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arxiv: 2605.20504 · v1 · pith:CBM7AGNSnew · submitted 2026-05-19 · 🧮 math.CO

Box Progressions, Abelian Power-Free Morphisms and A Sieve Technique for the Template Method

Sad{\i}k Eyido\u{g}an , Haydar G\"oral , Nihan Tan{\i}sal{\i} This is my paper

Pith reviewed 2026-05-21 06:15 UTC · model grok-4.3

classification 🧮 math.CO
keywords abelian power-free morphismsbinary alphabetsieve techniquetemplate methodfixed pointsarithmetic progressionsDekking's result
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The pith

There exist binary morphisms that are abelian 16-power free but not 15-power free, with abelian 14-power free fixed points, and morphisms that are not power-free but have 5-power free fixed points.

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

The paper connects a sequential ball-into-box allocation problem, where both ball labels and box labels must avoid k-term arithmetic progressions, to the construction of abelian power-free infinite words via morphisms on a binary alphabet. It gives sufficient conditions for a morphism to be abelian k-power free that extend Dekking's binary result and are more practical than Carpi's criterion. The authors then combine Dekking's result with the template method to create a sieve that checks far fewer candidate parent morphisms. This sieve produces two explicit examples: one morphism that avoids abelian 16-powers but not 15-powers, whose fixed point still avoids 14-powers, and another that fails to be abelian power-free yet whose fixed point avoids 5-powers. A sympathetic reader cares because the examples separate the avoidance strength of the finite rule from the infinite word it generates, directly supplying the power-free objects needed to bound the original progression question.

Core claim

We present sufficient conditions under which a morphism is abelian k-power-free, extending Dekking's result over a binary alphabet and offering a weaker yet more effective alternative to Carpi's. Combining Dekking's result with the template method of Currie and Rampersad, we develop a sieve technique that significantly reduces the number of parents that must be examined to establish abelian power-freeness. We then identify a binary morphism that is abelian 16-power free but not abelian 15-power free with an abelian 14-power free fixed point, and a binary morphism which is not abelian power-free yet has an abelian 5-power free fixed point.

What carries the argument

The sieve technique that reduces the number of parent morphisms examined when verifying abelian power-freeness via the template method combined with Dekking's result.

If this is right

  • The fixed point of a morphism can avoid strictly higher powers than the morphism itself.
  • The sieve makes verification of abelian power-freeness feasible for larger k by examining only a small fraction of possible parents.
  • These morphisms supply the abelian power-free words required to obtain finite bounds N on the ball-box arithmetic-progression question for given ℓ and k.
  • Abelian power-freeness of the morphism is not required for the same property to hold for its infinite fixed point.

Where Pith is reading between the lines

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

  • The same sieve could be run systematically to locate the smallest k at which a binary morphism and its fixed point first differ in their maximal avoided powers.
  • The technique may be adapted to decide the original box-progression question for concrete small values of ℓ and k by producing or ruling out the needed power-free fixed points.
  • Similar parent-reduction ideas might extend to other notions of power-freeness or to larger alphabets where exhaustive search is currently intractable.

Load-bearing premise

The sieve technique correctly identifies every relevant parent morphism without omitting any case that would produce an abelian power when the template method is applied.

What would settle it

Direct computation of the fixed point of the reported 16-power free morphism that uncovers an abelian 15-power inside it, or an abelian 16-power inside the morphism itself, would refute the central examples.

Figures

Figures reproduced from arXiv: 2605.20504 by Haydar G\"oral, Nihan Tan{\i}sal{\i}, Sad{\i}k Eyido\u{g}an.

Figure 1
Figure 1. Figure 1: An example of a 2-ball-box distribution Both the balls 3, 8, 13, 18, 23 and their box numbers 3, 6, 9, 12, 15 are 5-term arithmetic progressions, respectively. To refer to these type of progressions, we give the following definition. Definition 1.1. Given an ℓ-ball-box distribution, a sequence of positive integers h1, . . . , hk is called a k-term box progression (k-BP in short) if [PITH_FULL_IMAGE:figure… view at source ↗
Figure 2
Figure 2. Figure 2: An example of the correspondence between an abelian 4-power and a 5-BP [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Parents of T8 in Proposition 3.6 Proposition 3.6. Let Σ = {a, b} be a binary alphabet. Then, the fixed point h ω (a) of the morphism h : Σ∗ → Σ ∗ defined in Theorem 2.16 by h(a) = abaaaba, h(b) = babab contains abelian 8-powers. Proof. To find abelian 8-powers in our fixed point, we will find exactly 8 parent templates of T8 = [ϵ, ϵ, ϵ, ϵ, ϵ, ϵ, ϵ, ϵ, ϵ,(0, 0),(0, 0),(0, 0),(0, 0),(0, 0),(0, 0),(0, 0)] suc… view at source ↗
Figure 4
Figure 4. Figure 4: An abelian 8-power and the corresponding blocks As the Bi ’s are permutations of each other and f(h(a)) = f(h(b)) = 0, we deduce the following equalities: (9) f(v ′ 1 ) + f(v2) = f(v ′ 2 ) + f(v3) = · · · = f(v ′ 8 ) + f(v9). The above equalities in (9) yield that the sequence (f(vi)) for 1 ≤ i ≤ 9 is an arithmetic progression of length 9 in G. Now, we define the sequence (pi) for 1 ≤ i ≤ 9 by pi = f(vi). … view at source ↗
Figure 5
Figure 5. Figure 5: The blocks B′ i and Bi when j ̸= 7 Now, we focus on the non-trivial arithmetic progressions of length 9. There are exactly two such non-trivial arithmetic progressions. We denote them by L1 and L2 where L1 :(pi) 9 i=1 = {5, 1, 8, 4, 0, 7, 3, 10, 6}, L2 :(pi) 9 i=1 = {6, 10, 3, 7, 0, 4, 8, 1, 5}. Equation (7) shows that the fourth and the sixth terms of L1, namely 4 and 7, are images of two distinct prefixe… view at source ↗
Figure 6
Figure 6. Figure 6: An abelian 14-power and the corresponding blocks As the Bi ’s are permutations of each other and f(h(a)) = f(h(b)) = 0, we infer the following equalities: (12) f(v ′ 1 ) + f(v2) = f(v ′ 2 ) + f(v3) = · · · = f(v ′ 14) + f(v15). The equalities in (12) imply that the sequence (f(vi)) for 1 ≤ i ≤ 15 is an arithmetic progression of length 15 in G. For 1 ≤ i ≤ 15, we set pi = f(vi). Assume that (pi) 15 i=1 is a… view at source ↗
Figure 7
Figure 7. Figure 7: Maximal-length words avoiding bb and abelian 3-power Guided by this observation, we now detail the backtracking search used in our computa￾tions to prove the propositions in this section, see https://github.com/nihantanisali/ Abelian-Power-Free-Morphisms/blob/main/Section4. (1) Start with the empty word. (2) Extend the current word by appending (right-concatenating) a or b. (3) After each extension: • Chec… view at source ↗
Figure 8
Figure 8. Figure 8: A 2-ball-box distribution with no 6-BP Remark 5.3. In [PITH_FULL_IMAGE:figures/full_fig_p049_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Particle moves corresponding to the word aaaab aaaab aaaab aaaab abab obtained in Theorem 5.1 III. The concept of repetitive semigroups originated from a well-known result in com￾binatorics, namely van der Waerden’s theorem. This theorem laid the foundation for the study of repetitive structures in semigroups, where the concept of k-repetitiveness was introduced. Definition 6.1. [24] Let S be a semigroup, … view at source ↗
read the original abstract

Given balls and boxes both enumerated by the positive integers, we consider a sequential allocation of the balls into the boxes. We fix $\ell \ge 2$. Proceeding in increasing order of box labels, assign to each box the next $r$ smallest balls for some $ 1\leq r\leq\ell$. Given an integer $k\ge 3$, is there a natural number $N$ such that in any placement of $N$ balls into boxes, there exist $k$ balls whose labels and box labels each form a $k$-term arithmetic progression? We address this question by identifying abelian power-free fixed points of morphisms over a binary alphabet. We present sufficient conditions under which a morphism is abelian $k$-power-free. Our conditions extend Dekking's result over a binary alphabet and offer a weaker, yet more effective alternative to Carpi's. Combining Dekking's result with the template method of Currie and Rampersad, we develop a sieve technique that significantly reduces the number of parents that must be examined to establish abelian power-freeness. We then identify a binary morphism that is abelian 16-power free (but not abelian $15$-power free) with an abelian 14-power free fixed point, demonstrating the strength of our technique in verifying abelian power-freeness. Furthermore, we give a binary morphism which is not abelian power-free, yet has an abelian $5$-power free fixed point. These results offer novel examples of morphisms whose fixed points exhibit stronger abelian power-freeness than the corresponding morphisms.

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 addresses a question on the existence of k-term arithmetic progressions formed by ball and box labels in a sequential allocation process (with each box receiving between 1 and ℓ balls) by constructing abelian power-free morphisms over a binary alphabet. It extends Dekking's result by providing sufficient conditions for a morphism to be abelian k-power-free, develops a sieve technique that combines these conditions with the template method of Currie and Rampersad to reduce the number of parent morphisms requiring direct verification, and exhibits two concrete binary morphisms: one that is abelian 16-power free but not 15-power free and whose fixed point is abelian 14-power free, plus another that is not abelian power-free yet possesses an abelian 5-power free fixed point.

Significance. If the central claims hold, the manuscript supplies new separating examples in combinatorics on words, showing that a fixed point can be abelian power-free to a higher degree than the generating morphism itself. The sieve technique offers a concrete, effective reduction for applying the template method, which could streamline future verifications of abelian power-freeness. The link to the box-progression problem supplies motivation, while the explicit morphisms and the extension of Dekking's result constitute the primary technical contribution.

major comments (1)
  1. Sieve technique (as presented in the combination with Dekking's result and the template method): the claim that the sieve produces an exhaustive reduced set of parents for the concrete morphism used to certify abelian 16-power freeness is load-bearing for the existence result. Without an explicit enumeration of the parents examined, a bound on their number, or a proof that every possible parent outside the reduced set would violate the template conditions, it remains possible that an omitted parent produces an abelian 16-power in the fixed point, undermining the certificate.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and the constructive major comment. We address the point below and will strengthen the justification of the sieve technique in the revised version.

read point-by-point responses

  1. Referee: Sieve technique (as presented in the combination with Dekking's result and the template method): the claim that the sieve produces an exhaustive reduced set of parents for the concrete morphism used to certify abelian 16-power freeness is load-bearing for the existence result. Without an explicit enumeration of the parents examined, a bound on their number, or a proof that every possible parent outside the reduced set would violate the template conditions, it remains possible that an omitted parent produces an abelian 16-power in the fixed point, undermining the certificate.

    Authors: We agree that the exhaustiveness of the reduced set must be made fully explicit for the certificate to be rigorous. The sieve is obtained by first applying the sufficient conditions of our extension of Dekking's theorem to discard any parent morphism that necessarily produces an abelian k-power, then retaining only those parents compatible with the template conditions of Currie and Rampersad. In the revision we will add a dedicated subsection that (i) states a precise bound on the number of candidate parents (derived from the finite length of factors examined by the template method for fixed k), (ii) supplies a short lemma proving that every parent outside the reduced set violates either one of the abelian-power-free conditions or a template-matching requirement, and (iii) either lists the (finite) reduced set examined for the concrete 16-power-free morphism or describes the deterministic procedure that generates it. These additions will remove any ambiguity about omitted parents. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation relies on external theorems and explicit verification

full rationale

The paper extends Dekking's theorem on abelian power-free morphisms and combines it with the Currie-Rampersad template method plus a new sieve for case reduction. The existence claims rest on identifying concrete binary morphisms and verifying their properties via the combined technique, with the sieve presented as a sufficient reduction rather than a self-referential fit. No self-definitional equations, fitted inputs renamed as predictions, or load-bearing self-citations appear; all central steps cite independent prior results (Dekking, Carpi, Currie-Rampersad) that predate this work and have external grounding. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard properties of morphisms and abelian equivalence in word combinatorics; no free parameters are introduced, and no new entities are postulated. The sieve assumes the template method exhaustively covers cases when filtered.

axioms (2)
  • domain assumption Morphisms over a binary alphabet preserve abelian equivalence classes under concatenation in the manner required for power-freeness checks.
    Invoked when extending Dekking's result and applying the sieve to reduce parent examinations.
  • domain assumption The template method of Currie and Rampersad provides a complete enumeration of potential abelian powers when combined with the sieve filter.
    Central to the technique that significantly reduces the number of parents examined.

pith-pipeline@v0.9.0 · 5831 in / 1506 out tokens · 39274 ms · 2026-05-21T06:15:58.788338+00:00 · methodology

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Reference graph

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