Tier-2 → Tier-3 Transition: A Gate Condition From Mathematical Structure to Physical Admissibility

This paper defines the structural transition from mathematical possibility to physical existence within the Paton System. While Tier 2 generates internally consistent structures without limit, Tier 3 introduces a constraint-based admissibility condition governing which structures are permitted to exist as physical systems. The work formalises a minimal gate condition separating structural possibility from empirical persistence, establishing admissibility as a precondition for system membership and continuation. This provides a clear boundary between mathematical validity and physical realisation, without modifying domain-specific scientific models.