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arxiv: 2606.02071 · v1 · pith:WECABHN6new · submitted 2026-06-01 · 🧮 math.CO · cs.FL

On gapped repeats in a cyclic Fibonacci word

Takashi Horiyama , Yasuhide Numata , Kazuhisa Seto , Shuhei Tsujie This is my paper

Pith reviewed 2026-06-28 13:56 UTC · model grok-4.3

classification 🧮 math.CO cs.FL
keywords cyclic Fibonacci wordgapped repeatscombinatorics on wordsindex pairssubstring equalityrepetitions in wordsFibonacci substitution
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The pith

For a given length s, pairs of indices in a cyclic Fibonacci word where the s-length segments match are characterized and counted.

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

The paper focuses on cyclic indices in the Fibonacci word, meaning positions wrap around the end of the finite word. It defines pairs (ι, κ) for which the sequence of s symbols starting at ι is identical to that starting at κ. A complete characterization of all such pairs is presented for the Fibonacci case. The total number of these pairs is also computed. This work addresses questions about repetitions and overlaps in one of the most studied aperiodic words under a cyclic interpretation.

Core claim

We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.

What carries the argument

Pairs of cyclic indices (ι, κ) for which the factors of length s are equal in the cyclic Fibonacci word.

If this is right

  • The characterization identifies all positions where s-length repeats occur in the cyclic setting.
  • The exact number of such pairs is available for any fixed s.
  • These pairs correspond to the gapped repeats present in the word.

Where Pith is reading between the lines

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

  • This approach may extend to other morphic words or Sturmian words with similar substitution rules.
  • Applications could include string matching algorithms that exploit the known structure of Fibonacci words.
  • Further study might examine how these counts behave asymptotically as s grows.

Load-bearing premise

The cyclic Fibonacci word is constructed via the standard substitution rules and indices are taken modulo the length of the finite word, allowing the claimed characterization to hold.

What would settle it

For a concrete small Fibonacci word length and specific s, enumerate all index pairs and check if they match the proposed characterization and count; any discrepancy would disprove the result.

read the original abstract

In this article, we consider the words with cyclic indices. For given $s$, we consider the pair $(\iota,\kappa)$ of indices such that the word of length $s$ from $\iota$ is equal to the word of length $s$ from $\kappa$. We give a characterization of such pairs for a cyclic Fibonacci word, and give the number of them.

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

0 major / 1 minor

Summary. The paper considers words with cyclic indices. For a given length s, it characterizes pairs of indices (ι, κ) such that the factors of length s starting at ι and at κ are identical in the cyclic Fibonacci word, and provides the number of such pairs.

Significance. If the claimed characterization and count hold, the result would contribute a precise description of equal factors under cyclic shifts in the Fibonacci word, a canonical example of a Sturmian word. This could aid in understanding periodicity and repetitions in low-complexity infinite words, building on known properties such as the Fibonacci word's factor complexity being n+1.

minor comments (1)
  1. The abstract refers to 'cyclic Fibonacci word' and 'cyclic indices' without a preliminary definition or reference to the standard substitution φ: a→ab, b→a; this notation should be clarified in §1 or §2 for readers unfamiliar with the construction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and accurate summary of the manuscript. The significance assessment aligns with our goals of providing a precise characterization and enumeration for equal factors of length s under cyclic shifts in the Fibonacci word. The recommendation of 'uncertain' appears to reflect the absence of detailed concerns rather than identified issues; we are prepared to address any specific points if raised.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper states it gives a characterization of index pairs (ι,κ) for which the length-s factors are equal in the cyclic Fibonacci word (with indices modulo the word length) together with the count of such pairs. This is a direct combinatorial claim on the repetition structure of the standard infinite Fibonacci word under cyclic identification. No equations, definitions, or cited results are shown that reduce the claimed characterization or count to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The result is framed as an independent analysis resting on the substitution rules and modular indexing, which are external to the target count. The derivation is therefore self-contained against the known low-complexity properties of Fibonacci words.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard domain assumptions about the Fibonacci word construction and cyclic indexing; no free parameters or invented entities are indicated in the abstract.

axioms (2)
  • domain assumption The Fibonacci word is generated by the standard morphism 0 -> 01, 1 -> 0.
    This is the conventional definition used to study the word's properties in the cyclic setting.
  • domain assumption Indices are considered cyclically (modulo the length of the finite word).
    Directly stated in the abstract as the setup for pairs of indices.

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discussion (0)

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Reference graph

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