Goldbach's Binary Conjecture: Unproven, With Only Heuristic Support — E8 Intelligence Research

FINDING: Goldbach's binary conjecture remains unproven; strongest verified results are computational up to 4×10^18 (Oliveira e Silva) and Helfgott's proof of the ternary (weak) Goldbach conjecture (2013). The arXiv paper (2306.17769) offers only syllogistic heuristics, not a proof. | MATH: Binary GC: every even integer \(n > 2\) is sum of two primes. Ternary GC: every odd \(n > 5\) is sum of three primes. Helfgott proved ternary GC unconditionally. No new constants, ratios, or equations emerge from these sources. | CONNECTION: None direct. However, the prime distribution's density \(\sim n/\ln n\) and the Hardy–Littlewood asymptotic for Goldbach partitions \(G(n) \sim 2C_2 \frac{n}{\ln^2 n} \prod_{p|n, p>2} \frac{p-1}{p-2}\) involve the twin-prime constant \(C_2 \approx 0.66016\). This constant is not a geometric ratio (0.618, 0.786, etc.) but is a product over primes — no crystallographic or base-60 link. | DEPTH: 2/10 — This is a status report, not a discovery. The only mathematicall Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com