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. States that fail either condition cannot belong to the system regardless of whether they are conceivable, describable, or mathematically consistent. The theorem provides a domain-neutral definition of system membership applicable across physical, computational, biological, and formal systems.