A Restricted Weak Ternary Goldbach Theorem via Prime Anchoring and an Explicit Almost-All Bound with Effective Constants

We establish unconditional almost-all ternary results for the restricted weighted Goldbach sum W_(a,q) \( W_{a,q}(n) := \sum_{\substack{p_1+p_2+p_3=n \\ p_1 \equiv a \pmod{q}}} (\log p_1)(\log p_2)(\log p_3), \quad q \geq 1,\ \gcd(a,q)=1 \), together with conditional extensions under the Density Hypothesis and the Generalized Riemann Hypothesis. Main results. (A) Almost-all via prime anchoring [Proved]. For every \( A > 0,\#\{n \leq X,\ n \text{ odd} : W_{a,q}(n) = 0\} \ll_{A,q} X(\log X)^{-A}. \) (B) Almost-all with full asymptotic [Proved]. For all odd \( n \leq X \) outside an exceptional set of size \( O_{A,q}(X(\log X)^{-A}),W_{a,q}(n) = \frac{J_{3,a,q}(n)}{\varphi(q)}\, n^2 + O_{A,q}\!\left(\frac{n^2}{(\log n)^A}\right), \) where the ternary singular series \( J_{3,a,q}(n) > 0 \) is positive and effectively computable. For \( (a,q) = (3,4) \) the main term equals \( (C_2 S(n)/4)\,n^2 \) with \( C_2 \in [0.6601618157,\, 0.6601618160] \). The minor-arc bound \( K_{\min}(q, A) \leq 2.10/\sqrt{\varphi(q)} \) is proved in full, self-containedly, via the exact \( L^2 \)-orthogonality identity \( \|S_{a,q}\|_{L^2}^2 = \|S\|_{L^2}^2/\varphi(q) \) and a Hölder \( (L^2, L^\infty, L^2) \) factorization. The exceptional-set constant is \( C_{\text{ternary}}(A,q) \leq 4.41/\varphi(q) \cdot 2^A \) [Proved]. Articles 1, 2, and 3 by the same author of this work provide all the black box input data. No GRH, no density hypothesis, and no ternary sieve are used in Parts (A) and (B).