The cost rate of nonlinear remote stabilization on the Aubry--Andr\'e lattice: a reflected off-spectral exponent and the sharp identity for almost every phase
We study the exponential rate $r(\alpha,\lambda)$ of the energy $\mathcal{E}_N$ needed to steer a far site, at distance $N$, of an Aubry--Andr\'e chain $H_\lambda$ via one boundary actuator with closed-loop margin $\alpha$. An exact eigenbasis reduction writes $\mathcal{E}_N$ as a Cauchy quadratic form $\tilde b^\top C^{-1}\tilde b$ in the boundary-amplitude ratios, whose rate is the off-spectral Lyapunov exponent of the transfer cocycle at the reflected band edge $z^\star=2E_{\min}-2\alpha-E_{\max}$, giving $r(\alpha,\lambda)=\gamma_\lambda(z^\star)$. The rate lies in a bracket of width $\log_+(\lambda/2)$ whose ends coincide for $\lambda\le2$, the spectrum having logarithmic capacity $\max(1,\lambda/2)$. We prove the identity unconditionally, for every phase, on the whole metallic--critical range $0<\lambda\le2$: for $\lambda<2$ through subcritical almost reducibility as the sole external input, and at $\lambda=2$ because the Green's function there equals the Lyapunov exponent. For $\lambda>2$ the upper bound is unconditional, and the lower bound takes localization as its only external input: an inverse-free cocycle form makes $\mathcal{E}_N$ a cancellation-free positive sum, and a Christoffel--Darboux identity collapses its coefficients to $|c_k|=Q(\delta_k)(\hat\psi^{(k)}_N)^2$, where band-edge near-degeneracies cancel. With a three-distance lemma this yields $r=\gamma_\lambda(z^\star)$ at every Diophantine frequency and almost every phase, with gap $O(N^{-2/(2+\tau)})$ for type $\tau$ ($N^{-2/3}$ at bounded type), unconditionally for $\lambda\ge\lambda_1$ and under a polynomial-prefactor localization hypothesis for $2<\lambda<\lambda_1$. The relative gap $1-r/\gamma_\lambda(z^\star)$ vanishes at both ends of the localized phase, with $g_{\mathbb{C}\setminus\Sigma_\lambda}(z^\star)\to\operatorname{arccosh}3$ as $\lambda\to\infty$.