The Goldbach–Riemann Bridge for Shifted Primes. Analytic Structure, the Singular-Factor Constant <em>S</em>∞, Explicit Formula for Ψ*(x), and Extensive Computational Verification to p<6.79×10⁷, p~10³⁸, p~10¹⁵⁴, and RSA Scales up to ~10⁶¹⁷

The classical Goldbach conjecture asks whether every even integer n &gt;= 4 can be written as n = q + r with q, r prime. This paper studies a structurally distinct variant: for a prime p &gt;= 2, let N(p) = #{ (q, r) subset of primes: q &lt;= 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_∞ != 2 C_2. Proved (unconditional). The Euler product: S_∞ = ∏_{ℓ &gt; 2, ℓ prime} ( 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_∞ ⇔ Ĉ(x) → 2 C_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 |S̄(x) – S_∞| = O( S_∞ log x / √x ) is established (Theorem 3.6). Under GRH and Hardy‑Littlewood Conjecture B, N(p) &gt;= 1 for every prime p &gt; 11 is proved completely (Theorem 10.1). The parity obstruction is identified as the precise barrier to proving N(p) &gt;= 2 by current sieve methods (Proposition 10.3).Computationally verified: N(p) &gt;= 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 1000 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) – N̂₃(p)) / N̂₃(p) at significance level p &lt; 0.01 (range p in [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. Epistemic status: All claims carry explicit labels: [PROVED], [COND. PROVED], [COMP. VERIF.], [CONJECTURE], [NEW], [CORRECTED]. No claim is presented without its status. The central conjectures are open.