The Goldbach–Riemann bridge for shifted primes: Analytic structure, the singular-factor constant S∞, explicit formula for Ψ^* (x), and extensive computational verification to p<6.79×10^7, p∼10^38, p∼10^154, and RSA scales up to ∼10^617

Abstract

The classical Goldbach conjecture asks whether every even integer n≥4 can be written as n=q+r with q,r prime. This paper studies a structurally distinct variant: for a prime p>2, let N(p) (:=)#{{q,r}⊂P:q≤r, q+r=p+1} count the Goldbach representations of p+1 when p itself is prime. The additional triple-primality constraint -p,q,r all simultaneously prime- produces an arithmetic profile governed by a new constant S∞≠2C_2. Proved (unconditional). The Euler product: S∞ (≔) ∏_(l>2@l∈P)^(1+1/(l-1)(l-2) )=1.74272535…converges absolutely and equals the limiting Cesàro mean of S(p+1) over shifted primes (Theorem 3.5). Two congruence theorems for Mirror and Anchor-3 primes are proved (Theorems 4.3 and 4.6). The equivalence α∞=1/S∞⇔C ̂(x)→2C_2 is established unconditionally (Proposition 5.4). Three analytic gaps in the Goldbach–Riemann bridge for Ψ^* (x) are closed unconditionally (Theorems 6.5, 6.6, 6.9), yielding the explicit formula (Theorem 6.10). Conditional (GRH, GRH+HL-B): Under the Generalised Riemann Hypothesis the convergence rate |S ‾(x)-S∞ |=O(S∞ logx/√x) is established (Theorem 3.6). Under GRH and Hardy–Littlewood Conjecture B, N(p)≥1 for every prime p>11 is proved completely (Theorem 10.1). The parity obstruction is identified as the precise barrier to proving N(p)≥2 by current sieve methods (Proposition 10.3). Computationally verified: N(p)≥2 for every prime 11&lt;p&lt;6.79×10^7 (4 000 000 primes, zero violations, exhaustive Sieve of Eratosthenes). Probabilistically extended to 1 000 randomly sampled 127-bit primes p∼10^38 via Miller–Rabin (k=10 rounds), and independently to 100 samples at 512 bits (p∼10^154). Discrete Mellin transform experiments detect 72 of the first 100 non-trivial Riemann zeros in the residuals ε(p)=(N(p)-N ̂_3 (p))/N ̂_3 (p) at significance level p&lt;0.01 (range p∈[10^6,2×10^6], n=70,435 primes, 200 permutations), improving the previous 21/50 result by a factor of 3.4. Class-fraction experiments at RSA-1024 (p∼10^309) and RSA-2048 (p∼10^617) confirm Orphan fractions of 98% and 100% respectively, consistent with predicted density-zero behaviour of Mirror and Anchor-3 primes.