Admissibility Geometry: Structural Regions of Valid System States within the Paton System

Systems across scientific domains operate only within specific regions of valid states. Engineering systems function within structural tolerances, biological systems remain viable within physiological ranges, economic systems persist within financial constraints, and network systems maintain connectivity within resilience limits. These regions define the set of admissible states compatible with the governing constraints of the system. This paper introduces the concept of admissibility geometry within the Paton System framework. Admissibility geometry describes the structural shape of the state space within which systems can generate valid continuation. System stability corresponds to trajectories remaining within this admissible region, while instability or collapse occurs when trajectories exit the admissible structure. This interpretation provides a geometric perspective on stability and persistence across complex systems.