Goldbach Verified to 4×10^18 Without Analytic Proof — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: No new proof exists; Goldbach remains verified computationally to 4×10^18 with no analytic proof — the search returned only popular expositions and failed proof attempts. | MATH: Statement: ∀ even n > 2, ∃ primes p,q with n = p + q. Equivalently, the Goldbach partition function g(n) = #{(p,q): p+q=n, p≤q prime} satisfies g(n) ≥ 1. Known: Hardy–Littlewood asymptotic g(n) ~ 2C₂ · n / (ln n)² · ∏_{p|n, p>2} (p−1)/(p−2), where C₂ = ∏_{p>2} (1 − 1/(p−1)²) ≈ 0.6601618158 (twin prime constant). Best proven result: Chen (1973), every sufficiently large even n = p + P₂ (prime + semiprime). Verification bound: Oliveira e Silva et al., n ≤ 4×10^18. | CONNECTION: The singular series factor ∏(p−1)/(p−2) is a local-density correction — the same Euler-product structure that governs crystallographic lattice point counts and the Hardy–Littlewood k-tuple conjectures. C₂ ≈ 0.66016 is not a φ-ratio; no 0.618/1.618 signature appears in the constant. However, the multiplicative Euler product over p Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225049040