Topological Invariants for State Spaces of Dynamic Neural Networks: A Framework for Computational Algebraic Topology in Learning Systems

This work establishes a rigorous mathematical framework for analyzing artificial neural networks through topological characterization of their stratified state spaces. We develop computational methods based on stratified homological invariants to understand fundamental properties of convergence, generalization, and stability in deep learning architectures. Our framework addresses the computational complexity challenges inherent in high-dimensional topological analysis and provides realistic bounds for practical implementation. We demonstrate that certain topological properties of neural stratified spaces correlate with learning dynamics and provide stability guarantees under adversarial perturbations. AMS Subject Classification: 68T07 (Artificial Intelligence), 55N31 (Persistent Homology), 37N35 (Dynamical Systems in Machine Learning)