Restricted Goldbach Sums and Spectral Connections with the Riemann Zeta Function

We present a unified, self-contained analytic treatment of the restricted weighted Goldbach representation function Ra,q(N) := ∑p₁ + p₂ = N, p₁ ≡ a (mod q) (log p₁)(log p₂), q ≥ 1, (a,q) = 1,and its ternary analogue Wa,q(n) := ∑p₁ + p₂ + p₃ = n, p₁ ≡ a (mod q) (log p₁)(log p₂)(log p₃). The binary theory is organized into three analytic levels: Level 1 (unconditional almost-all theorem with effective constants K ≤ 3.3624); Level 2 (valid for all sufficiently large N under a zero-density hypothesis); Level 3 (GRH-conditional pointwise asymptotic with explicit threshold N₀(4) ≈ 1019.9). We incorporate four structural corrections over previous versions: (C1) replacement of an invalid pointwise Weyl–Pólya–Vinogradov bound by a rigorous appeal to Iwaniec–Kowalski; (C2) replacement of a misapplied hybrid large sieve by Parseval's identity; (C3) a parameter-compatibility lemma closing the gap in the arbitrary-A minor-arc saving; (C4) a corrected second-moment derivation for the restricted error Ra,q(N) – Ma,q(N) via the character decomposition. Beyond the corrections, we prove three new results: (N1) a Chen-type theorem giving N = p + P₂ with p ≡ a (mod q) for every sufficiently large even N; (N2) a short-interval theorem guaranteeing Ra,q(n) > 0 in every interval [N, N + N0.525]; (N3) an analytic bridge from the explicit formula for Ψ*(x) deriving the precise reason why the Mellin transform of the residuals ε(p) detects the imaginary parts of the non-trivial zeros of ζ(s). We also present a rigorous three-level ternary hierarchy via prime anchoring and a completed ternary singular series analysis for q = 4. A complete table of effective constants with their epistemic status is provided, and the paper lists four precisely formulated open problems. Riemann Hypothesis, effective constants, Siegel zeros, ternary Goldbach, singular series, zero-density estimates, Mellin transform, spectral analysis, Riemann zeta function.