Ghost-free higher-gradient Newtonian gravity from the Second Law of Thermodynamics
Higher-gradient modifications of Newtonian gravity remove the point-mass singularity, but their Lagrangian dynamics is haunted by the Ostrogradsky instability. We show that the same field equations follow from the Second Law of thermodynamics applied to a self-gravitating fluid with a weakly nonlocal state space, and that on this route the instability is absent by construction: the potential obeys a first-order relaxation equation that is a gradient flow, whose Lyapunov function is the entropy density, and whose relaxation spectrum is negative definite at every wavenumber precisely when the entropy is concave. The derivation also yields the pressure tensor and the energy current, an exact identity that reproduces Newton's force law without further input, and a dissipation-free cross-coupling to the bulk viscous pressure that no variational principle can produce. Regularity constrains the Yukawa-type corrections by an amplitude sum rule, $\sum_i\alpha_i=-1$, so that laboratory tests of the inverse-square law bound the largest internal length below $4\times10^{-5}$~m, and identify the complex-root regime as a classically consistent Lee--Wick-like sector.