Finite-step statistical mechanics of scale-invariant learning: spectral equations of state and curvature transport

Alexander Itin

Stationary learning admits an ideal-gas description under restrictive noise assumptions. In neural networks whose gradient noise departs from these assumptions, however, an entropy-scaling relation resembling a Maxwell relation remains approximately valid. We ask which thermodynamic-looking relations follow from scale symmetry rather than equilibrium, and how departures from the ideal-gas description can be quantified, for discrete stochastic gradient dynamics with memory. An exact rescaling of trajectories explains the entropy relation without assuming thermal equilibrium. Within a local harmonic approximation, self-adjusting weight norms yield a spectral equation of state whose virial coefficients are set by moments of the curvature spectrum, and coupled parameter groups can sustain stationary probability currents. Our central result beyond this approximation is an exact curvature-transport identity. Together with a finite-step loss balance, it decomposes corrections to the harmonic fluctuation relation into measurable curvature changes, finite-step remainders, gradient-memory correlations and finite-window drift terms. A related displacement-weighted curvature change provides…