{
  "claim_index": 1,
  "official_claim": "For approximately norm-minimizing interpolators in the Sobolev space W^{k,p}(R^d) with k in (d/p, 1.5d/p) and sufficiently large sample size, the conditional generalization error is bounded below by C*gamma^{-pd/(kp-d)}, a positive constant independent of n, establishing harmful (non-benign) overfitting (Section 3.2, Theorem 3.7).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`sobolev-harmful-overfitting`)\n\n> For approximately norm-minimizing interpolators in the Sobolev space W^{k,p}(R^d) with k in (d/p, 1.5d/p) and sufficiently large sample size, the conditional generalization error is bounded below by C*gamma^{-pd/(kp-d...\n\nSobolev interpolator certificate: near-interpolator train MSE=**2.24e-02** (n=200, d=2, p=2.0, k=1.2), population L\u00b2 error=**1589.9168** stays above lower-bound proxy C\u00b7\u03b3^{-pd/(kp\u2212d)}=**476.8372** (\u03b3=0.4) \u2014 harmful-overfitting style. Bump \u2016\u03c8\u2016 proxy=**0.9059** vs 1+\u03b4^{(d\u2212kp)/p}=**2.3195** (\u03b4=0.25).\n\n**Binding:** claim_sha14=`43dc236deecc1a` \u00b7 ORID=`OmmKEy9Ixf` \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",
  "certificate": {
    "orid": "OmmKEy9Ixf",
    "claim_index": 1,
    "cpu_only": true,
    "domain": "sobolev-harmful-overfitting",
    "title_hint": "Harmful Overfitting in Sobolev Spaces",
    "d": 2,
    "p": 2.0,
    "k": 1.2,
    "gamma": 0.4,
    "n": 200,
    "train_mse_interp": 0.02236250313625451,
    "pop_mse": 1589.9167596678774,
    "lower_bound_proxy": 476.83715820312557,
    "pop_above_lb": true,
    "bump_sobolev_norm": 0.90589855016253,
    "bump_bound": 2.319507910772894,
    "delta": 0.25,
    "claim_sha14": "43dc236deecc1a",
    "claim_snippet": "For approximately norm-minimizing interpolators in the Sobolev space W^{k,p}(R^d) with k in (d/p, 1.5d/p) and sufficiently large sample size, the conditional generalization error is bounded below by C*gamma^{-pd/(kp-d..."
  },
  "domain": "sobolev-harmful-overfitting",
  "orid": "OmmKEy9Ixf",
  "space_id": "neonforestmist/harmful-overfitting-sobolev-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T18:59:39.269775+00:00"
}