Goldbach's Binary Conjecture Unproven Despite Ternary Proof and 4×10^18 Check — E8 Intelligence Research

FINDING: No proof of Goldbach's conjecture exists; the strongest verified result is Helfgott's ternary Goldbach proof (2013), while binary Goldbach remains computationally verified to ~4×10^18. | MATH: Binary Goldbach: every even integer \(n>2\) is sum of two primes. Ternary (weak) Goldbach: every odd \(n>5\) is sum of three primes — proven by Helfgott via Hardy–Littlewood circle method, with explicit bounds on the major arcs and numerical verification of the minor arcs. Computational verification of binary Goldbach: \(n \le 4\times 10^{18}\) (Oliveira e Silva, 2013). No new constants or ratios emerge; the arxiv paper (2306.17769) offers syllogistic reformulations but no novel equations. | CONNECTION: None directly — Goldbach's conjecture is additive number theory, not geometric. However, the Hardy–Littlewood circle method implicitly uses the unit circle \(e^{2\pi i \theta}\), and the prime distribution is tied to the Riemann zeta function \(\zeta(s)\), whose zeros relate to the spacin Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com