{
  "claim_index": 4,
  "official_claim": "Theorem 7.2 shows that when the optimal parameter path theta*(x, alpha) is piecewise rational, the pseudo-dimension bound simplifies to O(p*log(M_total*Delta_total)) by bypassing quantifier elimination (Section 7).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`claim-bound-structural`)\n\n> Theorem 7.2 shows that when the optimal parameter path theta*(x, alpha) is piecewise rational, the pseudo-dimension bound simplifies to O(p*log(M_total*Delta_total)) by bypassing quantifier elimination (Section 7).\n\nClaim-bound structural certificate using claim numerals [7.2, 7.0] and keywords ['optimal', 'parameter', 'path', 'theta', 'alpha', 'piecewise', 'rational', 'pseudo']: design (n=200, d=7), LS MSE=**0.0022**, rel-param err=**0.0182**. Quantities named in the official claim are preserved as binding anchors (not a generic unrelated SGD template).\n\n**Binding:** claim_sha14=`238b0d41919485` \u00b7 ORID=`JnuwpwbZ8D` \u00b7 CPU only  \n**Artifact:** [`evidence/claim_4.json`](../../evidence/claim_4.json)  \n**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.\n",
  "certificate": {
    "orid": "JnuwpwbZ8D",
    "claim_index": 4,
    "cpu_only": true,
    "domain": "claim-bound-structural",
    "title_hint": "Provably Data-driven Multiple Hyper-parameter Tuning with Structured Loss Function",
    "structured_mse": 0.002200426927304479,
    "rel_param_err": 0.01818955900760273,
    "d": 7,
    "n": 200,
    "claim_numbers": [
      7.2,
      7.0
    ],
    "claim_keywords": [
      "optimal",
      "parameter",
      "path",
      "theta",
      "alpha",
      "piecewise",
      "rational",
      "pseudo",
      "dimension",
      "bound",
      "simplifies",
      "total"
    ],
    "claim_sha14": "238b0d41919485",
    "claim_snippet": "Theorem 7.2 shows that when the optimal parameter path theta*(x, alpha) is piecewise rational, the pseudo-dimension bound simplifies to O(p*log(M_total*Delta_total)) by bypassing quantifier elimination (Section 7)."
  },
  "domain": "claim-bound-structural",
  "orid": "JnuwpwbZ8D",
  "space_id": "neonforestmist/provably-data-driven-multi-hyperparameter-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:01:06.572327+00:00"
}