Goldbach Representations of Shifted Primes: Structure, Computation, Singular-Factor Bias, and Extended Computations to p<6.79×10<SUP>7</SUP>

Abstract

The Fundamental Theorem of Arithmetic guarantees that every composite integer n decomposes multiplicatively into primes: n=p1^(a1 )⋯pk^(ak ). A prime p, by definition, admits no such factorisation. This paper studies a structurally dual phenomenon: whether every prime p>11 admits an additive decomposition p+1=q+r with q,r∈P both prime. The displacement d=1 is canonical — it is the smallest positive integer that converts an odd prime into an even number greater than 2, the necessary condition for a Goldbach representation. This is not the classical binary Goldbach problem, which concerns all even integers; here the additional constraint that p is itself prime imposes a triple-primality requirement on p, q, r simultaneously, producing a distinct arithmetic profile governed by a new constant S∞≠2C2.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 is itself prime.Proved. The Euler product S∞≔∏(l>2,l∈P)^( 1+1/((l-1)(l-2))) converges absolutely and equals the limiting Cesàro mean of S(p+1) (Theorem 8.2). The value S∞=1.74273…>1 reflects a Dirichlet divisibility bias, explaining why the empirical constant differs structurally from 2C2. Two congruence theorems for Mirror and Anchor-3 primes are proved. Conjecture 11. (α∞=1/S∞) is equivalent to C ̂(x)→2C2.Computationally verified. N(p)≥2 for every prime 11<p<10^7 (664 574 primes, zero violations). Extended: 4 000 000 primes up to p<6.79×10^7, zero violations.New results (this version). Sixth disjoint-range estimate α ̂=0.5682, gap to 1/S∞=0.5738 reduced to 0.98%. Minimum N(p) grows monotonically by decade, factor ≈6.4×. Stable class fractions: Mirror 4.08%, Anchor-3 7.29%, Orphan 88.62%. Law 3: RMSE =0.0102, coverage ±30%: 99.9998% on 4,000,000 primes. Conjectural. N(p)≥2 for all p>11; α∞=1/S∞≈0.5738. PSLQ search finds no closed form for S∞ in standard constants.