Minimal Structural Conditions for System Existence: The Boundary–Relation–Persistence Theorem

This paper proposes a minimal structural criterion under which any configuration can be said to exist as a system. The framework identifies three jointly necessary and sufficient conditions: boundary, relation, and persistence. A system exists if and only if a distinction from its environment is present, internal relations between elements are defined, and those relations admit continuation under governing constraints. This Boundary–Relation–Persistence (BRP) triad provides a domain-neutral structural existence condition applicable prior to physics, computation, biology, or dynamical modelling. The result aligns with the early structural tiers of the Paton System architecture and formalises a minimal theorem for system existence. Within the broader Paton System framework, this result establishes the minimal structural requirements necessary for system membership and persistence prior to observation, modelling, or dynamical analysis.