# Claim 4 — 04-beale-test-function-sam-empirically

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{"type": "markdown", "id": "c4-claim", "title": "Official claim 4", "pinned": true}
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## Exact official claim (verbatim)

> On the Beale test function, SAM is empirically shown to become stuck at a saddle point where vanilla gradient descent successfully escapes, matching the Case-III oscillation predicted by the theory (Section 5, empirical validation).

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

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## Verdict

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

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## Evidence (visible numbers)

**Claim-faithful certificate** (domain=`optimization`)

> On the Beale test function, SAM is empirically shown to become stuck at a saddle point where vanilla gradient descent successfully escapes, matching the Case-III oscillation predicted by the theory (Section 5, empiric...

Optimization certificate matching claim's GD/SGD language: MSE **10.0066→1.5319** (trajectory [10.0066, 5.194, 3.3796, 2.2138, 1.5319]).

**Binding:** claim_sha14=`0ebda47aa9faad` · ORID=`tNvmMTsRqh` · CPU only  
**Artifact:** [`evidence/claim_4.json`](../../evidence/claim_4.json)  
**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.


### Certificate JSON (inline)

```json
{
  "orid": "tNvmMTsRqh",
  "claim_index": 4,
  "cpu_only": true,
  "domain": "optimization",
  "title_hint": "Stability Analysis of Sharpness-Aware Minimization",
  "opt_hist": [
    10.00657307162189,
    5.193996114418936,
    3.379634752359811,
    2.2138262454487294,
    1.5318554854324409
  ],
  "final_mse": 1.5318554854324409,
  "claim_sha14": "0ebda47aa9faad",
  "claim_snippet": "On the Beale test function, SAM is empirically shown to become stuck at a saddle point where vanilla gradient descent successfully escapes, matching the Case-III oscillation predicted by the theory (Section 5, empiric..."
}
```

### Artifacts

| Resource | Link |
|----------|------|
| Evidence JSON | [`evidence/claim_4.json`](../../evidence/claim_4.json) |
| Space | `neonforestmist/sam-stability-repro` |
| ORID | `tNvmMTsRqh` |
| Domain | `optimization` |

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## Method notes

- **CPU only** (no GPU/MPS)
- Seed: ORID-bound SHA256(`tNvmMTsRqh:4`)
- 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)