GOLDBACH'S BINARY CONJECTURE VIA VAUGHAN'S IDENTITY AND THE SYMMETRY CORRELATION FUNCTION

We prove the strong Goldbach conjecture for almost all even integers using Vaughan’s identity combined with the symmetry correlation function, the Abelsummation method, and Gallagher’s larger sieve. We further establish the conditional equivalence between the Generalised Riemann Hypothesis and the complete elimination of the exceptional set.The proof proceeds by expressing the Goldbach condition n = p + q as the positivity of the symmetry correlation function S(n) = Pm−1 d=0 Λ(m−d)Λ(m+d), where m = n/2.Vaughan’s identity decomposes the von Mangoldt function Λ into Type I (small divisors) and Type II (large divisors) components.Three critical innovations make the proof work:(1) Abel summation for Type I: The discrete sum over Diophantine solutions is evaluated exactly using the Abel summation formula (Euler–Maclaurin). Thisyields the main term S(n)n(log2 n−2 log n+C0) with a deterministic logarithmic shift arising from the geometry of the integration limits.(2) Gallagher’s larger sieve for Type II: The pointwise estimation of Type II sums is replaced by a mean-square bound via Gallagher’s inequality. Expandingthe inner sum and applying the Barban–Davenport–Halberstam theorem separates diagonal and off-diagonal contributions. The Möbius oscillation suppresses the off-diagonal terms, yielding the mean-square bound PX<n≤2X |Serror(n)|2 =OA(X2(log X)−A) for any A > 0.(3) Density-based classification: The error bound implies that the exceptional set E = {n ∈ 2N : S(n) ≤ 0} satisfies |E ∩ [1, X]| = OA(X(log X)−A). Thus, the strong Goldbach conjecture holds for almost all even integers with a superpolynomially sparse exceptional set.We formalise the algebraic concept of the additive rank ρ(n)—the minimal number of primes required to represent n as a sum. We prove a density-based classification theorem and the multiplicative compression law ρ(mn) ≤ min(mρ(n), nρ(m))−Ψ(gcd(m, n)), where the additive defect Ψ depends on the common divisors of m and n.Finally, we establish the conditional equivalence between the Generalised Riemann Hypothesis for Dirichlet L-functions and the emptiness of the exceptional set via the effective Bombieri–Vinogradov theorem. Under GRH, the strong Goldbach conjectureholds for all even n ≥ 4 without exception.