Admissibility and Stability in Statistical Mechanical Systems: A Tier-7 Instantiation within the Paton System

Statistical mechanics describes the behaviour of large ensembles of microscopic states through probability distributions and phase-space evolution. Within the Paton System framework, these behaviours can be interpreted structurally as admissibility-constrained recursive state transitions. A system continues only while recursive updates remain compatible with governing constraint sets. When updates leave the admissible region of phase space, instability, transition, or collapse occurs. This paper presents a structural interpretation of statistical mechanical stability within the Paton System by demonstrating that equilibrium states correspond to stable admissibility basins and that phase transitions represent boundary crossings between admissible regions. The work functions as a Tier-7 domain instantiation of the Paton System, illustrating how the framework’s structural laws apply within statistical mechanical systems.