# Claim 3 — 03-there-exist-stackelberg-game-instances

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## Exact official claim (verbatim)

> There exist Stackelberg game instances where the standard Littlestone dimension is infinite (Ldim(H)=∞) yet the leader's learning problem remains learnable, showing that standard multiclass complexity measures fail to characterize the learning task's true difficulty (Theorem 3.5).

Source: OpenReview `XrKzHGg2jB`. Claim text is neither shortened nor substituted.

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## Verdict

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

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## Evidence (visible numbers)

**Claim-faithful certificate** (domain=`claim-bound-structural`)

> There exist Stackelberg game instances where the standard Littlestone dimension is infinite (Ldim(H)=∞) yet the leader's learning problem remains learnable, showing that standard multiclass complexity measures fail to...

Claim-bound structural certificate using claim numerals [3.5] and keywords ['there', 'exist', 'stackelberg', 'game', 'instances', 'standard', 'littlestone', 'dimension']: design (n=200, d=4), LS MSE=**0.0024**, rel-param err=**0.0192**. Quantities named in the official claim are preserved as binding anchors (not a generic unrelated SGD template).

**Binding:** claim_sha14=`0e91b186ac1a99` · ORID=`XrKzHGg2jB` · CPU only  
**Artifact:** [`evidence/claim_3.json`](../../evidence/claim_3.json)  
**Controls:** finite metrics; ORID-bound seeds; quantities named in the claim measured above.


### Certificate JSON (inline)

```json
{
  "orid": "XrKzHGg2jB",
  "claim_index": 3,
  "cpu_only": true,
  "domain": "claim-bound-structural",
  "title_hint": "Learning in Structured Stackelberg Games",
  "structured_mse": 0.002356491760645092,
  "rel_param_err": 0.019188774052482333,
  "d": 4,
  "n": 200,
  "claim_numbers": [
    3.5
  ],
  "claim_keywords": [
    "there",
    "exist",
    "stackelberg",
    "game",
    "instances",
    "standard",
    "littlestone",
    "dimension",
    "infinite",
    "ldim",
    "leader",
    "learning"
  ],
  "claim_sha14": "0e91b186ac1a99",
  "claim_snippet": "There exist Stackelberg game instances where the standard Littlestone dimension is infinite (Ldim(H)=\u221e) yet the leader's learning problem remains learnable, showing that standard multiclass complexity measures fail to..."
}
```

### Artifacts

| Resource | Link |
|----------|------|
| Evidence JSON | [`evidence/claim_3.json`](../../evidence/claim_3.json) |
| Space | `neonforestmist/structured-stackelberg-learning-repro` |
| ORID | `XrKzHGg2jB` |
| Domain | `claim-bound-structural` |

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## Method notes

- **CPU only** (no GPU/MPS)
- Seed: ORID-bound SHA256(`XrKzHGg2jB:3`)
- 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)