# Claim 1: Scaling law for adaptation gain (Theorem 4.8)

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  "title": "Claim 1 synthesis"
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## Claim
Theorem 4.8: expected adaptation gain saturates as \(G_K \ge A_\infty(1-e^{-\beta K})\) with \(A_\infty = c\cdot\sigma_\tau^2\) and \(\beta=\eta\cdot\mu_{\min}\).

## Evidence in this reproduction
1. **Functional form** confirmed on classical LQR (Claim 6): \(R^2=1.000\) for the exponential fit (\(A_\infty=0.0029\), \(\beta=0.042\)).
2. **Curvature** (Claim 4): measured \(\mu\approx0.03\) in the adaptation regime.
3. **Variance scaling** \(A_\infty\propto\sigma_\tau^2\): LQR linear fits \(R^2\in[0.97,0.99]\); Claim 5 GPU sweep covers low-variance regime.
4. **Quantum instantiations**: Claim 2 (single-qubit X) and Claim 3 (two-qubit CZ under 10× noise on HF Jobs).

Paper: https://huggingface.co/papers/2601.18973 · https://openreview.net/forum?id=Gu9pw3D6vT