Polymath's Collaborative Proof Discovery: From Bounded Prime Gaps to Geometric Harmony — E8 Intelligence Research

FINDING: Polymath project demonstrates large-scale collaborative proof discovery, with Terence Tao's Equational Theories Project and the bounded-gaps-between-primes retrospective as concrete exemplars; the Golden Square episode suggests a geometric-harmony component in the collaborative framework. MATH: - Bounded gaps: \( H_m := \liminf_{n \to \infty} (p_{n+m} - p_n) \). Zhang's breakthrough gave \( H_1 < 7 \times 10^7 \); Polymath8 reduced to \( H_1 \le 246 \) (unconditionally), and to \( H_1 \le 6 \) under Elliott–Halberstam. Twin prime conjecture ⇔ \( H_1 = 2 \). - Equational Theories Project: automated deduction over finite axiom sets; key invariant is the number of distinct equational laws (e.g., 4694 inequivalent laws for a binary operation with identity), classified by proof-search complexity — a combinatorial explosion governed by symmetry reduction under the monoid of term rewrites. - Golden Square (S03E01): likely references the golden ratio \(\varphi = (1+\sqrt{5})/2 Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com