Many systems generate large spaces of possible configurations but permit continuation only for states that satisfy specific constraints. This paper formalises the Tier-3 “Eye of the Needle” within the Paton System as the final static admissibility configuration that must hold before dynamics are permitted. The Eye of the Needle represents a structural bottleneck where boundary, relation, and persistence conditions must simultaneously satisfy viability requirements. Comparable constraint bottlenecks appear across multiple domains, including gravitational collapse, phase transitions in statistical mechanics, evolutionary selection, embryonic development, algorithm validation, machine learning model screening, chemical reaction pathways, and neural attention filtering. Mathematically, similar narrowing structures arise in dynamical systems through attractors, bifurcations, and saddle points, where large trajectory spaces converge toward limited admissible continuations. Recognising this recurring bottleneck pattern reveals a domain-neutral structural principle: admissibility precedes dynamics. Systems must first satisfy structural viability before dynamical evolution, optimisation, or persistence can occur. The paper positions the Eye of the Needle as the Tier-3 admissibility bridge between constrained flow (Tier-2) and observable legibility (Tier-4) within the Paton System.