Geometry-Limited Policy Contrasts for Distribution-Valued Outcomes

Many interventions produce a distribution for each individual rather than a scalar response. Comparing only means or Wasserstein barycenters then discards population heterogeneity among individual distributions. We study population matching costs between the resulting policy-specific outer laws, allowing all matching mechanisms compatible with their marginals. Conditional Kantorovich duality gives pairwise sharp bounds. We construct a split-sample lower certificate by extending a finite transport dual to the full quantile cone. Under known assignment probabilities, global feasibility yields finite-sample validity uniformly over a policy class even when the outer-law learner is misspecified. Validity, however, need not imply information. With finitely many covariate cells and actions, one-dimensional outcomes, and cost $W_p^p$, the minimax information loss over unrestricted quantile outcomes is of order $(\log n{)}^{-(p\wedge 2)}$ for honest lower limits and point estimation. A cost-active high-probability quantization condition is sufficient for a polynomial upper rate. We also propagate uncertainty from finite within-individual samples. A semi-synthetic experiment built from real electricity-load quantile templates gives a positive simultaneous lower certificate when population heterogeneity changes despite equal scalar means and Wasserstein barycenters.