History-Dependent Gravity III: Spectral Origin of Time Memory and Cosmological Signatures

We demonstrate that the gravitational memory kernel in History-Dependent Gravity emerges from the spectral density of microscopic degrees of freedom. The renormalization group flow becomes an autonomous spectral flow, providing a dynamical origin for the arrow of time. Horizon thermodynamics and cosmological signatures (matter power spectrum modifications) are unified through a single spectral function, offering falsifiable predictions for Euclid and DESI surveys.