Bounded Gaps in Primes: Progress and Limits on the 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. A claimed arXiv proof (1708.07884) is not peer-validated and contains a flawed induction step. | MATH: Maynard–Tao theorem: \(\liminf_{n\to\infty} (p_{n+m} - p_n) \leq C_m\), with \(C_1 = 246\) (under Elliott–Halberstam, \(C_1 = 6\)). Twin prime conjecture: \(\#\{p : p, p+2 \text{ prime}\} = \infty\). No exact constant or ratio emerges from these results. | CONNECTION: None found. The bounded-gap constant 246 has no known link to 0.382, 0.618, 0.786, 1.618, 2.618, base-60, or crystallographic symmetry. The prime lattice is irregular; no harmonic ratio appears in the sieve bounds. | DEPTH: 3 — Important analytic number theory progress, but not a resolution; no geometric or harmonic structure revealed. The claimed "definitive proof" is not credible. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com