Sieve Proof Claims Twin Prime Infinity with Square-Root Lower Bound — E8 Intelligence Research

FINDING: A claimed proof of twin prime infinity via Eratosthenes sieve, with a specific bound: at least 3 new twin prime pairs appear when n increments by 1, using (6n+5)² as a sieve threshold. | MATH: Twin primes are pairs (p, p+2) with p ≡ 5 (mod 6) or p ≡ 1 (mod 6). The claim: for each n, the interval up to (6n+5)² contains at least 3 more twin prime pairs than up to (6n−1)². This implies a lower bound on the twin prime counting function π₂(x) ≥ 3·(√x − 5)/6 + C, i.e., π₂(x) ≳ (1/2)√x. No explicit error term or rigorous sieve estimate is given in the abstract; the "proof" is not peer-reviewed (arXiv v1, 2017). | CONNECTION: The modulus 6 is central — twin primes (except 3,5) live in residue classes ±1 mod 6, which are the two non-trivial sixth roots of unity. This is a 6-fold symmetry, related to the hexagonal lattice A₂ (root system of SU(3)), whose fundamental domain has area ratio 1/√3 ≈ 0.577. The bound √x is a square-root scaling, which in lattice terms corresponds to the radiu Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com