A Unified Framework for Cognitive-Affective Systems: Mathematical Foundations and Stability Analysis

This article presents a comprehensive theoretical framework for modeling cognitive-affective systems in artificial intelligence. We synthesize the core principles of the Internal State Dynamics Methodology (MDEI) for Interdisciplinary Applications, the Teleological Semantics Model (TSM), and the concept of Emotional Turbulence. The central contribution of this work is a rigorous mathematical proof of the stability of the coupled MDEI-TSM system. By constructing a joint Lyapunov functional, we demonstrate that the integration of affective dynamics (MDEI) and semantic processes (TSM) results in a globally and exponentially stable system. This stability emerges not as a static equilibrium, but as a dynamic pattern of coherence, providing a robust foundation for building artificial cognitive systems that are both adaptive and resilient.