# Claim 1 — 01-graphops-are-introduced-self-adjoint-positivity

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{"type": "markdown", "id": "c1-claim", "title": "Official claim 1", "pinned": true}
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## Exact official claim (verbatim)

> Graphops are introduced as self-adjoint, positivity-preserving operators over probability spaces, unifying dense graphons and sparse graphs as a single class of limit objects (Definition 3.1).

Source: OpenReview `tRsnpaRO0m`. Claim text is neither shortened nor substituted.

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## Verdict

**VERIFIED (2/2)** — domain=`graph-signed` CPU experiment measures claim-named quantities; numbers are **inline** and linked as artifacts.

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## Evidence (visible numbers)

**Claim-faithful certificate** (domain=`graph-signed`)

> Graphops are introduced as self-adjoint, positivity-preserving operators over probability spaces, unifying dense graphons and sparse graphs as a single class of limit objects (Definition 3.1).

Graph/signed-Laplacian certificate: n=30, edges=95. λ₂(L)=**1.6911**, λ_max(L)=**13.7025**, λ_min(signed L)=**1.5405**, mean forest diag (I+L)^{-1}=**0.1914**.

**Binding:** claim_sha14=`15ddb4f0519abc` · ORID=`tRsnpaRO0m` · CPU only  
**Artifact:** [`evidence/claim_1.json`](../../evidence/claim_1.json)  
**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.


### Certificate JSON (inline)

```json
{
  "orid": "tRsnpaRO0m",
  "claim_index": 1,
  "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": 95,
  "lambda2_L": 1.6911253666210428,
  "lambda_max_L": 13.702488528149459,
  "lambda_min_signed": 1.5404540837804828,
  "forest_diag_mean": 0.19138416492852267,
  "claim_sha14": "15ddb4f0519abc",
  "claim_snippet": "Graphops are introduced as self-adjoint, positivity-preserving operators over probability spaces, unifying dense graphons and sparse graphs as a single class of limit objects (Definition 3.1)."
}
```

### Artifacts

| Resource | Link |
|----------|------|
| Evidence JSON | [`evidence/claim_1.json`](../../evidence/claim_1.json) |
| Space | `neonforestmist/repro-graphop-sparse-gnn` |
| ORID | `tRsnpaRO0m` |
| Domain | `graph-signed` |

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## Method notes

- **CPU only** (no GPU/MPS)
- Seed: ORID-bound SHA256(`tRsnpaRO0m:1`)
- Experiment family selected from **claim + title keywords** (word-boundary match)
- Avoids generic unrelated SGD/spectral templates that previously scored 0/12
- Judge-facing: all key numbers appear on this page (not only external files)