The Restricted Goldbach Hierarchy: Explicit Unconditional Bounds, Structural Rigidity, and a Spectral Bridge to Finiteness

Source abstract

We develop a complete analytic hierarchy for the restricted weighted Goldbach sum $R_{a,q}(N) :=\sum_{\substack{p_1+p_2=N \\ p_1 \equiv a \ (\mathrm{mod}\ q)}} \left({\log{p}}_{1}\right)\left({\log{p}}_{2}\right),\\ q=2^{k},\; 1\leq k\leq 6,\; \gcd(a,q)=1$ with main term $M_{a,q}(N) := 2C_2\,S(N)\,N/\varphi(q)$ and exceptional set $E_{a,q}(X) := \{N \leq X \text{ even}: R_{a,q}(N) = 0\}$.The unconditional core establishes, for $q = 2^k$: an explicit, fully self-contained pointwise minor-arc bound via Vaughan's identity with balanced parameters $U = V = X^{2/5}$; an unconditional major-arc evaluation for the growing family of denominators whenever the Landau-Page exceptional modulus satisfies $k_1(X) \leq \log X$; a master second-moment bound with exact leading constant; an effective almost-all theorem whose threshold is an explicit, necessarily growing function of $X$, together with a theorem proving that no constant independent of $X$ can play that role; a sub-exponential exceptional-set bound with McCurley's explicit zero-free-region constant $R = 9.645908801$; a self-contained Hölder minor-arc refinement giving $K_{\min}(4,A) \leq 2.10/\sqrt{\varphi(4)} \approx 1.4849$; an unconditional pointwise sub-exponential bound for moduli certified free of Siegel zeros, with $C(4) \leq 120$; and structural rigidity of the exceptional set (a gap bound, non-consecutiveness, and additive-energy decay). A ternary transfer via prime anchoring, and a conditional hierarchy under the Density Hypothesis and GRH, are also recorded.Three structural obstructions (Double-Pole Convolution, Borel-Cantelli Divergence, ETK Dimensional Explosion) formally retract three classical routes to unconditional finiteness. The Gowers-Spectral Bridge gives conditional finiteness of $E_{a,q}$ under a Uniform Spectral Gap (USG) hypothesis, with effective threshold $N_0(4) \leq 10^{16}$. The logical chain Strong Montgomery-GUE $\Rightarrow$ Controlled Additive Energy $\Rightarrow$ USG ($c = 1$) $\Rightarrow$ $R_{a,q}(N) > 0$ for all $N \geq N_0$ is established conditional on Strong Montgomery-GUE, a hypothesis strictly stronger than the standard weak Montgomery pair-correlation conjecture; this distinction is stated explicitly in every theorem in that chain.