Heuristic Bounded Prime Gaps via a Chaotic Multidimensional Sieve and Random Matrix Theory

TL;DR

This paper introduces EMCHS, a probabilistic framework combining chaos theory and RMT to improve bounds on prime gaps, surpassing previous results with heuristic and partial rigorous evidence.

Contribution

The paper develops EMCHS, integrating chaotic perturbations and RMT to heuristically improve prime gap bounds beyond existing methods, with some rigorous analytic components.

Findings

Unconditional prime gap bound suggested: 180

Conditional prime gap bound under EHC: 8

Numerical evidence supports the heuristic bounds up to 10^18

Abstract

We present the Enhanced Multidimensional Chaotic Heuristic Sieve (EMCHS), a novel probabilistic framework that integrates chaotic perturbations and random matrix theory (RMT) to suggest improved bounds on prime gaps. Building upon the foundational sieves of Goldston-Pintz-Yildirim and Maynard, EMCHS heuristically suggests unconditional gaps of at most 180 and conditional gaps of at most 8 under a partial Elliott-Halberstam conjecture (EHC) with delta = 0.3. These heuristic suggestions surpass Maynard's unconditional bound of 246 through refined polytope optimizations and probabilistic enhancements. We provide rigorous proofs for certain analytic components (such as bounding chaotic perturbations via ergodic theory) and explicitly distinguish which arguments and conclusions are heuristic or conjectural. Numerical evidence for primes up to 10^18 supports the framework, and we discuss limitations and avenues for future rigorous work.