Polymath's Bounded Prime Gaps: From Zhang to \(H_1 \le 246\) — E8 Intelligence Research

FINDING: The Polymath Project's most mathematically substantive output is the retrospective on bounded prime gaps, defining \(H_m := \liminf_{n\to\infty}(p_{n+m}-p_n)\), with twin prime conjecture equivalent to \(H_1=2\). | MATH: \(H_m = \liminf_{n \to \infty} (p_{n+m} - p_n)\); twin prime: \(H_1 = 2\); Zhang's breakthrough gives finite bound for \(H_1\) (later improved via Polymath to \(H_1 \le 246\) under Elliott–Halberstam, or \(H_1 \le 6\) under generalized EH). | CONNECTION: No direct geometric ratio (0.382, 0.618, etc.) appears. However, the structure of prime gaps relates to the distribution of primes modulo small integers — a lattice/arithmetic progression structure in \(\mathbb{Z}\), and the sieve methods involve combinatorial designs with symmetries akin to crystallographic root systems (e.g., admissible tuples as lattice points in a simplex). The number 2 (twin prime gap) is the smallest even integer, reflecting the fundamental parity symmetry of primes. | DEPTH: 7 — This is Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com