Exact Policy Certificate Loss Under Normalization Coarsening

Robust policy certificates lower-bound policy improvement under a sensitivity model for unobserved confounding. Group-level inverse-weight normalization can weaken these certificates, even when the policy and full-history propensity reference remain fixed. In a one-step binary-action model, we characterize this loss for arbitrary measurable normalization interfaces: it equals the excess weighted pinball risk incurred by requiring histories in the same group to share a threshold. For sensitivity radii greater than one, grouping is lossless if and only if a measurable group-level threshold belongs to the conditional treated-reward quantile set at almost every history, including with discrete rewards and nonunique quantiles. To certify fitted thresholds with estimated nuisance quantities, we develop rectangular augmented certification (\RAC), combining nuisance correction, confidence-aware fitting and independent betting. For binary rewards and an always-treat target, simultaneous full-history nuisance envelopes yield a finite-sample-valid lower confidence bound on policy improvement after fitting. Experiments show that augmentation strengthens the matched protected weak dual, while confidence-aware fitting improves average certificate strength over empirical-mean fitting across tested positive-margin conditions; fixed thresholds remain competitive. Grouping trades structural loss against statistical uncertainty. Under correctly specified low-dimensional nuisance models, structured calibration reduces protection costs at larger history supports.