Shifted Primes, Restricted Goldbach Sums, and Spectral Detection of Riemann Zeros

This paper consolidates, corrects, and extends a research programme on the shifted-prime problem $p = q + r - 1$ with $p, q, r$ prime and its connections to the binary Goldbach conjecture and the non-trivial zeros of the Riemann zeta function $\zeta(s)$. New material over Version 6. The principal addition is a rigorous three-level treatment of the restricted Goldbach sum \[ P_{R_{3,4}}(N) = \sum_{\substack{p+q=N,\\ p\equiv 3\ (\mathrm{mod}\,4)}} (\log p)(\log q). \] At Level 1 [PROVED] (unconditional), the ``almost-all'' theorem of Montgomery--Vaughan type shows that the exceptional set of even integers $N\leq X$ for which $|R_{3,4}(N) - \tfrac{1}{2}C_2 S(N)N|$ exceeds $CN/(\log N)^3$ has measure $O_A\bigl(X/(\log X)^A\bigr)$ for every $A>0$. At Level 2 [PROVED] (unconditional), a transfer inequality bounds $|R_{a,q}(N)-\phi(q)^{-1}R(N)|$ in terms of twisted sums $S_\chi(N)$ with mean-square control. At Level 3 [COND. PROVED, GRH], for all sufficiently large even $N$ one has $R_{3,4}(N)=\tfrac{1}{2}C_2 S(N)N + O(N^{1/2+eps})$. Anderson's original claim of an explicit unconditional constant $K\leq 28.65$ for all $N$ is identified as relying on the Hardy--Littlewood binary asymptotic for each individual $N$, which is itself a conjecture; the claim is accordingly downgraded and the gap stated precisely. Retained from Version 6. Five analytical gaps (A--E) in the Goldbach--Riemann bridge for $\Psi^*(x)$ are fully closed unconditionally (Gaps D1, D2, D3, E) or under GRH (Gap C). The corrected spectral-detection results stand: $\lambda_1/\lambda_2 = 182.63$ ($n=892\,206$); 129/200 Riemann zeros detected at $p<0.01$ ($n=1\,310\,763$); Mellin--Lomb--Scargle concordance 29/30 versus 0/30; 9/10 direct Pearson correlations significant; heteroscedasticity of $eps(p)$ formally confirmed ($p=4.7\times 10^{-14}$). Principal corrections retained from Version 6. The $k=3$ existence problem is equivalent to binary Goldbach (open). The permutation-test bug in scripts~6.py--8.py is corrected ($199/200\to 129/200$). The formula for $S_\infty^{(k)}$ is corrected for $k\geq 3$. None of these results constitutes a proof of the Riemann Hypothesis. All claims carry explicit epistemic labels.