Restricted Goldbach Sums in Arithmetic Progressions II: Almost-All Theorems and a Shifted-Prime Limit Law

For \( q≥1 \) and \( gcd(a,q)=1 \) we study the restricted weighted Goldbach sum \( R_{a,q}(N)=\sum_{p_{1}+p_{2}=N,\; p_{1}\equiv a\,(q)} (logp_{1})(logp_{2}) \), the sum being over primes. An elementary local obstruction shows that if an odd prime \( \ell \) divides \( q \), then on the set of even \( N≡a (mod \ell) \), of density \( 1/\ell \), the sum collapses to \( O_{q}(logN) \); hence a main term of size \( N/φ(q) \) can hold for almost all even \( N \) only for \( q=2^{k} \), and for general \( q \) the correct main term carries the admissibility condition \( gcd(N-a,q)=1 \). For \( q=2^{k} \) we prove, by the circle method with Vaughan’s minor-arc bound, the Siegel–Walfisz theorem on the major arcs and an exact evaluation of the singular-series coefficients, the mean-square estimate \( \sum_{N\leq X} |R_{a,q}(N)-M_{a,q}(N)|^{2}{\ll }_{A,q}X^{3}(logX)^{-A} \) for every \( A>0 \), an exceptional set of size \( O_{A,q}(X(logX)^{-A}) \), and an asymptotic formula with positive singular series for the restricted ternary sum, hence the representation of every large odd integer as a sum of three primes the first of which lies in a prescribed odd class modulo \( 2^{k} \). For general moduli we record the almost-all theorem for the corrected main term, conditional on a mean-value theorem of Tolev, and the local densities of the restricted ternary problem at every prime power dividing q. We then prove that the odd part \( {\mathfrak{S}}_{0}(p+h) \) of the singular series along shifted primes converges in distribution to the explicit random Euler product \( Y_{h}=\prod_{\mathrm{\ell }\gt 2,\; \mathrm{\ell }\nmid h} (\frac{\mathrm{\ell }-1}{\mathrm{\ell }-2})^{B_{\mathrm{\ell }}} \), with independent Bernoulli variables \( B_{\mathrm{\ell }} \) of mean \( 1/(\ell-1) \), by a self-contained argument based on the Brun–Titchmarsh inequality; all integer moments converge to absolutely convergent Euler products, the Mellin transform is entire of order at most one, the law has no atoms, unbounded support and a superpolynomial tail, and the moments are enclosed in rigorous interval arithmetic. Combining the two parts, we show that \( \varphi (q)R_{a,q}(p+h)/(2C_{2}(p+h)) \) converges in distribution to \( Y_{h} for q=2^{k} \) and odd \( h \). The relation to the additive richness of a prime-generation graph is established under a stated Hardy–Littlewood hypothesis, and an inverse richness–hub correlation of about -0.79 over the first 5000 primes is recorded as a computational finding. All implied constants coming from the Siegel–Walfisz theorem are ineffective, and the paper proves neither Goldbach’s conjecture nor finiteness of any exceptional set.