{
 "artifacts": null,
 "category": "theory",
 "description": "Construct a neural acceptance or abstention set from calibration samples together with an explicit boundary map selecting the samples that determine the set. If the map is proper projective and its cross-sample complexity profile is stable, the conditional violation risk has an exact beta law indexed by boundary size rather than network parameter count. This provides a falsifiable, distribution-free certificate for neural selective classifiers and learned safety filters.",
 "download_zip": "https://synthcore.org/download/1152",
 "formulas_latex": [
  "$$R(S)=\\mathbb{P}_{Z\\sim P}\\left\\{Z\\notin A(S)\\mid S\\right\\},\\qquad K(S)=\\left|B(S)\\right|.$$",
  "$$z_i\\in A(S\\setminus\\{z_i\\})\\quad\\Longleftrightarrow\\quad B(S\\setminus\\{z_i\\})=B(S).$$",
  "$$x\\in A(S),\\ x\\notin B(S)\\quad\\Longrightarrow\\quad B(S\\setminus\\{x\\})=B(S).$$",
  "$$R\\mid K=k\\sim\\operatorname{Beta}(k,n+1-k)\\quad\\text{if the cross-sample profile is stable};\\qquad f_{R\\mid K=k}(r)=\\sum_c w(c\\mid k)f_{R\\mid c}(r)\\quad\\text{otherwise}.$$"
 ],
 "id": 3002,
 "implementation": "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.",
 "math_summary": "Let S=(z_1,...,z_n) be an exchangeable calibration sample, A(S) the acceptance set for a fresh point z, and B(S) a boundary map. Define R(S)=P(z is not in A(S) given S) and K(S)=|B(S)|. Proper projectivity requires that a held-out point z_i is accepted by the leave-one-out rule exactly when the full-sample boundary is retained, and that deleting any accepted non-boundary point does not change B(S). If the cross-sample complexity profile q_k is stable, exchangeability gives a beta law, with indexing determined by the boundary convention; a common convention is R given K=k distributed as Beta(k,n+1-k). If q_k varies, conditioning only on K is invalid and the conditional law must be represented as a mixture over boundary configurations c with weights w(c given k).",
 "math_tags": [
  "probability",
  "statistics",
  "combinatorics",
  "optimization",
  "learning-theory"
 ],
 "ml_areas": [
  "mlp",
  "transformer",
  "regularization"
 ],
 "paper": {
  "arxiv_id": "2609.01355",
  "arxiv_url": "https://arxiv.org/abs/2609.01355",
  "summary_what_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.",
  "title": "Exact Risk-Complexity Laws for Projective Boundaries in Scenario Optimization and Distribution-Free Certification",
  "year": "2026"
 },
 "ratings": {
  "difficulty": 5,
  "novelty": 7,
  "usefulness": 8
 },
 "solves": [
  "generalization",
  "stability"
 ],
 "title": "Projective Boundary Certificates for Neural Selective Prediction",
 "url": "https://synthcore.org/idea/3002/projective-boundary-certificates-for-neural-selective-predic",
 "verification": {
  "peer_reviewed": false,
  "status": "running_mechanism",
  "status_label": "Checking mechanism…",
  "updated_at": "2026-09-02T16:40:53",
  "verdict_source": "deterministic test code (paired-seed permutation statistics)",
  "verification_axes": {
   "benchmark_mechanism": {
    "confirmed": null,
    "tested": false
   },
   "practical_benchmark": {
    "beats_baseline": null,
    "tested": false
   },
   "toy_mechanism_gate": {
    "confirmed": null,
    "tested": false
   }
  }
 }
}