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arxiv: 2607.00674 · v1 · pith:N5NXBTPDnew · submitted 2026-07-01 · 🧮 math.CO

Relaxation of Square-Freeness

Hiroki Shibata , Takuya Mieno , Dominik K\"oppl , Shunsuke Inenaga This is my paper

Pith reviewed 2026-07-02 10:59 UTC · model grok-4.3

classification 🧮 math.CO
keywords square-free wordsparameterized squaresorder-preserving squaresmorphic constructionsavoidance problemsnonrepetitive sequencescombinatorics on words
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The pith

Morphic constructions produce infinite ternary words avoiding parameterized squares of length six or more and binary words avoiding order-preserving squares of the same length.

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

The paper extends classical results on square-free sequences by relaxing equality checks to parameterized equivalence and order-preserving equivalence. It introduces ℓ⁺-squares as repetitions whose total length is at least 2ℓ and focuses on the case ℓ=3. Explicit morphic substitutions are used to build an infinite ternary word free of all 3⁺-parameterized squares and an infinite binary word free of all 3⁺-order-preserving squares. These constructions demonstrate that the relaxed repetition notions permit infinite avoidance on alphabets where standard square-freeness is impossible or requires more symbols. The authors also compute the longest finite ℓ⁺-square-free words for several equivalence relations and parameter values.

Core claim

Using morphic constructions, we obtain an infinite 3⁺-parameterized-square-free ternary word and an infinite 3⁺-order-preserving-square-free binary word. In addition, we report the longest ℓ⁺-square-free words across several equivalence relations.

What carries the argument

Morphic substitutions that map letters to finite words and generate infinite strings whose every factor avoids ℓ⁺-squares under the chosen equivalence relation.

If this is right

  • Infinite avoidance holds for 3⁺-parameterized squares on a three-letter alphabet.
  • Infinite avoidance holds for 3⁺-order-preserving squares on a two-letter alphabet.
  • Maximal lengths of finite ℓ⁺-square-free words can be determined for multiple equivalence relations.
  • The relaxed equivalences permit square avoidance on smaller alphabets than the classical equality case.

Where Pith is reading between the lines

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

  • The same style of morphism might be tested for avoidance of shorter squares or other power types.
  • Order-preserving avoidance on binary alphabets could link to pattern-avoidance questions in sequences or permutations.
  • Growth rates or repetition thresholds under these equivalences may differ from the standard square-free case.

Load-bearing premise

The given morphic substitutions generate infinite words in which every factor satisfies the avoidance condition under the respective equivalence.

What would settle it

A concrete prefix of one of the constructed words that contains a factor forming a 3⁺-square under parameterized or order-preserving equivalence.

Figures

Figures reproduced from arXiv: 2607.00674 by Dominik K\"oppl, Hiroki Shibata, Shunsuke Inenaga, Takuya Mieno.

Figure 1
Figure 1. Figure 1: Illustration for Lemma 3.2. In this figure, the two occurrences of [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration for Lemma 3.2. In the figure, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustrations of examples for Case 3-(b) (top) and Case 3-(c)-(ii) (bottom) of Lemma 4.1. In the first [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustrations of examples for Case 3-(c)-(i) of Lemma 4.1. In the first figure, the word [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
read the original abstract

We extend the analysis of nonrepetitive sequences of Entringer et al. [Journal of Combinatorial Theory, 1974] to relaxations of equality testing under nonstandard equivalence relations, in particular parameterized equivalence and order-preserving equivalence. For this setting, we introduce $\ell^+$-squares, defined as squares whose total length is at least $2\ell$. Using morphic constructions, we obtain an infinite $3^+$-parameterized-square-free ternary word and an infinite $3^+$-order-preserving-square-free binary word. In addition, we report the longest $\ell^+$-square-free words across several equivalence relations.

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.

Referee Report

1 major / 0 minor

Summary. The paper extends the study of nonrepetitive sequences to relaxed notions of repetition under parameterized equivalence and order-preserving equivalence. It defines ℓ⁺-squares (squares of total length at least 2ℓ) and claims to construct, via morphic substitutions, an infinite 3⁺-parameterized-square-free ternary word and an infinite 3⁺-order-preserving-square-free binary word; it additionally reports the lengths of the longest finite ℓ⁺-square-free words under several equivalences.

Significance. If the morphic constructions are shown to generate words in which every factor avoids the stated 3⁺-squares under the respective equivalence, the results would supply explicit infinite examples of repetition avoidance under strictly weaker relations than equality, extending classical square-freeness results. The tabulated maximal finite lengths supply concrete comparative data across equivalences.

major comments (1)
  1. [Morphic constructions (as referenced in the abstract and main body)] The central existence claims rest on the morphic constructions. The manuscript must supply the explicit substitutions together with either (i) a proof that the morphisms preserve avoidance of 3⁺-squares under the parameterized (resp. order-preserving) equivalence or (ii) a computational certification that a prefix long enough for the iterated images to cover all possible factors contains no 3⁺-square; without one of these, the weaker equivalence leaves open the possibility that repetitions appear only in concatenations longer than the images checked during construction.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thorough review and for highlighting the need for explicit details on the morphic constructions. We address the major comment below and will revise the manuscript to strengthen the presentation of these results.

read point-by-point responses

  1. Referee: The central existence claims rest on the morphic constructions. The manuscript must supply the explicit substitutions together with either (i) a proof that the morphisms preserve avoidance of 3⁺-squares under the parameterized (resp. order-preserving) equivalence or (ii) a computational certification that a prefix long enough for the iterated images to cover all possible factors contains no 3⁺-square; without one of these, the weaker equivalence leaves open the possibility that repetitions appear only in concatenations longer than the images checked during construction.

    Authors: We agree that the original manuscript did not provide sufficient explicit detail on the substitutions or their verification. In the revised version, we will include the explicit morphisms for the infinite 3⁺-parameterized-square-free ternary word and the 3⁺-order-preserving-square-free binary word. We will also supply a proof that these morphisms preserve the respective avoidance properties, ensuring that no 3⁺-squares arise under the equivalences even in longer concatenations of images. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit morphic constructions with external verification

full rationale

The paper's central results are obtained by defining specific morphisms on finite alphabets and asserting that their infinite iterates avoid ℓ⁺-squares under parameterized or order-preserving equivalence. These are direct constructive claims rather than derivations that reduce to fitted parameters, self-defined quantities, or load-bearing self-citations. The 1974 Entringer et al. reference is external and historical. No equations equate a prediction to its own input by construction, and the avoidance property is an independent combinatorial check on the generated language, not a renaming or ansatz smuggled via prior author work. The derivation chain is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only abstract available; no explicit free parameters, axioms, or invented entities can be extracted.

pith-pipeline@v0.9.1-grok · 5634 in / 1017 out tokens · 33367 ms · 2026-07-02T10:59:57.516523+00:00 · methodology

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