A Unified Mathematical Framework for Neural Network Architectures: Structure, Optimization, and Cognitive Evolution

This paper establishes a unified mathematical framework for neural networks using measuretheoretic representations.We prove that any L-layer network can be expressed as a hierarchical integral transform (Theorem 3.1) and develop a dynamic optimization theory with guaranteed convergence (Theorem 3.4).Novel contributions include: (1) Neural-symbolic differential logic with recursive satisfaction degrees, (2) Quantum-classical hybrid architecture with O(1/) gradient complexity, and (3) Experimental validation showing 42% L 2 error reduction versus baselines.The framework provides mathematical foundations for next-generation cognitive AI systems.