Restricted Goldbach Sums in Arithmetic Progressions, and a Limit Law for the Singular-Series Bias on Shifted Primes

For \(q \geq 1\) and \(\gcd(a,q)=1\), consider the restricted weighted Goldbach sum \[ R_{a,q}(N):=\sum_{\substack{p_1+p_2=N\\p_1\equiv a\pmod q}}(\log p_1)(\log p_2). \] We give a fully rigorous, self-contained treatment of the restricted binary, ternary and quaternary Goldbach problems in this setting. We first prove an elementary but decisive local obstruction: if an odd prime \(\ell\) divides \(q\), then \(R_{a,q}(N)\) collapses to \(O_q(\log N)\) on a positive-density set of even \(N\), so no main term of size \(\asymp N/\varphi(q)\) can hold uniformly; for \(q=2^k\) this obstruction is absent, and we identify the correct local main term \[ M_{a,q}(N)=\frac{C_2}{\varphi(q)}S(N)N, \] Within this scope we prove a qualitative almost-all theorem, with a complete major/minor-arc derivation, showing the exceptional set has density zero. We record a conditional impossibility theorem showing that no \(X\)-independent threshold can upgrade this to an effective almost-all statement once a matching second-moment lower bound is granted, and we isolate, as honestly labelled structural cautions rather than theorems, four classical routes that fail to upgrade the almost-all theorem to unconditional finiteness. We add a fully unconditional restricted Chen-type theorem obtained from the Selberg--Chen sieve and the classical Bombieri--Vinogradov theorem, a ternary prime-anchoring transfer, and positivity of the restricted quaternary singular series through explicit local densities. As a companion study, we then prove that the variable factor \[ S_0(n):=\prod_{\ell\nmid n}\frac{\ell-1}{\ell-2} \] of the Hardy--Littlewood singular series, evaluated along shifted primes \(n=p+h\) for fixed \(h\neq 0\), converges in distribution to an explicit random Euler product \[ Y_h=\prod_{\ell>2,\ell\nmid h}\left(\frac{\ell-1}{\ell-2}\right)B_\ell \] with independent Bernoulli local factors \(P(B_\ell=1)=1/(\ell-1)\); we identify its Mellin transform as an entire function of order one, prove convergence of every integral moment, and establish that the law is non-atomic, has unbounded support, and superpolynomially decaying tails, strictly amplified relative to generic integers. We then connect the two studies: since \(S(N)=S_0(N)\) is exactly the amplitude of the restricted Goldbach main term \(M_{a,q}(N)\), the limit law furnishes a rigorous probabilistic description of how that amplitude fluctuates as \(N\) ranges over the shifted-prime sequence \(N=p+h\), a connection neither source study states. Throughout, every asserted theorem is unconditional and every numerical constant is independently certified via partial Euler products with explicit tail bounds; statements retracted at an earlier stage of this programme after failing independent verification are recorded only as open problems.