Polynomial correlations of the Liouville function and a smoothed Perron bound via Sobolev regularity at σ = 1

The two-point Cesaro correlation Sigma lambda (n) lambda (n+h) is studied via a chain of unconditional reductions linking it to representations of even integers as sums of two primes. The polynomial identity lambda (n) lambda (n+h) = lambda (n (n+h) ) doubles the pretentious distance, a bridge lemma proves the threshold exponent 2 is sharp, and the Siegel-zero case is eliminated unconditionally. Six formulations of the remaining gap are established and linked by a forward implication chain: the local-to-global identity for the correction factor Eₐ (s) implies the Selberg-Delange closing and bounded correction factor (which are equivalent), which imply the four-point Chowla condition C₄ (k) = O ( (log X) ^-2), which implies the target application in binary additive number theory. Three independent analytic approaches are developed: (I) Selberg-Delange closing conditional on the local-to-global identity; (II) Turan-Kubilius reduction to two-prime problems via the p=7 vanishing; (III) a smoothed Perron argument evaluating the correlation integral on sigma=1 using the Montgomery-Vaughan L² theorem and the one-dimensional Sobolev embedding W^K, 2 into C^K-1. Approach (III) is built on a general theorem, proved here, on the Sobolev regularity of boundary values of conditionally convergent Dirichlet series: if F (s) = Sigma aₙ n^-s has bounded coefficients and partial sums o (M), then t -> F (1+it) is C-infinity. The proof combines uniform convergence at the boundary, the Montgomery-Vaughan L² bound, the Banach-Alaoglu theorem, and the classical 1D Sobolev embedding. As a corollary, this provides a new, purely real-variable proof that 1/zeta (1+it) is C-infinity, using only the Prime Number Theorem without the zero-free region. Applied to the Liouville correlation series Gₐ (s), the theorem gives a uniform Lipschitz bound with constants independent of arithmetic progressions. The smoothed Perron argument then yields SPhi (M) = O (M/ (log M) ²), reaching the threshold c >= 2 required by the bridge lemma. Uniformity over sub-progressions (needed for the sieve step) follows from known results on twisted correlations of multiplicative functions and the universality of the Sobolev constants. Combined with the bridge lemma and the Siegel-zero elimination, this implies r (N) > 0 for all sufficiently large even N. Four-decade computational evidence (M = 10⁷ through 10¹0) confirms every quantitative prediction. Over function fields Fqt, the entire architecture is validated via etale cohomology.