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

For \(q \ge 1\) and \(\gcd(a,q)=1\) we study the restricted weighted Goldbach sum \(R_{a,q}(N) = \sum_{p_1+p_2=N,\ p_i\equiv a(q)} (\log p_1)(\log p_2)\) for even \(N\), with candidate main term \(M_{a,q}(N) = 2C_2\varphi(q)^{-1}\Xi_0(N)N\), where \(C_2\) is the twin-prime constant and \(\Xi_0(N)\) is the odd part of the Hardy--Littlewood singular series. We first prove an elementary local-obstruction theorem: if \(q\) has an odd prime factor \(\ell\), then \(R_{a,q}(N) = O_q(\log N)\) for every \(N\equiv a\ (\ell)\), a set of positive density, so no asymptotic of size \(\asymp N/\varphi(q)\) can hold; the restriction to \(q=2^k\) in the rest of the paper is therefore forced by the truth of the statement. For \(q=2^k\) we evaluate the major arcs through an exact additive-shift decomposition of the progression and a factorisation of the resulting singular-series coefficients, which yields the main term \(M_{a,q}\) with an explicit coefficient, and, combined with the Siegel--Walfisz theorem and Vaughan's bound on the minor arcs, the mean-square estimate \(\sum_{N\le X}|R_{a,q}(N)-M_{a,q}(N)|^2 \ll_{A,q} X^3(\log X)^{-A}\) for every fixed \(A>0\). This gives an exceptional set \(\#\{N\le X \text{ even}: R_{a,q}(N)=0\} \ll_{A,q} X(\log X)^{-A}\). An elementary transfer principle then shows that the same bound controls \(R_{a,q}\) along any injective map into the even integers of at most linear growth. Consequently, for every residue class ("ray") \(c\) coprime to \(q\), almost all primes \(p_n\equiv c\ (q)\) have \(p_n+p_{n+1}\), \(p_n+p_{n+k}\) and \(p_n+h\) (\(h\) odd) representable with first summand in the class \(a\). For a modulus with an odd prime factor \(\ell\) we prove that \(N_n=p_n+p_{n+1}\equiv a\ (\ell)\) is a condition on the pair of rays of \(p_n\) and \(p_{n+1}\), that it holds infinitely often by a theorem of Shiu, and, for \(q=3\), we verify computationally up to \(6\times10^6\) that \(R_{1,3}(N_n)=0\) exactly when \(N_n\equiv1\ (3)\) and \(N_n-3\) is composite (only the implication from this condition to the vanishing of the sum is proved). We document by exact computation below \(10^8\) the digit-repulsion bias between consecutive primes, the repetition of rays according to the size of the gap, the doubling of the relative variance of \(R_{a,q}\) on the class \(N\equiv2a\ (q)\), and the out-of-sample predictability of the next ray; the ray-gap identity and the symmetry \(p_1\leftrightarrow p_2\) behind this doubling are proved. The same circle-method machinery gives the restricted ternary asymptotic \(W_{a,q}(n)=\Xi_3(n)n^2/(2\varphi(q))+O_{A,q}(n^2(\log n)^{-A})\) and positivity for all large odd \(n\). Finally, the amplitude \(\Xi_0(p+h)\) of the main term along shifted primes converges in distribution to an explicit random Euler product \(Y_h\), with all moments converging to absolutely convergent Euler products, an entire moment generating function with \(\log\mathbb E Y_h^s \le s\log\log s+O(s)\), a doubly exponentially thin upper tail that is sharp on a doubly logarithmic scale, and no atoms. Every theorem is unconditional; computations are labelled with their exact range.