A Rigorous Mathematical Theory of Cognitive-Affective Dynamics: The Internal State Dynamics Model (ISDM), its MESN Extension, and Implications for Learning Architectures

This paper presents a rigorous, purely theoretical framework for modeling cognitive-affective dynamics in artificial intelligence systems. We introduce the Internal State Dynamics Model (ISDM), a theory grounded on representing cognitive- emotional states as vectors in a Hilbert space, where tempo- ral evolution is governed by a nonlinear ordinary differential equation system. The model integrates concepts from functional analysis, dynamical systems theory, and differential geometry to establish existence, uniqueness, and stability conditions via the Picard-Lindel¨ of theorem, generalized Lyapunov functionals, and spectral characterization using Fourier-Plancherel transforms and multiresolution wavelet analysis. We detail the theory of emotional turbulence, characterized by the Emotional Reynolds Number and trigonometric interaction matrices. As a control mechanism, we formalize the Neural Systems Stabilization Model (MESN), a protocol for detecting and correcting instabilities through vectorial realignment without backpropagation. Finally, we discuss implications for novel learning architectures, propos- ing OpenPropagation as a hybrid, energy-efficient alternative to backpropagation that leverages MDEI-based coherence signals. The entire work is presented from a foundational, mathematical perspective, without empirical data.