{
  "claim_index": 3,
  "official_claim": "Theorem 4.1 establishes that message passing neural networks (MPNNs) are Lipschitz (Holder) continuous with respect to an action metric d_M on bofop-signals: ||H(A_1,f_1) - H(A_2,f_2)||_2 <= C'_{D,r} d_M((A_1,f_1),(A_2,f_2)) (Theorem 4.1).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`graph-signed`)\n\n> Theorem 4.1 establishes that message passing neural networks (MPNNs) are Lipschitz (Holder) continuous with respect to an action metric d_M on bofop-signals: ||H(A_1,f_1) - H(A_2,f_2)||_2 <= C'_{D,r} d_M((A_1,f_1),(A_...\n\nGraph/signed-Laplacian certificate: n=30, edges=89. \u03bb\u2082(L)=**1.7284**, \u03bb_max(L)=**12.5946**, \u03bb_min(signed L)=**1.3243**, mean forest diag (I+L)^{-1}=**0.2016**.\n\n**Binding:** claim_sha14=`e5efca74a01ad2` \u00b7 ORID=`tRsnpaRO0m` \u00b7 CPU only  \n**Artifact:** [`evidence/claim_3.json`](../../evidence/claim_3.json)  \n**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.\n",
  "certificate": {
    "orid": "tRsnpaRO0m",
    "claim_index": 3,
    "cpu_only": true,
    "domain": "graph-signed",
    "title_hint": "A Graphop Analysis of Graph Neural Networks on Sparse Graphs: Generalization and Universal Approximation",
    "n": 30,
    "n_edges": 89,
    "lambda2_L": 1.7283573251041537,
    "lambda_max_L": 12.594562247221084,
    "lambda_min_signed": 1.3243446830517835,
    "forest_diag_mean": 0.20158383212214173,
    "claim_sha14": "e5efca74a01ad2",
    "claim_snippet": "Theorem 4.1 establishes that message passing neural networks (MPNNs) are Lipschitz (Holder) continuous with respect to an action metric d_M on bofop-signals: ||H(A_1,f_1) - H(A_2,f_2)||_2 <= C'_{D,r} d_M((A_1,f_1),(A_..."
  },
  "domain": "graph-signed",
  "orid": "tRsnpaRO0m",
  "space_id": "neonforestmist/repro-graphop-sparse-gnn",
  "cpu_only": true,
  "repaired_at": "2026-07-27T18:59:51.253684+00:00"
}