Half-flips are 5-avoidable

Pascal Ochem

arxiv: 2605.18811 · v1 · pith:FRHRLOZTnew · submitted 2026-05-12 · 🧮 math.CO · cs.FL

Half-flips are 5-avoidable

Pascal Ochem This is my paper

Pith reviewed 2026-05-20 22:18 UTC · model grok-4.3

classification 🧮 math.CO cs.FL

keywords half-flipsavoidabilitypure morphic wordscombinatorics on wordspattern avoidanceinfinite wordsmorphism

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The pith

A pure morphic word over five letters avoids 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 that contains no half-flips in any of its factors. A half-flip is a pair of factors uv and vu where u and v are nonempty and have identical length. This settles the open question of the smallest alphabet size permitting infinite half-flip avoidance by supplying a constructive example over five letters, where prior work had only a non-constructive existence proof and a conjecture for five letters. The same approach also yields smaller alphabets once the minimum block length is increased.

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, whose iterated images are shown to contain no factors of the form uv and vu with equal lengths.

If this is right

Half-flips are avoidable over an alphabet of size 5.
Half-flips with blocks of length at least 2 are avoidable over an alphabet of size 3.
Half-flips with blocks of length at least 4 are avoidable over an alphabet of size 2.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The explicit five-letter morphism could be tested computationally for avoidance of other local patterns in combinatorics on words.
The pattern of smaller alphabets working for longer minimum block lengths suggests a possible trade-off that might be quantified in related avoidance problems.

Load-bearing premise

The chosen morphism must generate an infinite word in which no finite factor ever contains both uv and vu with |u| equal to |v|.

What would settle it

Finding any finite factor of the form uv vu (with |u|=|v|) inside the infinite word produced by repeated application of the morphism would disprove avoidance.

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 explicit 5-letter pure morphic word avoiding half-flips and settles the minimal alphabet conjecture.

read the letter

Ochem has given an explicit pure morphic word over 5 letters that avoids half-flips. This confirms the prior conjecture and answers the question on the smallest alphabet for the property. The paper also shows half-flips with |u| at least 2 are 3-avoidable and those with |u| at least 4 are 2-avoidable. Those extra statements are useful but sit apart from the main 5-letter result. The construction is the real advance. Earlier work had a non-constructive existence proof and then a conjecture for a morphic word on 5 letters. Delivering the actual morphism moves the claim from guess to object that can be examined. Pure morphic words are a standard tool here, so the choice fits the literature. The verification that the infinite word has no uv and vu pairs of equal length is the part that needs to hold. For these problems the check is usually finite once you fix the morphism, and the stress-test note sees no hidden assumption or inconsistency in the structure. If the full paper spells out that argument clearly, the central claim stands. This paper is for people who work on word avoidability and explicit constructions in combinatorics on words. A reader who wants to see the morphism or extend the technique will get direct value from it. It deserves a serious referee because it resolves a concrete open question with a new explicit result rather than additional theory.

Referee Report

0 major / 2 minor

Summary. The paper presents an explicit pure morphic word over a 5-letter alphabet that avoids half-flips (factors uv and vu with |u|=|v|). It also proves that half-flips with |u|≥2 are 3-avoidable and those with |u|≥4 are 2-avoidable, providing a constructive answer to the minimal alphabet size question posed by Fici and confirming the 5-letter case conjectured by Currie and Rampersad.

Significance. If the avoidance claim holds, the result is significant for combinatorics on words: it supplies the first explicit pure morphic construction over 5 letters, establishing 5-avoidability and improving on the prior 8-letter construction. The additional restricted avoidability results clarify how the parameter |u| affects the minimal alphabet size. The explicit morphism is a strength, as it supports direct verification and potential extensions.

minor comments (2)

In the section defining the morphism, state explicitly whether the half-flip avoidance is established by a finite reduction argument or by exhaustive enumeration up to a computable length bound; this would strengthen reproducibility of the central claim.
The abstract could briefly indicate the 5-letter alphabet (e.g., {a,b,c,d,e}) used in the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report, accurate summary of our contributions, and recommendation of minor revision. We appreciate the recognition that our explicit pure morphic construction improves on the prior 8-letter example and addresses the questions of Fici, Currie, and Rampersad.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper offers an explicit pure morphic construction (a specific 5-letter morphism) together with a direct verification that the generated infinite word contains no half-flip factors. This is a standard constructive argument in combinatorics on words that relies on checking the morphism's action on a finite set of factors rather than on any fitted parameter, self-referential definition, or load-bearing self-citation. The claims for 3-avoidability and 2-avoidability of longer half-flips are independent extensions and do not feed back into the central 5-letter result. The derivation chain is therefore self-contained and does not reduce to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on standard results about morphic words and factor avoidance; no free parameters, invented entities, or ad-hoc axioms are apparent from the abstract.

axioms (1)

standard math Standard properties of pure morphic words and their factors in combinatorics on words.
The construction and avoidance claims rely on established theory of morphisms and infinite words.

pith-pipeline@v0.9.0 · 5654 in / 1109 out tokens · 65965 ms · 2026-05-20T22:18:29.517472+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery and embed_injective unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We use the following 95-uniform morphism: m(0)=c1301240... and prove avoidance by showing 2p ≡ 0 (mod 95) then reducing to a smaller half-flip in the preimage.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

1 extracted references · 1 canonical work pages

[1]

Currie and N

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

work page 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

work page 2025