Analytic continuation, Phragmén–Lindelöf, and Turán–Kubilius for the correction factor Ea(s)

Three complementary approaches to the correction factor Ea(s)=Ga(s)/Gaapprox(s)E_a(s) = G_a(s)/G_aᵃᵖᵖʳᵒˣ(s) Ea(s)=Ga(s)/Gaapprox(s) in the Goldbach reduction chain. (I) The approximate Euler product factors as ζ(s)−2Pw(s)2H0(s)ζ(s)⁻² P_w(s)^2 H_0(s) ζ(s)−2Pw(s)2H0(s) with a double zero at s=1s=1 s=1; Abel summation from Tao's S(M)=o(M)S(M)=o(M) S(M)=o(M) gives continuity of GaG_a Ga on σ≥1σ ≥ 1 σ≥1. (II) The Phragmén–Lindelöf method gives ∣Fa(1+it)∣≪exp⁡(c(log⁡∣t∣)1/3)|F_a(1+it)| exp(c(log|t|)1/3) ∣Fa(1+it)∣≪exp(c(log∣t∣)1/3) unconditionally; the Selberg–Delange closing follows conditionally on the local-to-global identity (LTG). (III) The Turán–Kubilius product expansion, using the E7=0E_7=0 E7=0 vanishing, reduces four-point Chowla to a coherent sum of two-prime interactions; each is bounded by O(X2/3p1/3)O(X2/3 p1/3) O(X2/3p1/3) via CRT–truncation (correcting an earlier claim that TT25 applies — it does not, since the local components χpχ_p χp are pretentious). The LTG identity and four-point Chowla are proved equivalent. Two large-scale computations at Xmax⁡=1010Xₘₐₓ=10¹⁰ Xmax=1010 show (i) four-point decay exponents α≈8α ≈ 8 α≈8–1818 18, far exceeding the Goldbach threshold α≥2α ≥ 2 α≥2, and (ii) two-prime TK interactions ∣A7,p∣≈1.4×10−8|A7,p| ≈ 1.4 × 10⁻⁸ ∣A7,p∣≈1.4×10−8, with empirical scaling X−0.845X-0.845 X−0.845 far beyond current theory.