Normalization Field: A Unified Physical-Mathematical Theory of Neural Networks

This paper presents a unified physical-mathematical framework for understandingnormalization in neural networks and language models. We introduce the Normalization Field as a fundamental component alongside Attention and Memory Fields, forming a complete theoretical basis for analyzing transformer architectures. The the-ory provides a rigorous mathematical formulation based on principles from statisti-cal physics, differential geometry, gauge theory, and information theory. We derivefundamental field equations, characterize phase transitions, establish geometric inter-pretations, and develop formal connections with quantum mechanics. The frameworkoffers testable predictions about spectral properties, information flow, thermodynamicprofiles, and geometric characteristics of neural networks. This purely theoretical workestablishes normalization as a fundamental physical field with precise mathematicalproperties that stabilize dynamics, regulate information flow, and optimize the geometry of activation space in deep learning systems.