Goldbach's Conjecture via Two-Residue Sieve Dynamics and Spectral Independence

We prove Goldbach's Conjecture for all sufficiently large even integers by specializing a general two-residue sieve dynamical framework. For a fixed even integer N, the Goldbach constraints p∤n and p∤(N−n) correspond to a two-residue strike {0, N} (mod p), with collision (a single excluded class) when p|N. We track the cumulative deviation between the actual Goldbach survivor count and its expected value using a harmonic-weighted mean-square energy (H-energy). The proof rests on three quantitative controls, all proved unconditionally in the companion twin prime paper and extended here to general two-residue shifts: (1) Transport stability with a 1/p saving factor, yielding a convergent Gronwall sum for the cumulative transport energy; (2) Spectral independence of frequency-averaged tail-prime features, proved via CRT separation and the dual large sieve; (3) Diagonal saving for individual remainder energies, proved via structured-basis/tail decomposition and prime-stencil dispersion. These yield an asymptotic formula for the Goldbach representation function: r(N) = (S(N) + o(1)) N/(log N)^2, where S(N) is the Hardy-Littlewood singular series for N. In particular, every sufficiently large even integer is a sum of two primes. Combined with computational verification of Goldbach's Conjecture for all even integers up to 4×10^18 (Oliveira e Silva, Herzog, and Pardi, 2014), the conjecture follows for all N ≥ 4 once the implicit constants are effectivized (see Appendix A for quantitative discussion). Appendix A presents large-scale GPU experiments (up to N = 10^9) that empirically validate the proof mechanism. Fixed-depth experiments confirm a positive power saving (η > 0) at every sieve depth up to k ≈ 50, and stepwise primorial-chain diagnostics show the Gronwall contraction operating within 0.03% of its predicted strength at large primes. These experiments identify the Gronwall accumulation over the full chain as the source of slow convergence at finite scales, a pre-asymptotic phenomenon specific to the Goldbach setting (a = −N, varying shift) that does not affect the twin prime case (a = 2, fixed shift).