Removing the logarithmic loss in the Li--Wu KdV integrator

Li and Wu proved that their unfiltered integrator for periodic KdV converges in $L^2$ with error $O(\tau^\gamma\log(1/\tau))$ for real, mean-zero $H^\gamma$ data, $0<\gamma\leq1$. We remove the logarithmic factor for the same time-semidiscrete algorithm, with constants uniform on bounded $H^\gamma$ sets. The main estimate concerns the exact averaged cubic remainder. Retaining both ratios of its two oscillatory phases bounds the dual four-frequency multiplier by the reciprocal of the third largest frequency. A dyadic four-linear estimate sums this majorant without a logarithm and includes a mixed $L^2$ endpoint. That endpoint controls the remainder at the numerical state without a numerical Sobolev bound. We then prove the full global estimate using the continuous trilinear bounds of Li and Wu, a stepwise residual calculation, low--high frequency stability, a restarted bootstrap, and density. Finite Fourier experiments check the implemented update, the remainder, and accumulated step defects.