{
  "claim_index": 5,
  "official_claim": "Taking the median of three independently trained interpolators achieves expected cutoff loss O(d_\u03b3/n), yielding optimal sample complexity O(d_\u03b3/\u03b5) that matches the lower bound (Theorem 3.11).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`sobolev-harmful-overfitting`)\n\n> Taking the median of three independently trained interpolators achieves expected cutoff loss O(d_\u03b3/n), yielding optimal sample complexity O(d_\u03b3/\u03b5) that matches the lower bound (Theorem 3.11).\n\nSobolev interpolator certificate: near-interpolator train MSE=**4.21e-02** (n=200, d=2, p=2.0, k=1.2), population L\u00b2 error=**5601.5455** 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=`695f978b5ed9b7` \u00b7 ORID=`qXlovWytwg` \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": "qXlovWytwg",
    "claim_index": 5,
    "cpu_only": true,
    "domain": "sobolev-harmful-overfitting",
    "title_hint": "The Interplay Between Interpolation and Aggregation in Regression: Optimal Sample Complexity",
    "d": 2,
    "p": 2.0,
    "k": 1.2,
    "gamma": 0.4,
    "n": 200,
    "train_mse_interp": 0.042103327210951884,
    "pop_mse": 5601.54547725074,
    "lower_bound_proxy": 476.83715820312557,
    "pop_above_lb": true,
    "bump_sobolev_norm": 0.90589855016253,
    "bump_bound": 2.319507910772894,
    "delta": 0.25,
    "claim_sha14": "695f978b5ed9b7",
    "claim_snippet": "Taking the median of three independently trained interpolators achieves expected cutoff loss O(d_\u03b3/n), yielding optimal sample complexity O(d_\u03b3/\u03b5) that matches the lower bound (Theorem 3.11)."
  },
  "domain": "sobolev-harmful-overfitting",
  "orid": "qXlovWytwg",
  "space_id": "neonforestmist/interpolation-aggregation-regression-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:00:57.114275+00:00"
}