Admissibility Thresholds in System Formation

Systems across physics, biology, computation, and social organisation often appear when structural conditions cross a critical threshold. These events are traditionally described as phase transitions, emergence, or self-organisation. However, such interpretations typically focus on behaviour after the transition rather than the structural condition that permits the system to exist at all. Within the Paton System, system continuation requires admissibility: each successive state must remain compatible with the constraints governing the system. This paper extends that framework by identifying the threshold at which admissibility first becomes possible. A system forms when configuration space first contains a minimal admissible structure capable of sustaining recursive continuation. This threshold condition unifies system formation across domains. Phase transitions in physics, self-sustaining networks in biology, executable states in computation, and stable governance structures in organisations can all be interpreted as instances where configuration space first admits a structure capable of persistent evolution. By identifying this structural threshold, the Paton System provides a unified framework for understanding how systems begin.