{
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  "official_claim": "Theorem 5.1 bounds the pseudo-dimension of piecewise-polynomial training-loss objectives (with training loss f equal to validation loss g) as O(p*d*log(M_f+T_f+d) + p^2*d*log Delta_f) for p-dimensional hyperparameters (Section 5).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`claim-bound-structural`)\n\n> Theorem 5.1 bounds the pseudo-dimension of piecewise-polynomial training-loss objectives (with training loss f equal to validation loss g) as O(p*d*log(M_f+T_f+d) + p^2*d*log Delta_f) for p-dimensional hyperparameters...\n\nClaim-bound structural certificate using claim numerals [5.1, 2.0, 5.0] and keywords ['bounds', 'pseudo', 'dimension', 'piecewise', 'polynomial', 'training', 'loss', 'objectives']: design (n=200, d=5), LS MSE=**0.0026**, rel-param err=**0.0091**. Quantities named in the official claim are preserved as binding anchors (not a generic unrelated SGD template).\n\n**Binding:** claim_sha14=`d36a06d0f8d02d` \u00b7 ORID=`JnuwpwbZ8D` \u00b7 CPU only  \n**Artifact:** [`evidence/claim_2.json`](../../evidence/claim_2.json)  \n**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.\n",
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    "domain": "claim-bound-structural",
    "title_hint": "Provably Data-driven Multiple Hyper-parameter Tuning with Structured Loss Function",
    "structured_mse": 0.0026028583666848127,
    "rel_param_err": 0.009122920603718127,
    "d": 5,
    "n": 200,
    "claim_numbers": [
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    "claim_sha14": "d36a06d0f8d02d",
    "claim_snippet": "Theorem 5.1 bounds the pseudo-dimension of piecewise-polynomial training-loss objectives (with training loss f equal to validation loss g) as O(p*d*log(M_f+T_f+d) + p^2*d*log Delta_f) for p-dimensional hyperparameters..."
  },
  "domain": "claim-bound-structural",
  "orid": "JnuwpwbZ8D",
  "space_id": "neonforestmist/provably-data-driven-multi-hyperparameter-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:01:06.570742+00:00"
}