The Continuation Mechanism: Structural Conditions for State Transition in the Paton System

This paper defines the continuation mechanism within the Paton System, establishing the structural condition under which system states transition from one step to the next. While admissibility determines whether a state is permitted and the operator layer defines how transformations occur, this work formalises how executed states become persistent states. The framework introduces no predictive, optimisation, or dynamic modelling functions. It provides a pre-theoretical rule governing stepwise persistence under constraint, defining continuation as conditional on admissibility after execution. A domain-neutral formulation is presented in which system evolution proceeds through the sequence: admissibility, execution, and continuation. Persistence is not assumed and is evaluated locally at each step. Termination occurs at the first failure of admissibility. The paper establishes continuation as a minimal structural condition within the Paton System, completing the transition chain between execution and boundary.