Twin Prime Proof Claim Unverified; Maynard's Bound Remains Best — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve bounds gaps, but no proof exists. One arXiv claim (1708.07884v1) asserts a constructive proof via sieve of Eratosthenes, but is not peer-validated. | MATH: Twin primes: pairs \((p, p+2)\) with \(p\) prime. Maynard's theorem: \(n→∞ (pₙ₊₁ - p_n) ≤ 246\) (unconditionally; 6 with Elliott–Halberstam). Conjecture: infinitely many \(n\) with \(pₙ₊₁ - p_n = 2\). The arXiv paper claims: for each \(n\), at least three new twin pairs appear between \((6n+5)^2\) and \((6(n+1)+5)^2\) — a density claim, not a proof of infinitude. | CONNECTION: Twin primes modulo 6: all pairs > 3 are \((6k-1, 6k+1)\). This is a residue class symmetry in \(Z/6Z\) — a cyclic group of order 6, related to hexagonal lattice symmetry (root system \(A_2\)). The gap 2 is the smallest nonzero difference in the even lattice \(2Z\). No direct golden-ratio or base-60 link appears in the given s Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com