Curvature-Regularized Free-Energy Dynamics: A Unified Lyapunov Framework for Boundary-Coupled Systems

抄録

This paper develops a curvature-regularized formulation of free-energy dynamics that unifies inference-only and boundary-coupled systems within a single Lyapunov framework. We introduce a curvature field κ(x, y) that quantifies the local deformation of the free-energy landscape induced by external structural gradients. This curvature arises when internal inference interacts with boundary-coupled dynamics. The composite potential V(x, y) = F(x, y) + (λ/2)κ(x, y)^2 serves as a Lyapunov function guaranteeing V̇ ≤ 0 under standard regularity assumptions. The special case κ = 0 recovers pure inference, whereas κ > 0 describes boundary-regularized inference consistent with dynamic Bayesian stability.