The Admissibility → Observation Theorem: Why Observation Requires Structural Permission in the Paton System

This paper introduces the Admissibility → Observation Theorem within the Paton System. The theorem states that observation cannot occur unless a system state has first satisfied the structural admissibility conditions defined at Tier-3. Observation therefore does not create system states; it registers states that have already passed the governing constraints of the system. The paper formalises the structural relationship between admissibility and observation in the Paton System architecture. Tier-3 determines whether structural configurations are permitted within a system, while Tier-4 provides the observational interface through which permitted states become legible. The theorem is expressed formally as: Observation(s) ⇒ Admissible(s) and conversely: ¬Admissible(s) ⇒ ¬Observable(s) This establishes admissibility as a necessary precondition for observable states. The result clarifies that observation functions as a registration interface rather than a generative mechanism. Cross-domain examples from physics, computation, biology, and cognition illustrate how observable states consistently follow admissibility conditions. The theorem therefore explains why the observation interface cannot display states that violate system constraints. By formalising the relationship between Tier-3 filtering and Tier-4 observation, the Admissibility → Observation Theorem clarifies the structural logic underlying observable systems within the Paton System.