Bounded Gaps in Primes: Progress, Not Proof, on Twin Prime Conjecture — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress via Maynard's sieve methods shows infinitely many primes with bounded gaps (≤246), but no proof of infinitely many pairs with gap exactly 2. The arXiv paper claims a constructive sieve proof but is not peer-validated. | MATH: Maynard–Tao theorem: \(n→∞ (pₙ₊ₘ - p_n) ≤ C_m\), with \(C_1 = 246\) (unconditional, under Elliott–Halberstam \(C_1 = 6\)). Twin prime conjecture: \(#\{p : p, p+2 prime\} = ∞\). The arXiv paper (1708.07884) asserts at least 3 new twin prime pairs per increment of \(n\) when \((6n+5)^2\) is used as a sieve bound — but this is unverified and likely flawed (no rigorous density argument). | CONNECTION: The modulus 6 in the sieve (\(6n±1\) forms for all primes >3) reflects the hexagonal lattice structure of integers modulo 6 — a crystallographic symmetry (hexagonal close packing). The gap 2 corresponds to the nearest-neighbor distance in that lattice. No golden-ratio Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com