An Almost-All Theorem for a Restricted Goldbach Sum over Arithmetic Progressions with Explicit Unconditional Constants

In this paper, which is entirely unconditional, we prove a sharpened almost-all theorem with fully explicit effective constants for the restricted weighted Goldbach sum$$R_{a,q}(N)\mspace{6mu}: = \mspace{6mu}\sum_{\substack{p_{1} + p_{2} = N \\ p_{1} \equiv a\ (mod\ q)}}^{}(\log p_{1})(logp_{2}),\quad q \geq 1,\quad gcd(a,q) = 1,$$whose expected main term is $M_{a,q}(N) = C_{2}S(N)N/\varphi(q)$, where $C_{2} = 0.6601618\ldots$ is the twin-prime constant and $S(N)$ is the binary singular series.Our results are organised around four pillars. (I) We give a complete character-pair decomposition of the second moment of the error $E(N): = R_{a,q}(N) - M_{a,q}(N)$, extracting the exact diagonal constant $G/(2\varphi(q))$, where $G = \prod_{p > 2}^{}(1 + (p - 1)^{- 2}) \in \lbrack 1.41320886,1.41320899\rbrack$ is the Gallagher--Goldston constant. (II) We establish a uniform minor-arc $L^{4}$ bound$$\int_{\mathfrak{m}}^{}|S(\alpha)|^{4}\, d\alpha\mspace{6mu} \leq \mspace{6mu}\kappa_{safe} \cdot 2^{A} \cdot \frac{X^{3}}{(logX)^{A}},\quad\quad\kappa_{safe} = 4.40,$$by combining the complete Vaughan identity with the Bombieri--Vinogradov theorem in integral form, giving an explicit derivation of $\kappa_{explicit} = C_{V}^{2}c_{L^{2}} = 4.004$ before applying a rigorous $10\%$ safety margin. (III) We derive the effective almost-all theorem$$\#\left\{ N \leq X\text{ even}:|R_{a,q}(N) - M_{a,q}(N)| > C(A,q)\, N(logN)^{- 3} \right\}\mspace{6mu} \ll_{A,q}\mspace{6mu} X(logX)^{- A},$$with the explicit constant $K: = 2C(1,4) \leq 38.02$, obtained from $C(1,4) \leq 19.01$ via a Stechkin-type optimisation. (IV) We prove a Pintz-type exceptional-set bound on $\{ N \leq X:R_{a,q}(N) = 0\}$.Every statement in the main body carries the tag \[PROVED\]. No Generalised Riemann Hypothesis, no zero-density hypothesis, no ternary sum $W_{a,q}(n)$, no spectral input, and no Chen-type sieve are used anyware.