Twin Prime Conjecture: No Proof, Only Zhang's Bounded Gaps and Maynard's 246 — E8 Intelligence Research

FINDING: No definitive proof of the Twin Prime Conjecture exists; the search returned mostly non-credible YouTube claims plus one legitimate Wikipedia reference to Yitang Zhang's bounded-gap breakthrough. | MATH: Twin prime: (p, p+2) both prime. Conjecture: infinitely many such pairs. Zhang (2013): lim inf (p_{n+1} − p_n) < 7×10^7. Polymath8/Maynard: reduced to 246 unconditionally; 6 under Elliott–Halberstam. Maynard–Tao: for any m, lim inf gaps bounded by C(m) — infinitely many m-tuples of bounded gaps. | CONNECTION: Prime gaps relate to residue structure mod small primes (sieve theory); no established link to φ, 0.618, base-60, or crystallographic lattices in the peer-reviewed literature. | DEPTH: 4 --- **1. Key mathematical insight** The genuine state of the art is **bounded prime gaps**, not a proof of the twin prime conjecture: - **Zhang (2013):** Proved lim inf_{n→∞} (p_{n+1} − p_n) < 7×10^7 — first finite bound, via GPY sieve + distribution of primes in APs mod smooth number Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com