Uniformity, Local Structure and Boundary Correlations in Goldbach Representations

We study uniformity in Goldbach representations and the signed correlations arising in a reflected-prime comparison. From classical analytic inputs we derive a common exceptional set for all compatible residue prescriptions modulo sufficiently small fixed powers of the location parameter. We also prove weighted Type I distribution, convex factor regularity, a mixed prime–rough estimate approaching the square-root scale, a balanced rough-pair asymptotic and an uncentered two-prime upper bound. For a sharp bilinear boundary component, we compute the exact unit mean and coefficient moments, remove a repeated-prime contribution with its variance cross terms, and obtain a restricted Möbius product identity. A lower bound for the rowwise absolute character majorant shows why that particular relaxation cannot deliver arbitrary logarithmic savings. Multiplicative prime-factor markers and a rational-target CRT kernel clarify the remaining distribution requirements. Additional results treat almost-all prime-product representations, a conditional implication for real zeros, and finite paired-congruence criteria. All exceptional sets and unproved hypotheses are retained. The results do not establish the required full signed correlation estimate or resolve the binary Goldbach conjecture.