Free-volume origin of diverging direct correlations in hard crystals: insights from an exact one-dimensional model

What the math gives to ML

The paper gives a constructive free-volume mechanism for singular second functional derivatives in an inhomogeneous periodic hard-rod system. The minimum insertion volume is A_min = Δ + (1 − Δ)n_vac, producing c^(1) proportional to log(A_min) and c^(2) proportional to −1/A_min; the apparent 1/n_vac divergence occurs only when vacancies dominate the residual localization volume. A transferable neural-network version treats under-utilized experts, tokens, or memory slots as vacancies and uses inverse free volume as an adaptive load-balancing force or curvature estimate. The key falsifiable signature is a crossover from inverse-vacancy scaling to saturation at the localization floor Δ.