Modular Periodicity of Random Initialized Recurrences

Marc T. Pudelko

arxiv: 2510.24882 · v5 · submitted 2025-10-28 · 🧮 math.NT · math.CO

Modular Periodicity of Random Initialized Recurrences

Marc T. Pudelko This is my paper

Pith reviewed 2026-05-18 02:34 UTC · model grok-4.3

classification 🧮 math.NT math.CO

keywords Fibonacci sequencePisano periodsmodular arithmeticquadratic reciprocityperiodic sequencesrandom recurrencescyclotomic recurrences

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The pith

The Fibonacci recurrence modulo m shows perfect mirror symmetry with its parity transform for every initialization.

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

Classical studies of Fibonacci sequences focus on periodicity modulo m using only the standard start values, yet this paper maps the full periodic behavior arising from every possible pair of initial conditions in the integers modulo m. It identifies an exact mirror symmetry between the usual addition recurrence and the version that negates the previous term, together with self-similar fractal patterns that appear when the modulus is raised from a prime to a prime power. The symmetries are organized by quadratic reciprocity, a symmetric distribution of minimal periods is introduced via Lucas ratios, and counting formulas are proposed for cyclotomic cases, pointing toward broader links with random recurrences and arithmetic functions.

Core claim

For any modulus m the complete set of sequences generated by a_n = a_{n-1} + a_{n-2} from all m squared initial pairs in (Z/mZ)^2 exhibits exact mirror symmetry with the sequences produced by the parity transform a_n = -a_{n-1} + a_{n-2}; this mirror structure lifts fractally to prime-power moduli while preserving weight, and the distribution P(n) of minimal periods satisfies the symmetry P(n) = P(1-n).

What carries the argument

the complete periodic structure generated by all m squared initial pairs for the Fibonacci recurrence and its parity transform

Load-bearing premise

The observed periodic structures and symmetries hold uniformly across all m squared initializations and extend via quadratic reciprocity classification without exceptions.

What would settle it

A single modulus m together with an initial pair whose period length or sequence values under the standard recurrence fail to mirror those under the parity transform would disprove the symmetry.

read the original abstract

Classical studies of the Fibonacci sequence focus on its periodicity modulo $m$ (the Pisano periods) with canonical initialization. We investigate instead the complete periodic structure arising from all $m^2$ possible initializations in $(\mathbb{Z}/m\mathbb{Z})^2$. We discover perfect mirror symmetry between the Fibonacci recurrence $a_n = a_{n-1} + a_{n-2}$ and its parity transform $a_n = - a_{n-1} + a_{n-2}$ and observe fractal self-similarity in the extension from prime to prime power moduli. Additionally, we classify prime moduli based on their quadratic reciprocity and demonstrate that periodic sequences exhibit weight preservation under modular extension. Furthermore, we define a minima distribution $P(n)$ governed by Lucas ratios, which satisfies the symmetric relation $P(n)=P(1-n)$. For cyclotomic recurrences, we propose explicit counting functions for the number of distinct periods with connections to necklace enumeration. These findings imply potential connections to Viswanath's random recurrence, modular forms and L-functions.

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

The paper claims novel symmetries and classifications for Fibonacci recurrences modulo m across all initial conditions, but only the abstract is available so nothing can be checked.

read the letter

The main thing to know is that this paper shifts focus from the standard Pisano period of the Fibonacci sequence modulo m to the periodic structures that arise from all possible initial conditions in (Z/mZ)^2. It claims a perfect mirror symmetry with the parity-transformed recurrence and fractal self-similarity when moving to prime power moduli. What is new here is the complete classification for every one of the m^2 starting pairs, plus the use of quadratic reciprocity to sort the primes and the definition of a minima distribution P(n) that obeys P(n) = P(1-n). The proposal of explicit counting functions for distinct periods in cyclotomic recurrences also stands out, with a nod to necklace enumeration. The paper does well in drawing potential connections to Viswanath's random recurrence, modular forms, and L-functions. These links, if substantiated, could give the results broader interest in number theory. The soft spots are significant because only the abstract is available. Without any proofs, explicit constructions, or even sample computations, it's impossible to confirm whether the symmetries hold uniformly or if there are exceptions in the extension to higher powers. The assumption of no exceptions via quadratic reciprocity classification needs direct checking. This work is aimed at number theorists studying recurrence relations and their periodicity properties. A reader interested in generalizations of classical results on Fibonacci sequences modulo m would find the scope interesting, provided the full paper supports the claims with evidence. I recommend putting it through peer review. The claims are concrete enough that a referee could assess their validity and see if the organizing principles actually work.

Referee Report

3 major / 2 minor

Summary. The manuscript examines the periodic properties of Fibonacci-like recurrences modulo m for all possible initial pairs. It reports perfect mirror symmetry between the standard and parity-transformed recurrences, fractal self-similarity from primes to prime powers, quadratic reciprocity-based classification, weight preservation, a symmetric minima distribution P(n), and counting functions for cyclotomic recurrences, hinting at connections to random recurrences and L-functions.

Significance. If substantiated, these results could significantly advance the understanding of modular periodicity beyond canonical cases, offering insights into self-similar structures and symmetries that may relate to broader number-theoretic phenomena. The comprehensive approach to all m^2 initializations is a strength, but without proofs or data, the potential impact remains speculative.

major comments (3)

[Abstract] The claim of perfect mirror symmetry between the Fibonacci recurrence and its parity transform is presented without any supporting equation, example, or proof, yet it is central to the paper's main discovery and must be substantiated to support the overall conclusions.
[Abstract] The fractal self-similarity in the extension from prime to prime power moduli and the classification of prime moduli based on quadratic reciprocity are key observations, but the abstract provides no concrete data, theorem statement, or verification method, which is load-bearing for these claims.
[Abstract] The minima distribution P(n) governed by Lucas ratios satisfying P(n)=P(1-n) is introduced without the explicit definition or derivation of P(n), making it impossible to assess its validity or connection to the periodic sequences.

minor comments (2)

[Abstract] The term 'weight preservation' is used without prior definition or context, which could confuse readers unfamiliar with the specific usage in this work.
[Abstract] A reference or brief explanation for 'Viswanath's random recurrence' would help situate the implications for readers.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for their insightful comments, which highlight areas where the abstract could be improved for better clarity and substantiation. Below we respond to each major comment and indicate the revisions we will make to the manuscript.

read point-by-point responses

Referee: [Abstract] The claim of perfect mirror symmetry between the Fibonacci recurrence and its parity transform is presented without any supporting equation, example, or proof, yet it is central to the paper's main discovery and must be substantiated to support the overall conclusions.

Authors: We agree that the abstract would benefit from more immediate support for this central claim. In the revised version, we will add a concise example of the mirror symmetry, such as the observed period equality for complementary initial conditions modulo a small prime, along with the basic relation between the two recurrences. The detailed observation and any supporting data will be expanded in the introduction section. revision: yes
Referee: [Abstract] The fractal self-similarity in the extension from prime to prime power moduli and the classification of prime moduli based on quadratic reciprocity are key observations, but the abstract provides no concrete data, theorem statement, or verification method, which is load-bearing for these claims.

Authors: We acknowledge this point and will revise the abstract to include a specific instance of the fractal self-similarity, for example noting the pattern repetition from modulus p to p^k for a small prime p, and briefly state the classification criterion based on quadratic reciprocity. A theorem statement will be referenced, with full details and verification in the main text. revision: yes
Referee: [Abstract] The minima distribution P(n) governed by Lucas ratios satisfying P(n)=P(1-n) is introduced without the explicit definition or derivation of P(n), making it impossible to assess its validity or connection to the periodic sequences.

Authors: We accept that the abstract should define P(n) more explicitly. We will update it to specify that P(n) is the probability distribution of the shortest period lengths across all initial pairs, derived using ratios from the associated Lucas sequences, and explain briefly the origin of the symmetry P(n) = P(1-n) from the mirror property. The derivation will be detailed in the body of the paper. revision: yes

Circularity Check

0 steps flagged

No circularity identified from abstract alone

full rationale

Only the abstract is available, which states observational claims such as mirror symmetry between recurrences, fractal self-similarity under modular extension, classification via quadratic reciprocity, and a minima distribution P(n) satisfying P(n)=P(1-n) governed by Lucas ratios. No equations, parameter fittings, derivation steps, or self-citations are presented that could be inspected for reduction to inputs by construction. The claims are presented as discoveries without methodological details, so no load-bearing circular steps exist and the derivation chain cannot be evaluated as non-self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only; no free parameters, axioms or new entities explicitly introduced or detailed.

pith-pipeline@v0.9.0 · 9115 in / 1029 out tokens · 52184 ms · 2026-05-18T02:34:02.683140+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

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echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

We discover perfect chiral symmetry between the Fibonacci recurrence a_n = a_{n-1} + a_{n-2} and its parity transform a_n = -a_{n-1} + a_{n-2} ... governed by Lucas ratios, which satisfies the symmetric relation P(n)=P(1-n)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

periods of length 8 at every third modulus ... fractal self-similarity in the extension from prime to prime power moduli

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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.