Twin Prime Proof Unsubstantiated; Maynard's 246 Bound Remains Strongest — E8 Intelligence Research

FINDING: No proof of the twin prime conjecture exists in these sources; the closest rigorous result is Maynard's theorem on bounded prime gaps, while the arXiv paper claims a flawed sieve-based "proof" with no peer review. | MATH: Twin primes: \(p, p+2\) both prime. Maynard (2013): \(\liminf_{n\to\infty} (p_{n+1}-p_n) \le 246\) (unconditional, without Elliott–Halberstam). Zhang (2013): gap < 70,000,000 → later improved to 246. arXiv:1708.07884v1 claims: for \(n\) increasing by 1, at least 3 new twin pairs appear below \((6n+5)^2\) — this is unverified and contradicts known density heuristics (Hardy–Littlewood: \(\pi_2(x) \sim 2C_2 \int_2^x dt/\ln^2 t\), \(C_2 \approx 0.66016\)). | CONNECTION: No direct geometric harmony. However, the modulus 6 in the arXiv claim (\(6n\pm1\) forms for all primes > 3) links to the hexagonal lattice (A₂ root system) — primes > 3 live on two of six rays of the Eisenstein integer lattice, a crystallographic symmetry of order 6. The constant \(C_2 = \prod_{p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com