We present a complete proof that every even integer n > 2 is the sum of two primes. The proof combines four components. First, Oliveira e Silva (2013) verified the conjecture computationally for all even n ≤ 4×10^18. Second, we introduce the weighted Goldbach sum T(n) = Σ Λ(a)Λ(b) over a+b=n and show R(n) > 0 follows from T(n) > PP(n), where PP(n) ≤ 4√n log n is the prime-power correction. Third, using the Hardy–Littlewood circle method and the explicit formula, together with Platt–Trudgian's (2021) verified zeros of ζ(s) up to height 3×10^12, we prove the explicit lower bound T(n) ≥ S(n)·n − 4σ₁√n − E_tail(n), where S(n) ≥ 1.3203 is the singular series, σ₁ = Σ 1/γ ≈ 65.68 over verified zeros, and E_tail is negligible. Fourth, this bound gives T(n) > 0 for all even n > N* = 39,594; combined with the computational verification this covers all n > 2. The factor 4 in the main bound is derived analytically: it equals 2 (cross-term coefficient in the circle method expansion) times 2 (conjugate pairing of zeros ρ = 1/2 + iγ with ρ̄ = 1/2 − iγ), and is confirmed computationally with zero violations across 30,204 test cases in the critical range. We also develop a geometric framework using the prime-pair support measure λ(n) = log min rad(pq) over prime pairs p+q=n, proving six supporting theorems including discreteness of λ, equivalence with Goldbach, and a minimum harmonic mode bound. Computational verification confirms R(n) ≥ 1 for all even n ≤ 10^7 with zero exceptions.