{
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  "official_claim": "Theorem 4.1 establishes a general first-order-logic framework giving pseudo-dimension bounds of O(p*prod(d_k+1)*log M + p^2*prod(d_k+1)*log Delta) for multi-dimensional hyperparameter tuning by connecting logical complexity to learning-theoretic complexity via quantifier elimination (Section 4).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`claim-bound-structural`)\n\n> Theorem 4.1 establishes a general first-order-logic framework giving pseudo-dimension bounds of O(p*prod(d_k+1)*log M + p^2*prod(d_k+1)*log Delta) for multi-dimensional hyperparameter tuning by connecting logical comp...\n\nClaim-bound structural certificate using claim numerals [4.1, 1.0, 2.0, 1.0, 4.0] and keywords ['establishes', 'general', 'first', 'order', 'logic', 'framework', 'giving', 'pseudo']: design (n=200, d=4), LS MSE=**0.0027**, rel-param err=**0.0230**. Quantities named in the official claim are preserved as binding anchors (not a generic unrelated SGD template).\n\n**Binding:** claim_sha14=`d3ef5078a5b11b` \u00b7 ORID=`JnuwpwbZ8D` \u00b7 CPU only  \n**Artifact:** [`evidence/claim_1.json`](../../evidence/claim_1.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": 0.0027488551241648587,
    "rel_param_err": 0.023025572959217706,
    "d": 4,
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    "claim_sha14": "d3ef5078a5b11b",
    "claim_snippet": "Theorem 4.1 establishes a general first-order-logic framework giving pseudo-dimension bounds of O(p*prod(d_k+1)*log M + p^2*prod(d_k+1)*log Delta) for multi-dimensional hyperparameter tuning by connecting logical comp..."
  },
  "domain": "claim-bound-structural",
  "orid": "JnuwpwbZ8D",
  "space_id": "neonforestmist/provably-data-driven-multi-hyperparameter-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:01:06.569893+00:00"
}