Spectrally local geometric response at the onset of many-body quantum chaos

What the math gives to ML

The paper introduces an energy-resolved measure of eigenstate sensitivity: fidelity susceptibility assigns a perturbative response to each eigenstate, and the spectral density B_lambda(E) reveals where that response is concentrated. The transferable idea is not the quantum Hamiltonian itself, but the combination of eigenmode sensitivity and spectral localization: a system can become unstable in a narrow spectral band before instability spreads globally. Neural-network loss Hessians and Jacobians have analogous eigenmodes, so this construction can become a diagnostic and adaptive damping rule that targets only curvature bands with large eigenvector rotation. The main engineering challenge is estimating the required spectral quantities cheaply with Lanczos or stochastic trace probes.