Journal of Economic Intelligence and Mathematics

ABSTRACT

We develop a unified and fully audited analytic hierarchy for the restricted weighted Goldbach sum

$R_{a,q}\left(N\right):={\sum }_{\begin{array}{c}\hfill p_1+p_2=N\hfill \\ \hfill p_1\equiv a\left(modq\right)\hfill \end{array}}^{}\left(log{p}_1\right)\left(log{p}_2\right),q\ge 1,gcd(a,q)=1,$

with expected main term $M_{a,q}\left(N\right):=C_2𝔖\mathfrak{(}N)N/\phi \left(q\right)$, and exceptional set ${ℰ}_{a,q}\left(X\right):=\{N\le X,Neven:R_{a,q}\left(N\right)=0\}$. 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 $L^4$ minor-arc route, with explicit constant $K=2C(1,4)\le 38.82$. Level 1.5 is a sub-exponential exceptional-set bound $\#{ℰ}_{a,q}\left(X\right){\ll }_{q}Xexp(-\sqrt{logX/R})$ 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 $K_{new}\le 9.80$ and, for moduli $q\le 200$ certified free of Siegel zeros, an unconditional pointwise sub-exponential bound with $C\left(4\right)\le 120$ and $log{N}_0\left(4\right)\le 42$.

Three structural obstructions (Double-Pole, Borel-Cantelli, ETK Dimensional Explosion) formally retract three classical routes to unconditional finiteness. Under DH and GRH, conditional hierarchies (with $\theta \left(A\right)=1-2/(A+2)$ and $log{N}_0\left(4\right)=45.93$) are recorded. The Gowers-Spectral Bridge gives conditional finiteness under the Uniform Spectral Gap (USG) hypothesis with effective threshold $N_0\left(4\right)\le {10}^{16}$.

Keywords: Goldbach problem; arithmetic progressions; restricted Goldbach sums; almost-all theorem; explicit constants; Hölder minor-arc bound; sub-exponential exceptional set; Siegel-zero absorption; Stechkin zero-free region; circle method; ternary Goldbach; Density Hypothesis; GRH; Gowers-Spectral Bridge; Uniform Spectral Gap; structural obstructions.