Free-Energy Geometry and Dynamical Regime Switching under Feasibility Constraints A Transferable Dynamical Motif for Nonequilibrium Systems

We develop a statistical-mechanical framework for nonequilibrium systems that exhibits regime switching under feasibility constraints. The framework is organized around a freeenergy– like evaluation geometry on a thermodynamic-style U–S plane, where U summarizes mismatch-/cost-like contributions and S summarizes uncertainty-/openness-like contributions. We define Z(λ) = U − λS not as a scalar objective required to decrease monotonically, but as a parametrized evaluative ruler whose rotation with λ changes the relative weighting of the two contributions. A central claim is that the relevant type asymmetry is structural: one evaluative term is present-conditioned and point-like, whereas the other is future-oriented and trajectory-based. Selection is therefore formulated by a feasibility-aware “first-touch” rule: among admissible branches, the preferred regime is identified by the first feasible contact between an iso-Z ruler and the candidate bundle. This geometry yields delayed switching, hysteresis, and path dependence without requiring a global optimization narrative. To show that the construction is compatible with a dynamical system perspective, we also present a concrete mechanical realization that implements the same rotation–contact– regime-change logic as a minimal periodic dynamics. The result is a transferable dynamical motif for constrained nonequilibrium systems in which free-energy geometry, feasibility boundaries, and regime switching can be analyzed within a common descriptive framework.