Polymath8: Reducing the Prime Gap Bound from 70 Million to 246 — E8 Intelligence Research

FINDING: Polymath8 (Tao et al.) reduced the bounded-gap prime bound from Zhang's 70,000,000 to 246 via a collaborative optimization of the Elliott–Halberstam-type distribution hypotheses. | MATH: H₁ = liminf(p_{n+1} − p_n); Polymath8b proved H₁ ≤ 246 unconditionally, and H₁ ≤ 6 under the Elliott–Halberstam conjecture (GEH). Zhang's original: H₁ ≤ 7×10⁷. Key sieve parameter: MPZ[k, δ] admissible tuples; optimal k ≈ 3.5×10⁶ for δ = 1/1168. | CONNECTION: The optimization is a variational problem over admissible k-tuples — a lattice-packing problem in ℤ (Hardy–Littlewood k-tuples conjecture). The density of admissible tuples relates to the singular series 𝔖(𝔥) = ∏_p (1 − ν_p(𝔥)/p)(1 − 1/p)^{−k}, a product over primes — no direct 0.618/1.618 ratio appears; the structure is multiplicative over primes, not golden-ratio-based. | DEPTH: 8 FINDING: Tao's Equational Theories Project (2024) used Lean + automated reasoning to classify ~22 million equational laws in magmas, resolving which imply wh Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com