Bounded Gaps Between Primes: Zhang's 70 Million Bound Reduced to 246 — E8 Intelligence Research

Abstract

FINDING: Bounded gaps between primes proven via sieve methods and modular arithmetic clustering, revealing hidden structure in prime distribution. | MATH: Zhang's bound: gaps between consecutive primes are infinitely often less than 70 million (later reduced to 246 via Polymath8). Key constant: \( \liminf_{n \to \infty} (p_{n+1} - p_n) \leq 246 \). No exact ratio or constant emerges from the gaps themselves, but the method relies on the Elliott–Halberstam conjecture and the Bombieri–Vinogradov theorem, which involve exponential sums and modular arithmetic mod \( q \). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the clustering of primes in residue classes mod \( q \) hints at lattice-like periodicity in the integers, reminiscent of crystallographic symmetry in 1D (e.g., the set of primes modulo a fixed integer forms a pattern akin to a quasiperiodic lattice). The use of the Chinese remainder theorem and moduli \( q \) with many prime factors echoes roo Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com