Exact Risk-Complexity Laws for Projective Boundaries in Scenario Optimization and Distribution-Free Certification

What the math gives to ML

The paper gives a constructive explanation for when finite-sample violation risk has an exact beta distribution: the acceptance rule must have a proper projective boundary map, meaning that held-out acceptance is equivalent to retaining the full-sample boundary and accepted non-boundary points can be deleted without changing that boundary. The transferable asset is a distribution-free certification layer for neural abstention rules, learned safety filters, and selective predictors, where effective boundary size replaces parameter count as the relevant complexity. A stable cross-sample boundary profile predicts an exact beta law, whereas a varying profile requires a profile-aware mixture certificate and shows that observed boundary size alone is insufficient.