Projective Boundary Certificates for Neural Selective Prediction

1. Integration point: train a neural predictor on a training split, then construct an acceptance rule on an independent calibration split. For selective classification, let s_theta(x,y) be a nonconformity score and accept when s_theta(x,y) is at most a calibrated threshold t(S). For a safety filter, use several scores and a vector threshold. Store an explicit boundary map consisting of points attaining the active order statistic or active Pareto/support constraints. 2. Pseudocode: compute calibration scores; construct A(S) and B(S); record K; for each calibration index i, delete z_i and recompute the rule; test whether deleting i preserves the boundary and whether z_i is accepted exactly when the boundary is preserved. Repeat over random subsets to estimate the cross-sample profile. If the profile is stable, calculate the beta upper quantile BetaInv(1-delta;k,n+1-k). If it is unstable, estimate mixture weights over boundary configurations and use the corresponding mixture quantile. 3. Taken from the paper: projectivity, deletion invariance, boundary-size conditioning, and the beta-versus-mixture distinction. Estimated empirically: profile stability, deletion invariance, and mixture weights. 4. First experiment: use an MLP on MNIST with an independent calibration split and top-score abstention. Compare ordinary conformal calibration, the beta certificate, and a naive boundary-size certificate. Repeat over at least 1,000 test batches. The prediction is that scalar order-statistic rules will match the beta CDF, while tied or competing vector boundaries will deviate from it but match the estimated profile mixture; certified violation frequency should remain below delta.