A Baseline $T\log^2 T$ Upper Bound for KL-Regularized Prime--Zero Optimal Transport

Abstract

We prove that $\mathsf{OT}_\eta(T)\ll T\log^2 T$ unconditionally via a band-limited test scheme with Fej\'er averaging. The approach normalizes $\|\widehat f\|_1=\Theta(T^{-1})$ to ensure $\|h\|_1\asymp T$ and $\|\widehat h\|_1\ll 1$, and applies an $L^1$-controlled smoothed explicit formula to bound the error terms. As a result, the prime--zero optimal transport distance admits the baseline $T\log^2 T$ upper bound without additional assumptions.