Local Analytic LU Charts and Nonlinear Logarithmic Factor Flows in Weighted Wiener Algebras

We construct a local analytic LU chart near the identity of the weighted Wiener algebra 𝒲R of absolutely summable Laurent series on the circle. The chart is gauge-fixed by the scalar condition P₀ (log f) = 0; on that slice the lower and upper factors are recovered explicitly by logarithmic splitting: f = L (f) U (f), L (f) = exp (P_↓ log f), U (f) = exp (P_↑ log f). We prove a generator-identification theorem for nonlinear logarithmic factor flows: if g (t) evolves in the zero-mode slice by an analytic Banach-space ODE ġ = N (g), then f (t) = e^ (g (t) ) admits a factorized evolution whose lower and upper logarithmic generators are exactly the projections P_↓ N (g) and P_↑ N (g). Conversely, any differentiable symbol flow ḟ = qf remaining in the logarithmic chart recovers its factor generators by projection of q = f⁻¹ ḟ. A central-block exactness result for finite-section matrices is established, and reproducible numerical experiments are included.