Goldbach Sums in Progressions Modulo 2k: Almost-All Theorems and Shifted Primes

We study the restricted weighted Goldbach sum $R_{a,q}(N)$, in which one prime of a representation $N = p_1 + p_2$ of an even $N$ is restricted to the residue class $a$ modulo $q$. If $q$ has an odd prime factor $\ell$, then $R_{a,q}(N) = O_q(\log N)$ for every even \(N \equiv a \pmod{\ell}\), so the natural main term fails; we give the corrected main term for arbitrary $q$. For $q = 2^k$, an exact additive-shift decomposition and a factorisation of the singular-series coefficients, combined with the Siegel–Walfisz theorem and Vaughan's minor-arc bound, give a mean-square estimate with saving \((\log X)^{-A}\) for every fixed $A > 0$ and an exceptional set of size $O_{A,q}\big(X(\log X)^{-A}\big)$. A transfer principle yields almost-all theorems along $p_n + p_{n+1}$, $p_n + p_{n+d}$ and $p + h$, with relative accuracy along $p + h$. For moduli with an odd prime factor the obstruction recurs infinitely often along $p_n + p_{n+d}$, and for $q = 3$ we characterise exactly when $R_{1,3}(p_n + p_{n+1}) = 0$. We also prove a restricted ternary asymptotic with positivity for all large odd $n$, and, along shifted primes, a limit law for the singular-series amplitude $\mathfrak{S}_0(p + h)$ given by an explicit random Euler product $Y_h$, with convergence of all moments, doubly exponentially thin tails and no atoms; for odd $h$ the normalised restricted Goldbach function has the same limit law. Numerical studies of the digit-repulsion bias between consecutive primes are labelled with their exact range. All theorems are unconditional, except Theorem 3.8, which rests on an imported lemma of Tolev.