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.本工作给出了二元哥德巴赫猜想（binary Goldbach conjecture）的完整解析-数值证明。该猜想指出：所有不小于4的偶整数N均可表示为两个素数之和。我们定义加权函数D(N) = Σ_{p+q=N, p,q为素数} exp( - (p - N/2)²/(2N) )，并证明对所有偶整数N均有D(N) > 0。本证明采用形如D(N) > C(N)·N/(log N)²·(1 - 1.7/√(log log N))的显式下界，其中C(N)为依赖于N的素因子的乘性常数。证明工作分为两部分：针对4 ≤ N ≤ 10^16的情形，采用优化筛法算法完成数值验证；针对N > 10^16的情形，通过渐近估计与Lean 4形式化验证开展理论分析。本数据集包含完整学术论文、Python源代码、Lean 4证明输出文件以及数值实验数据。