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) Eₐ (s) = Gₐ (s) /Gₐ^approx (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) (s) ^-2 Pw (s) ² H₀ (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 GaGₐ Ga on σ≥1 1 σ≥1. (II) The Phragmén–Lindelöf method gives ∣Fa (1+it) ∣≪exp⁡ (c (log⁡∣t∣) 1/3) |Fₐ (1+it) | (c (|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=0E₇=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 (X^2/3 p^1/3) O (X2/3p1/3) via CRT–truncation (correcting an earlier claim that TT25 applies — it does not, since the local components χpₚ χp are pretentious). The LTG identity and four-point Chowla are proved equivalent. Two large-scale computations at Xmax⁡=1010X_=10^10 Xmax=1010 show (i) four-point decay exponents α≈8 8 α≈8–1818 18, far exceeding the Goldbach threshold α≥2 2 α≥2, and (ii) two-prime TK interactions ∣A7, p∣≈1. 4×10−8|A₇, ₏| 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.