Goldbach's Conjecture: Verified Computationally, Proven Only in Ternary Case — E8 Intelligence Research

FINDING: No proof of Goldbach's conjecture exists; current state is computational verification up to ~4×10¹⁸ (binary) and Helfgott's 2013 proof of the ternary case (every odd n ≥ 7 is sum of three primes). | MATH: Binary GC: ∀ even n>2, ∃ primes p,q: n=p+q. Ternary GC (proven): ∀ odd n≥7, ∃ primes p₁,p₂,p₃: n=p₁+p₂+p₃. Helfgott's proof uses the Hardy–Littlewood circle method, with explicit error bounds via the Vinogradov theorem; verified for n < 10²⁷ by computation. No new constants or ratios emerge. | CONNECTION: The circle method partitions the unit circle into major arcs (near rationals with small denominators) and minor arcs — this is a Fourier-analytic decomposition on the torus ℝ/ℤ, whose structure is a 1D lattice. The major arcs correspond to rational points with denominators up to ~log N, echoing base-60's preference for highly composite denominators (2,3,5). No golden ratio or crystallographic symmetry appears. | DEPTH: 3 — This is a major computational and analytic milestone Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com