Here is a polished English abstract: We study the minimax problem of estimating the two-sample Wasserstein distance (W_p(P,Q)) over the unrestricted class of Borel probability measures on ([0,1]^d). In the supercritical regime (d>2p), the theorem-level picture has remained incomplete: the best known lower bound in the balanced case (n=m=N) is ((N\log N)^{-1/d}), whereas the empirical plug-in estimator attains only the upper bound (N^{-1/d}). We prove that the lower-envelope scale is already exact on the full Euclidean neighborhood of the diagonal [ \mathcal L_{A,N}={(P,Q): W_p(P,Q)\le A(N\log N)^{-1/d}}, ] for a suitable constant (A=A(d,p)). On this class, the minimax absolute and squared risks are of orders ((N\log N)^{-1/d}) and ((N\log N)^{-2/d}), respectively, while the empirical plug-in estimator remains strictly suboptimal. We then develop several theorem-level off-diagonal results. For fixed positive baseline radius (r), we prove minimax upper and lower envelopes on full separated Euclidean product neighborhoods. On full two-ball mixed-scale classes, we establish an exact first-plus-second-order decomposition of (W_p^p), identify the residual term with a uniformly comparable quadratic transport cost, and derive a phase transition at the critical width (s_{\mathrm{crit}}(N,r)\asymp rN^{2/d-1/2}): below this scale the problem is generically parametric, whereas above it the problem is generically nonparametric up to the same logarithmic gap as in smooth-cost transport estimation. We also construct translated thin-annulus classes away from the diagonal on which direct estimation still improves over empirical plug-in by a logarithmic factor. Beyond these geometric subclasses, we prove that every fixed finite-template positive-baseline multicluster family is estimable at or below the global target scale, and that rigid shrinking matched-template families obey the same mixed-scale phase diagram as the two-ball model. For growing matched-template families, we obtain explicit complexity-sensitive upper bounds showing that such classes remain below the global target scale provided their effective branching complexity is sufficiently controlled. Finally, we combine these results with critical-grid and piecewise-affine linearization reductions, which sharpen the remaining open core of the unrestricted problem: after eliminating the diagonal neighborhood, fixed positive-distance neighborhoods, full two-ball geometries, thin annuli, and all fixed finite-template positive-baseline families, the unresolved difficulty is concentrated in a genuinely global branching regime in which microscopic diagonal nonsmoothness interacts with collapsing or proliferating macro-geometry.