The Paton System: A Unified Framework for AI State Evolution, Continuity, and Validation

This paper introduces the Paton System, a novel framework for understanding the membership and evolution of states within AI systems. By combining admissibility and reachability, the framework provides a formalized approach to understanding how AI states are defined, evolve, and persist within a system. The recursion principle from Tier-3 is central to modeling state evolution, providing a new lens through which AI models can be analyzed for stability and growth. Through this framework, we show how AI states emerge as discrete, admissible structures from an initial configuration, and how they maintain persistent trajectories over time. This paper demonstrates the utility of the Paton System in offering a rigorous method for system validation, model optimization, and long-term stability in real-world AI applications, with a focus on neural networks and reinforcement learning.