Predictable Wasserstein Critics for Two-Sample Testing

A Wasserstein critic learns distributional differences that respect a chosen metric. Reusing past observations to train successive critics preserves out-of-sample evaluation, but leaves a random variance scale that can invalidate Gaussian studentization. We resolve this difficulty for a finite Lipschitz dictionary by letting entropy regularization vanish more slowly than training noise. The learned critic then has vanishing null variance, while retaining the dictionary's population discrepancy under alternatives. Adding independent random signs with positive amplitude gives a studentized Gaussian test under a $(2+\delta )$-moment null. The same fresh-pair scores also admit a closed-form half-normal e-process, valid at arbitrary stopping times for any predictable critic without moment assumptions. Approximate empirical Kantorovich-Rubinstein optimization recovers full or restricted population drift under first moments and fixed-alternative consistency under second moments, without requiring the fitted functions to converge. Simulations show earlier detection than online Newton-step betting in the studied alternatives, and a stratified penguin analysis illustrates learned projection critics.