Restricted Goldbach Sums in Arithmetic Progressions: Analytic Hierarchy, Sub-Exponential Bounds, and Riemann Zero Detection

Abstract

We develop a unified and fully audited analytic hierarchy for the restricted weighted Goldbach sum, in which the primes are weighted by their logarithms and the first prime is constrained to lie in a fixed residue class modulo q. The expected main term is built from the twin-prime constant and the Hardy-Littlewood singular series, divided by Euler's totient of q. The paper consolidates and supersedes preprint version 3, integrating results from Papers 1, 9 and 14 of the Anderson Series, with all documented corrections applied. The unconditional core establishes three nested levels. Level 1 is an effective almost-all theorem via the standard fourth-moment minor-arc route, with explicit constant K = 38.82 or smaller. Level 1.5 is a sub-exponential exceptional-set bound, of the form X times exp(-sqrt(log X / R)), with Stechkin's constant R = 9.6459, proved unconditionally by absorbing any potential Siegel zero into a modified main term. Level 1.5+ is a Hölder minor-arc refinement giving the improved constant K_new of at most 9.80 and, for moduli q up to 200 certified free of Siegel zeros, an unconditional pointwise sub-exponential bound with C(4) of at most 120 and log N_0(4) of at most 42. Three structural obstructions (Double-Pole, Borel-Cantelli, ETK Dimensional Explosion) formally retract three classical routes to unconditional finiteness. Under the Density Hypothesis and GRH, conditional hierarchies are recorded, including an exceptional-set exponent of 1 minus 2 over (A+2) and a GRH threshold of log N_0(4) = 45.93. The Gowers-Spectral Bridge gives conditional finiteness under the Uniform Spectral Gap (USG) hypothesis, with an effective threshold N_0(4) of at most 10^16.