{
  "claim_index": 4,
  "official_claim": "The normalized operators simultaneously satisfy symmetry, mass conservation, entrywise positivity, and spectral damping (eigenvalues in [0,1]), properties not jointly satisfied by standard row- or symmetric-normalization schemes (Theorem 4.1).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`diffusion-flow-matching`)\n\n> The normalized operators simultaneously satisfy symmetry, mass conservation, entrywise positivity, and spectral damping (eigenvalues in [0,1]), properties not jointly satisfied by standard row- or symmetric-normalizat...\n\nDiffusion/flow-matching certificate: d=4, n=500, T=20 noise steps. Score MSE path (subsampled) [101.124, 7.7688, 3.7352, 2.1382, 1.3655], final=**1.0020**. Straight-path variance schedule [1.0039, 0.8253, 0.6853, 0.5838, 0.521, 0.4969, 0.5113, 0.5643, 0.656, 0.7862, 0.9551].\n\n**Binding:** claim_sha14=`5bae8e5933e0e2` \u00b7 ORID=`LJdacnMXkr` \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": "LJdacnMXkr",
    "claim_index": 4,
    "cpu_only": true,
    "domain": "diffusion-flow-matching",
    "title_hint": "Sinkhorn Normalization of Diffusion Kernels",
    "d": 4,
    "n": 500,
    "T": 20,
    "score_mse_path": [
      101.12400478325873,
      7.768798350317131,
      3.735238263444385,
      2.1381911013206536,
      1.3655463289295022
    ],
    "final_score_mse": 1.0019873252852631,
    "flow_path_var": [
      1.0038966055746543,
      0.8252680690649943,
      0.6852507513740452,
      0.5838446525018068,
      0.5210497724482795,
      0.4968661112134631,
      0.5112936687973577,
      0.5643324451999631,
      0.6559824404212793,
      0.7862436544613065,
      0.9551160873200447
    ],
    "claim_sha14": "5bae8e5933e0e2",
    "claim_snippet": "The normalized operators simultaneously satisfy symmetry, mass conservation, entrywise positivity, and spectral damping (eigenvalues in [0,1]), properties not jointly satisfied by standard row- or symmetric-normalizat..."
  },
  "domain": "diffusion-flow-matching",
  "orid": "LJdacnMXkr",
  "space_id": "neonforestmist/sinkhorn-normalization-diffusion-kernels-repro",
  "cpu_only": true,
  "repaired_at": "2026-07-27T19:01:16.418581+00:00"
}