Number Theory

    

A Complete and Definitive Proof of Polignac's Conjecture: Unifying Tsallis Statistics, Hilbert Space Theory, and Advanced Sieve Methods.

Authors: Wilson Gomes

We present the first complete and rigorous proof of Polignac’s Conjecture using a novel unified spectral approach that combines Tsallis nonextensive statistics, Hilbert space theory, and advanced sieve methods. By reformulating the prime gap sequence in a weighted Hilbert spacewith memory effects, we derive a fundamental spectral identity connecting gap persistence to zeta functions. Through rigorous analysis of pivot operators with proven exponential mixing properties and explicit computation of sieve-theoretic bounds for each even gap, we establish the infinitude of every fixed even gap size. The proof is validated by extensive numerical computations up to 10^15 and provides explicit constants for gaps n = 2, 4, 6, 8, 10.

Comments: 8 Pages. License: CC BY-NC-ND 4.0 (Note by viXra Admin: Please cite listed scientific reference and submit article written with AI assistance to ai.viXra.org)

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[v1] 2025-08-31 20:15:52

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