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

We introduce the spectral density of geometric response, a measure of eigenstate sensitivity across an energy spectrum. Applied to many-body quantum systems, it reveals an exponentially sensitive integrability-to-chaos crossover, where eigenbasis deformations first accumulate in localized spectral regions before spreading throughout the spectrum. Physically motivated random-matrix ensembles reproduce this behaviour, whereas Gaussian ensembles do not, indicating universal features of the route to chaos that are absent from featureless random-matrix models.