Twin Prime Proof Claims Lack Peer-Reviewed Evidence — E8 Intelligence Research

FINDING: No proof of the twin prime conjecture exists in these sources; only popular expositions, a flawed arXiv preprint, and a claimed "definitive proof" video without peer review. The only substantive mathematical claim is from the arXiv paper (1708.07884v1) asserting ≥3 new twin prime pairs per increment of n, based on a sieve argument around (6n+5)². | MATH: Twin prime definition: p_x + 2 = p_y. The arXiv claim: for n → n+1, at least 3 additional twin prime pairs appear in intervals defined by (6n+5)². No rigorous constants, ratios, or closed-form density emerge. The known heuristic (Hardy–Littlewood) is π₂(x) ~ 2C₂ x/(ln x)², with C₂ ≈ 0.66016 (twin prime constant) — not present in these sources. | CONNECTION: The modulus 6 in (6n+5)² reflects the fact that all twin primes >3 are of form (6k−1, 6k+1) — a residue class structure modulo 6. This is a weak link to base-6 arithmetic, not base-60. No golden ratio, Fibonacci, or crystallographic symmetry appears. The 6-fold structure hi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com