pages/claim-1-scaling-law-for-adaptation-gain-theorem-4-8/page.md

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

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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