The Paton System: Minimal Structural Condition on System Evolution

This note presents the minimal structural condition governing system evolution within the Paton System. All system transitions are expressed as the interaction between a generative transition function and an admissibility operator. A candidate state is proposed by a transition function F(Sₙ) and evaluated by an admissibility operator G(Sₙ, Mₙ). Continuation occurs only when the admissibility condition is satisfied. Otherwise, system evolution terminates. This formulation defines a minimal, domain-independent condition beneath all physical, computational, biological, and abstract systems. It does not replace domain-specific laws but specifies the structural requirement under which they operate. The condition unifies prior Paton System components, including admissibility (Tier 3), observation (Tier 4), continuation (Tier 5), and formal structural representation (Tier 6), while remaining consistent across all Tier 7 domain instantiations. This document serves as a compressed canonical statement of the Paton System’s core evolutionary condition and is intended as a companion reference to “The Admissibility Operator” and “The Unified Transition Equation.”