Structural Notes on Viability and Constraint Geometry

We present a structural analysis of admissible viability regions for open systems under perturbation.Let $S$ denote a system state space and let $V_0 \subset S$ denote its physically admissible viable region.Under standard assumptions drawn from viability theory and stochastic perturbation theory, we observethat admissible viability regions can be characterized purely through exclusionary (negative) invariants.We further distinguish between positive invariants that are reducible to collapse-state exclusion andthose that impose additional persistent constraints on admissible states. Under sufficiently mixing perturbations, non-reducible positive invariants strictly contract theadmissible viable region and induce monotone upper bounds on first-exit probabilities and expectedsurvival times. A compatibility remark is given showing that, for finite systems with bounded freeenergy, protocols enforcing non-vanishing asymptotic work extraction correspond to such non-reducibleconstraints. The analysis introduces no new dynamics and is intended as an organizational perspectiveon viability, first-passage, and admissibility frameworks.