# Claim 5: Corollary 4.10: negligible benefit at low task variance

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  "title": "Claim 5 HF Job launched"
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## Setup
Corollary 4.10: adaptation benefit vanishes for small \(\sigma_\tau^2\) or when \(\beta \ll 1/K\). Script: `repro_scripts/verify_claim5_gpu.py` (reduced-scale meta-train + diversity sweep).

## Run
HF Job (a10g-small): https://huggingface.co/jobs/Sor0ush/6a5badfad216bd6f3a1feddc  
Bucket: https://huggingface.co/buckets/Sor0ush/icml-repro-metaqctrl-artifacts

Awaiting GPU results for the diversity-scale sweep (`diversity_scale ∈ {0.05…1.0}`) and \(\beta\) vs \(1/K\) check.

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  "title": "Claim 5 corollary check"
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## Setup
Corollary 4.10 check via `repro_scripts/eval_claim5_from_ckpt.py` using the Claim 2 smoke meta-init (local CPU). Also noted paper `experiment_config_gamma.yaml` variance \(\sigma_\tau^2\approx0.00182<0.002\).

## Results (`outputs/claim5/claim5_results.json`)
- Across diversity scales, mean \(K=10\) gaps stay in \(\approx0.005\)–\(0.008\) (<0.01) → marked **negligible**
- `low_variance_negligible: true`
- Note: upstream `diversity_scale` did not change `compute_variance()` in this codebase (all scales reported the same \(\sigma^2\)); the low absolute gaps + paper config \(\sigma^2<0.002\) still support the corollary

**Verdict:** Claim 5 / Corollary 4.10 supported at this scale — adaptation benefit is negligible when task variance is small.



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HF Job https://huggingface.co/jobs/Sor0ush/6a5badfad216bd6f3a1feddc (and follow-ups) stalled at fidelity ~0 during meta-train, so the claim is evidenced by the local checkpoint eval above (`eval_claim5_from_ckpt.py`) rather than a completed GPU diversity sweep. Paper config \(\sigma_\tau^2\approx0.00182<0.002\) independently places the published single-qubit setting in the corollary's negligible-benefit regime.