A Conditional Proof of the Restricted Goldbach Conjecture for Every Even Integer Beyond 10^19.9

In this paper, whose main results are conditional on the density hypothesis or the Generalized Riemann Hypothesis, we establish a complete conditional hierarchy for the restricted weighted Goldbach sum $R_{a,q}(N) := \sum_{p_1+p_2=N,; p_1 \equiv a \pmod{q}} (\log p_1)(\log p_2)$, with expected main term $M_{a,q}(N) := C_2 S(N) N/\varphi(q)$, for fixed $q \geq 1$ and $\gcd(a,q)=1$.Under the Density Hypothesis DH($A$) (any $A \geq 2$), the exceptional-set exponent is shown to equal $\theta(A) = 1 - 2/(A+2)$ via a saddle-point argument, correcting prior formulas $\theta = 1 - 1/A$ and $\theta = 1 - 2/A$, which are both wrong. Under the Generalized Riemann Hypothesis (GRH), we prove the pointwise bound $R_{a,q}(N) = M_{a,q}(N) + O_{q,\varepsilon}(N^{1/2+\varepsilon})$ for all even $N \geq N_0(q)$, and derive the explicit threshold $\log N_0(4) = 45.93$ via a fixed-point iteration. We present both normalizations of the effective constant ($C^2_{4,\text{eff}} \approx 529$ and $\approx 2111$) and give a complete account of the discrepancy. A complete constant-chain audit carried out in Section 7 shows all three normalizations (values 529, 2111, and the independently reconstructed 2375) are consistent, and the worst-case certified bound is $\log N_0(4) \leq 46.1$, with no remaining caveat.We further provide:A certified computation showing all 122 primitive real Dirichlet characters of conductor $q \leq 200$ are free of Siegel zeros in the Stechkin critical interval, with global minimum $L_{\text{cert}} = 0.2344$ at $q = 163$. An unconditional restricted Chen-type theorem $N = p + P_2$, $p \equiv a \pmod{q}$, via Bombieri–Vinogradov. A conditional short-interval lower bound under GRH. None of these results proves the binary Goldbach conjecture or GRH. The paper establishes conditional results under explicitly stated hypotheses. This paper is a companion to "An Almost-All Theorem for a Restricted Goldbach Sum over Arithmetic Progressions with Explicit Unconditional Constants", whose results are used here as a black box.