Spectral Signatures of the Riemann Zeta Function in Shifted-Prime Residuals: Amplification Factor <span class="math-tex">$S_∞$, High-Precision Numerical Verification, and the Generalised Amplification Conjecture

Source abstract

Let $P$ denote the set of rational primes. For a prime $p>2$, define N(p): = #{{q,r}⊂P:q≤r, q+r=p+1}, the number of Goldbach representations of the even integer p+1 subject to the additional constraint that p itself is prime. The triple-primality condition distinguishes this shifted-prime problem from the classical Goldbach problem and from twin primes. We prove unconditionally that the bridge function , where R(n)=∑a+b=nΛ(a)Λ(b) is the von Mangoldt convolution, satisfies an explicit formula whose residue coefficients at the non-trivial zeros ρk=1⁄2+iγk of ζ(s) are amplified by the constant S∞=∏l>2, l∈P(1+1⁄(l-1)(l-2)) =1.74272535… relative to all classical Goldbach–Riemann bridges (Fujii 1991, Bhowmik–Schlage-Puchta 2010, Goldston–Suriajaya 2023). This amplification originates from a systematic Dirichlet divisibility bias. Using 334,351 primes in [106,6×106] and a 500-permutation test, we detect 123 of the first 200 non-trivial zeros at p<0.01 (61.5%), and 102 at p<0.001 (51%). The maximum z-score is 104.12 at γ1=14.1347. The spectral amplitude decay law |Mk |∝1/|γk| is confirmed with log-log slope -1.0182 (R2=0.516, p<10-4). We correct a previously conjectured generalised Euler product S∞(k), showing it grows as Θ(2k) (not linearly), with the error reaching 71% at k=5. Amplification constants for Mirror-prime and Anchor-3-prime subsequences are computed analytically as C(M)=C(A)=S∞/(3/2)≈1.162. The Generalised Amplification Conjecture is proposed: every arithmetically restricted prime subsequence carries its own Euler-product amplification constant. Supplementary results include the monotone growth of minN(p) up to 10^8, the empirical trajectory of C ̂(x)→2C2, and stable class ratios for Mirror, Anchor-3, and Orphan primes.