Twin Prime Conjecture: Bounded Gaps Proven, Infinite Pairs Unresolved — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress centers on bounded gaps (Maynard–Tao), while a claimed proof via sieve density analysis (arXiv:1708.07884) is not peer-validated. | MATH: Twin primes: pairs (p, p+2) with p prime. Conjecture: infinitely many such pairs. Key proven result (Maynard, 2013): lim inf_n (pₙ₊₁ − p_n) ≤ 246 (unconditionally; 6 with Elliott–Halberstam). No exact constant or ratio emerges from these sources. The arXiv paper claims: for n → n+1, at least 3 new twin prime pairs appear when considering (6n+5)² — this is a density heuristic, not a proof. | CONNECTION: No direct geometric harmony (0.382, 0.618, 0.786, 1.618, 2.618) appears. However, the structure of twin primes modulo 6 is strict: all twin prime pairs > (3,5) are of form (6k−1, 6k+1). This is a crystallographic constraint — a 6-fold lattice symmetry in the integers (hexagonal lattice root system A₂). The gap 2 is the smallest nonzero vector in that lattice. | DEPTH: 4/10 — The findin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com