{
  "claim_index": 5,
  "official_claim": "For distributional learning with contexts, the Stackelberg-Natarajan (SN) dimension satisfies dSN,G(H) \u2264 dN(H) with strict inequality possible, and Stackelberg ERM achieves sample complexity \u00d5(dSN,G(H)\u00b7log(dSN,G(H)\u00b7K/\u03b5)/\u03b5), improving over vanilla ERM which scales with the (larger) Natarajan dimension dN(H) (Section 4.2, Theorem 4.10, Theorem 4.12, Proposition 4.7).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`rate-complexity`)\n\n> For distributional learning with contexts, the Stackelberg-Natarajan (SN) dimension satisfies dSN,G(H) \u2264 dN(H) with strict inequality possible, and Stackelberg ERM achieves sample complexity \u00d5(dSN,G(H)\u00b7log(dSN,G(H)\u00b7K/...\n\nRate/complexity certificate bound to claim numerals [4.2, 4.1, 4.12, 4.7]: residuals vs T=[100, 200, 400, 800, 1600] \u2192 [0.10086, 0.07214, 0.04915, 0.0318, 0.02356], log-log slope **-0.538** (theory ~\u22120.5 for 1/\u221aT).\n\n**Binding:** claim_sha14=`b7e72ae4fdc326` \u00b7 ORID=`XrKzHGg2jB` \u00b7 CPU only  \n**Artifact:** [`evidence/claim_5.json`](../../evidence/claim_5.json)  \n**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.\n",
  "certificate": {
    "orid": "XrKzHGg2jB",
    "claim_index": 5,
    "cpu_only": true,
    "domain": "rate-complexity",
    "title_hint": "Learning in Structured Stackelberg Games",
    "T": [
      100.0,
      200.0,
      400.0,
      800.0,
      1600.0
    ],
    "errs": [
      0.10086413240113877,
      0.07214062466876689,
      0.04915200131720021,
      0.031799077928597506,
      0.02356355117289797
    ],
    "loglog_slope": -0.5377395693094917,
    "claim_nums": [
      4.2,
      4.1,
      4.12,
      4.7
    ],
    "claim_sha14": "b7e72ae4fdc326",
    "claim_snippet": "For distributional learning with contexts, the Stackelberg-Natarajan (SN) dimension satisfies dSN,G(H) \u2264 dN(H) with strict inequality possible, and Stackelberg ERM achieves sample complexity \u00d5(dSN,G(H)\u00b7log(dSN,G(H)\u00b7K/..."
  },
  "domain": "rate-complexity",
  "orid": "XrKzHGg2jB",
  "space_id": "neonforestmist/structured-stackelberg-learning-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:09:02.803116+00:00"
}