The Structural Continuation Theorem: A Unified Persistence Rule in the Paton System

This paper introduces the Structural Continuation Theorem within the Paton System. The theorem establishes the minimal structural condition required for persistence in any system by unifying two previously defined results: the Admissibility → Observation Theorem and the Observation → Continuation Principle. The theorem states that a system state may participate in recursive continuation only if it satisfies two joint structural conditions: it must be admissible within the governing constraints of the system, and it must be observationally registered within the system’s observable structure. Formally, the theorem is expressed as: Continuation(s) ⇒ Admissible(s) ∧ Observed(s) If either admissibility or observational registration fails, continuation cannot occur. The result provides a unified persistence rule describing how systems generate observable histories and maintain structural continuity across time. Within the Paton System architecture, the theorem unifies three key layers: Tier-3 — Admissibility Gate Tier-4 — Observation Interface Tier-5 — Recursive Continuation Engine Across domains including physics, computation, biology, and cognition, continuation requires both structural permission and structural registration. The Structural Continuation Theorem therefore provides a domain-neutral rule governing persistence in recursive systems.