Bounded Gaps ≤246 Proven, But Twin Prime Conjecture Remains Open — E8 Intelligence Research

FINDING: Twin prime conjecture remains unproven; recent progress (Maynard) establishes infinitely many primes with bounded gaps ≤246, but no proof of infinitely many pairs with gap exactly 2. | MATH: Let \(p_n\) be the \(n\)-th prime. Maynard–Tao: \(n→∞ (pₙ₊ₘ - p_n) ≤ C_m\), with \(C_1 = 246\) (unconditional, Polymath8b). Twin prime conjecture: \(n→∞ (pₙ₊₁ - p_n) = 2\). The arXiv paper (1708.07884v1) claims a sieve-based proof of at least three new twin prime pairs per unit increment of \(n\) beyond \((6n+5)^2\), but this is not peer-reviewed and contradicts known density bounds — likely flawed. | CONNECTION: No direct geometric harmony ratio appears. However, the prime gaps modulo 6: all twin primes >3 are of form \((6k-1, 6k+1)\) — this is a base-6 (not base-60) residue symmetry, related to the cyclic group \(Z_6^× Z_2\). The bounded-gap result uses the Selberg sieve and GPY method, which rely on combinatorial we Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com