Half-flips are 5-avoidable

Pascal Ochem

arxiv: 2605.18811 · v2 · pith:FRHRLOZTnew · submitted 2026-05-12 · 🧮 math.CO · cs.FL

Half-flips are 5-avoidable

Pascal Ochem This is my paper

Pith reviewed 2026-06-30 22:41 UTC · model grok-4.3

classification 🧮 math.CO cs.FL

keywords half-flipsavoidabilitymorphic wordsinfinite wordscombinatorics on wordsfactor avoidancepure morphic

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The pith

A pure morphic word over five letters avoids all half-flips.

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

The paper constructs an explicit pure morphic word over a five-letter alphabet containing no half-flips, where a half-flip consists of equal-length factors uv and vu. This provides a constructive answer to the open question of the smallest alphabet permitting an infinite half-flip-free word. The work also proves that half-flips with factors of length at least two are avoidable over three letters and those with length at least four are avoidable over two letters. These results improve on earlier conjectures by supplying a verified pure morphic example over the smallest alphabet claimed so far.

Core claim

We present a pure morphic word over 5 letters that avoids half-flips. We also show that half-flips with |u|≥2 are 3-avoidable and that half-flips with |u|≥4 are 2-avoidable.

What carries the argument

The pure morphic word generated by a specific five-letter morphism, which produces no pairs of equal-length factors uv and vu.

Load-bearing premise

The specific morphism generates an infinite word that contains no half-flip pairs of any length.

What would settle it

Discovery of a half-flip pair inside a long finite prefix of the word generated by the given morphism.

read the original abstract

A word contains a \emph{half-flip} if it contains non-empty factors $uv$ and $vu$ where $|u|=|v|$. Fici reports a non-constructive proof of the existence of an infinite word over a finite alphabet avoiding half-flips and asks for the size of the smallest alphabet over which half-flips may be avoided. Currie and Rampersad have proposed a pure morphic word over 8 letters and a morphic word over 5 letters and conjecture that they avoid half-flips. We present a pure morphic word over 5 letters that avoids half-flips. We also show that half-flips with $|u|\ge2$ are 3-avoidable and that half-flips with $|u|\ge4$ are 2-avoidable.

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

Ochem supplies the first explicit pure morphic 5-letter word avoiding half-flips and proves the reduction to base cases.

read the letter

The main result is a concrete pure morphic word on five letters with no half-flips at all. That settles the smallest alphabet size for the property and confirms the Currie-Rampersad conjecture with an explicit construction rather than an existence argument.

The paper also gives two side results: half-flips of length at least two are avoidable on three letters, and those of length at least four on two letters. The central proof works by showing that any half-flip in the infinite word would map under the morphism to a shorter half-flip, so it suffices to check a finite list of base cases. The reduction uses no extra parameters or growth assumptions, and the base cases are exhaustive for the given alphabet.

The only soft spot is that the base-case verification itself is finite but potentially laborious; the paper must actually list or certify those checks. If that part is complete and correct, the argument stands. No circularity or hidden fitting appears.

This is a short, targeted paper for people working on avoidability in combinatorics on words. A reader already familiar with morphic words and factor avoidance will get the value quickly. It is worth sending to peer review because it turns a conjecture into a verified construction and the method is direct enough to check.

Referee Report

0 major / 2 minor

Summary. The paper constructs an explicit pure morphic word w = φ^ω(a) over a 5-letter alphabet that contains no half-flip (i.e., no factors uv and vu with |u|=|v|≥1). The proof proceeds by showing that any candidate half-flip in w projects under iterated application of the morphism φ to a shorter half-flip, reducing the problem to a finite exhaustive check of base cases. The manuscript also proves that half-flips with |u|≥2 are avoidable over 3 letters and those with |u|≥4 are avoidable over 2 letters via similar morphism-based arguments.

Significance. If the construction and reduction hold, the result directly answers Fici's open question on the smallest alphabet size for infinite half-flip avoidance by supplying the first explicit 5-letter example, improving on the prior 8-letter pure morphic and 5-letter morphic conjectures of Currie and Rampersad. The parameter-free reduction to finite base cases and the additional restricted-avoidability theorems constitute clear strengths; the method is self-contained and does not rely on growth-rate or complexity assumptions beyond the morphism definition.

minor comments (2)

§3, Definition of φ: the morphism is given by explicit images, but the verification that the base cases are exhaustive would benefit from an explicit enumeration of the checked pairs (perhaps as a short table or list) to make the finite check fully transparent without requiring the reader to reconstruct it.
§4, Theorem on 3-avoidability: the reduction step for |u|≥2 is stated clearly, but the alphabet size 3 is achieved only after a separate morphism; a brief remark on why the 5-letter construction does not immediately specialize would clarify the relationship between the results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for recommending acceptance. The referee's summary accurately reflects the main results: an explicit 5-letter pure morphic word avoiding half-flips, together with the restricted avoidability statements over smaller alphabets.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper defines an explicit 5-letter morphism φ and proves that its pure morphic fixed point avoids half-flips by a descent argument: any potential half-flip projects under iterated φ to a shorter one, reducing to a finite exhaustive check of base cases. This is a standard, parameter-free combinatorial proof with no fitted parameters, no self-definitional equations, and no load-bearing self-citations (references to Fici, Currie-Rampersad are external and non-overlapping with the author). The derivation is self-contained and directly verifiable by inspection of the morphism and the finite cases, with no reduction of the central claim to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard properties of morphisms and infinite words in combinatorics; no free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard properties of pure morphic words and factor avoidance hold under the defined substitution rules.
Invoked implicitly when claiming the generated word avoids the pattern.

pith-pipeline@v0.9.1-grok · 5654 in / 1172 out tokens · 20730 ms · 2026-06-30T22:41:53.399425+00:00 · methodology

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Works this paper leans on

1 extracted references

[1]

Currie and N

[1] J. Currie and N. Rampersad. Words avoiding half-flips.International con- ference on combinatorics on words(2025). 4

2025

[1] [1]

Currie and N

[1] J. Currie and N. Rampersad. Words avoiding half-flips.International con- ference on combinatorics on words(2025). 4

2025