Restricted Goldbach Sums over Arithmetic Progressions: Unconditional Theory, Conditional Refinements, and Ternary and Quaternary Extensions

We develop a self-contained analytic theory of the restricted weighted Goldbach sum \( R_{a,q}(N) := \sum_{\substack{p_1+p_2=N \\ p_1 \equiv a \ (\mathrm{mod}\ q)}} (\log p_1)(\log p_2), \qquad q \geq 1, \ \gcd(a,q)=1, \) with expected main term \( M_{a,q}(N) := C_2\,\mathfrak{S}(N)\,N/\varphi(q) \), where \( C_2 \) is the twin-prime constant and \( \mathfrak{S}(N) \) is the binary singular series. We establish, unconditionally, an effective almost-all theorem with explicit constant \( K \leq 38.92 \), a sub-exponential bound on the exceptional set \( \mathcal{E}_{a,q}(X) := \{N \leq X \text{ even} : R_{a,q}(N)=0\} \) with Stechkin constant \( R=9.6459 \), and a collection of structural rigidity results (a short-interval gap bound, non-consecutiveness, and additive-energy decay of the exceptional set). Under the Density Hypothesis \( \mathrm{DH}(A) \) we obtain the exceptional-set exponent \( \theta(A) = 1 - 2/(A+2) \), and under the Generalized Riemann Hypothesis (GRH) we obtain a fully explicit pointwise threshold \( \log N_0(4) \leq 46.1 \) together with a certification that all 122 primitive real Dirichlet characters of conductor \( q \leq 200 \) are free of Siegel zeros.We extend the theory to the restricted ternary sum \( W_{a,q}(n) \) proving an unconditional almost-all theorem with a fully explicit, self-contained minor-arc bound \( K_{\min}(q,A) \leq 2.10/\sqrt{\varphi(q)} \) obtained via an \( (L^2,L^\infty,L^2) \) Hölder factorization. Finally we determine the local densities of the restricted quaternary singular series at every prime dividing the modulus, prove their unconditional positivity, and confirm the resulting predictions empirically by direct Fourier-convolution computation and by exhaustive verification of two universal additive representations. Every result stated without qualification is proved unconditionally; every conditional result is labelled with the hypothesis on which it depends.