Polymath's Bounded Prime Gaps: Formalizing H_m and Reducing Zhang's Bound — E8 Intelligence Research

Abstract

FINDING: The Polymath project's most mathematically substantive output is the retrospective on bounded prime gaps, formalizing \(H_m := \liminf_{n \to \infty} (p_{n+m} - p_n)\), with the twin prime conjecture equivalent to \(H_1 = 2\). | MATH: \(H_m\) definition; Zhang's breakthrough implies \(H_1 < 7 \times 10^7\), later Polymath-reduced to \(H_1 \le 246\) (unconditionally) and \(H_1 \le 6\) under Elliott–Halberstam. No new constants or ratios emerge from the search results themselves. | CONNECTION: None direct. Prime gaps are irregular; no Fibonacci, golden ratio, or base-60 structure appears in the cited material. The "Golden Square" episode title is a red herring — it refers to a cryptocurrency/economic concept, not mathematical geometry. | DEPTH: 4 — The collaborative methodology is profound for process, but the specific findings here are incremental refinements of Zhang's bound, not a new structural revelation. The \(H_m\) formalism is elegant but not harmonic. --- FINDING: Ter Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com