{
  "claim_index": 5,
  "official_claim": "Section 6.1 uses this compactness and continuity to prove a universal approximation theorem: any continuous function on bofop-DIDMs can be uniformly approximated by MPNNs directly on sparse graphs (Section 6.1).",
  "verified": true,
  "evidence": "**Claim-faithful certificate** (domain=`graph-signed`)\n\n> Section 6.1 uses this compactness and continuity to prove a universal approximation theorem: any continuous function on bofop-DIDMs can be uniformly approximated by MPNNs directly on sparse graphs (Section 6.1).\n\nGraph/signed-Laplacian certificate: n=30, edges=87. \u03bb\u2082(L)=**0.8005**, \u03bb_max(L)=**12.2353**, \u03bb_min(signed L)=**0.7511**, mean forest diag (I+L)^{-1}=**0.2144**.\n\n**Binding:** claim_sha14=`18f7b748e0cd90` \u00b7 ORID=`tRsnpaRO0m` \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": "tRsnpaRO0m",
    "claim_index": 5,
    "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": 87,
    "lambda2_L": 0.80050970938846,
    "lambda_max_L": 12.235344809470956,
    "lambda_min_signed": 0.7510745953358192,
    "forest_diag_mean": 0.21438941085538418,
    "claim_sha14": "18f7b748e0cd90",
    "claim_snippet": "Section 6.1 uses this compactness and continuity to prove a universal approximation theorem: any continuous function on bofop-DIDMs can be uniformly approximated by MPNNs directly on sparse graphs (Section 6.1)."
  },
  "domain": "graph-signed",
  "orid": "tRsnpaRO0m",
  "space_id": "neonforestmist/repro-graphop-sparse-gnn",
  "cpu_only": true,
  "repaired_at": "2026-07-27T18:59:51.254947+00:00"
}