Signed Prime Correlations and Local Centering in the Binary Goldbach Problem

We develop the coupled sieve and correlation candidate for binary Goldbach with the fixed Vaughan cut . The selected product coefficient is reconstructed from its two original signed windows, including repeated prime factors. The full complementary operator is retained in the completion identity. We prove that fixed-modulus local centering changes the selected error by a negligible amount, and give an exact residue-class decomposition modulo 15 with both windows and their complete prescribed center. This identifies what a local obstruction can and cannot imply for the required one-sided prime correlation. For the quadratic diagnostic, we compute the local factors of the three forms , , and . Their off-diagonal singular-series comparison has an exact expansion in nonnegative primitive-character energies, up to a negligible diagonal correction. Elementary tail estimates and the multiplicative large sieve reduce this arithmetic comparison to moduli at most . The first surviving prime modulus already gives a concrete test with the actual coefficients. Ordinary Siegel–Walfisz cancellation controls its linear contribution but does not establish the much smaller quadratic budget. We explain why this limitation is not a necessary condition for Goldbach, and why existing results on quadratic polynomials or structured progressions do not automatically provide the missing transfer to the genuine prime correlations. Status. The exact identities and local arithmetic lemmas below are proved. The Type-I main term is an explicitly identified input from Gold-44. The two global analytic inequalities are open. This manuscript gives a conditional completion criterion and a sharper diagnosis of the remaining work; it does not claim an unconditional proof of binary Goldbach.