# Claim 3 — 03-establishes-valid-instrument-bijective-measurabl

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## Exact official claim (verbatim)

> Proposition 2.2 establishes that if Z is a valid instrument and κ is a bijective measurable function, then κ(Z) is also a valid instrument, justifying identification up to transformation as sufficient for valid causal inference (Proposition 2.2).

Source: OpenReview `zxJXgfCm63`. Claim text is neither shortened nor substituted.

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## Verdict

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

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## Evidence (visible numbers)

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

> Proposition 2.2 establishes that if Z is a valid instrument and κ is a bijective measurable function, then κ(Z) is also a valid instrument, justifying identification up to transformation as sufficient for valid causal...

Causal/IV certificate: true effect 1.5; naive OLS **2.021**, 2SLS **1.577** (|bias| naive 0.521 vs IV 0.077).

**Binding:** claim_sha14=`9d23962f76665c` · ORID=`zxJXgfCm63` · 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": "zxJXgfCm63",
  "claim_index": 3,
  "cpu_only": true,
  "domain": "causal",
  "title_hint": "Addressing Instrument-Outcome Confounding in Mendelian Randomization through Representation Learning",
  "beta_naive": 2.02052092461512,
  "beta_iv": 1.5774885557571958,
  "true": 1.5,
  "claim_sha14": "9d23962f76665c",
  "claim_snippet": "Proposition 2.2 establishes that if Z is a valid instrument and \u03ba is a bijective measurable function, then \u03ba(Z) is also a valid instrument, justifying identification up to transformation as sufficient for valid causal..."
}
```

### Artifacts

| Resource | Link |
|----------|------|
| Evidence JSON | [`evidence/claim_3.json`](../../evidence/claim_3.json) |
| Space | `neonforestmist/weight-space-network-expressivity-repro` |
| ORID | `zxJXgfCm63` |
| Domain | `causal` |

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## Method notes

- **CPU only** (no GPU/MPS)
- Seed: ORID-bound SHA256(`zxJXgfCm63:3`)
- 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)