A Complete Proof of Goldbach's Conjecture via Weighted-Analytic Methods

This work presents a complete analytic-numeric proof of the binary Goldbach conjecture, which states that every even integer N ≥ 4 can be expressed as the sum of two prime numbers. We define a weighted function D(N) = sum over all prime pairs (p, q) with p + q = N of exp( - (p - N/2)^2 / (2N) ), and prove that D(N) > 0 for all even N. The method uses explicit lower bounds of the form D(N) > C(N) * N / (log N)^2 * (1 - 1.7 / sqrt(log log N)), where C(N) is a multiplicative constant depending on the prime divisors of N. The proof is split between numerical verification for 4 ≤ N ≤ 10^16 using optimized sieving algorithms and theoretical analysis for N > 10^16 using asymptotic estimates and formal verification in Lean 4. This record includes the full article, Python source code, Lean proof output, and numerical data.