presents a comprehensive mathematical framework for advanced cognitive modules, aiming to establish theoretical foundations for dynamic, self-maintaining cognitive architectures. The document outlines a system that integrates world modeling, which includes latent structure extraction and multi-scale prediction with uncertainty quantification, with adaptive neural network maintenance techniques like synaptic health modeling and intelligent pruning. It also details multi-order relational reasoning for understanding complex relationships, encompassing hierarchical, causal, temporal, spatial, and conceptual structures. Furthermore, the framework explores integrated cognitive architecture components such as a meta-controller and memory integration, alongside theoretical properties like stability and convergence, and an analysis of computational complexity and various optimization objectives. Ultimately, the text provides a rigorous blueprint for developing intelligent systems capable of continuous learning, adaptation, and sophisticated cognitive functions.