Ternary Goldbach Verified, Not Binary: Helfgott's Proof and Lean Formalization — E8 Intelligence Research

FINDING: No new proof of binary Goldbach; the verified frontier is Helfgott's 2013 proof of the *ternary* (weak) Goldbach conjecture, now being machine-formalized in Lean. | MATH: Ternary Goldbach: every odd n > 5 = p₁+p₂+p₃. Helfgott's proof rests on: (i) the circle method with minor-arc bounds, (ii) verification of the Generalized Riemann Hypothesis to height ~10³⁰⁰ for Dirichlet L-functions of conductor ≤ 300,000, (iii) explicit major-arc estimates. Binary Goldbach remains open; best results are Chen's theorem (n = p + P₂, P₂ = semiprime) and the Hardy–Littlewood asymptotic r₂(n) ~ 2C₂ · n/(ln n)² · ∏_{p|n, p>2} (p−1)/(p−2). | CONNECTION: The Hardy–Littlewood singular series constant C₂ = ∏_{p>2} (1 − 1/(p−1)²) ≈ 0.6601618158... is the *twin-prime constant* — the same constant governing twin primes, revealing that Goldbach and twin primes share a single arithmetic invariant. No direct link to φ = 1.618, 0.618, or base-60; the structure here is multiplicative over primes, not geometr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com