Restricted Goldbach Sums in Arithmetic Progressions II: Almost-All Theorems for General Moduli

For \( q≥1 \) and \( gcd(a,q)=1 \) we study the restricted weighted Goldbach sum \( R_{a,q}(N)=\sum_{p_{1}+p_{2}=N,\; p_{1}\equiv a\,(q)} (logp_{1})(logp_{2}) \). We show that an odd prime divisor of \( q \) forces \( R_{a,q}(N)=O_{q}(logN) \) on a set of even \( N \) of positive density, so that a main term of size \( ≍N/φ(q) \) can hold on almost all \( N \) only for \( q=2^{k} \). For \( q=2^{k} \) we state 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 \( A>0 \), with \( M_{a,q}(N)=2C_{2}\varphi (q)^{-1}\mathfrak{S}(N)N \), and derive from it that the exceptional set \( \{N\leq X:R_{a,q}(N)=0\} \) has size \( {\ll }_{A,q}X(logX)^{-A} \) and that \( |R_{a,q}(N)-M_{a,q}(N)|\leq N(logN)^{-3} \) for all but \( O_{A,q}(X(logX)^{-A}) \) even \( N≤X \). A ternary anchoring transfer and an unconditional quaternary existence statement via Helfgott’s theorem follow. Part II proves an unconditional limit law: for every fixed integer \( h≠0 \), the odd part \( {\mathfrak{S}}_{0}(n)=\prod_{\mathrm{\ell }\mid n,\mathrm{\ell }\gt 2} (\mathrm{\ell }-1)/(\mathrm{\ell }-2) \) of the Hardy–Littlewood singular series, evaluated at the shifted primes \( n=p+h \), converges in distribution to an explicit random Euler product \( Y_{h}=\prod_{\mathrm{\ell }\gt 2,\mathrm{\ell }\nmid h} a_{\mathrm{\ell }}^{B_{\mathrm{\ell }}} \) with independent Bernoulli factors \( \mathbb{P}(B_{\mathrm{\ell }}=1)=1/(\mathrm{\ell }-1) \); we establish convergence of all integer moments, an entire Mellin transform of order at most one, non-atomicity, unbounded support, a superpolynomial upper tail, and strict amplification relative to generic integers. Part III applies this law to the amplitude of the restricted main term along \( N=p+h \) and, under a stated Hardy–Littlewood-type hypothesis, to the additive richness \( R(p) \) of the prime-generation DAG. Part IV records, as computational findings with stated ranges, an inverse correlation \( corr(R,H)≈-0.7935 \) between richness and hub count over the first 5000 primes, of which conditioning on \( {\mathfrak{S}}_{0}(p+1) \) explains about 8%. All numerical constants are enclosed by truncation of the relevant Euler products at P={10}^{7} with an explicit tail bound. The paper does not prove Goldbach’s conjecture or finiteness of the exceptional set.