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

We study the restricted weighted Goldbach sum Ra,q(N), in which one prime of a representation N = p₁ + p₂ of an even N is restricted to the residue class a modulo q. If q has an odd prime factor , then Ra,q(N) = Oq(log N) for every even \(N ≡ a {}\), so the natural main term fails; we give the corrected main term for arbitrary q. For q = 2ᵏ, 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 OA,q(X(log X)-A). A transfer principle yields almost-all theorems along pₙ + pₙ₊₁, pₙ + pn+d and $p + h$, with relative accuracy along $p + h$. For moduli with an odd prime factor the obstruction recurs infinitely often along pₙ + pn+d, and for $q = 3$ we characterise exactly when R1,3(pₙ + pₙ₊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 S₀(p + h) given by an explicit random Euler product Yₕ, 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.