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We establish the unrestricted fixed-dimensional minimax theory for estimating the two-sample Wasserstein distance $W_p(P,Q)$ over the full class of Borel probability measures on $[0,1]^d$ in the supercritical regime $d>2p$. For every $p\ge 1$, every $d>2p$, and arbitrary sample sizes $n,m$ with $N:=n\wedge m$, we prove that the exact minimax absolute and squared risks are $(N\log N)^{-1/d}$ and $(N\log N)^{-2/d}$. The argument is entirely Euclidean. First, we show that the lower-envelope scale is already exact on a full diagonal neighborhood and that the empirical plug-in estimator remains locally suboptimal there. Second, we identify the critical finite model by reducing the balanced supercritical problem to the Euclidean problem on the dyadic grid with $|X_J|\asymp N\log N$ atoms, and we determine the generic off-diagonal geometry through mesoscopic annuli, adaptive linearization, and full positive-distance neighborhoods. Third, a critical linearization of the $p$-cost separates a parametric first-order affine layer from normalized second-order residuals; semiconcave branching bounds and support-complexity eliminations then reduce the unresolved part of the problem to a finite nearest-neighbor macrograph on active shells. Finally, that terminal layer is closed by three different mechanisms: uniform semiconcavity (with sub-dimensional entropy) for the semiconcave wedge $1\le p<\frac{2d}{d-1}$, a quadratic-cost reduction for the superquadratic band $p\ge\frac{2d}{d-1}$, and a sharp $L_1$-covering argument together with the shell-active structure and no-go theorem for adaptive shell transfer.
Чурилов М. В. 2026. Minimax Wasserstein Estimation in the Supercritical Regime. PREPRINTS.RU. https://doi.org/10.24108/preprints-3114797
