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.
Andrew John Paton (Wed,) studied this question.