Goldbach's Binary Conjecture Unproven; Ternary Proof Stands Alone — E8 Intelligence Research

Abstract

FINDING: No proof of Goldbach exists; current state is computational verification up to large bounds and Helfgott's ternary proof (2013), with a weak syllogistic survey paper adding no new results. | 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 for all \(n\) (effective bounds, no unproven heuristics). Computational verification: binary Goldbach verified for \(n \leq 4 \times 10^{18}\) (Oliveira e Silva et al., 2014). No new constants, ratios, or equations emerge from these sources. | CONNECTION: None found. No geometric ratios (0.382, 0.618, 0.786, 1.618, 2.618), no base-60, no crystallographic or root-system structure. The conjecture is purely additive number theory — prime distribution, not symmetry. | DEPTH: 2/10 — The findings are meta-commentary (YouTube videos, a survey with no new theorems). The only mathematically substantive item is Helfgott's proof of the tern Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com