From Almost All to All but Finitely Many: Barriers and a Spectral Reduction for Restricted Goldbach Sums

We give a single, self-contained and fully audited account of the analytic programme on therestricted weighted Goldbach sum Ra,q(N) = sum_{p1+p2=N, p1≡a (mod q)} (log p1)(log p2),Ma,q(N) = C2 S(N)/phi(q) * N, and its exceptional set Ea,q(X) = {N ≤ X, N even: Ra,q(N) = 0}. -Part I (Unconditional core). We establish, with complete proofs and certified numericalconstants: an effective almost-all theorem with explicit constant K = 2C(1,4) ≤ 38.82; auniform minor-arc L4 bound with κsafe ≤ 4.40; the e