Goldbach Representations of Shifted Primes: Structure, Computation, and Singular-Factor Bias

For a prime p>2, let N(p):=#{{q,r}⊂P:q≤r, q+r=p+1}, counting Goldbach representations of p+1 when p is itself prime. We study the arithmetic, computational, and heuristic behaviour of N(p) on the shifted-prime subsequence {p+1:p∈P}.Our principal result proves that the Euler product S_∞ := ∏_(l>2,l∈P)^( 1+1/((l-1)(l-2)))=1.74273… converges absolutely and equals the limiting Cesàro mean of the Hardy–Littlewood singular factor S(p+1) on this subsequence. This value, strictly greater than 1, reflects a divisibility bias from Dirichlet’s theorem and explains why the empirical constant differs structurally from 2C_2.We verify N(p)≥2 for every prime 11<p<10^7 (664 574 primes, zero violations) and classify primes into Mirror, Anchor-3, and Orphan types, proving two congruence theorems. Law 3 achieves RMSE 13 times smaller than the classical formula. Five disjoint ranges support α_∞=1/S_∞≈0.5738 over the alternative α_∞=1/2.