Polymath's Collaborative Proof Narrows Prime Gap Bound to 246 — E8 Intelligence Research

FINDING: Polymath project's most mathematically substantive result is the collaborative refinement of Zhang's prime gap theorem, yielding explicit bounds on \(H_m\) (liminf gaps between consecutive primes). | MATH: \(H_m := \liminf_{n\to\infty}(p_{n+m}-p_n)\); twin prime conjecture ⇔ \(H_1=2\). Zhang proved \(H_1 < 70,000,000\); Polymath8 reduced this to \(H_1 \le 246\) (unconditionally), and \(H_1 \le 6\) assuming the Elliott–Halberstam conjecture. Key constants: 70,000,000 → 246 → 6. | CONNECTION: Prime gaps are irregular, but the *liminf* structure reveals a hidden lattice-like regularity — the admissible \(k\)-tuples form combinatorial designs (Sieve theory's "admissible sets" relate to covering systems and finite geometries). The ratio 246/70,000,000 ≈ 0.0000035 has no harmonic significance; however, the *method* (GPY sieve) uses weights that optimize via a quadratic form whose eigenvalues relate to Bessel functions — whose zeros connect to \(j_{0,1} \approx 2.4048\), not directly Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com