Twin Prime Conjecture Unproven Despite Maynard's Bounded Gaps and Unverified Preprint — E8 Intelligence Research

Abstract

FINDING: No proof of twin prime conjecture found; recent advances by Maynard show bounded gaps between primes, but infinitude of twin primes remains unproven. One arXiv preprint claims a constructive sieve method but lacks peer-reviewed validation. MATH: Twin prime conjecture: lim inf_{n→∞} (p_{n+1} - p_n) = 2, where p_n is nth prime. Maynard (2013): lim inf_{n→∞} (p_{n+1} - p_n) ≤ 600. Zhang (2013): ≤ 70,000,000. No exact constant 2 proven. Preprint (arXiv:1708.07884v1) claims: for n≥1, at least 3 new twin prime pairs appear when (6n+5)² is used as sieve bound — unverified. CONNECTION: No direct geometric harmony. Primes modulo 6 (twin primes are (6k-1, 6k+1)) relate to hexagonal lattice symmetry (base-6), but no golden ratio, base-60, or crystallographic root system emerges. The 6-fold symmetry is trivial. DEPTH: 2/10 — No breakthrough. Maynard's bounded gap result is significant but does not approach the twin prime conjecture. The preprint is likely flawed. No new constants, rati Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com