Fibonacci Minima Under Bounded Initialization

Abstract

A uniform count bound refutes the proposed one-quarter limit for Fibonacci minima.

Definition 1.1 (The bilateral Fibonacci recurrence).

$$\forall x\in \mathbb{Z},y\in \mathbb{Z},a\in \mathbb{Z}\to \mathbb{Z},(\mathrm{IsBilateralFibonacci}\left(x,y,a\right))\iff (((a\left(0\right)=x)\wedge \left((a\left(1\right)=y)\wedge (\forall k\in \mathbb{Z},a\left(k+2\right)=a\left(k+1\right)+a\left(k\right))\right)))$$

Formalization. D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.IsBilateralFibonacci (✓ std3).

Citation. Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Commentary.

A sequence belongs to the bilateral Fibonacci recurrence when its values at indices zero and one are x and y and the Fibonacci recurrence holds at every integer index. Thus both negative and positive indices are part of the same recurrence.

Definition 1.2 (A global minimum at position zero).

$$\forall x\in \mathbb{Z},y\in \mathbb{Z},(min0\left(x,y\right))\iff ((\forall a\in \mathbb{Z}\to \mathbb{Z},(\mathrm{IsBilateralFibonacci}\left(x,y,a\right))\Rightarrow ((\forall k\in \mathbb{Z},\left|a\left(0\right)\right|\le \left|a\left(k\right)\right|))))$$

Formalization. D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.min0 (✓ std3).

Citation. Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Commentary.

The predicate allows ties: position zero need only be one global minimizer of the absolute values for every sequence satisfying the bilateral recurrence. The index quantifier ranges over all integers rather than a finite observation window.

Definition 1.3 (The bounded-initialization probability).

$$\forall N\in \mathbb{N},boundedMin0Probability\left(N\right)=\frac{(\left|\{(x,y)\in {\mathbb{Z}}^2\mid ((-(N:\mathbb{Z})\le x)\wedge \left((x\le (N:\mathbb{Z}))\wedge \left((-(N:\mathbb{Z})\le y)\wedge \left((y\le (N:\mathbb{Z}))\wedge (min0\left(x,y\right))\right)\right)\right))\}\right|:\mathbb{Q})}{(2\cdot N+1:\mathbb{Q}{)}^2}$$

Formalization. D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.boundedMin0Probability (✓ std3).

Citation. Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Commentary.

The numerator counts integer pairs in the inclusive square [-N,N]^2 for which zero is a global minimizer. The denominator is the square’s cardinality (2N+1)^2, and both quantities are coerced to rational numbers before division.

Definition 1.4 (The proposed one-quarter limit).

$$(claim)\iff ((\forall epsilon\in \mathbb{Q},(0<epsilon)\Rightarrow ((\exists Nzero\in \mathbb{N},\forall N\in \mathbb{N},(Nzero\le N)\Rightarrow ((\left|boundedMin0Probability\left(N\right)-\frac{1}{4}\right|<epsilon))))))$$

Formalization. D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.claim (✓ std3).

Citation. Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Commentary.

This is the epsilon-threshold form of convergence of the bounded probability to one quarter. It is the bounded-initialization reading of the paper’s formula at minimum position zero for the Fibonacci recurrence.

Theorem 1.5 (The proposed limit is false).

$$\neg claim$$

Proof. Machine-checked in Lean as D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.result (✓ std3). ∎

Resolves. Problems/pudelko-fibonacci-minimum-limit-refutation (refuted) by D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.result.

Source. Repository-derived.

Acknowledgement. Marc T. Pudelko (2025). Modular Periodicity of Random Initialized Recurrences. URL: https://arxiv.org/abs/2510.24882v5.

Commentary.

For t at least three, the count bound gives probability at most 2/9, which is separated from 1/4 by more than 1/72. After any proposed threshold, choosing t as the maximum of three and that threshold supplies a later index N=6t and contradicts the required epsilon bound.

References

• Truth anchor: D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.IsBilateralFibonacci
• Truth anchor: D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.boundedMin0Probability
• Truth anchor: D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.claim
• Truth anchor: D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.min0
• Truth anchor: D5/S1/Recurrence/PudelkoFibonacciMinimumLimitRefutation.result