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

Abstract We develop a statistical-mechanical framework for nonequilibrium systems governed by two structurally distinct variational evaluation terms under feasibility constraints. The formulation is motivated by a fundamental type asymmetry: one term is present-conditioned and point-like (local in state), whereas the other is future-oriented and inherently trajectory-based (indexed by admissible policies and horizons). Rather than collapsing these contributions into a single scalar objective, we embed them into a shared evaluation space that preserves their asymmetry and makes constraint-driven switching explicit. Under a minimal bridging assumption (embedding into a canonical family), we introduce a thermodynamic-style U–S plane, where U summarizes mismatch-/cost-like contributions and S summarizes uncertainty-/openness-like contributions in an operational sense. On this plane we define Z(λ) = U −λS, not as a quantity required to decrease monotonically, but as a parametrized evaluative ruler whose rotation with λ encodes changes in evaluative attitude. Selection is then 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. Introducing inertia yields delayed switching and hysteresis, and constraint-induced breakdown is predicted as geometric loss of feasible contact. To reduce the risk of purely interpretive readings, we provide a concrete mechanical realization that implements the same rotation–contact–regime-change logic as a minimal periodic dynamics. This serves as an existence proof that the proposed dynamical motif is physically realizable.