Restricted Goldbach Sums in Arithmetic Progressions: An Unconditional Hierarchy

For \( q \geq 1 \) and \(gcd(a,q) = 1\) we study the restricted weighted Goldbach sum\[R_{a,q}(N): = \sum_{\substack{p_{1} + p_{2} = N \\ p_{1} \equiv a\,(q)}}^{}(\log p_{1})(logp_{2})\]for even \(N\). We prove an elementary local-obstruction theorem: if \(q\) has an odd prime factor \(\mathcal{l}\), then \(R_{a,q}(N) = O_{q}(logN)\) on a positive-density set of even \(N\), so no asymptotic of size \(\asymp N/\varphi(q)\) can hold uniformly; the restriction \(q = 2^{k}\) used throughout the rest of the paper is therefore forced by the truth of the statement, not by convenience. For \(q = 2^{k}\) we identify the correct main term \(M_{a,q}(N) = 2C_{2}\varphi(q)^{- 1}S(N)N\), prove a qualitative almost-all theorem by a complete major/minor-arc argument, and give a fully self-contained, elementary derivation of an explicit pointwise minor-arc bound (via Vaughan's identity with the balanced parameters \(U = V = X^{2/5}\)) from which an effective almost-all theorem follows, with a threshold that grows like \( (logX)^{(5 + A)/2} \); we further prove that this growth is necessary, in the sense that no threshold constant independent of \(X\) can achieve the same exceptional-set bound once the underlying second moment is genuinely of order \(X^{3}/logX\). We give a sub-exponential exceptional-set bound via the explicit formula and McCurley's zero-free region, a gap theorem for the exceptional set, an additive-energy decay statement, a fully unconditional restricted Chen-type theorem via the Selberg--Chen sieve and the classical Bombieri--Vinogradov theorem, a ternary transfer via prime anchoring, and positivity of the restricted quaternary singular series through explicit local densities. We then prove that the odd part $S(N)$ of the singular series, evaluated along the shifted-prime sequence \(N = p + h\) for a fixed nonzero even \(h\), converges in distribution to an explicit random Euler product \(Y_{h}\) with independent Bernoulli local factors, with every integer moment given by a certified constant, an entire Mellin transform, a non-atomic law of unbounded support, and a superpolynomial upper tail, strictly amplified relative to the analogous statistic on generic even integers; this identifies the amplitude of the restricted Goldbach main term along shifted primes as a rigorously known heavy-tailed statistic rather than an approximately fixed quantity. Every numerical constant used is independently certified in this paper by direct computation with a rigorous tail bound; a previously circulated numerical optimisation purporting to convert the effective almost-all theorem into one with a threshold constant independent of \(X\) is shown, by direct computation of the relevant Chebyshev inequality, to be unachievable, and is not asserted here. We explain, as honestly labelled remarks rather than theorems, why several heuristic routes to unconditional finiteness of the exceptional set do not currently constitute proofs. Every result in this paper is unconditional; no Generalized Riemann Hypothesis, density hypothesis, or other unproved conjecture is used anywhere.