The Paton System: Minimal Structural Condition on System Evolution

This note presents the minimal structural condition governing system evolution within the Paton System. All state transitions are expressed as the interaction between a generative transition function and an admissibility operator. A candidate state is proposed by a transition function and evaluated by an admissibility condition. Continuation occurs only when admissibility is satisfied; otherwise, evolution terminates. This formulation defines a minimal, domain-independent condition underlying physical, computational, biological, and abstract systems. It does not replace domain-specific laws but specifies the structural requirement under which they operate. The note serves as a compressed canonical statement of system evolution within the Paton framework.