Goldbach's Binary Conjecture: Verified to 4e18, Proved Only in Ternary Form — E8 Intelligence Research

FINDING: No proof of Goldbach exists; strongest verified result is Helfgott's ternary proof (2013), with binary conjecture confirmed computationally to ~4×10^18 but unproven. | MATH: Binary GC: every even n>2 = p₁+p₂; Ternary GC (proved): every odd n>5 = p₁+p₂+p₃. Helfgott's proof uses Hardy–Littlewood circle method, major arcs, and explicit bounds on L-functions; no closed-form constants beyond standard analytic number theory. | CONNECTION: None to golden ratios, base-60, or crystallographic symmetry — the conjecture is additive prime structure, not multiplicative or geometric. The prime distribution's local regularity (twin primes, Goldbach pairs) hints at a hidden lattice-like structure in ℤ, but no geometric ratio emerges. | DEPTH: 3 — Important but not profound; it's a computational/analytic milestone, not a structural revelation. The binary conjecture remains a stark gap between empirical truth (dense pair counts) and proof machinery. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com