bibkey: pudelko2025modular authors: Marc T. Pudelko year: 2025 title: Modular Periodicity of Random Initialized Recurrences doi: null url: https://arxiv.org/abs/2510.24882v5 claim: Conjecture 8 and equation (14) predict that the bounded-initialization probability of a Fibonacci absolute minimum at position zero converges to one quarter. strata_touched:

• D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation license: citation-only triage: anchor

Modular periodicity of random initialized recurrences

Pudelko studies the Fibonacci recurrence and its parity analogue under random integer initialization. Conjecture 8 states a proposed distribution for the position of the absolute minimum, with value P(0)=1/4 for position zero. The following remark makes the probability operational for bounded integer initializations in [-N,N]^2: equation (14) expects P_N(n)=P(n)+epsilon(N) with epsilon(N) tending to zero as N tends to infinity. The present formal statement uses this bounded-square reading for the Fibonacci recurrence and position zero. A global minimum is allowed to be tied, which gives position zero the largest count among the usual tie rules.

The repository refutes that clause by an unbounded family. For every natural t, the number of initial pairs in the square of radius 6t for which zero is a global absolute-value minimizer is at most 30t^2+10t+1. For t>=3, the corresponding probability is at most 2/9, so it cannot converge to 1/4. The bounded literature audit through 17 September 2026 found no later independent proof or refutation of this clause. Exhaustive publication priority remains unverified.

Verified locator

• URL: https://arxiv.org/abs/2510.24882v5
• arXiv: 2510.24882v5, section 3, Conjecture 8, equation (13), and the bounded-initialization remark containing equation (14).