Goldbach's Binary Conjecture Unproven; Ternary Form Solved, Verification to 4×10^18 — E8 Intelligence Research

FINDING: No proof of Goldbach exists; current state is computational verification up to ~4×10^18 (binary) and Helfgott's 2013 proof of the weak/ternary form. The arXiv paper (2306.17769) offers only conditional or weaker results, not a full resolution. | MATH: Binary Goldbach: every even integer >2 is sum of two primes. Verified computationally to 4×10^18 (Oliveira e Silva et al.). Ternary/weak Goldbach: every odd integer >5 is sum of three primes — proven by Helfgott (2013) using circle method, with explicit bounds: for n > 10^27, the result holds unconditionally; finite check covers below. No new constants or ratios emerge. | CONNECTION: None to golden ratio, base-60, or crystallographic symmetry. The circle method (Hardy–Littlewood) uses exponential sums over primes, which relate to the zeros of the Riemann zeta function — a spectral/lattice-like structure, but no direct geometric harmony ratio appears. The prime distribution's local structure (twin primes, gaps) shows no simple rat Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com