Polynomial correlations of the Liouville function with an application to an additive problem

The two-point Cesàro correlation of the Liouville function, Σ_{n≤X} λ(n)λ(n+h), is studied via a chain of reductions of the binary Goldbach conjecture. The polynomial identity λ(n)λ(n+h) = λ(n(n+h)) doubles the pretentious distance from D² to 2D², yielding the current unconditional bound O(X/(log X)^c) for a small absolute constant c > 0; the Siegel-zero branch of the correlation analysis is closed unconditionally through the two-point Chowla theorem of Tao and Teräväinen in the presence of an exceptional zero. A bridge to the additive problem is constructed on the pointwise weighted inequality of Wu, with the count side evaluated at the level-of-distribution results of Lichtman at the residue N itself and the twisted side consumed by two sifted cancellation hypotheses stated at the moduli actually used; under these hypotheses, every sufficiently large even N satisfies r(N) ≥ 0.9 C(N) N (log N)^{-2}, the margin of order one and carried entirely by published sieve inputs. Six formulations of the residual gap between the current bound and the correlation targets are established, three of them proved equivalent and the remainder linked by unconditional implications; a Turán–Kubilius decomposition further reduces the four-point Chowla conjecture to a coherent sum of two-prime problems with explicit truncation bounds. Over the function fields F_q[t] every step of the architecture is validated, with exponent 109/72 via the analytic route and square-root cancellation via the geometric route; computational evidence at X ≤ 10^10 supports every quantitative prediction. This version supersedes the previous deposit and should be cited in its place.