Admissible Regions and Boundary Geometry: The Structural Shape of Constraint-Compatible Systems

This paper defines admissible regions as the subset of state space within which system states satisfy governing constraints. It introduces boundary geometry as the structural description of the limits of admissibility, where continuation becomes impossible. Building on the Admissibility Field, Viability Gradient, Admissibility Curvature, and Admissible Trajectories, this work formalises the geometry of constraint space itself. The framework shows that system behaviour is governed not only by motion within admissible regions but by the structure of the boundaries that define them. This provides a complete geometric interpretation of persistence, transition, and collapse within the Paton System.