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arxiv: 2607.00814 · v1 · pith:BY3KWZRBnew · submitted 2026-07-01 · 🧮 math.CO

Wythoff-Fibonacci Sequences and a Perturbed Greedy Almost-involution

Luis Mart\'inez , Iker Malaina This is my paper

Pith reviewed 2026-07-02 10:37 UTC · model grok-4.3

classification 🧮 math.CO
keywords Wythoff sequencesFibonacci numbersalmost-involutionpermutationsgreedy algorithmsnatural number partitionsBeatty sequences
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The pith

A signed Fibonacci correction to the Wythoff sequences produces two complementary sequences that define an almost-involution on the integers.

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

The paper defines the lower and upper Wythoff-Fibonacci sequences by adding the term epsilon to the classical Wythoff sequences a(n) and b(n), where epsilon equals (-1) to the k at the k-th Fibonacci number and zero elsewhere. It proves that the resulting sequences together contain every natural number exactly once. Using this partition the authors obtain an explicit formula for the sequence q star j, a permutation of the non-negative integers that satisfies q star of q star j equals j for all j at least 5. The sequence also has the property that its successive differences q star j minus j include every integer exactly once, and it arises from a greedy rule with the first term fixed at 3.

Core claim

Defining the lower Wythoff-Fibonacci sequence to begin 1, 3 and then equal a(n) plus epsilon(n) for n at least 3, and the upper to begin 2 and then equal b(n) plus epsilon(n), yields two sequences whose union is exactly the natural numbers. This partition supplies an explicit formula for q star j, which is a permutation of the non-negative integers obeying q star subscript q star j equals j for every j at least 5.

What carries the argument

The correction epsilon(j) that equals (-1)^k precisely when j is the k-th Fibonacci number and equals zero otherwise, added to the classical Wythoff sequences a(n) and b(n).

If this is right

  • The lower and upper Wythoff-Fibonacci sequences are complementary and partition the natural numbers.
  • q star is a permutation of the non-negative integers.
  • q star satisfies q star of q star j equals j for all j greater than or equal to 5.
  • The differences q star j minus j contain every integer exactly once.
  • A second greedy algorithm generates the same sequence q star.

Where Pith is reading between the lines

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

  • The explicit formula allows computation of any term of q star without iterating the greedy rule.
  • The same style of index-dependent correction could be tested on other pairs of complementary Beatty sequences.

Load-bearing premise

The specific correction epsilon preserves the property that the adjusted lower and upper sequences remain disjoint and together cover every natural number.

What would settle it

Direct computation of the first several hundred terms of both sequences revealing any repeated number or any natural number absent from both would disprove the partition claim.

Figures

Figures reproduced from arXiv: 2607.00814 by Iker Malaina, Luis Mart\'inez.

Figure 1
Figure 1. Figure 1: Plot of the first 100 values of the sequence [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
read the original abstract

We introduce the lower and upper Wythoff-Fibonacci sequences, obtained from the classical Wythoff sequences by a Fibonacci correction. Specifically, if we put $$\epsilon(j)=\begin{cases}(-1)^k, & \text{if }j=F_k\text{ for some }k\\ 0, & \text{in other case}\end{cases},$$ where $F_k$ is the $k$-th Fibonacci number, then we define the general terms of the lower and upper Wythoff-Fibonacci sequences by $$LWF(n)=\begin{cases} 1, & \text{if }n=1,\\ 3, & \text{if }n=2,\\ a(n)+\epsilon(n), & \text{if }n\geq 3.\end{cases}$$ and $$UWF(n)=\begin{cases} 2, & \text{if }n=1,\\ b(n)+\epsilon(n), & \text{if }n\geq 2,\end{cases}$$ respectively. We show that these sequences partition the set of natural numbers and use them to give an explicit formula for a sequence $q^{\star}_j$, defined from a greedy construction studied by the first author and his coauthors in a previous paper, but with the additional condition that $q^{\star}_1=3$, instead of being defined by the greedy rule. This sequence is a permutation of the set of non-negative integers and has the property that every integer appears exactly once in the sequence of differences $q^{\star}_j-j$. We prove that $q^{\star}_{q^{\star}_j}=j\ \forall j\geq 5$, so that $q^{\star}_j$ is an almost-involution. We also give another greedy algorithm generating $q^{\star}_j$.

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 / 3 minor

Summary. The paper introduces lower Wythoff-Fibonacci (LWF) and upper Wythoff-Fibonacci (UWF) sequences by perturbing the classical Wythoff sequences a(n) and b(n) with the correction term ε(n), which equals (-1)^k at Fibonacci indices F_k and zero otherwise. Special cases are given for the first few terms. The manuscript proves that LWF and UWF partition the natural numbers, derives an explicit formula for the sequence q*_j (a modification of a prior greedy construction with q*_1 fixed to 3), shows that the differences q*_j - j are all distinct (hence q* permutes the non-negative integers), and establishes that q* satisfies q*_{q*_j} = j for all j ≥ 5, making it an almost-involution. A second greedy algorithm for generating q*_j is also provided.

Significance. If the partitioning proof and the derivation of the explicit formula hold, the work supplies a concrete link between a Fibonacci-indexed perturbation of Beatty sequences and the explicit characterization of a greedy almost-involution. The explicit formula derived from the new sequences, together with the independent second greedy algorithm, strengthens the result by offering two distinct constructions. This is of interest in combinatorial number theory for understanding when perturbations preserve partitioning properties and for constructing permutations with controlled difference sets.

minor comments (3)
  1. [§2] §2, definition of ε(j): the Fibonacci indexing (whether F_1=1, F_2=1 or F_1=1, F_2=2) is not stated explicitly; this affects the sign pattern of the correction at the first few indices and should be fixed for reproducibility.
  2. [Abstract, §4] Abstract and §4: the almost-involution property is stated only for j≥5; a brief remark explaining the failure (or different behavior) for j=1–4 would improve clarity without altering the main claim.
  3. The manuscript should include a short table or explicit computation of the first 10–15 terms of LWF, UWF, and q* to allow immediate verification of the partitioning and the q*_{q*_j}=j relation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive and accurate summary of our manuscript on the Wythoff-Fibonacci sequences and the almost-involution q*. The description correctly captures the perturbation via ε(n), the partitioning result, the explicit formula, the distinct differences property, the almost-involution condition for j ≥ 5, and the second greedy algorithm. We appreciate the significance assessment and the recommendation of minor revision. We will incorporate any editorial suggestions in the revised version.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper defines ε(j) explicitly, constructs LWF(n) and UWF(n) from classical Wythoff sequences plus this correction, proves the partition property directly, and derives the explicit formula for q*_j from those sequences. The almost-involution q*_{q*_j}=j (j≥5) is then proved from the partition. Although q* was introduced via greedy construction in a prior self-cited paper, the present work supplies an independent closed-form expression and verification that does not reduce to the prior definition by construction or by load-bearing self-citation. No self-definitional steps, fitted inputs renamed as predictions, or ansatzes imported via citation appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claims rest on the standard partitioning property of the classical Wythoff sequences and the standard recursive definition of Fibonacci numbers. No free parameters are fitted and no new entities are postulated beyond the explicit sequence definitions.

axioms (2)
  • standard math The classical Wythoff sequences a(n) and b(n) partition the natural numbers.
    Base sequences to which the correction is applied.
  • standard math Fibonacci numbers F_k are defined by the standard recurrence F_1=1, F_2=1, F_k=F_{k-1}+F_{k-2}.
    Used to locate the positions where ε(j) is nonzero.

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Reference graph

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