The Admissibility Operator: A Universal Condition on State Transitions

This paper formalises the admissibility operator as a universal condition governing state transitions within the Paton System. Prior work defines admissibility as the condition for system membership and continuation as persistence within admissible space. However, the mechanism by which admissibility regulates system evolution has not been expressed in a unified mathematical form. This paper introduces the admissibility operator as a gated transition structure acting on state evolution. The operator combines a generative transition function with an admissibility condition that determines whether proposed state transitions are permitted. This formulation provides a minimal, domain-independent condition for continuation applicable across physical, computational, biological, and abstract systems.