The Paton Persistence Law: A Minimal Structural Expression of System Existence

The Paton System provides a structural framework for understanding how systems begin, persist, and collapse under governing constraints. Previous work established admissibility as the condition for continuation, defined origin thresholds for system formation, described admissible regions as constraint manifolds, and identified resolution ceilings beyond which structural instability occurs. This paper compresses these results into a minimal formal law of system existence. A system persists if and only if three simultaneous conditions are satisfied: an admissible origin exists, the evolving state remains within the admissible constraint manifold, and structural resolution remains below the governing ceiling. This minimal expression unifies origin, continuation, and collapse within a single structural condition. The result functions as a canonical minimal formulation of the Paton System and applies across physical, biological, computational, and organisational domains.