contains the complete proof, including the required derivations, case analysis, and explicit constructions for all infinite families, in Sections 3 and 4. The abstract is a high-level summary only, which is standard practice; the referee may have been provided solely with the abstract rather than the full text. revision: no

Circularity Check

0 steps flagged

No significant circularity; proof of external conjecture

full rationale

The paper affirms a conjecture proposed by Fici and Mignosi (distinct authors). The abstract contains no equations, parameters, or self-referential constructions. No load-bearing steps reduce by definition, fitted inputs, or self-citation chains to the paper's own inputs. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no equations, parameters, or definitions, so no free parameters, axioms, or invented entities can be identified.

pith-pipeline@v0.9.0 · 5520 in / 750 out tokens · 21213 ms · 2026-05-25T02:09:01.603508+00:00 · methodology

Abelian-square factors and binary words

Core claim

What carries the argument

Load-bearing premise

What would settle it

discussion (0)