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. The operator is defined as a gating structure acting on system evolution, combining a generative transition function F with an admissibility condition G that determines whether proposed state transitions are permitted. System evolution is expressed as: Sₙ₊₁ = F(Sₙ) · G(Sₙ, Mₙ) Where F proposes a candidate state and G evaluates whether that transition satisfies governing constraints. If G = 1, the system continues. If G = 0, the transition is not permitted and the system stops. This formulation provides a minimal, domain-independent condition for continuation applicable across physical, computational, biological, and abstract systems. The paper does not replace domain-specific laws or models. It defines the structural condition under which such models are permitted to operate. Within the Paton System, the admissibility operator functions as the universal gate on state transitions.