Origin Threshold Formal Operator: A Mathematical Definition of Admissibility Onset in the Paton System

This paper introduces the Origin Threshold Operator, a mathematical construct that identifies the structural moment at which admissibility first becomes possible within a system. Previous work in the Paton System identified system initiation as the point at which a configuration becomes compatible with the constraints governing recursive continuation. The Origin Threshold Operator formalises this transition by defining an operator that evaluates compatibility between system state and governing constraints. By expressing admissibility onset as a formal operator, the framework converts system origin from a descriptive concept into a mathematically testable condition. The model provides a unified structural interpretation of system initiation across physical, biological, computational, and organisational domains.