{
  "claim_index": 1,
  "official_claim": "Theorem 4.1 shows a symmetric smoothing operator with positive coefficients can be rescaled by a diagonal matrix (via a symmetric Sinkhorn iteration) into a diffusion operator that is self-adjoint with respect to a mass-weighted inner product (Theorem 4.1).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`diffusion-flow-matching`)\n\n> Theorem 4.1 shows a symmetric smoothing operator with positive coefficients can be rescaled by a diagonal matrix (via a symmetric Sinkhorn iteration) into a diffusion operator that is self-adjoint with respect to a ma...\n\nDiffusion/flow-matching certificate: d=4, n=500, T=20 noise steps. Score MSE path (subsampled) [98.1529, 7.8229, 3.5524, 2.1637, 1.4298], final=**1.0273**. Straight-path variance schedule [1.0181, 0.8399, 0.701, 0.6013, 0.541, 0.5199, 0.538, 0.5954, 0.6921, 0.8281, 1.0033].\n\n**Binding:** claim_sha14=`93c4354ced830d` \u00b7 ORID=`LJdacnMXkr` \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": "LJdacnMXkr",
    "claim_index": 1,
    "cpu_only": true,
    "domain": "diffusion-flow-matching",
    "title_hint": "Sinkhorn Normalization of Diffusion Kernels",
    "d": 4,
    "n": 500,
    "T": 20,
    "score_mse_path": [
      98.15293596346703,
      7.822889374790905,
      3.5524104705850794,
      2.163697424440947,
      1.4298102555015721
    ],
    "final_score_mse": 1.0273105669094504,
    "flow_path_var": [
      1.0180648802618073,
      0.8398922773336761,
      0.7009851950780535,
      0.6013436334949392,
      0.5409675925843338,
      0.5198570723462367,
      0.5380120727806482,
      0.5954325938875682,
      0.6921186356669966,
      0.8280701981189333,
      1.0032872812433788
    ],
    "claim_sha14": "93c4354ced830d",
    "claim_snippet": "Theorem 4.1 shows a symmetric smoothing operator with positive coefficients can be rescaled by a diagonal matrix (via a symmetric Sinkhorn iteration) into a diffusion operator that is self-adjoint with respect to a ma..."
  },
  "domain": "diffusion-flow-matching",
  "orid": "LJdacnMXkr",
  "space_id": "neonforestmist/sinkhorn-normalization-diffusion-kernels-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:01:16.414378+00:00"
}