The Minimal System Existence Theorem: Structural Conditions for System Membership in the Paton System

This paper introduces the Minimal System Existence Theorem within the Paton System. The theorem defines the minimal structural conditions under which a system state can be considered a valid member of a system. A state must satisfy two requirements: structural admissibility and admissible reachability from a permitted origin. The theorem establishes that conceivable, describable, or mathematically consistent states do not automatically belong to a system. System membership instead requires satisfaction of governing constraints together with the existence of an admissible path from a permitted starting condition. Formally, the theorem is expressed as: State ∈ System ⇔ Admissible(state) ∧ Reachable(state) If either admissibility or reachability fails, the state cannot belong to the system. This rule provides a domain-neutral definition of system membership applicable across physical, computational, biological, and formal systems. Within the Paton System architecture, the theorem defines the foundational boundary of valid system states before observational registration or recursive continuation can occur. The Minimal System Existence Theorem therefore establishes the structural conditions required for system existence and defines the boundary within which system dynamics may occur.